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Annuities and Retirement by lifemate

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									                   Annuities and Retirement∗
             Eytan Sheshinski, Department of Economics
                The Hebrew University of Jerusalem
                         First Draft: January 1999
                        This Version: September 1999


                                        Abstract
            This paper examines the interaction between the market for an-
       nuities and retirement and consumption decisions in the presence of
       lifetime uncertainty. We focus on two aspects of the demand for annu-
       ities: the timing of annuitization and the information available to the
       issuers of annuities with regard to purchasers’ survival probabilities.
       The First-Best is attained by continuous annuitization of savings, with
       a guaranteed lump-sum payment to beneficiaries upon death. Annu-
       itization of savings at retirement, and a-fortioti no annuitization, are
       inferior and lead to distortions in retirement and consumption deci-
       sions. Applying a ‘Stochastic Dominance’ approach, we show how
       these decisions depend on individuals’ degree of risk aversion. Un-
       der imperfect information, we analyze Pooling Equilibria and compare
       them with the First-Best.




   ∗
    This paper was written while I was visiting Princeton University in the Fall of 1998.
I wish to thank my colleagues and friends at Princeton for their hospitality. I also wish to
acknowledge useful comments received from Kenneth Arrow, Peter Diamond and Sergiu
Hart.

                                             1
1       Introduction

     Retirement plans involve numerous uncertainties. Foremost is the un-
certainty with respect to lifetime. Not only is lifetime duration uncertain
but, particularly early in life, survival probabilities are difficult to predict.
These depend largely on health conditions which only unfold over time. With
these uncertainties, lifetime consumption and savings plans depend crucially
on the availability of insurance markets. The market most relevant for the
length of life uncertainty is the market for annuities. These financial instru-
ments provide payments to annuity holders starting on a specified date and
contingent on survival.1

    By pooling mortality risks of many individuals, issuers of annuities can
fully insure against individual lifetime uncertainty. The terms at which annu-
ities are offered in competitive markets depend on the information available
to issuers. Purchasers belong to one of a number of ‘risk classes’, each con-
sisting of individuals with the same survival function. When suppliers can
identify the ‘risk class’ to which customers belong, competitive annuity prices
differ by ‘risk class’ and a First Best allocation is attained (Yaari, 1965).

   When information on survival probabilities is private, or only imperfectly
available to firms, annuity prices are based on ‘pooled’ (average) survival
rates of several ‘risk classes.’ Pooling leads to adverse selection, that is, an-
nuities are purchased primarily by groups with high survival probabilities
while those with low survival probability prefer to purchase fewer annuities.2
In extreme cases, some ‘risk classes’ may choose not to purchase any an-
nuities. In a Separating Equilibrium of this kind, the actions of individuals
reveal (partially or fully) their characteristics (Rothschild and Stiglitz, 1976).

    Whenever annuity prices are based on pooling of some ‘risk classes’ the al-
location is not First Best. These prices imply transfers (‘cross-subsidization’)
    1
     These are called ‘single annuities.’ ‘Joint annuities’ provide, in addition, specified
payments to spouses or children after the death of the annuity holder. Section 6 below
analyzes this type of annuities.
   2
     Convincing evidence on the different mortality probabilities of annuity purchasers
compared to the population average is presented in Mitchell, Poterba, Warshawsky, and
Brown (1999).



                                            2
across groups and these entail, in turn, distortions in consumption and in re-
tirement decisions. The objective of this paper is to explore the interaction
between these decisions and the characteristics of the annuity market.

    At an optimum retirement age, the benefits and costs of a marginal post-
ponement of retirement are equal. The instantaneous benefits of such post-
ponement are the wage rate times the marginal utility of consumption. The
level of consumption depends, in turn, on the structure of the insurance
markets, in particular the market for annuities. Costs are the instantaneous
disutility of labor. When benefits decrease and costs increase with age, there
is a unique optimum retirement age.

    This paper focuses on the timing of the annuitization of savings. When
the purpose of savings is the provision of consumption during retirement
whose duration is uncertain, it is most efficient to annuitize savings continu-
ously, i.e. to purchase ‘deferred annuities’ which start payments, contingent
on survival, upon retirement. Age-dependent annuity prices are cheaper for
young purchasers, reflecting the probability that the annuity holder may die
prior to retirement. Annuitization of savings at retirement is inferior to con-
tinuous annuitization (the same amount of cumulative savings yields a lower
post-retirement flow of payments), because insurance firms are not able to
pull mortality risks prior to retirement. Consequently, retirement is delayed
and consumption during retirement is lower, compared to the First-Best.
This assumes that individuals place no (or low) value on non-annuitized as-
sets left behind when death occurs before retirement.

    When individuals have a bequest motive, the First-Best allocation is at-
tained with continuous annuitization and a guaranteed lump-sum payment
upon death to designated beneficiaries (which entails correspondingly lower
retirement benefits). Annuitization at retirement leads to excessive (com-
pared to the First-Best) accumulation of assets during the working phase,
whereby part of these assets is annuitized upon retirement and part is left for
bequest. As with no bequest motive, retirement is delayed, post-retirement
consumption and bequests are lower, compared to the First-Best.

    All individuals are shown to purchase some annuities at any price. Hence,
no Separating Equilibrium exists. A Pooling Equilibrium distorts consump-
tion and retirement decisions. Specifically, if individuals are sufficiently risk-


                                      3
averse (coefficient of relative risk aversion larger than one), retirement ages
and consumption are ‘pooled’ together compared to the First-Best.

    It has been argued that annuity markets in the U.S. are expensive and
inefficient (e.g. Friedman and Warshawsky (1990), Mitchell, et. al. (1999),
Walliser (1998)). It is not clear how much of this inefficiency can be at-
tributed to the existence of a mandatory Social Security system which pro-
vides (real) annuities at pooled rates and leaves little room for residual de-
mand. We do not discuss directly the merits of different reform proposals
for the U.S. Social Security system. Our underlying assumption is that a
competitive, decentralized and voluntary annuity market is feasible and we
analyze its potential efficiency under alternative informational structures.

