64 •Chapter 8
Now let
r(z)=pF1(z)r1(z)+(1 −p)F2(z)r2(z)
pF1(z)+(1 −p)F2(z)
=δr1(z)+(1 −δ)r2(z),M≤z≤T,(8.25)
where
δ=δ(z)=pF1(z)
pF1(z)+(1 −p)F2(z),0<δ(z)<1.(8.26)
The future instantaneous rate of return at any age z≥Mon long-term
annuities held at age Mis a weighted average of the risk-class rates of
return, the weights being the fraction of each risk class in the population.5
Inserting (8.25) into (8.24), the latter becomes
pT
M
F1(z)(w(z)−c1)dz +(1 −p)T
M
F2(z)(w(z)−c2)dz
+M
0
F(z)(w(z)−c)dz =0.(8.27)
From (8.27) it is now straightforward to draw the following conclu-
sion: The unique solution to (8.22) and (8.23) that satisfies (8.1), with
r(z)given by (8.25), is c =c1=c2=c∗and R∗
1=R∗
2=R∗,where
(c∗,R∗)is the First-Best solution (8.3) and (8.4).
A separating competitive equilibrium with long-term annuities sup-
ports the first-best allocation. Individuals are able to insure themselves
against uncertainty with respect to their future risk class by purchasing
long-term annuities early in life. In equilibrium these annuities yield at
every age a rate of return equal to the population average of risk class
rates of return. The returns from these annuities provide an individual
with a consumption level that is independent of risk-class realization.
The transfers across states of nature necessary for the first-best
allocation are obtained through the revaluation of long-term annuities.
The stochastically dominant risk class obtains a windfall because the
annuities held by individuals in this class are worth more because of the
5The change in r(z)is˙
r(z)=δr1
δr1+(1 −δ)r2f′
1(z)
f1(z)+(1 −δ)r2
δr1+(1 −δ)r2f′
2(z)
f2(z).
The sign of this expression can be negative or positive. The change in the hazard
rate, f(z)
F(z),is equal to f(z)
F(z)f′(z)
f(z)+f(z)
F(z).A nondecreasing hazard rate implies that
−f′(z)
f(z)≤f(z)
F(z)but does not sign f′(z)
f(z)(for the exponential function, f′(z)<0,and this
inequality becomes an equality).
Uncertain Future Survival Functions •65
higher life expectancy of the owners. The other risk class experiences a
loss for the opposite reason.
Another important implication of the fact that in equilibrium con-
sumption is independent of the state of nature is the following. From
(8.20) it is seen that when ci=c∗,i=1,2,the solution to (8.20)
has ˙
ai(z)=ai(z)=0,M≤z≤T.Thus: The market for risk
class annuities after age M (sometimes called “the residual market”) is
inactive. Under full information, the competitive equilibrium yields zero
trading in annuities after age M. As argued above and seen from (8.21),
the interpretation of this result is that the flow of returns from annuities
held at age Mcan be matched, using the relevant risk-class survival
function of the holder of the annuities, to finance a constant flow of
consumption:
c∗=R∗
MFi(z)w(z)dz +a∗(M)T
MFi(z)r(z)dz
T
MFi(z)dz ,(8.28)
where a∗(M) is the optimum level of annuities at age M:
a∗(M)=1
F(M)M
0
F(z)(w(z)−c∗)dz.
8.5Example: Exponential Survival Functions
Let F(z)=e−αz,0≤z≤M,and Fi(z)=e−αMe−αi(z−M),M≤z≤∞,
i=1,2.Assume further that wages are constant; w(z)=w.
With a constant level of consumption, c,before age M, the level of
annuities held at age Mis
a(M)=1
F(M)M
0
F(z)(w−c)dz =w−c
αeαM−1.
