Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=rael20

Download by: [95.146.221.172] Date: 12 October 2015, At: 19:14

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

Published online: 05 Oct 2010.

Submit your article to this journal

Article views: 52

View related articles

Citing articles: 2 View citing articles

I. INTRODUCTION

Deaton (1987) has shown that the permanent incomehypothesis under certainty equivalence (the PIH-CE model)implies that the optimal consumption path is more volatilethan the labour income path when labour income is positivelyautocorrelated in first differences. This prediction is incon-sistent with one of the principal raisons d’etre of thepermanent income hypothesis, namely that consumption issmooth because permanent income is smoother than labourincome. Moreover, this prediction is contradicted by a strikingfeature of consumption behaviour; observed consumption issmooth relative to labour income. Thus, the PIH-CE model isrejected because actual consumption is excessively smooth.This excess smoothness phenomenon is called the Deatonparadox.

One potential way to resolve this paradox is to relaxcertainty equivalence by analysing precautionary savingbehaviour. Such saving is used to self-insure against theuncertainty of future labour income. However, Caballero(1990) has found that some restrictive parametrizations of thepermanent income hypothesis under precautionary saving (thePIH-PS model) yield the same excessive smoothness asobtained from the PIH-CE model. This letter demonstrates thatseveral flexible, but reasonable, parametrizations of the PIH-PS model fully account for the excess smoothness problem.

II. THE DEATON PARADOX

It is assumed that a representative consumer solves thefollowing; problem:

maxfCt‡jg

Et

X1

jˆ0

…1 ¡ † Ct‡j

1 ¡‡ t‡j …1 ‡ r†¡j …1†

s:t: Et

X1

jˆ0

Ct‡j…1 ‡ r†¡j ˆ At ‡ Et

X1

jˆ0

Yt‡j…1 ‡ r†¡j ˆ Wt

…2†

Yt ˆ 0 ‡ 1 Yt¡1 ‡ t …3†

where Et and represent the conditional expectation and thefirst difference operators, r is the time preference rate (equal tothe constant interest rate), Ct is consumption, At is financialwealth, Wt is expected total wealth, and Yt corresponds to anon-insurable stochastic (after-tax) labour income. Also,Equation 1 involves a hyperbolic absolute risk aversion(HARA) utility function where > 0; ‰Ct…1 ¡ †¡1 ‡ tŠ > 0for non-satiation, 6ˆ 1, and t is time-varying in order toaccommodate consumption growth (Merton, 1971). This is avery flexible specification which nests quadratic ( ˆ 2),exponential ( ˆ ‡1 and t ˆ ), isoelastic ( t ˆ 0) andlog ( ˆ t ˆ 0) utility functions. Equation 2 is the budgetconstraint. Equation 3 is the labour income process used byDeaton (1987).

The quadratic utility function ( ˆ 2) yields the traditionalPIH-CE model (Hall, 1978). For this model, the ratio ( u= )can be derived analytically – where u and are standarddeviations of the innovations ut in consumption and t inlabour income. The smaller the ratio, the smoother theconsumption relative to labour income. By calibrating thelabour income process (Equation 3) from Blinder and

Applied Economics Letters, 1997, 4, 187–190

1350–5851 © 1997 Routledge 187

Precautionary saving and the Deaton paradox

MICHEL NORMANDIN

Department of Economics and Research Center on Employment and EconomicFluctuations, Universite du Quebec a Montreal, CP 8888, Succ. Centre-Ville, Montreal,Quebec, Canada H3C 3P8

Received 12 March 1996

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

Dow

nloa

ded

by [

95.1

46.2

21.1

72]

at 1

9:14

12

Oct

ober

201

5

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 ismore volatile than labour income. However, this model isrejected 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). Forthis model, an explicit solution for ( u= ) can be obtainedonly with an exponential utility function. Caballero (1990) hasshown that such preferences produce the same smoothness asinduced by the PIH-CE model. Thus, this particular para-metrization of the PIH-PS model cannot resolve the Deatonparadox.

For the HARA utility function, ( u= ) must be foundnumerically. This is done by applying the method developedby Den Haan and Marcet (1990). More precisely, the expectedfuture marginal utility, the non-resolvable term, is firstapproximated by an exponential function of a second-orderpolynomial in Wt, so the Euler equation becomes

Ct

1 ¡‡ t

¡1

ˆ EtCt‡1

1 ¡‡ t‡1

¡1

exp 0 ‡ 1 log Wt ‡ 2…log Wt†2

…4†

Then, Equation 4 yields the following approximation of theconsumption function (for which the analytical solution isunknown):

Ct ˆ …1 ¡ † exp0 ‡ 1 log Wt ‡ 2…log Wt†2

¡ 1

" #¡ t

( )

…5†

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

interest rate, and labour income process (required toconstruct expected future labour incomes involved in Wt)as in Deaton (1987). Finally, ( u= ) is computed bycalibrating from the OLS estimate of Equation 3 and byevaluating u from the constructed time seriesut ˆ Ct ¡ Et¡1Ct where Et¡1Ct is obtained from a second-order Taylor series expansion (evaluated at known variablesin t ¡ 1) on Equation 5.