   The use of annuities to insure against lifetime uncertainty has been in-
troduced by Yaari (1965). A number of papers (Merton (1981), Sheshinski
and Weiss (1981), and Abel (1993)) excluded by assumption private annuity
markets and focused on insurance offered by the government. The first paper
that examined the different types of annuity market equilibria is Eckstein,
Eichenbaum and Peled (1985). The issue of the timing of annuity purchases
was analyzed by Brugiavini (1993).3 These papers used a two or three-period
model and did not examine endogenous retirement and its interaction with
the annuity market.



2       Optimum Consumption and Retirement
        with Lifetime Uncertainty

     Consider an individual who has to decide on his or her optimum con-
sumption at different ages and the age of retirement in the presence of uncer-
tainty about the length of life. Suppose that this is the only uncertainty that
the individual faces. Later we analyze the effects of additional uncertainties
with respect to future income and survival rates.

   Let T , T > 0, be the planning horizon, i.e. maximum lifetime (it is
possible to allow T = ∞). Lifetime uncertainty is represented by a survival
    3
    I became aware of Brugiavini’s paper and of the timing of annuitization from a lecture
note by Peter Diamond.

                                            4
distribution function F (z), which is non-increasing in age, z, 0 ≤ z ≤ T ,
with F (0) = 1, F (T ) = 0.

     Denote consumption at age z by c(z), c(z) ≥ 0. Utility of consumption at
different ages is separable and independent of age and there is no subjective
discount rate. Instantaneous utility, u(c), displays risk-aversion (u0 (c) > 0,
u00 (c) < 0) and, to ensure an interior solution, satisfies the end-conditions
u0 (0) = ∞ and u0 (∞) = 0.

   When working, the individual provides one unit of labor. Disutility of
work, a(z) > 0, 0 < z < T , is independent of consumption and nondecreasing
with age, a0 (z) ≥ 0. Again, to ensure an interior solution, assume that
a(0) = 0 and a(T ) = ∞. Contingent on survival, the individual works
between ages 0 and R, 0 < R < T , i.e. retirement occurs at age R.

       The individual’s objective is to maximize expected utility, V ,

                            ZT                      ZR
                      V =        F (z)u(c(z))dz −        F (z)a(z)dz.               (1)
                            0                       0



   The choice of the optimum consumption path and age of retirement will
depend on the insurance options available in the market.


2.1       No Annuities

    Consider first the case when no insurance is available. Let wages at age
z be w(z), w(z) ≥ 0. Savings, w(z) − c(z), earn a zero rate of interest.4
With no initial assets, the individual’s assets at age z, B(z), are equal to
cumulative savings:
                                          Zz
                       B(z) = W (z) −          c(z)dz, 0 ≤ z ≤ T,                   (2)
                                          0

   4
     It is well-known how the results are modified when the rate of interest and the sub-
jective discount rate are positive.


                                               5
                    R
                  min(z,R)
where W (z) =                w(z)dz, 0 ≤ z ≤ T , are cumulative wages. Feasible
                     0
consumption plans must have non-negative assets at all ages:5

                                         B(z) ≥ 0.                                 (3)

   Maximization of (1) subject to (3) with respect to c(z) and R yields
                                      ˆ
F.O.C. for an interior solution (ˆ(z),R)
                                 c

                         F (z)u0 (ˆ)(z)) − λ = 0, 0 ≤ z ≤ T,
                                  c                                                (4)


                                       ˆ      ˆ
                                     a(R) − ψ(R) = 0,                              (5)

where ψ(R) = u0 (ˆ(R))w(R) for all R, 0 < R < T , and λ, λ > 0, is a
                   c
constant. Condition (4) indicates that optimum expected marginal utility is
constant over age. Denote the solution to (4) by c(z) and write c(z) = c0 h(z),
                                                 ˆ              ˆ
                                0
where h(0) = 1 and, from (4), u (c0 ) = λ. Optimum consumption is seen to
decrease with age,

                               c0 (z)
                               ˆ        h0 (z)    1 f (z)
                                      =        =−         ,                        (6)
                               ˆ
                               c(z)     h(z)      σ F (z)
                              00
where σ = σ(z) = − uu0(c)c > 0, is the elasticity of the marginal utility,
                           (c)
termed Coefficient of Relative Risk Aversion, f (z), f ≥ 0, is the density of
              f
1 − F (z) and F(z) is the conditional probability of dying at age z, termed the
                (z)
Hazard-Rate. By (4) and the assumption that u0 (0) = ∞, c(T ) = h(T ) = 0.
                                                              ˆ

    With no bequest motive, constraint (3) is binding at T . This condition
is used to solve for initial consumption, c0 ,

                                                   R
                                                   R
                                                        w(z)dz
                                                   0
                                   c0 = c0 (R) =                                   (7)
                                                   RT
                                                        h(z)dz
                                                   0

   5
    Since optimum consumption is shown below to strictly decrease with age, (3) implies
that B(z) > 0, R ≤ z < T . The constraint B(z) ≥ 0 may bind the optimum consumption
path for 0 ≤ z < R.


                                             6
    The condition which determines the optimum retirement age, (5), has an
intuitive interpretation. A marginal postponement of retirement yields an ad-
ditional income of w(R). It’s value in utility terms is ψ(R) = u0 (ˆ(R))w(R).
                                                                   c
The costs of postponement in utility terms are the disutility of labor, a(R).
At the optimum, costs and benefits are equalized.

   The sign of ψ0 (R) cannot be established in general and hence there is a
possibility of multiple solutions to (5).6 An example is provided below. We
                                                                        ˆ
shall not dwell on this question and assume a unique solution, denoted R.7
                                         ˆ
       Optimum expected utility, denoted V , is given by

                              ZT                        ZR
                        ˆ
                        V =        F (z)u(ˆ(z))dz −
                                          c                  F (z)a(z)dz            (8)
                              0                         0




2.2        Continuous Annuitization

      An annuity is a financial instrument which promises certain payments at
specified ages, contingent on survival of the holder of the annuity.8 The issue
of annuities can promise certain payments by pooling the mortality risks of
many individuals. When individuals save in order to finance consumption
during retirement, it is efficient to convert savings continuously during the
working phase into annuities which will start payments upon retirement.
The purchase of such ‘deferred-annuities’ provides perfect insurance against
lifetime uncertainty and the individual takes advantage of the lower prices
   6
       Differentiating ψ(R) with respect to R,

                          ψ0 (R)   f (R)          w(R)     w0 (R)
                                 =       − σ(R) R        +
                          ψ(R)     F (R)       R           w(R)
                                                  w(z)dz
                                                    0


   7 ˆ                                                                       ˆ
     R satisfies second order conditions if a(R) − ψ(R) < (>)0 for R < (>) R whenever
(5) is satisfied.
   8
     Annuities that also provide payments to beneficiaries after the death of the annuity
holder are discussed in Section 6.



