We back out an estimate of a personal discount rate of between 3 and 4 percent for a person with a life expectancy of 74 years who dies at age 30 (or 40) and has a value of statistical life of $6.3 million. Central to these calculations is the series generated by Murphy and Topel of value of life years (the dollar value of consumption plus the dollar value of leisure, with some smoothing for income in retirement). We employ the Makeham "model" of life expectancy in our calculations.Discount rate, Value of life

John Hartwick

English

Research Papers in Economics

QED
Queen’s Economics Department Working Paper No. 1191
The Discount Rate and the Value of Remaining Years of Life
John Hartwick
Queen’s University
Department of Economics
Queen’s University
94 University Avenue
Kingston, Ontario, Canada
K7L 3N6
11-2008The Discount Rate and the Value of Remaining
Years of Life
John M. Hartwick￿
Economics, Queen￿ s University, Kingston, Ontario
July 3, 2009
Abstract
We back out an estimate of a personal discount rate of between 3 and
4 percent for a person with a life expectancy of 74 years who dies at age
30 (or 40) and has a value of statistical life of $6.3 million. Central to
these calculations is the series generated by Murphy and Topel of value of
life years (the dollar value of consumption plus the dollar value of leisure,
with some smoothing for income in retirement). We employ the Makeham
"model" of life expectancy in our calculations.
￿ key words: discount rate, value of life
￿ JEL classi￿cation: J170, I120
1. Introduction
We back out an estimate for a representative individual￿ s discount rate, given
a value of statistical life of $6.3 million, the Makeham "model" of life expectancy,
and the schedule of "value of life years" reported in Murphy and Topel [2006;
￿Morten Nielsen provided helpful guidance as I commenced calculation of life expectancy.
1Figure 2b].1 For a person dying suddenly at age 30, we ￿nd that a discount rate
of just under 4% yields a present value of future expected years of life of $6.3 and
for a person dying suddenly at age 40, our backed-out discount rate is somewhat
lower, but still between 3% and 4%. We link values of life years to values of
remaining years of life via the Makeham "model" and various trials for a discount
rate. Our "best" trial discount rate yields an expected value of remaining years
of life at $6.3 (the value of a statistical life favored by Murphy and Topel).
The Makeham function for arriving at an expected age at death derives directly
from the well-known Gompertz function and includes an additional parameter.2
Since the probability of a person of age k dying over the current period is eas-
ily obtained from survey data, one can estimate the parameters for the "force
of mortality" function (a hazard rate function) quite easily. These parameters
also parameterize the survival function in turn and one thus has a convenient,
empirically based life expectancy "model" to work with.
2. Makeham￿ s Formulation of Life Expectancy
The Makeham hazard function is h(t) = B + ￿e￿t and the survival function is
S(t) = e
(￿Bt￿ ￿
￿ (e￿t￿1)): Then minus the derivative of the survival function is the
density function f(t) = [B + ￿e￿t]e
(￿Bt￿ ￿
￿ (e￿t￿1)): This function3 is of course a
product of the hazard rate and the survival function. Expected age at death, for
1We will report on what a "value of life year" is below and what estimates Murphy and Topel
came up with for this central series.
2For the very special case of a constant "death rate" (hazard rate) D; the expected age at
death, starting from age 0 is
R 1
0 tDe￿Dtdt which equals 1=D: Here e￿Dt is the survival function
and
R 1
0 De￿Dtdt = 1. The variance for this case is
R 1
0 (t ￿ 1
D)2De￿Dtdt which equals 1
D2:
3Bebbington, Lai and Zitikis [2007] reports on the Gompertz and Makeham density functions
for life expectancy madeling. They also report on an approach to re￿ning functions of the
Gompertz sort in order to capture better real world data on life expectancy.a person of age Z, is now
A(Z) =
Z 1
Z
t ￿ h(t) ￿
S(t)
S(Z)
dt:
and the variance is V (Z) =
R 1
Z (t￿A(Z))2 ￿h(t)￿
S(t)
S(Z)dt: One needs appropriate
parameter values for B;￿ and ￿: The issue now is to ￿t observed data on age-