This exercise is performed for the reasonable parametriza-tion ( < 1; 0 < r 10) of the HARA utility function, where

is recovered from and the relative risk aversion r. Therestrictions on r are consistent with most empirical work. Therestriction on produces precautionary saving and decreasingabsolute risk aversion (i. e. the consumer is willing to pay lessto avoid a given bet as wealth increases). For all thereasonable parametrizations, the Den Haan and Marcet(1994) test indicates that Equation 5, is an accurateapproximation of the consumption function, i.e. the inducedinnovations in marginal utility and in consumption areorthogonal to the agent’s past information.

Table 1 reveals that the isoelastic utility function( ˆ ¡9; r ˆ 10) increases the volatility of consumptionsince … u= † ˆ 3:65 is larger than obtained from the PIH-CEmodel, which is 1.77. This result holds for all isoelastic utilityfunctions, for the log-utility function and for all HARA utilityfunctions having 0. Consequently, these parametrizationsaccentuate the Deaton paradox. In contrast, HARA utilityfunctions involving 0 < < 1 almost always yield smoothconsumption paths relative to labour income (i .e.… u= † < 1).

Next it is verified whether the smoothness induced by themost promising parametrizations (0 < < 1; 0 < r 10)matches that found in the data. This exercise requires theratio ( u= ) associated with Blinder and Deaton’s (1985) dataset. Once again, is calibrated by estimating Equation 3using OLS. And u is evaluated from the time series ut, which

M. Normandin188

Table 1. Smoothness induced by the PIH-PS model

r

0.1 0.5 0.9 1.0 2.0 5.0 10.0

¡90 – – – – – – 3.65 (1.00)¡40 – – – – – 2.01 (1.00) 7.59 (1.00)¡1:0 – – – 1.95 (1.00) 5.99 (1.00) 6.07 (1.00)

0.0 – – – 1.92 (1.00) 2.69 (1.00) 4.29 (1.00) 4.62 (1.00)0.1 – – 1.89 (1.00) 1.83 (1.00) 1.01 (0.38) 0.86 (0.17) 0.61 (0.05)0.5 – 1.79 (1.00) 1.16 (0.59) 1.11 (0.53) 0.98 (0.23) 0.81 (0.07) 0.57 (0.01)0.9 1.64 (1.00) 1.01 (0.31) 0.93 (0.22) 0.89 (0.18) 0.69 (0.05) 0.54 (0.01) 0.51 (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. Entriesin 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 arecomputed by performing a Monte Carlo experiment with T ˆ 127 (the actual sample size) and 1200 replications. Numbers in bold correspond to isoelastic utilityfunctions for 6ˆ 0 and to the log-utility function for ˆ 0. For the PIH-CE model. ( u= † ˆ 1:77.

Dow

nloa

ded

by [

95.1

46.2

21.1

72]

at 1

9:14

12

Oct

ober

201

5

is approximated via the following ARIMAX process (esti-mated by OLS):

Ct ˆ ‡X

jˆ0j Ct¡j ‡

X

jˆ0j Wt¡j ‡ ut …6†

where Ct and Wt are stationary variables and, 0 ˆ 0 ˆ 0. Asusual, consumption is measured by expenditures on non-durables and services. Expected total wealth is computed bysumming the financial wealth …1 ‡ r†=r times capital income(the difference between total income and labour income) – andthe present value of expected future labour incomes,constructed using Equation 3. Four specifications of Equation6 are analysed. S1 : Ct ˆ Ct; Wt ˆ Wt and ˆ 0;S2 : Ct ˆ Ct; Wt ˆ Wt and ˆ 5; S3 : Ct ˆ Ct=…1 ‡ †t

;Wt ˆ Wt=…1 ‡ †t and ˆ 0; and S4 : Ct ˆ Ct=…1 ‡ †t

;Wt ˆ Wt=…1 ‡ †t and ˆ 5. These various specificationssuggest that the ratios ( u= ) associated with the data arebetween 0.32 and 0.48. For comparability, simulated con-sumption and expected total wealth are also used to estimate(by OLS) each specification of Equation 6 in order to computethe ratios ( u= ) associated with the PIH-PS.

