DIFFERENTIAL MORTALITY, UNCERTAIN

DIFFERENTIAL MORTALITY, UNCERTAIN

MEDICAL EXPENSES, AND THE SAVING OF

ELDERLY SINGLES

Mariacristina De NardiFederal Reserve Bank of Chicago, NBER,

and University of Minnesota

Eric FrenchFederal Reserve Bank of Chicago

John Bailey JonesUniversity at Albany, SUNY

May 10, 2006Saving of Singles: May, 2006 – p. 1/36

What we do

Formulate and

estimate

a structural model of savings after retirement allowing forheterogeneity in

life expectancy

medical expenses

Saving of Singles: May, 2006 – p. 2/36

What are we trying to understand?


Figure 1: AHEAD data


Why our model?

Data show considerable heterogeneity in

life expectancy

medical expenses

by:

Sex

Permanent income

Health


Heterogeneity implications

For saving behaviorDifferential mortality⇒ heterogenous saving rates,with high PI people and women saving more.Medical expenses rise quickly with age⇒ keep assetsfor old age.Medical expenses rising with PI⇒ high PI people saveat higher rate.

For observed samplemortality bias


Figure 2: Median assets by birth cohort, AHEAD data


How we do it

First step: estimate mortality and medical expenses as afunction of age, sex, health and permanent income.

Second step: use first step results to estimate our modelwith method of simulated moments.


Contributions

Estimate medical expenses using better data and moreflexible functional forms.

Medical expenses rise quickly with age and PI.

Estimate mortality probabilities by age, sex, permanentincome, and health.

Variation is large.

Construct and estimate a rich model of saving.Reasonable parameter estimatesModel fits the data extremely well.

Find that medical expenses and social insurance are key tounderstanding the elderly’s savings.


Related Literature (Subset)

Gourinchas and Parker (2001), Cagetti (2003):Savingprior to retirement⇒ mortality not a big issue, also lackmedical expense risk.

Hurd (1989, 1999):Only risk is uncertain mortality.

Palumbo (1999):Considers medical expense risk, notdifferential mortality.


Model

Singles only,abstract from spousal survival.

Householdsmaximize total expected lifetime utility.

Flow utility from consumption (CRRA). Utility can varywith health.

Rational expectations.Beliefs about mortality rates, healthcost distribution, etc., are estimated from the data.

No bequest motive


Uncertainty

Health status uncertainty: Health status follows age-,gender- and permanent-income-specific Markov chain.

Survival uncertainty: Mortality rates depend on gender,age, health status, and permanent income.

Medical expense uncertainty:Persistent and transitory shocksDistributions shift with age, gender, health status andpermanent income.


Constraints

Budget constraint:

at+1 = at + y(rtat + yt, τ) + trt − hct − ct

= xt − ct.

y(.) = post-tax income;yt = “non-interest” income;trt = government transfers;hct = medical expenses;xt = “cash-on-hand”.

Transfers support a consumption floor:

xt ≥ cmin.

Borrowing constraint:

at+1 ≥ 0 ⇔ ct ≤ xt.Saving of Singles: May, 2006 – p. 13/36

Recursive Formulation

Vt(xt, g, I,mt, ζt) = maxct,xt+1

{

[1 + δmt]c1−νt

1 − ν+

βsg,m,I,tEt

(

Vt+1(xt+1, g, I,mt+1, ζt+1))

}

mt = health status (0 ⇒ bad,1 ⇒ good)g = genderI = permanent incomext = cash-on-handζt = persistent health cost shock


Constraints in Detail

xt+1 = max{xt − ct + y(

r(xt − ct) + yt+1, τ)

− hct+1, cmin},

yt+1 = y(g, I, t+ 1),

xt ≥ cmin,

ct ≤ xt,

ln(hct+1) = hc(g,mI,t+1, t+ 1, I) + σ(g,mI,t+1, I, t+ 1)ψt+1,

ψt+1 = ζt+1 + ξt+1.


Method of Simulated Moments

Our approach: Match median assets by permanent incomequintile, cohort and age.

Consider HHi of birth cohortc in calendar yeart,belonging to theqth permanent income quintile.

Let aqct denote the model-predicted median asset level.

Moment condition for GMM criterion function:

E (I{ait ≤ aqct} − 1/2|q, c, t,hh alive att) = 0.


Econometric Problem 1: Cohort Effects

Older HHs are born in earlier years and have lower lifetimeincomes⇒ understate asset growth and saving.

Our solution: Cohort- and permanent income-specificmoments


Econometric Problem 2: Mortality Bias

Sample composition changes (High PI people live longer)

+ →Our solution: Allow mortality rates to depend on permanentincome and gender.


AHEAD Data

Household heads aged 70 or older in 1993

Consider only the retired

Follow-up interviews in 1995, 1998, 2000, 2002

Asset data begins in 1995 (1993 asset data faulty), uses2,793 individuals

Use full, unbalanced panel


Results from first step estimation


050

0010

000

1500

020

000

2500

0

70 80 90 100age

_20th_percentile _40th_percentile_60th_percentile _80th_percentile

1998

dol

lars

Medical Expenses, by Permanent Income Percentile, Women in Good Health

Figure 3: Average medical expenses, AHEAD data


Income Healthy Unhealthy Healthy UnhealthyPercentile Male Male Female Female All

20 8.2 6.2 13.8 11.9 12.040 9.1 7.0 14.8 12.9 13.060 10.1 7.9 15.9 14.1 14.180 11.2 9.1 17.0 15.5 15.2Men 10.2Women 15.0Healthy 15.3Unhealthy 11.9

Table 1: Life expectancy at age 70


0.1

.2.3

.4.5

70 80 90 100age


Probability of Death, by Permanent Income Percentile, Men, Bad Health

Figure 4: Mortality probabilities, AHEAD data


Results from second step estimation


Estimated structural parameters

Baseline δ = 0 cmin = 5KParameter (1) (2) (3)ν: coeff. relative risk aversion 4.15 4.35 6.951

(0.90) (1.03) (1.73)β: discount factor 0.976 .948 0.950

(0.07) (0.08) (0.09)δ: pref. shifter, bad health -0.228 0.0 -0.251

(0.21) – (0.18)cmin: consumption floor 2904 2661 5000

(319) (468) –


Figure 5: Median assets by cohort and PI quintile: data and bench-

mark modelSaving of Singles: May, 2006 – p. 26/36

Mortality Bias

Figure 6: Left panel→ AHEAD data; right panel→ benchmark

model


Findings from estimated structural model

Fix preference parameters at baseline estimates, vary otherparameters.

Lowering the consumption floor to $500 has a big effect onsavings, even for the rich.

Eliminating medical expense risk has small effects.

Eliminating out-of-pocket medical expenditures has bigeffects.

Life expectancy matters.


Figure 7: Benchmark and model with a $500 consumption floor


Figure 8: Benchmark and model with no medical expenditure risk


Figure 9: Benchmark and model with no medical expenditures


Figure 10: Benchmark and model in which everyone has the life ex-

pectancy of a healthy woman at the top 20% PISaving of Singles: May, 2006 – p. 32/36

Conclusions

Medical expenses important.

Consumption floor important.

To correctly evaluate any policy reform affecting theelderly’s saving decisions, need to model both theconsumption floor and the way in which medicalexpenditures by age and PI.


7000

1000

015

000

2000

0

70 80 90 100age


1998

dol

lars

Income, by Permanent Income Percentile, Healthy Women

7000

1000

015

000

2000

0

70 80 90 100age


1998

dol

lars

Income, by Permanent Income Percentile, Unhealthy Women

Figure 11: Average income


0.1

.2.3

70 80 90 100age


Pro

b(he

alth

=ba

d)

In Good Health 1 Year Ago, MenProbability of Being in Bad Health, by Permanent Income Percentile,

.7.8

.9

70 80 90 100age


Pro

b(he

alth

=ba

d)

In Bad Health 1 Year Ago, MenProbability of Being in Bad Health, by Permanent Income Percentile,

Figure 12: Health transition probabilities


Figure 13: Median consumption by cohort and PI quintile: bench-

mark modelSaving of Singles: May, 2006 – p. 36/36

DIFFERENTIAL MORTALITY, UNCERTAIN

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