is easily shown to apply to the effects of a change in current wealth as well. Table 1 thus generally applies to this section as well.
The existence of a market for actuarial notes, however, does exert an indepen- dent influence on the value of life saving, as is apparent from a comparison of the
Ž .
Ž .
first components of Eqs. 3.5 and 3.5a , representing the willingness to pay for a marginal reduction just in the risk of mortality. The direction of that influence,
seemingly ambiguous at this point, will be clarified in Section 6 below, following our more general analysis of the issue in the next two sections.
5. The model with complete insurance markets and bequest preferences
The market for actuarial notes creates opportunities to insure separately the financial hazards associated with ‘‘living too long’’ and ‘‘living too short’’. A
positive demand for life insurance requires, however, the existence of a bequest motive or altruism toward dependents.
As in the preceding case of insurance against living too long, insuring depen- dents against the consequences of one’s premature death can be achieved through
Ž .
either market insurance i.e., conventional life insurance or self-insurance through ordinary savings, which are inherently transferable to survivors. In this context,
however, neither form of insurance dominates the other. Indeed, both are subject to identical opportunity costs on the margin, since life insurance policies are
purchased implicitly by selling an actuarial note and buying instead the right to a
Ž .
regular saving note see Yaari, 1965 . Thus, the mix of life insurance and regular savings is in principle indeterminate. The critical individual choice concerns the
sum of the two, i.e., the optimal amount of bequest or ‘‘total life insurance’’. To facilitate the derivation of an analytical solution for this general, but more
complex case, it is necessary to introduce some simplifications outlined below. The solution generalizes earlier specifications of the comparably defined ‘‘ value
of life saving’’.
7
5.1. Introducing bequest Self-protection is assumed to affect exclusively mortality risk. The impact of
protection on morbidity risk, hence current earnings or utility, is thus ignored. Although designed to simplify the analysis, this assumption is in line with the
7
Ž .
Ž .
Ž .
See, in particular, the studies by Thaler and Rosen 1975 , Conley 1976 , Viscusi 1978, l978a , Ž
. Ž
. Ž
. Arthur 1981 , Shepard and Zeckhauser 1984 , and the articles in Jones Lee 1982 . Some of these
studies have discussed the effect of specific variables on the value of life saving, and have attempted to link the latter with the discounted value of labor income. But none has developed a comprehensive
‘‘indirect’’ value function, which accounts specifically for the role of bequest preferences, borrowing and saving decisions, age, components of wealth, and insurance markets in affecting the value of a
marginal reduction in mortality risk.
literature on the value of life saving, which has generally ignored morbidity risks. The analysis of Section 4 is generalized in an important way, however, by
Ž .
incorporating in the objective function a utility-of-bequest function, V B,t with the following properties:
V 0,t F 0; V B,t 0, V
B,t - 0, B G 0, 5.1
Ž .
Ž .
Ž .
Ž .
B B B
where B stands for the amount of bequest. Under actuarially fair pricing of personal annuities and life insurance, the force
Ž Ž .. of mortality
f t represents the unit opportunity cost of regular saving and life
Ž .
insurance hence, bequest as well as the unit premium on annuities. The instanta- neous wealth constraint becomes:
2
A s r q f t A t q wh t y cI
t y Z t y f t B t , 5.2
Ž . Ž .
Ž . Ž .
Ž . Ž . Ž .
Ž .
Ž .
1 1
where A represents here the sum of accumulated ordinary savings and annuities,
1
or financial wealth, subject to the restriction that a person cannot reach the end Ž .
point of the planning horizon with negative financial wealth, or A D G 0, by the
1
institutional constraints on actuarial notes. These constraints also restrict bequest not to exceed the person’s total wealth, i.e.,
0 F B t F A t q L t G 0, 5.3
Ž . Ž .
Ž . Ž
.
1
Ž . where L t denotes the discounted value of potential net life-time earnings stream,
or human wealth.
8
They further require that total wealth cannot be negative, as borrowing cannot exceed human wealth.
The maximized expected utility function in this formulation is given by:
D
J A t ,t ;a s Max
E exp yr s y t
U Z s , h s d s
Ž . Ž
. Ž . Ž .
Ž .
Ž .
H
1 1
Z , B , I
½
t
w
qexp y r D y t V B D ,t ,
5.4
Ž .
Ž . Ž
.
Ž .
5
Ž .
Ž .
subject to Eqs. 5.2 and 5.3 . Again, by the stochastic dynamic optimization approach, the optimal solution must satisfy:
2 U
U U
U
yJ s yr J q U Z q J
r q f A q wh y c I y Z y fB
Ž .
Ž .
Ž .
1 t 1
1 A 1
U
q f V B ,t y J ,
5.5
Ž .
Ž .
1
Ž Ž .
. Ž
U
Ž . .
subject to
the boundary
conditions J
A D , D; a s V B D , D s
1 1
8
Ž .
D
w Ž
. Ž
.xw Ž .
2
Ž .x That is, L t s
H exp y r uy t y m t,u wh u y cI
u du. This expression represents the discounted value of net potential future labor income — i.e., the discounted gross value of all healthy
time net of life protection costs, including the opportunity costs of time spent at health preservation.
Ž Ž .
. Ž .
U U
U
V A D , D and A D G 0. The optimal control variables I , Z , and B are
1 1
then solved from:
U U
U U
2 cI s 1rJ J
A t ,t ;a y V B q B y A Õ
t 5.6
Ž .
Ž . Ž
. Ž .
Ž .
Ž .
1 A 1
1 1
1
U Z
U
s J 5.7
Ž .
Ž .
Z 1 A
B
U
s 0 if J V
X
B
U
Ž .
1 A
B
U
s ´ 0, A t q L t , if J
s V
X
B
U
Ž . Ž .
Ž .
Ž .
1 1 A
B
U
s A t q L t , if J -
V
X
B
U
. 5.8
Ž . Ž .
Ž .
Ž .
1 1 A
A corner solution for optimal bequest, and hence for total life insurance, is obtained where the marginal value of bequest either falls short of or exceeds the
expected marginal utility of wealth over the remaining life span. A no-bequest, thus no-life-insurance solution, is more likely, of course, if concern for dependents
is small, as may be the case for unrelated individuals. But in this case, the existence of the self-protection alternative will be shown to increase the person’s
demand for it. An interior solution for bequest and life insurance is the more representative case, however, and the following analysis focuses on this case and
its behavioral implications.
5.2. Explicit solutions To obtain an explicit solution for the maximized life-time expected utility
Ž .
function J in Eq. 5.5 , the instantaneous utility of consumption function is
1
specialized to exhibit ‘constant relative risk aversion’, U Z s 1rk Z
k
, 5.9
Ž . Ž
. Ž
.
Ž .
where k - 1 to assure concavity, and d s 1 y k denotes the degree of relative risk aversion. The utility of bequest function is similarly specialized as:
k
V B,t s n t rk B , 5.10
Ž .
Ž . Ž
.
Ž . with n t representing the intensity of utility derived by the individual from capital
Ž Ž
.. to be bequeathed to heirs see Richard 1975 . Finally, the optimal consumption
and bequest choices are now taken to be conditional on a predetermined path of optimal self-protection outlays.
These simplifications permit an explicit solution for the partial differential Eq. Ž
. 5.5 . Consider the following indirect expected utility function:
k
J A t ,t ;a s a t rk
A t q L t ,
5.11
Ž . Ž .
Ž . Ž .
Ž .
Ž .
1 1
1
as a solution candidate. Using the optimality conditions for consumption and Ž
. Ž
. bequest under fair insurance, Eqs. 5.7 and 5.8 , the utility functions specified in
Ž .
Ž .
Ž .
Eqs. 5.9 and 5.10 , and the boundary conditions associated with Eq. 5.5 , by Ž .
Ž . Ž
. which a D s n D , one can verify that Eq. 5.11 does indeed provide a unique
solution for the optimal value function, with the marginal expected utility of a unit Ž .
of wealth while one is alive, a t , given by:
D
Ž .
Ž .
1rd 1rd
a t s exp y
x u d u n D
Ž . Ž .
Ž .
H
½ 5
u D
q y u exp y
x s d s d u, 5.12
Ž . Ž .
Ž .
H H
½ 5
t
Ž . Ž .
w Ž
. Ž .x
Ž . Ž .w Ž .x
Ž1r d .
where x u f u q r y rk r 1 y k , and y u 1 q f u n u
. Ž
. Ž
. Eqs. 5.11 and 5.12 provide the explicit solution for the indirect expected
utility of the remaining life span we have sought. It thus becomes possible to derive explicit solutions for the optimal consumption, bequest, and the value-of-
life-saving functions as well. The latter variable is the shadow price of a marginal
U
Ž .
reduction in mortality risk, Õ , as defined in Eq. 5.6 :
1 U
Õ t s E J
A t ,t ;a rE f t rJ
A t ,t ;a , or
Ž . Ž .
Ž . Ž .
Ž .
Ž .
1 1 t
1 1 A
1 U
U
Õ t s J
A t ,t ;a y V B t ,t
rJ A t ,t ;a
Ž . Ž .
Ž . Ž .
Ž .
Ž .
Ž .
1 1
1 1 A
1
q B
U
t y A t . 5.6a
Ž . Ž .
Ž .
1
Ž .
Eq. 5.6a summarizes, in value terms, the three basic outcomes of a marginal reduction in mortality risk in the presence of complete and actuarially fair
Ž . insurance markets:
1 an increase in personal welfare by the extent of the
9
Ž .w Ž
. Ž
U
.x difference between the utility from living and dying,
1rJ J A y V B
;
1 A 1
1
Ž . 2 a reduction in the total return on investment in actuarial notes, if the individual
Ž . is a net saver, due to the fall in the risk premium paid on such notes, yA ; and 3
1
Ž a reduction in the opportunity cost of bequest
whether in the form of life .
Ž .
insurance or regular saving due to the fall in the insurance premium qB . A more useful summary of the net effect of these outcomes is obtained by substitut-
Ž . Ž
. Ž .
Ž .
Ž .
ing Eqs. 5.8 ,
5.9 , 5.10
and 5.11
directly in Eq. 5.6a
to produce a closed-form solution for the value of life saving function we have sought:
U
Õ t s z t
A t q L t y A t
5.13
Ž . Ž .
Ž . Ž .
Ž . Ž
.
1 1
1
Ž . w
Ž . Ž .x
Ž1r d .
Ž .
w Ž .
Ž .x
Ž1r d .
4 where z t s n t ra t
q 1rk 1 y n t ra t .
Ž .
Eq. 5.13 represents an explicit solution for the private value of life saving as a function of personal and market-dictated parameters. This solution takes as given
Ž .
optimal self-protection outlays, although Eq. 5.6 implies that these outlays, in Ž
. turn, are determined by the optimal value of life in Eq. 5.13 . It is possible,
however, to relax the assumption that optimal self-protection outlays are given,
9
Ž Ž . .
Ž
U
Ž . . The solution must also satisfy the stability condition J A t ,t;a V B
t ,t , t - D, by which
1
the utility from living must exceed that from dying during all periods of life. Of course, by the
U
Ž . Ž .
Ž . boundary conditions, Õ s 0 at t s D, since J D sV B , and B s A D .
1 1
1
Ž .
Ž .
and proceed with a simultaneous solution of Eqs. 5.6
and 5.13
through Ž
. simulation analyses. Ehrlich and Yin
1999 pursue such simulations, which
confirm the validity of the closed-form solutions for the value of life saving in Eq. Ž
. 5.13 , as well as for all the other control variables of the model summarized
below.
10
The explicit solutions for optimal consumption and bequest are similarly Ž
. Ž .
Ž .
obtained by exploiting Eqs. 5.7 , 5.8 and 5.11 as follows:
Ž .
1rd U
Z t s 1ra t
A t q L t ,
5.7a
Ž . Ž .
Ž . Ž .
Ž .
1
and
Ž .
1rd U
B t s n t ra t
A t q L t .
5.8a
Ž . Ž .
Ž . Ž .
Ž . Ž
.
1
These solutions specify both consumption and bequest to be strictly proportional to current expected wealth at any point in time. Furthermore, by inserting the
Ž .
explicit solution for bequest in Eq. 5.8a , the closed-form solution for the optimal value of life saving can be rewritten as:
U U
Õ t s 1rk L t q
1 y k rk A t y B
t .
5.13a
Ž . Ž
. Ž . Ž
. Ž .
Ž . Ž
.
1 1
Ž .
U
Ž . Ž .
The term A t y B t Q t represents the amount of actuarial notes owned by
1
the individual. This alternative measure of the value of life saving has the further merit of being estimable empirically using available data on individual human
wealth, the net amount of actuarial notes, and independent assessments of the Ž
. coefficient of relative risk aversion d s 1 y k .
5.3. BehaÕioral propositions Certain restrictions apply to the parameters of the demand and value functions
just introduced. As long as life insurance and annuities are available at actuarially Ž .
Ž .
fair terms, the magnitude of z t in Eq. 5.13 will necessarily exceed unity in all periods of life. Indeed, by combining the implicit stability and optimality condi-
Ž tions, which require that the utility from living exceed that from dying see
. Footnote 9 and that the marginal utilities of consumption and bequest be equal at
any period of life, it is easy to show that:
11
B
U
F A q L and nra F 1 if 0 F k - 1; 5.14
Ž .
Ž .
1
B
U
A q L and nra 1 if k - 0.
Ž .
1
Ž .
U
Ž .
U
Ž . Ž
. Ž
. But the wealth constraint 4.3 restricts Z
t and B t in Eqs. 5.7a and 5.8a
10
Ž .
An elaborate calibrated simulation analysis is pursued in Ehrlich and Yin 1999
whereby
U
Ž . self-protection, I
t , and the corresponding path of mortality risks and current human wealth are
U
Ž .
U
Ž .
U
Ž . simultaneously determined with Õ
t , Z t , and B
t . These simulations produce unique and stable
1
Ž . Ž
. Ž
.
U
Ž . dynamic solutions for Eqs. 5.13 , 5.7a and 5.8a , as well as for I
t , thus proving the validity of the closed-form solutions summarized by these equations.
11
Ž .
k
Ž .w
x
k ky1
w x
ky 1
The conditions n r k B - ar k
A q L and nB
s a A q L together imply that
1 1
Ž .
Ž .
Ž . Ž
. Ž
. Br k - A q L r k, so that B -
A q L if k - 0.
1 1
Ž . Ž .
to be strictly less than A t q L t in all periods of life. Therefore, only a value
1
of 0 F k - 1 is permissible under fair insurance. Put differently, a stable solution under actuarially fair market insurance restricts
both the total and the marginal utility from a dollar of own wealth not to fall short w
Ž . x
w Ž .
x Ž .
Ž . of that from bequeathed wealth, or a t rk G n t rk and a t G n t , at all
12
Ž . time periods.
The first restriction alone guarantees that z t G 1, i.e., that the
U
Ž . value of life saving, Õ
t , remain non-negative in all periods of life, regardless of
1
Ž . the magnitude of human wealth, L t . A negative value of life would be
inconsistent with a stable equilibrium because it would generate an incentive to engage in ‘‘negative self-protection’’, i.e., precipitate an increase in the risk of
mortality, which also rules out as optimal any positive expenditures on self-protec- tion in earlier periods of life.
Ž .
There is a clear link, according to Eq. 5.13 , between the relative bequest preference and the value of life saving. In the rare case where a dollar bequeathed
w Ž .
x w
Ž . x
Ž yields the same utility as a dollar enjoyed while alive, or n t rk s a t rk
the .
‘‘infinitely lived household’’ case , optimal bequest will exhaust the individual’s wealth constraint, and the value of life will just equal the discounted value of
Ž . expected future labor income, L t , which had served as a conventional measure
Ž of value of life in the early literature on human capital see Dublin and Lotka
Ž ..
1946 . The intuitive reason is that in this case a person derives equal satisfaction from a given amount of financial wealth, whether directly or vicariously through
survivors, but forgoes the benefits of expected future labor income in the event of death.
Ž .
The analysis is consistent with the proposition of Conley 1976 that the value of life saving exceeds one’s expected human wealth. It shows this to be the case
under actuarially fair insurance even when individuals have a strong bequest Ž .
Ž .
motive, provided they are net savers, or A t 0 but see Footnote 12 . If a
1
Ž Ž .
. person is a net borrower A t - 0 , however, the optimized value of life is seen
1
Ž .
to approach one’s expected net labor income from above as borrowing ap- Ž .
Ž . proaches its upper bound A t s yL t , regardless of bequest preferences. In
1
12
If an identical loading factor q 0 were applied to both lending and borrowing transactions in Ž
. Ž Ž
.. Ž
.. actuarial notes, raising the risk premium for both from f to 1q q f see Eq. 5.5 , Eq. 5.12
would Ž .
Ž . Ž .
Ž . Ž .w
Ž . x
remain valid except that the functions x u and y u would become x u 1r d f u 1y 1q q f
1
4 Ž .
Ž .
y Ž k r d .
Ž . Ž .
Ž1r d .
Ž .
q r y rk and y u 1q 1q q f u n u
, respectively, and Eq. 5.13 would become
1
Ž .
U
Ž . Ž .w
Ž . Ž .x
Ž .
Ž . Ž .
w x
y Ž k r d .
w Ž .
Ž .x
Ž1r d .
Eq. 5.13a Õ t s z t
A t q L t y 1q q A t , where z t 1q q n t r a t
q
2 1
1 1
1
Ž .
w x
y Ž k r d .
w Ž .
Ž .x
Ž1r d .
4 Ž
. 1r k 1y 1q q
n t r a t . By the reasoning underlying the derivation of Eq. 5.15 ,
Ž .
Ž .
w Ž .
Ž .x Ž
. one cannot rule out in this case the possibility that n r k ar k , provided that n t r a t F 1q q ,
U
Ž . w Ž .
Ž .x Ž .4
Ž1r d .
w Ž .
Ž .x which guarantees the equilibrium condition that B
t s n t r a t r 1q q A t q L t -
1
Ž . Ž .
Ž . Ž .
Ž . Ž .
A t q L t . In this case, z t would be less than unity and Õ t could be less than L t if A t 0
1 1
2 1
Ž Ž
. .
see Eq. 5.13a above . This result may be of limited empirical value, however, since the existence of transaction costs in insurance suggests that loading factors of opposite signs should be applied to
borrowing and lending transactions. No general solutions may be obtained under such insurance terms.
contrast, the analysis shows that in periods where the relative bequest preference Ž .
Ž . n t ra t is negligible, the value of life may exceed the sum of both human and
Ž .
non-human wealth if the degree of relative risk aversion in Eq. 5.9 is sufficiently Ž
. high, or 1r2 F d - 1. Formally,
Õ t s L t , if n t ra t s 1
Ž . Ž .
Ž . Ž .
1
Õ t s z t
A t q L t y A L t ,
Ž . Ž .
Ž . Ž .
Ž .
1 1
1
if n t ra t -1 and A t G0
Ž . Ž .
Ž .
1
Õ t f L t , if n t ra t - 1 and A t f yL t
Ž . Ž .
Ž . Ž .
Ž . Ž .
1 1
Õ t s 1rk
L u q 1 y k A u , if n u ra u f 0.
5.15
Ž . Ž
. Ž .
Ž .
Ž . Ž .
Ž . Ž
.
1 1
The private value-of-life measure a la Schelling is thus linked parametrically with more traditional measures based on the assessment of human and non-human
wealth. The specific link is shown to depend largely, however, on relative bequest preferences, the availability and actuarial fairness of the insurance terms, and the
status of the individual as a net lender or borrower.
Ž .
The value-of-life-saving measure given in Eq. 5.13
is expected to vary systematically over the life cycle as a function of personal characteristics and
market opportunities.
13
In general, persons with relatively high bequest prefer- Ž
. Ž .
ences nra will have a lower value of z t , and hence of life saving. Assuming Ž .
Ž .
that z t does not vary considerably with age, Eq. 5.13 also implies that one’s Ž
. value of life would generally be rising with especially young age, as long as the
Ž .
accumulation of non-human assets or the settling of past debts rises sufficiently faster than the eventual decline in the value of human wealth due to the
contracting life expectancy, but it must ultimately fall at a sufficiently old age. Ž
. Indeed, the boundary conditions in Eq. 5.5 imply that as one reaches the final
Ž .