                                                7
of early annuity purchases which reflect the probability that the purchaser
does not survive to retirement.9

    All individuals are assumed to have the same survival distribution func-
tion, F (z). With a zero rate of interest, lending and borrowing is made at a
discount rate equal to this common rate. Hence, without loss of generality,
we can take annuity payments to be synchronized with consumption during
retirement.

    With continuous conversion of savings, w(z) − c(z), into annuities which
start payments at R, competition implies a zero expected profits condition:

                  ZT                   ZR
                       F (z)c(z)dz −        F (z)(w(z) − c(z))dz = 0
                   R                   0

or
                        ZT                   ZR
                             F (z)c(z)dz −        F (z)w(z)dz = 0.                    (9)
                        0                    0



   Maximization of (1) subject to (9) with respect to c(z) and R, yield
F.O.C. for an interior solution (c∗ (z), R∗ ):

                             u0 (c∗ (z)) − λ = 0, 0 ≤ z ≤ T                          (10)


                                  a(R∗ ) − φ(R∗ ) = 0                                (11)

where φ(R) = u0 (c∗ (R))w(R) for all R, 0 ≤ R ≤ T , and λ, λ > 0, is a
constant.10
     9
      Age-dependent prices assume, of course, full infomration on the purchasers age. Pur-
chasing deferred annuities is equivalent to investing in a large pension fund which dis-
tributes the assets of deceased members to survivors of the same age cohort.
   10
      The functional form of φ, (11), and ψ, (5), is the same, but their dependence on R
differs because of the different consumption paths, c∗ (z) and c(z), respectively. We use
                                                                ˆ
thoughout the same notation for the Lagrangan constants, λ, but their values differ, of
course, for each maximization.



                                             8
   It is seen from (10) that optimum consumption is constant: c∗ (z) = c∗ ,
0 ≤ z ≤ T . By (9), for all R, 0 ≤ R ≤ T ,

                                                                W (R)
                                       c∗ = c∗ (R) =                                                (12)
                                                                  z
               R
               T
where z =          F (z)dz is life-expectancy,11 and
               0

                    ZR                          ZR
        W (R) =          F (z)w(z)dz =               f (z)W (z)dz + (1 − F (R))W (R).               (13)
                    0                           0

is expected total wages until retirement.

    Condition (11) determines the optimum retirement age, R∗ . As before,
it balances the benefits and costs of a marginal postponement in retirement.
When w0 (R) ≤ 0, φ(R) decreases with R and (11) determines a unique
solution.12

       Optimum expected utility, denoted V ∗ , is

                                            ZT                      Z∗
                                                                    R

                        V ∗ = u(c∗ )                F (z)dz −            F (z)a(z)dz =
                                            0                       0
                                       µ                ¶       Z∗
                                                                R
                                           W (R∗ )
                                   u               z−                   F (z)a(z)dz.                (14)
                                             z
                                                                0



   Fully insuring against lifetime uncertainty, continuous annuitization is the
First-Best allocation. Hence,

  11
                               R
                               T                    R
                                                    T
       Integrating by parts:       F (z)dz =            zf (z)dz where f (z) is the density of 1 − F (z),
                               0                    0
when it exists.
 12
    For all R, 0 ≤ R ≤ T ,

           φ0 (R)   w0 (R)        w(R)                        u00 (c∗ (R))c∗ (R)
                  =        − σ(R)       F (R), where σ(R) = −                    >0
           φ(R)     w(R)          W (R)                            u0 (c∗ (R))



                                                            9
                        ˆ
   Proposition 1: V ∗ > V .

   Proof. By concavity, u(c) < u(c∗ ) + u0 (c∗ )(c − c∗ ), for any (c, c∗ ), c 6= c∗ .
Let c = c(z) be a function of z. Multiplying by F (z) and integrating,

                  ZT                                   ZT
                                                   ∗
                          F (z)u(c(z))dz < u(c )                F (z)dz
                  0                                        0
                             T                        
                             Z                ZT
                  +u0 (c∗ )  F (z)c(z)dz − c∗ F (z)dz 
                                      0                         0
                                                                                                 
                             ZT                            ZT                     ZR
               = u(c∗ )           F (z)dz + u0 (c∗ )           F (z)c(z)dz −          F (z)w(z)dz  , (15)
                             0                             0                      0

where R corresponds to c∗ = c∗ (R) by (12). Now take c(z) = c(z) and
                                                              ˆ
     ˆ                                                     ˆ
R = R, the optimum with no annuities. Setting w(z) = 0 for R < z ≤ T ,
     Rˆ
      R              R
                     T
write F (z)w(z)dz = F (z)w(z)dz. Integrating by parts, using condition
           0                              0
       R
       T                                               R
                                                       T                          Rˆ
                                                                                   R
(3),       F (z)(w(z) − c(z))dz ≥ 0, or
                        ˆ                                      F (z)ˆ(z)dz ≤
                                                                    c                  F (z)w(z)dz. It now
       0                                                   0                      0
follows from (15) that

                                 ZT                                       ZT
                                             c              ˆ
                                      F (z)u(ˆ(z))dz < u(c (R))     ∗
                                                                               F (z)dz.               (16)
                                 0                                        0


                      Rˆ
                       R
Subtracting                F (z)a(z)dz on both sides of (16), noting that V ∗ is maximum
                      0
                                              ˆ
at R = R∗ , it follows from (8) and (14) that V < V ∗ .                                   ¥




                                                           10
2.3       Annuitization at Retirement

    Suppose that savings are converted to annuities only upon retirement.13
The zero expected profits condition is now

                   ZT                          ZR
                        F (z)c(z)dz − F (R)         (w(z) − c(z))dz = 0            (17)
                   R                           0

    Maximization of (1) subject to (17) yields F.O.C. for an interior solution
 c       ˆ
(ˆR (z), RR ):

                     F (z)u0 (ˆR (z)) − λF (R) = 0,
                              c                             0≤z≤R                  (18)


                           u0 (ˆR (z)) − λ = 0,
                               c                        R<z≤T                      (19)

and
                                   ˆ          ˆ
                                 a(RR ) − φR (RR ) = 0                             (20)

where φR (R) = u0 (ˆR (R))w(R), for all R, 0 ≤ R ≤ T , and λ, λ > 0,
                     c
is a constant. It is seen from (18) and (19) that optimum consumption
decreases for 0 ≤ z ≤ R and is constant thereafter (Figure 1). Again, write
cR (z) = c0R h(z), 0 ≤ z ≤ R, where h(0) = 1 and, by (18), h(z) satisfies
ˆ
differential equation (6).

       Initial consumption, c0R , is solved from (17),

                                                    R
                                                    R
                                                        w(z)dz
                                                    0
                     c0R = c0R (R) =                                        .      (21)
                                       h(R)    R
                                               T               R
                                                               R

                                       F (R)
                                                   F (z)dz +       h(z)dz
                                               R               0

  13
    This is the case in many of the mandatory privately managed pension schemes, fol-
lowing the example of Chile. See Diamond and Valdez-Prieto (1994). Assets of deceased
members in each fund are transferred to designated beneficiaries. It is assumed here that
individuals put on value on these transfers. In Section 6 below, we introduce a bequest
motive.



                                           11
                                                                       ˆ
    Optimum expected utility with annuitization at retirement, denoted VR ,
is thus
                                                                  ˆ
             ZR                                     ZT            ZR
                                                                  R
      ˆ
      VR =        F (z)u(ˆR (z))dz + u(ˆR (R))
                         c             c                 F (z)dz − F (z)a(z)dz    (22)
             0                                      ˆ
                                                    RR              0



   Annuitization at retirement is inferior to continuous annuitization and
superior to no annuitization:
                         ˆ    ˆ
    Proposition 2: V ∗ > VR > V .

    Proof is straightforward.



3     Comparison of Optimum Retirement Ages

    We want to compare optimum retirement ages and levels of consumption
during retirement under the different insurance regimes. Without annuities,
optimum consumption during retirement decreases with age. In this case,
therefore, optimum retirement age is uniquely related only to the level of
consumption upon retirement.


3.1     Continuous Annuitization vs. Annuitization at
        Retirement

    First, we show that optimum retirement age is lower with continuous
annuitization than with annuitization at retirement:
                        ˆ           ˆ ˆ
    Proposition 3: R∗ < RR and c∗ > cR (RR ).
                                                              ˆ
    Proof. From conditions (11) and (20) it is seen that R∗ < RR if c∗ (R) >
cR (R) for all R, 0 ≤ R ≤ T. >From (3),
ˆ
                         ZR                         ZR
                 F (R)        (w(z) − cR (z))dz <
                                      ˆ                  F (z)(w(z) − cR (z))dz
                                                                      ˆ           (23)
                         0                          0


                                              12
for all R, 0 ≤ R ≤ T . Hence, by (17) and (24)
            ZR                             ZT                ZR
                 F (z)ˆR (z)dz + cR (R)
                      c          ˆ               F (z)dz ≤        F (z)w(z)dz   (24)
             0                             R                 0

Since, by (18), cR (z) > cR (R), 0 ≤ z < R, it follows from (24) and (12) that
                ˆ        ˆ
                               R
                               R
                                   F (z)w(z)dz
                    ˆ
                    cR (R) <   0
                                                   = c∗ (R).         ¥          (25)
                                   R
                                   T
                                       F (z)dz
                                   0



    This result is intuitive. Without insurance during the working phase and
facing decreasing survival rates, the individual chooses decreasing levels of
consumption so as to keep expected marginal utility constant. Consequently
consumption during retirement is lower than under complete insurance, i.e.
continuous annuitization.

    In the absence of annuitization, consumption is uninsured both before and
after retirement. The decision on optimum retirement depends therefore both
on the survival distribution function and on the individual’s risk aversion.
Before we analyze this case it will be useful for subsequent discussions to
link it to the familiar notion of ‘Stochastic Dominance’.


3.2    Ranking of Survival Functions

     Consider two survival distribution functions, F1 (z) and F2 (z), 0 ≤ z ≤
T . Clearly, Fi (0) = 1, Fi (T ) = 0 and Fi (z) non-increases in z, i = 1, 2.

    Definition 1. (‘Single Crossing’ or ‘Stochastic Dominance’): The func-
tion F1 (z) is said to (strictly) stochastically dominate F2 (z) if the ‘Hazard-
Rates’ satisfy

                         f2 (z)   f1 (z)
                                >        ,          0 ≤ z ≤ T.                  (26)
                         F2 (z)   F1 (z)

                                            13
                                                                                                  d`nF (z)
    In words, the rate of decrease of survival probabilities,                                        dz
                                                                                                                 f
                                                                                                             = − F(z) ,
                                                                                                                   (z)
is smaller at all ages with distribution 1 than with 2.

   Two implications of this definition are important. First, consider the
functions R Fi (z) , 0 ≤ z ≤ T , i = 1, 2. Being positive and their integral over
          T
                      Fi (z)dz
                  0
(0, T ) equal to one, they must intersect (cross) at least once over this range.
At any such crossing, when R F1 (z) = R F2 (z) , condition (26) implies that
                              T           T
                                                           F1 (z)dz            F2 (z)dz
                                                    0                  0


d     T F1 (z)  >              d     T F2 (z) . Hence, there can be only a ‘single crossing’.
dz       R                       dz     R
             F1 (z)dz                         F2 (z)dz
         0                                0
That is, there exists an age zc , 0 < zc < T , such that (Figure 2),

                                              F1 (z)                  F2 (z)
                                                             S                      as z S zc .                   (27)
                                       R
                                       T                         R
                                                                 T
                                              F1 (z)dz                F2 (z)dz
                                       0                         0

   Intuitively, (27) means that the dominant distribution has higher (lower)
survival rates, relative to life expectancy, at older (younger) ages.

     Second, since Fi (0) = 1, i = 1, 2, it follows from (27) that

                                                ZT                      ZT
                                      z1 =           F1 (z)dz >                F2 (z)dz = z 2 ,                   (28)
                                                 0                         0

i.e., stochastic dominance implies higher life expectancy.


3.3           Effects of Annuitization

    We are now ready to compare the no-annuities case with those where
annuitization is available, either continuously or at retirement.

    With no insurance against lifetime uncertainty, the individual’s consump-
tion and retirement plans depend on his or her attitude towards risk, repre-
sented by the coefficient of (relative) risk-aversion, σ.

                                                                      14
                                                           ˆ    ˆ
  Proposition 4: If σ ≥ 1 for all z, 0 ≤ z ≤ T , then R∗ < RR < R and
    ∗
c > cR > c.
    ˆ    ˆ

    Proof. Recall that with no annuities, optimum consumption is c(z) =
                                                                      ˆ
c0 h(z), where c0 is given by (7), h(0) = 1 and h(z) obeys the differential
equation (6). Note that, by (6), if σ = 1 for all z, 0 ≤ z ≤ T , then h(z) =
F (z). We now want to show that, when σ > 1, h(z) stochastically dominates
F (z).

        When σ > 1, then by (6),

                                         h0 (z)    1 f (z)    f (z)
                                                =−         >−                                                             (29)
                                         h(z)      σ F (z)    F (z)
                                                                                                                                        

Hence, whenever R h(z)                   =       F (z)
                                                           , (29) implies that            d     T h(z)  >         d     T F (z) ,
                T                            R
                                             T                                            dz     R                  dz     R
                                h(z)dz           F (z)dz                                               h(z)dz                    F (z)dz
                            0                0                                                     0                         0
which, in turn, implies a ‘single-crossing’.

        Since h(z) is non-increasing, by (7),
                                                    R
                                                    R                           R
                                                                                R
                                                         w(z)dz                      h(z)w(z)dz
                                                    0                            0
                     c(R) = c0 h(R) =
                     ˆ                                                 h(R) <                              ,              (30)
                                                    RT                                R
                                                                                      T
                                                         h(z)dz                           h(z)dz
                                                     0                                0

for all 0 ≤ R ≤ T . An implication of ‘single-crossing’ is that
                     
RR
    T h(z) − T F (z)  dz < 0, for all R < T (because, for R = T the integral
     R        R
0           h(z)dz       F (z)dz
        0            0
is equal to zero and hence, by (27), it is negative for all R < T ).

   >From (30) and the identity h(z) = F (z) for σ = 1, we conclude that
whenever σ ≥ 1
                                    R
                                    R                            R
                                                                 R
                                         h(z)w(z)dz                    F (z)w(z)dz
                                    0                              0
                         ˆ
                         c(R) <                              ≤                                 = c∗ (R)                   (31)
                                         R
                                         T                              R
                                                                        T
                                             h(z)dz                         F (z)dz
                                         0                              0


                                                              15
for all 0 ≤ R ≤ T . Thus, by (5) and (11) ψ(R) > φ(R), 0 ≤ R ≤ T . Since
                                                  ˆ
a0 (R) ≥ 0, it follows from these conditions that R > R∗ .

   To show that cR (R) > c(R), for all 0 ≤ R ≤ T , compare c0R , given by
                    ˆ        ˆ
                                                   R F (z)
                                                   T          R h(z)
                                                              T
(22), to c0 , given by (7). We see that c0R S c0 as F (R) dz T h(R) dz. By
                                                                        R                R
(6), if σ ≥ 1,
       d`nh(z)       h0 (z)                                             h0 (z)
         dz
                 =   h(z)
                              = − σ F(z) ≤ − F(z) =
                                  1 f
                                      (z)
                                             f
                                               (z)
                                                      d`nF (z)
                                                         dz
                                                               ,   or   h(R)
                                                                                 ≥   F (z)
                                                                                     F (R)
                                                                                             for all R ≤
z ≤ T.

    Thus, c0R ≥ c0 (with equality when σ = 1), and the result follows (Figure
3).    ¥



4        Income Effects on Optimum Retirement

     Write wages at age z as θw(z), where θ, θ > 0, is a constant. A change
in θ is a proportionate change in wages at all ages.14

    Using (12), (7) and (22), it can be shown that the elasticities of c∗ (R),
cR (R) with respect to θ and cR (R) are all equal to one. Furthermore, from
ˆ                             ˆ
the definitions of φ(R), ψ(R), and φR (R), (11), (5) and (20), respectively, it
can be shown that
                                 θ ∂φ   θ ∂ψ    θ ∂φR
                                      =      =        =1−σ                                          (32)
                                 φ ∂θ   ψ ∂θ   φR ∂θ

                                            ˆ       ˆ
for all 0 ≤ R ≤ T . Consequently, R∗ , R and RR all decrease (increase)
when wages increase provided σ > (<)1. Intuitively, a large coefficient of
risk aversion implies that the negative income effect due to increased wages
(labor being a normal good) outweighs the positive substitution effect (due
to a larger opportunity cost to retire), and vice-versa.

  14
    In Section 7 we study a more general case, where θ is uncertain and occurs sometime
in the future.




                                                16
5       Example

       Consider the survival function
                              e−αz − e−αT
                 F (z) =                  , α > 0 a constant, 0 ≤ z ≤ T                        (33)
                               1 − e−αT

                                             ³                ´
                                         1       eαT −1−αT
      Expected lifetime is z =           α         eαT −1
                                                                  , which is inversely related to α.
                                                    1
The simplest case to explore is when T → ∞ (and z → α ).15

                    c1−σ
      Let u(c) =    1−σ
                         ,   σ 6= 1 or u(c) = `n(c). Assume that w(z) = w, 0 ≤ z ≤
T.

   With continuous annuitization, it can be calculated from (12) that c∗ (R) =
w(1 − e−αR ).
                                                                              0
      With no annuities, by (4) and (7), c(R) = α0 Rwe−α R , where α0 = α .
                                         ˆ                              σ

      With annuitization at retirement, by (18) and (22),
                                                          0
                                           α0 Rwe−α z
                             cR (z) =
                             ˆ                    ¡ 0 −α ¢ , 0 ≤ z ≤ R
                                        1 + e−α0 R α α

and cR (z) = cR (R) for R ≤ z.
    ˆ        ˆ

    It can be demonstrated that for α0 < α (i.e. σ > 1), we have c∗ (R) >
                                                ˆ    ˆ
cR (R) > c(R) for all R ≥ 0, implying that R∗ < RR < R.
ˆ         ˆ

    Notice that when α0 > α (σ < 1), cR (R) < c(R) and hence RR > R.
                                          ˆ         ˆ               ˆ     ˆ
                                                                 ∗
It can also be seen that for small σ it is possible that c(R) > c (R), which
                                                         ˆ
               ∗
implies that R > R.ˆ

    For α0 = α, i.e. σ = 1 (u(c) = `nc), then c∗ (R) > cR (R) = c(R) and
                                                            ˆ          ˆ
  ∗
R <R          ˆ
       ˆ R = R. For this case (Figure 4), φ(R) = w =         1
                                                                  is the inverse
                                                     c∗   1−e−α/R
of the ‘replacement-ratio’, i.e., the ratio of post-retirement consumption to
pre-retirement income.
 15
      All subsequent results hold also for finite T .

                                                     17
                       1                                                    1
   ψ(R) = φR (R) = αR eαR , is downward (upward) sloping for R < (>) α .
                                                                            1
Thus, if it is assumed that retirement age is earlier than life expectancy, α ,
uniqueness is guaranteed.

   Comparative statics also yield expected results. For example, for σ = 1
the elasticity of optimum retirement with respect to life expectancy is
                                                        ∗
                                                 αR∗ e−αR
                           α dR∗                  1−e−αR
                                                         ∗
                                 =      R ∗ α0 (R∗ )       ∗ −αR∗
                                                                                     (34)
                           R∗ dα                     + αR e ∗
                                          α(R∗ )        1−e−αR

                d dR∗
Hence, −1 ≤     R∗ dα
                        ≤ 0.



6      Bequest Motive

     It was assumed above that individuals have no bequest motive. Con-
sequently, savings are used solely to finance consumption during retirement.
With continuous annuitization, since all wealth is in the form of (deferred)
annuities, no resources are left behind when individuals die. In the absence
of a market for annuities, or when savings are annuitized at retirement and
death occurs earlier , some resources held to finance consumption during
retirement are left behind when death occurs, their level depending on the
age of the deceased. So far, we assumed that individuals’ consumption and
retirement plans place no value on these uncertain resources left behind.

    Suppose now that individuals have a bequest motive, that is, they have
a utility, v(B), from assets left behind as bequests B(≥ 0). Consider first
the complete insurance case, i.e. continuous annuitization of savings with
payments, contingent on survival, starting after retirement and, in addition,
a certain amount provided to beneficiaries upon death.16
  16
    This is similar to the ‘fixed payments certain’ type of annuities which ensure payments
to annuitants’ beneficiaries for a fixed period. This annuity, however, does not provide
complete insurance if the amount that beneficiaries receive depends on the time of death.
For a description of the type of annuities available in the U.S. see Poterba (1997).



                                           18
    The zero expected profits condition is
                                  ZT                                  ZR
                          B+           F (z)u(c(z))dz −                    F (z)w(z)dz − B0 = 0                 (35)
                                   0                                  0

where B0 is initial endowment. Maximization of expected utility
                                       ZT                                  ZR
                              V =           F (z)u(c(z))dz −                    F (x)a(z) + v(B)                (36)
                                       0                                   0

subject to (34) determines optimum consumption, c∗ , retirement age, R∗ ,
and bequests B ∗ , satisfying F.O.C.
                                                     u0 (c∗ ) − v 0 (B ∗ ) = 0                                  (37)

                                                a(R∗ ) − u0 (c∗ )w(R∗ ) = 0                                     (38)
and
                                                          W (R∗ ) + B0 − B ∗
                                                 c∗ =                                                           (39)
                                                                   z
where W (R) is given by (13). Differentiating (37)-(39), it is seen that con-
                                          dc∗                  ∗
sumption and bequests are normal goods, dB0 > 0 and 0 < dB0 < 1.17
                                                            dB

   Thus, in a dynamic setting, bequests can be viewed as converging to a
fixed-point or steady state, B0 = B ∗ .
   When savings are annuitized at retirement, the terms of the annuities can
again include a certain level of bequests (reflected, correspondingly, with a
lower level of payments). If however, an individual dies during the working
phase, the level of assets left behind depends on the age of death.
    Expected utility is given by
         ZT                                     ZR                              ZR
  V =               F (z)u(c(z))dz −                 F (z)a(z)dz +                   f (z)v(B(z))dz + F (R)v(BR )
            0                                    0                              0
                                                                                                                (40)
                                           00 ∗
  17 dB ∗                               h u (c00) ∗
     dB0        =                            u (c )w(R∗ )2 F (R∗ )
                                                                      i   and, by second-order conditions, a0 (R∗ ) −
                    u00 (c∗ )+v00 (B ∗ ) z− a0 (R∗ )−u0 (c∗ )w0 (R∗ )
u0 (c∗ )w0 (R∗ ) > 0.

                                                                    19
where BR is the (certain) level of bequest after retirement.

   The level of assets, B(z), during the working phase is given by (2). Assets
change is equal to savings,
                         B 0 (z) = w(z) − c(z),      0≤z≤R               (41)
with B(0) = B0 .

      The zero expected profits condition is
                                  R                         
          ZT                      Z
             F (z)c(z)dz − F (R)  (w(z) − c(z))dz − BR + B0  = 0.      (42)
           R                          0


    Maximization of (40) subject to (41) and (42) yields optimum consump-
                                                                    ˆ
tion, cR (z), the corresponding path of assets prior to retirement, B(z), 0 ≤
      ˆ
                           ˆ                                   ˆ
z ≤ R, retirement age, RR , and bequest after retirement, BR . As before,
c(z) = c(R), R ≤ z ≤ T , i.e. optimum level of consumption after retirement
ˆ       ˆ
is constant. All the optimum values depend on initial endowment, B0 .

    We want to compare continuous annuitization with annuitization at re-
tirement when both are in a steady-state. That is, when B0 = B ∗ and
      ˆ
B0 = BR , respectively.
                                                                 ˆ
    Proposition 5. In a steady-state with a bequest motive, R∗ < RR ,
                ˆ
cR (R) < c∗ and B R < B ∗ .
ˆ

      Proof. See Appendix A.

    The comparison of steady states eliminates, as seen for the budget con-
straints (35) and (42), income effects.

    Qualitatively, the previous results are seen to carry-over in the presence
of a bequest motive. Of course, without annuitization, lifetime uncertainty
makes bequests random at all ages and the appropriate model to analyze
optimum consumption and retirement is via the stationary asset distributions
generated by bequests.18

 18
      We plan to analyze this case in a companion paper.

                                           20
7     Lifetime and Income Uncertainty

    Suppose that individuals face lifetime uncertainty and are also uncertain
about their future income. Income uncertainty may reflect, for example,
the possibility of future disability. It is assumed that income uncertainty
cannot be insured, presumably due to ‘Moral-Hazard’. We model income
uncertainty by assuming that beyond a certain age M , prior to retirement,
wages become θw(z), where θ, θ > 0, is a constant which can assume different
                                                            Rθ
values, 0 < θ < θ, with probability p(θ), 0 ≤ p(θ) ≤ 1, i.e. p(θ)dθ = 1.
                                                                              0

   Consumption after the realization of θ (at M ), and the chosen retirement
age, depend on θ. Denote these c(θ, z) and R(θ), respectively.

    Expected utility is now

                   ZM

         V   =            F (z)u(c(z))dz +
                      0
                                                                                 
                   Zθ              ZT                          Z
                                                               R(θ)

                          p(θ)         F (z)u(c(θ, z))dz −            F (z)a(z)dz  dθ   (43)
                      0         M                              0



    Under continuous annuitization, a zero expected profits condition holds
for each θ,

              ZM                         ZT                      ZM

                   F (z)c(z)dz +              F (z)c(θ, z)dz −         F (z)w(z)dz −
              0                          M                         0
                  Z
                  R(θ)

              θ           F (z)w(z)dz = 0,             0 ≤ θ ≤ θ.                         (44)
                  M

    Maximization of (43) subject to (44) yields constant optimum consump-
tion prior to M , c∗ , and constant optimum consumption after age M depen-
dent on θ, c∗ (θ). These contingent optimum consumption levels satisfy the


                                                  21
condition,

                                        Zθ
                              0   ∗
                           u (c ) =          p(θ)u0 (c∗ (θ))dθ                         (45)
                                        0

   Optimum marginal utility of consumption prior to M is equal to expected
optimum marginal utility of consumption after M .

   For each θ, optimum retirement, R∗ (θ), is determined by the condition

                           a(R∗ (θ)) − φ(θ, R∗ (θ)) = 0                                (46)

where for any R, φ(θ, R) = θw(R)u0 (c∗ (θ, R)). The level of c∗ (θ, R) depends
on R through (44),

                          R
                          M                                    R
                                                               R
                              F (z)(w(z) − c∗ )dz + θ              F (z)w(z)dz
              ∗           0                                    M
             c (θ, R) =                                                                (47)
                                               R
                                               T
                                                   F (z)dz
                                               M



   To find the effect of θ, differentiate φ with respect to θ, given R,
                                      θ ∂φ
                                           = 1 − σε                                    (48)
                                      φ ∂θ

where, by (44) and (47),

                                                       R
                                                       R
                                                   θ       F (z)w(z)dz
         θ     ∂c∗ (θ, R)                              M
    ε= ∗                  = M                                                          (49)
      c (θ, R)     ∂θ       R                                        R
                                                                     R
                                      F (z)(w(z) − c∗ )dz + θ            F (z)w(z)dz
                                  0                                  M



    The effect of θ on φ depends on the product of the coefficient of risk
aversion, σ, and the elasticity of optimum consumption after M with respect
to θ, ε. The value of ε depends on the timing of the change in wages, M ,
relative to R.


                                             22
  By (47), ε > 0 for all 0 ≤ M ≤ R, ε = 1 when M = 0 and ε = 0 when
                                                       R
                                                       M
M = R.19 If annuitized savings prior to M are negative, F (z)(w(z) −
                                                                         0
c∗ )dz < 0 (in expectation of high wages afterward), ε may be larger than 1
for some M (Figure 5a).

   The effect of θ on φ and, by (46), on the optimum R, is seen to depend
on the sign of 1 − σε (Figure 5b). If σ > 1 for all z, then ∂φ < 0 for small
                                                            ∂θ
M and ∂φ > 0 for M close to R. This implies, by (46), that for small (large)
        ∂θ
M , an increase in wages decreases (increases) the optimum retirement age.
The explanation is that the earlier the wage increase occurs the larger is
the income effect, which, with σ > 1, outweighs the (opposite) substitution
effect. When σ < 1 the substitution effect works in the same direction as the
income effect to increase optimum retirement age.



8        Different ‘Risk Classes’: Full Information

     Suppose that the population consists of two homogeneous groups, i =
1, 2. Individuals in each group, called ‘risk-class’ by insurance firms, have
the same survival function, Fi (z).

   Assume that groups 1’s survival function stochastically dominates, ac-
cording to (26), that of group 2. In particular, group 1 has a higher life
expectancy.

    Assume further that firms can identify annuity purchasers according to
the group to which they belong. In a perfectly competitive market, annuities
are then priced differently for each group, the analysis in Section 2 applying
to each group separately.

   We focus on the effect of heterogeneity in survival rates and hence assume
that wages are the same for both groups.

  19
                                                                             R
                                                                             M
       In (47), c∗ also depends on M through (46). Furthermore, as M → R,        F (z)(w(z) −
                                                                             0
c∗ )dz > 0 since c∗ (θ, R) → c∗ . The case M = 0 has been discussed in Section 4.



                                            23
     Proposition 6:               With full information and continuous annuitization,
 ∗      ∗
R1   > R2 .
                                                                                   R
                                                                                   R
                                                                                        Fi (z)w(z)dz
     Proof. Applying (12) to each group, c∗ = c∗ (R) =
                                          i    i
                                                                                    0
                                                                                        R
                                                                                        T              , i = 1, 2.
                                                                                            Fi (z)dz
                                                                                        0
     By (27),
                             R
                             R                              R
                                                            R
                                  F1 (z)w(z)dz                  F2 (z)w(z)dz
                              0                             0
                c∗ (R)
                 1       =                             <                           = c∗ (R),
                                                                                      2                       (50)
                                   R
                                   T                            R
                                                                T
                                       F1 (z)dz                     F2 (z)dz
                                   0                            0

for all R, 0 < R < T . It follows from (11) that R1 > R∗ .
                                                  ∗
                                                       2                                        ¥

   Individuals with higher life expectancy partially compensate for longevity
by retiring later, but their optimum consumption remains lower throughout.

    Without insurance, the effect of higher life expectancy on optimum re-
tirement is indeterminate. Recall from (7) that
                                           R
                                           R
                                               w(z)dz
                                           0
                             ci (R) =
                             ˆ                             hi (R)        i = 1, 2                             (51)
                                           R
                                           T
                                               hi (z)dz
                                           0

                                                                      h0 (z)
where hi (R) is defined by (6): hi (0) = 1 and                          i
                                                                      hi (z)
                                                                                   1 f
                                                                               = − σ Fii(z) . It now follows
                                                                                        (z)
from (27) that
                                  h1 (R)               h2 (R)
                                               S                     as R S Rc                                (52)
                             R
                             T                     R
                                                   T
                                  h1 (z)dz             h2 (z)dz
                             0                     0

for some Rc , θ < Rc < T . Recall that the benefit from a marginal postpone-
ment of retirement, (5), for a group i individual is ψ i (R) = w(R)u0 (ˆi (R)).
                                                                       c
                             ˆ              ˆ     ˆ            ˆ
In view of (51) and (52), if R1 < Rc then R1 > R2 , while if R1 > Rc , then
 ˆ    ˆ
R1 < R2 .

   The explanation of this indeterminacy is as follows. In the absence of
insurance and with the same age of retirement, individuals with higher life

                                                       24
expectancy consume less at early ages and more at advanced ages, compared
to individuals with lower life expectancy. Hence, the benefit from a marginal
postponement of retirement is higher for the former group, provided that
retirement takes place when their consumption is lower. The opposite holds
if retirement occurs when their consumption is higher.

   This argument is easily demonstrated with the help of a simple case from
Section 5. With u(c) = `nc, Fi (z) = e−αi z , i = 1, 2, and w(z) = w, one finds
that ψ i (R) = α1R eαi R . Figure 6 clearly displays the reversal possibility and
                 i
the critical switch point, Rc , for these functions.



9     Different ‘Risk Classes’: Pooling Equilib-
      rium

     Social security systems, pension funds and pension schemes offered by
employers, trade-unions and professional organizations offer annuities at prices
which are based on the ‘pooling’ of survival rates of annuity purchasers who
belong to different ‘risk-classes’. Sometimes the non-discriminatory pricing
of annuities is due to a legal constraint and often it reflects the imperfect
information available to insurers on purchasers’ risk characteristics. Presum-
ably, the costs of additional information are higher than the benefits from
selective pricing.

    When a variety of annuities are offered in terms of the time pattern of
payment (over age), different ‘risk classes’ will typically choose different annu-
ities, thereby potentially revealing their type. However, such ‘self-selection’
is not transparent when individuals purchase annuities from many insurers.
Rather than question the feasibility of pooling equilibria, we want to explore
their implications.

    Assume again that the population consists of two groups, such that all in-
dividuals who belong to a certain group who have the same survival function.
We continue to assume that the survival function of group 1 stochastically
dominates, according to (26), that of group 2. The relative size of group 1
in the population is g, 0 < g < 1.


                                       25
    Annuity prices are based on a uniform age survival function, G(z), which
is a weighted average of the two groups’ survival functions, F1 (z) and F2 (z).
Clearly individuals in group 1 benefit from being pooled with the shorter-
lived group 2. In contrast, individuals in group 2 pay higher prices than those
based on their own survival function. However, it can be shown that they
always prefer to annuitize some of their savings (see Appendix B). In par-
ticular, with continuous annuitization and the ability to adjust consumption
at the pooled rate, group 2 individuals choose to annuitize all their savings.
When annuities provide a constant flow of payments and consumption may
differ from these payments by using non-annuitized resources, group 2 indi-
viduals will choose to have a lump-sum available upon retirement in order
to supplement consumption over a certain period after retirement (Appendix
B). In any case, there is no Separating Equilibrium, i.e. an equilibrium in
which only group 1 individuals purchase annuities. With everybody purchas-
ing annuities, the pooled survival function has as weights the relative size of
the two groups in the population:
                          G(z) = gF1 (z) + (1 − g)F2 (z)                      (53)
   Each group’s annuities satisfy a zero expected profits condition:
              ZT                    Zi
                                    R

                   G(z)ci (z)dz −        G(z)w(z)dz = 0,        i = 1, 2, .   (54)
              0                     0

  Maximization of expected utility subject to (54) yields F.O.C. for opti-
                                             e
mum consumption, ei (z), and retirement age, Ri , of group i:
                 c
                           Fi (z)u0 (ci (z)) − λi G(z) = 0                    (55)

                                  e      e e
                                a(Ri ) − φi (Ri ) = 0                         (56)
      e
where φi (R) = u0 (ei (R))w(R) for all R, 0 ≤ R ≤ T , and λi , λi > 0, constant.
                   c
As before, write ei (z) = c0i hi (z), hi (0) = 1.
                 c
   Solving for c0i from (54),
                                             R
                                             R
                                                 G(z)w(z)dz
                                             0
                          c0i = c0i (R) =                                     (57)
                                             R
                                             T
                                                 G(z)hi (z)dz
                                             0


                                            26
Differentiating (55), h1 (z) and h2 (z) are seen to satisfy:
                                      ·                 ¸
                      h01 (z)   1 − β f2 (z)     f1 (z)
                              =               −           ,                                             (58)
                      h1 (z)      σ     F2 (z) F1 (z)
and
                                             ·                ¸
                                  h02 (z)   β f1 (z)   f2 (z)
                                          =          −          ,                                       (59)
                                  h2 (z)    σ F1 (z) F2 (z)
             gF1 (z)              gF1 (z)
where β =     G(z)
                       =   gF1 (z)+(1−g)F2 (z)
                                               ,     0 < β < 1.

   Since F1 (z) stochastically dominates F2 (z), (58) and (59) show that op-
timum consumption of group 1 increases, h01 (z) > 0, while that of group 2
decreases, h02 (z) < 0, 0 ≤ z ≤ T .

    Pooling distorts retirement decisions compared with the First-Best. