## Abstract

I estimate the effect of shocks to subjective mortality hazards on consumption expenditures of retired individuals using the Survey of Health, Ageing and Retirement in Europe. I measure mortality expectations with survey responses on survival probabilities. To create plausibly exogenous variation in mortality hazard, I use the death of a sibling as an instrument. My results show that survey responses contain economically relevant information about longevity expectations and confirm the predictions of life-cycle theories about the effect of these expectations on intertemporal choice.

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## Notes

At the aggregate level and in the long run, increasing longevity can affect the income level partly through intergenerational transfers (see Hock and Weil (2012) for such an analysis).

There are two decision variables:

*C*_{ i0}and \(T_{i}^{\ast }.\) Based on the assumption of exact depletion, \(C_{iT_{i}^{\ast }}=Y_{i}\), and \( C_{iT_{i}^{\ast }}=C_{i0}\left( S_{0}^{T_{i}^{\ast }}\right) ^{\frac{\eta _{i0}}{\gamma }}\left( \beta R\right) ^{\frac{T_{i}^{\ast }}{\gamma }}\) from the Euler equation. Thus, \(Y_{i}/C_{i0}=\left( S_{0}^{T_{i}^{\ast }}\right) ^{ \frac{\eta _{i0}}{\gamma }}\left( \beta R\right) ^{\frac{T_{i}^{\ast }}{ \gamma }}\), which shows that given income and initial consumption \( T_{i}^{\ast }\) has to decrease if the mortality hazard increases (i.e.*η*_{ i0}increases).This representation implies that the shock has a permanent effect in the sense that the survival probabilities to each future periods are affected. If instead only the short run expectations change, then the effect on consumption level has the same sign but smaller magnitude.

If wealth is allowed to be depleted before time

*T*_{ i }, then the solution of the consumption model can be found only numerically. Numerical results indicate that the effect of an upward hazard shock on consumption expenditures is positive, and an upward hazard shock might decrease \( T_{i}^{\ast }\). The upward shift in the optimal consumption level depends on the parameters in the model. Holding the income fixed, if the wealth level is higher, then the optimal consumption level is more sensitive to the hazard shock. The sensitivity is not a monotone function of the coefficient of relative risk aversion, but, at higher relative wealth holdings, the effect of shock decreases with the risk aversion coefficient.Using

*dh*_{ i1}in Eq. 10 is also a simplification, which can support the interpretability of the empirical results. Apart from hazard shocks, the hazard increases due to ageing between two dates of observation, which effect is also included in*dh*_{ i1}, but not in \(-\eta _{i1}\ln S_{1}^{t}+\eta _{i0}\ln S_{1}^{t}\). The sign of the estimated effect of changing hazard is robust to this simplification. If the two indicators of changing hazard are cleared from the effect of ageing, then their estimated effects become stronger.This paper uses data from SHARE release 2.3.1, as of July 29, 2010. SHARE data collection in 2004–2007 was primarily funded by the European Commission through its fifth and sixth framework programmes (project numbers QLK6-CT-2001- 00360; RII-CT- 2006-062193; CIT5-CT-2005-028857). Additional funding by the US National Institute on Aging (grant numbers U01 AG09740-13S2; P01 AG005842; P01 AG08291; P30 AG12815; Y1-AG-4553-01; OGHA 04-064; R21 AG025169) as well as by various national sources is gratefully acknowledged (see http://www.share-project.org for a full list of funding institutions).

The wording of the question is the following: “Thinking about the last 12 months: about how much did your household spend in a typical month on food to be consumed at/outside home?” This amount is multiplied by 12 to generate the annual amount.

Alternatively, the empirical analysis could also be based on the HRS merged with the Consumption and Activities Mail Survey. These data include subjective survival probability measures and more general consumption measures, as well. However, the identification strategy applied here does not work with these alternative data. Based on the Rand HRS data file (version L, waves 5–10), the death of a sibling does not have significant effect on the first differenced subjective hazard, and its effect on the binary indicator of hazard shock is positive but small with a

*p*value of 0.08. This difference from the SHARE findings can be due to more severe measurement errors, different reporting styles, or less tight family relationships among siblings.Source: http://apps.who.int/whosis/database/life_tables/life_tables.cfm. These are period (or current) and not cohort life tables. Period life tables might underestimate the survival probabilities to old ages. Since the life tables are used only for adjusting the reported probabilities, using period life tables does not cause bias in the estimates.

Based on the WHO life tables, the survival probabilities can be determined only for 5-year age ranges. In order to calculate the survival probability to any age, I make the simplifying assumption that the number of people alive from a given cohort declines linearly within the given 5-year intervals.

The rest of the included regressors have mostly the expected sign. Due to measurement errors (as also reflected by the focal responses), the estimation results of the models of changing hazard are noisier than the results of the hazard level model, and the explanatory power of the regressions is small. There are some unexpected results, e.g. the negative coefficient of a new diagnosis with diabetes. The negative sign can be due to changing health behaviour after the diagnosis (e.g. healthier diet) but can be also caused by measurement errors among the 300 newly diagnosed respondents.

The instrumental variable models are estimated and tested by the

*ivreg2*command of Stata, as provided by Baum et al. (2007).I present in Table 9 in the Appendix the detailed IV estimation results, restricting the sample to those with positive wealth. I test the joint significance of the control variables other than the hazard and age indicators. The

*p*value of the chi-squared statistic is 0.61 (if the differenced hazard is included) and 0.37 (if the binary indicator of increasing hazard is included).The estimated effect of increasing hazard is a “local average treatment effect” if there is a systematic difference between those who update their expectations after a sibling’s death and those who do not. In this paper, I use the simplifying assumption that the effect of a hazard shock on the logarithmic consumption expenditures is constant across the analysed population.

These calculations are based on the German life table. It is assumed that before the hazard shock, the subjective survival probability of this individual was equal to the life table survival probability.

The indicator of noresponse is set to one if the differenced survival probability is missing. The control variables besides the country dummies are age, age squared, gender, marital status, education level, income and self-reported health status. I also control for the interviewer’s observation of declining willingness to answer during the interview. This is reasonable since the expectation questions are in the final block of the SHARE questionnaire.

Fuller’s estimator is a member of the

*k*-class estimators. If the structural model is*Y*=*Xβ*+*u*, then the*k*-class estimator is \(\hat{ \beta}=\left( X^{\prime }(I-kM_{Z})X\right) ^{-1}X^{\prime }(I-kM_{Z})Y\). Here,*Z*is the vector of first-stage regressors, and \(M_{Z}=I-P_{Z}=I-Z \left( Z^{\prime }Z\right) ^{-1}Z^{\prime }\). The OLS estimator is obtained if*k*= 0, the 2SLS is obtained if*k*= 1. The LIML estimator is obtained if*k*=*λ*, where*λ*is the smallest eigenvalue of the matrix \( W^{\prime }P_{Z}W(W^{\prime }M_{Z}W)^{-1}\) with*W*= [*Y*,*X*].In Fuller’s estimator

*k*=*λ*−*a*/(*N*−*K*), where*N*is the number of observations, and*K*is the number of regressors in the first-stage model. If*a*= 1, then the model is approximately unbiassed; if*a*= 4, then there is bias, but the mean squared error is smaller. Further details about these estimation methods are provided by Davidson and MacKinnon (1993) and Hahn et al. (2004).The

*jive*command of Stata written by Poi (2006) is applied in the jackknife estimation.The following simplifying assumptions are made in the two-period life-cycle model. The utility of bequest has the same functional form as that of consumption, but multiplied with an individual specific multiplicator (

*B*_{ i }). This term indicates the strength of the bequest motive. In the first period, the individual decides on the current consumption level, and in the second period, he/she either consumes all the remaining wealth (if survives) or leaves bequest (if dies).Under these assumptions, the sign of the effect of subjective hazard on the optimal consumption level is the same as the sign of \(\left( 1-B_{i}\right)\), provided that the credit constraint is not binding. Therefore, the partial effect of mortality hazard is smaller if bequest motives are stronger.

An alternative approach could be to use the reported subjective probability of leaving inheritance. However, it is not obvious that this measure indicates bequest motives; it suffers from measurement errors similarly to the subjective survival probability, and again, the majority (around 75 %) of the respondents indicate positive probability.

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## Acknowledgements

An earlier version of this paper is a part of my PhD thesis written at the Central European University. I am grateful for comments received from the anonymous referees, Gábor Kézdi, Miklós Koren, from seminar participants at CEU, HECER, IHS, ROA, The University of Edinburgh, and at the European Winter Meeting of the Econometric Society in Rome.

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*Responsible editor:* Junsen Zhang

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Bíró, A. Subjective mortality hazard shocks and the adjustment of consumption expenditures.
*J Popul Econ* **26**, 1379–1408 (2013). https://doi.org/10.1007/s00148-012-0461-5

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DOI: https://doi.org/10.1007/s00148-012-0461-5

### Keywords

- Consumption decisions
- Subjective longevity
- IV estimation