Table 2 indicates that the ratios ( u= ) obtained from thedata do not fall in the 95% probability intervals of the ratiosinduced 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 otherhand, the parametrization ( ˆ 0:1; r 2:0), ( ˆ 0:5;

r 0:9), and ( ˆ 09; 0:5 r < 10:0) of the HARA utilityfunction induce a smoothness which matches that found in thedata. These results are robust to the various specifications ofthe ARIMAX process (Equation 6).

III. CONCLUSION

The analysis of the smoothness of consumption was performedfor the environment studied by Deaton (1987) with amodification to the structure of preferences. This exercisereveals that log and isoelastic utility functions cannot resolvethe Deaton paradox. In contrast, several reasonable parame-trizations of the HARA utility function imply that the PIH-PSmodel yields consumption paths which are smoother thanlabour income and the relative smoothness statisticallymatches that found in the data.

ACKNOWLEDGEMENTS

I am grateful to John Galbraith, Allan Gregory, ThomasMcCurdy, Gregor Smith, Tony Smith and Paul Storer forhelpful suggestions. I would like to thank Alan Blinder andAngus Deaton for supplying me with data. I acknowledgefinancial support from Fonds pour la Formation de Chercheurset l’Aide a la Recherche (FCAR). I retain full responsibilityfor any errors.

REFERENCES

Blinder, A.S. and Deaton, A. (1985) The time series consumptionfunction revisited, Brookings Papers on Economic Activity, 2,465–511.

Caballero, R.J.(1990) Consumption puzzles and precautionarysavings, Journal of Monetary Economics, 25, 113–36..

Precautionary saving and the Deaton paradox 189

Table 2. The 95% probability intervals of the smoothness induced by the PIH-PS model

r

Specifications 0.1 0.5 0.9 2.0 5.0 10.0

0.1 S1 – – (2.21, 2.76) (0.20, 2.01) (0.17, 1.14) (0.14, 0.90)S2 – – (2.11, 2.65) (0.17, 1.66) (0.13, 1.05) (0.09, 0.82)S3 – – (1.66, 2.11) (0.13, 1.15) (0.10, 0.71) (0.07, 0.57)S4 – – (1.59, 2.02) (0.12, 1.04) (0.09, 0.67) (0.07, 0.53)

0.5 S1 – (1.55, 1.91) (0.23, 2.25) (0.21, 1.08) (0.17, 0.77) (0.15, 0.50)S2 – (1.72, 1.82) (0.21, 1.93) (0.17, 1.03) (0.12, 0.74) (0.10, 0.48)S3 – (1.34, 1.67) (0.15, 1.29) (0.14, 0.75) (0.10, 0.54) (0.08, 0.36)S4 – (1.28, 1.62) (0.14, 1.22) (0.13, 1.22) (0.10, 0.51) (0.08, 0.34)

0.9 S1 (1.54, 1.90) (0.38, 1.03) (0.27, 0.91) (0.18, 0.67) (0.14, 0.54) (0.14, 0.25)S2 (1.40, 1.90) (0.25, 1.24) (0.24, 0.88) (0.10, 0.89) (0.09, 0.51) (0.09, 0.22)S3 (1.16, 1.45) (0.29, 0.75) (0.20, 0.66) (0.12, 0.48) (0.08, 0.39) (0.08, 0.17)S4 (1.10, 1.40) (0.28, 0.72) (0.19, 0.63) (0.11, 0.46) (0.07, 0.38) (0.07, 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 byestimating (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.

Dow

nloa

ded

by [

95.1

46.2

21.1

72]

at 1

9:14

12

Oct

ober

201

5

Deaton, A. (1987) Life-cycle models of consumption: is the evidenceconsistent with the theory, in Advances in Econometrics FifthWorld Congress, Vol. 2, ed. T.F. Bewley, Cambridge UniversityPress, Cambridge, pp. 121–48.

Den Haan, W.J. and Marcet, A. (1990) Solving the stochastic growthmodel by parameterizing expectations, Journal of Business andEconomic 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–permanentincome hypothesis: theory and evidence, Journal of PoliticalEconomy, 86, 971–87.

Leland, H.E., (968) Saving and uncertainty: the precautionarydemand for saving, Quarterly Journal of Economics, 82, 465–73.

Marcet, A., (1991) Solving non-linear stochastic models byparameterizing expectations: an application to asset pricingwith production, Economics Working Paper 5, UniversitatPompeu Fabra.

Merton, R.C., (1971) Optimum consumption and portfolio rules in acontinuous time model, Journal of Economic Theory, 3, 373–413.

M. Normandin190

Dow

nloa

ded

by [

95.1

46.2

21.1

72]

at 1

9:14

12

Oct

ober

201

5