speci￿c death rates to the function, B + ￿e￿t with three parameters. There is
a sizable literature on this issue. Gavrilov and Gavrilova [1991; p.76] report
values based on Swedish men from 1926-1930 of B = 0:00376; ￿ = 0:0000274 and
￿ = 0:104: These parameter values yield an expected age at death of A = 64:17
years and a corresponding variance of 495:72: The plot of the death rate (hazard
function) stays ￿ at along the time axis up to about age 45 and then rises fairly
rapidly up past 80 years. Hence we have a theoretical model of expected life
span that works and is ultimately derived from data on observed death rates. Of
interest is that the standard deviation of 22.25 years is quite large, a datum that
empirically-minded researchers do not appear to have focused their attention on.
The coe¢ cient of variation is 22:25
64:17 = 0:347:
Gavrilov and Gavrilova (p. 77) emphasize that the data on the increase in life
expectancy of the Swedish men satisfy best a Makeham equation with a simple
shift down in the value of parameter B4. Hence we proceeded to "create" a base
equation with B = 0:0000000006; ￿ = 0:0000274 and ￿ = 0:104: This yields a life
expectancy of 73.72 years and a variance of 150.27 years.5 We have plotted the
mortality schedule in Figure 1 and the density function in Figure 2.
4As B gets smaller, remaining positive, the mortality schedule in Figure 1 shifts to the right.
The Gompertz case of B = 0 corresponds to the mortality schedule implausibly "far" to the
right.
5There is a problem here. Gavrilov and Gavrilova require us to change parameter B in orderFigure 1: Hazard Rate
Recall that 73.72 was the expected age at death for this case, with age zero
the "base year".
Figure 2: Density Function
With age 30 the base year (our person has attained age 30) for our choice of
parameters above, we observe that life expectancy is 74.015 and the variance is
to track an increase in life expectancy. For B very small we are stuck with a life expectancy
of about 73.7 years. In the limit of B = 0; we retrieve of classic Gompertz equation which
does not work well empirically. For life expectancies above 74 years, we must either change
an additional parameter, which we have done as exercises, or appeal to another model. This
Makeham-Gompertz model responds fairly sluggishly to parameter changes. For ￿ reduced from
0.104 to 0.076 (B = 0:00376; ￿ = 0:0000274), we observe that the expected age at death rises
to 80.74 and the variance rises to 1261.43. The coe¢ cient of variation rises to 0:44; a sizable
jump from 0:35:135.27. With age 40 the base year, we observe life expectancy at 74.434 and the
variance is 120.55.
3. Value of Remaining Life
We proceed now to estimate the value-loss to a person dying at age 30, say in an
auto accident, using the "value per life-year" estimates from Figure 2b in Murphy
and Topel [2006]. Murphy and Topel de￿ne the "value of life year" (p. 881), m(t)
as the representative agent￿ s value of her current "full consumption" weighted by
"surplus per dollar of full consumption" plus "full income". On page 880 "full
income and consumption" are de￿ned by "adding the shadow value of non-market
time to each". In addition lifetime income is smoothed so that the is a positive
￿ ow of income to a person in retirement (post age 65). The value of a life year isthen current consumption in dollars plus current value of leisure in dollars. The
series is constructed from actual life-cycle wage and consumption data and the
series is calibrated to ￿t with a value of a statistical life of $6.3 million. The
ultimate series (Figure 2b) starts at $200,000 at age 20, rises to $360,000 at age
50, and declines to $200,000 at about age 75. It then declines slowly to about
$80,000 at age 110. For our calculations we obtained values for life-years by simply
reading them o⁄ the schedule in Figure 2b of Murphy and Topel.
We proceeded to calculate wealth (value of remaining uncertain years) W(30;r;74) =
R 1
30 L(t)￿h(t)￿
S(t)
S(30)dt for a person with an expected life of 74 years at birth, us-
ing the Makeham function for S(t) (and h(t)) and Murphy-Topel annual values of
m(t) in L(t) =
R t
30 m(z)e￿rzdz . We made exploratory runs with di⁄erent values of
the discount rate. See the detail in Table 1. (Parameter values for the Makeham