Ž . Ž .
phase of the planning but not necessarily actual horizon, n D s a D in Eqs. Ž
. Ž
. Ž
U
Ž . Ž .
. 5.10 and 5.11 , and thus the ‘ value of life’ becomes nil Õ
D s L D s 0 . As in the analysis of Section 4, a greater endowment of non-human capital is
Ž Ž
. .
shown to increase the value of life saving unambiguously as long as nra - 1 because it generates a sufficiently large increase in the value of the remaining life
Ž Ž .
. span to more than offset the fall in annuity income if A t 0 , or the reduction
1
Ž Ž .
. in debt service gains if A t - 0 , because of the decline in the risk-premium
1
Ž . yield on actuarial notes. An increase in human capital L , in contrast, is expected
to cause an even larger increase in one’s valuation of life, technically because the increase in wealth and utility it generates is not offset by any change in the capital
Ž .
loss component of Eq. 5.13 .
13
Ž .
Although the comparative dynamic predictions below are obtained directly from Eq. 5.13 , which takes optimal self-protection expenditures as given, they have been confirmed through simulation
analyses that allow a simultaneous determination of self-protection and all other control and co-state Ž
. variables see Footnote 10 .
The intuition behind this result is straightforward: Non-human capital survives its owner, while human capital does not. Given the opportunity to purchase
actuarial notes, one can surrender one’s accumulated non-human assets to an insurance company in return for an added annuity premium, proportional to one’s
conditional risk of mortality, which reflects the undisturbed market value of the surrendered assets in the event of the original owner’s death. No such option is
available to the owner of human capital since the market value of the latter is conditional on the future earnings it generates. The only way to buy an actuarial
note with human capital is by selling an actuarial note drawn against it. This is why a reduction in mortality causes, in part, a capital loss to the owner of a
non-human asset, but it generates a gain to the owner of a human asset. Put differently, the only way to realize the market value of the latter asset is by
protecting longevity. The existence of a market for actuarial notes, which estab-
Ž .
lishes the uncertain market value of human capital, also establishes its greater importance relative to non-human wealth as a determinant of the private value of
life protection. This relative importance of human wealth exists regardless of any bequest motive, but it becomes larger the higher the bequest preferences.
14
A larger health endowment, as indicated by a lower biological risk of mortality Ž .
Ž .
j t , generally has an ambiguous effect on the value of life saving see Section 3 . Ž .
It depends, in part, on whether the level of j t per se has any systematic effect on the productivity of self-protection efforts. The value of life saving is directly
Ž . affected, however, by the personal level of human wealth, L t , which is an
unambiguously decreasing function of mortality risks. By this analysis, the value of life saving, and thus expenditures on life protection, is likely to be higher the
Ž .
lower the mortality odds. Eq. 5.13 also indicates that a higher degree of relatiÕe Ž
. risk aÕersion d s 1 y k unambiguously raises the value of life saving.
The incentives to buy market insurance and to provide self-protection are interrelated by their responsiveness to specific parameters of the model. For
Ž .
Ž .
example, Eqs. 5.8a and 5.13 indicate that an exogenous increase in wealth raises the optimal value of bequest and the value of life saving, and thus,
Ž Ž
.. generally, the demand for both life insurance and self-protection see Eq. 5.6 .
An increase in relative bequest preferences, in contrast, increases the incentive to provide life insurance, but decreases the incentive for self-protection. Thus, life
insurance and self-protection are likely to be complements with respect to a
Ž .
change in wealth regardless of its source and substitutes with regard to a change in bequest preference.
14
If an asset is transferable, its owner suffers a smaller utility loss from the prospect of death the higher is his bequest preference. Indeed, the differentially greater impact of human relative to
Ž . w Ž . x
Ž .
non-human wealth on the value of life saving, as measured by z t r z t y1 in Eq. 5.13 , becomes Ž .
Ž . even larger the higher the person’s relative bequest preferences, n t r a t .
6. The role of annuities and life insurance