For the above survival function, the risk-class rates of return at age z≥M
are αi,i=1,2.We assume that risk class 1 stochastically dominates risk
class 2, α1<α
2.Annuities yield a rate of return, r(z),that is a weighted
average of these returns: r(z)=δ(z)α1+(1 −δ(z))α2,where
δ(z)=pe−α1(z−M)
pe−α1(z−M)+(1 −p)e−α2(z−M).(8.29)
The weight δ(z) is the fraction of risk class 1 in the population. It
increases from pto 1 as zincreases from Mto ∞.Accordingly, r(z)
decreases with zfrom pα1+(1 −p)α2at z=M,approaching α1as
z→∞(figure 8.2).
66 •Chapter 8
Figure 8.2. The rate of return on long-term annuities.
Consumption after age Mfor a risk-class-iindividual is con-
stant, ci,and the budget constraint is wR
MFi(z)dz −ciT
MFi(z)+
aMT
MFi(z)r(z)dz =0.For our case it is equal to
we−αM
αi1−e−αi(Ri−M)−ci
αi
e−αM
+w−c
α1−e−αMT
M
e−αi(z−M)r(z)dz =0,i=1,2.
(8.30)
Multiplying (8.13) by pfor i=1 and by 1 −pfor i=2,and adding,
it can be seen that the unique solution to (8.30) is c∗=c1=c2and
R∗=R1=R2,where
c∗=
w1
α(eαM−1) +p
α1
(1 −e−α1(R∗−M))+1−p
α2
(1 −e−α2(R∗−M))
1
α(eαM−1) +p
α1
+1−p
α2
.
(8.31)
CHAPTER 7
Moral Hazard
7.1Introduction
The holding of annuities may lead individuals to devote additional
resources to life extension or, more generally, to increasing survival
probabilities. We shall show that such actions by an individual in a
competitive annuity market lead to inefficient resource allocation. Specifi-
cally, this behavior, which is called moral hazard, leads to overinvestment
in raising survival probabilities. The reason for this inefficiency is that
individuals disregard the effect of their actions on the equilibrium rate
of return on annuities. The impact of individuals disregarding their
actions on the terms of insurance contracts is common in insurance
markets. Perhaps moral hazard plays a relatively small role in annuity
markets, as Finkelstein and Poterba (2004) speculate, but it is important
to understand the potential direction of its effect.
Following the discussion in chapter 6, assume that survival functions
depend on a parameter α, F(z,α).A decrease in αincreases survival
probabilities at all ages: ∂F(z,α)/∂α ≤0.Individuals can affect the level
of αby investing resources, whose level is denoted by m(α),such as med-
ical care and healthy nutrition. Increasing survival requires additional
resources, m′(α)<0, with increasing marginal costs, m′′ (α)>0.
7.2Comparison of First Best and Competitive Equilibrium
Let us first examine the first-best allocation. With consumption constant
at all ages, the resource constraint is now
cT
0
F(z,α)dz −R
0
F(z,α)w(z)dz +m(α)=0.(7.1)
Maximizing expected utility, (4.1), with respect to c,R, and αyields
the familiar first-order condition
u′(c)w(R)−e(R)=0(7.2)
52 •Chapter 7
Figure 7.1. Investment in raising survival probabilities.
and the additional condition
(u(c)−u′(c)c)T
0
∂F(z,α)
∂α dz +R
0
∂F(z,α)
∂α (u(c)w(z)
−e(z)) dz −u′(c)m′(α)=0,
(7.3)
where, from (7.1),
c=c(R)=R
0F(z,α)w(z)dz −m(α)
T
0F(z,α)dz .(7.4)
Conditions (7.1)–(7.3) jointly determine the efficient allocation
(c∗,R∗,α
∗).Denote the left hand sides of (7.1) and (7.3) by ϕ(c,α,R)
and ψ(c,α,R),respectively. We assume that second-order conditions are
satisfied and relegate the technical analysis to the appendix. Figure 7.1
holds the optimum retirement age R∗constant and describes the condi-
tions ϕ(c,α,R∗)=0 and ψ(c,α,R∗)=0.
Under competition, it is assumed that the level of expenditures on
longevity, m(α),is private information. Hence, annuity-issuing firms
cannot condition the rate of return on annuities on the level of these
expenditures by annuitants. Let the rate of return faced by individuals at