function: B = 0:0000000006; ￿ = 0:0000274; and ￿ = 0:104:)Table 1
r 0.01 0.02 0.03 0.04
W(30;r,74) $10,790,648 $8,857,493 $7,383,426 $6,243,112.3
SD $2,068,605 $1,492,942 $1,113,090 $840,037
Observe ￿rst that a discount rate just under 0.04 yields an expected value of
future life equal to $6.3. Standard deviations are listed below in the SD row. We
repeated our calculations for the case of a person dying accidentally at age 40
rather than age 30. The results in Table 2 are for the same parameter values.
Table 2
r 0.01 0.02 0.03 0.04
W(40;r,74) $8,660,002 $7,598,688 $6,547,183 $5,759,806
SD $2,039,565 $1,582,626 $1,252,408 $1,003,362
A discount rate slightly above 0.03 yields a value of a statistical life at $6.3
million. This suggests the inference (a) that a discount rate of between 0.03 and
0.04 is quite robust in the sense that the "appropriate" value of a statistical life
emerges and (b) that a personal discount rate declining slowly in middle age gets
support from this analysis.6
A typical series of terms, L(t)￿h(t)￿
S(t)
S(30) in our sum rose to about age 80 and
then slowly declined to age 110. Of interest is that L(t) is always rising over time
while h(t)￿
S(t)
S(30) rises to about 80 and then falls o⁄ rapidly. For an discount rate
of 3.0% (favored by Maurphy and Topel p. 898) our values of remaining years
are $7.383 and $6.547 million above for base age 30 and 40 respectively. These
compare with $7 and $6.2 million from Figure 3 ("Value of remaining life") of
Murphy and Topel. We are unable to say more in this comparison since there
6Azfar [1999] provides an analysis justifying a personal discount rate declining over a lifetime.is no detail in Murphy and Topel about their construction of the series in their
Figure 3.7
Our two inferences (discount rate between 0.03 and 0.04, and discount rate
declining slowly between age 30 and 40) received support when we repeated our
analysis above for two very di⁄erent selections of parameters for the Makeham
function. Each of these two parameter selections (B = 0:00376; ￿ = 0:104 and
￿ = 0:0000034; and B = 0:00376; ￿ = 0:076 and ￿ = 0:0000274) yielded an
expected age at death of 80 years and each yielded expected values for remaining
life of $6.3 million from ages 30 and 40 with a discount rate between 0.03 and
0.04. In addition between base years 30 and 40 the appropriate discount rate
declined slightly for each case. These parameter values have the merit of yielding
reasonable life expectancies but are quite far o⁄ those endorsed by Gavrilov and
Gavrilova. Hence one should interpret the outputs from these "extra" analyses as
weak support for the our two main inferences.
References
[1] Azfar, O. [1999] "Rationalizing Hyperbolic Discounting" Journal of Economic
Behavior and Organization, 38, pp. 245-52.
[2] Bebbington, Mark, Chin-Diew Lai and Ricardas Zitikis [2007] "Modeling Hu-
man Mortality using Mixtures of Bathtub-shaped Failure Distributions", Jour-
nal of Theoretical Biology, 245, pp. 528-538.
7With regard to Firgure 3 (Value of remaining life) we have the comment: "The e⁄ects of
discounting and future mortality are apparent: the value of remaining life reaches $7 million
near age 30 and then falls, but Figure 2b showed that the value of a life-year rises until age 50."
Since Murphy and Topel are explicit that the "value of life years" series is calibrated to a value
of statistical life of $6.3 million, it seems reasonable to assume that the two series (Figures 2
and 3) were generated simultaneously.[3] Gavrilov, L.A. and N.S. Gavrilova [1991] The Biology of Life Span: A Quan-
titative Approach, Chur: Harwood.
[4] Murphy, Kevin M., Robert H. Topel, [2006] "The Value of Health and
Longevity", Journal of Political Economy, vol. 114, no. 5, October, pp. 871-
904.

Chin-Diew Lai and Ricardas Zitikis [2007] &quot;Modeling Human Mortality using Mixtures of Bathtub-shaped Failure Distributions&quot;,

Rationalizing Hyperbolic Discounting&quot;

The Value of Health and Longevity&quot;,

The Discount Rate and the Value of Remaining Years of Life

http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1191.pdf

The Discount Rate and the Value of Remaining Years of Life

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics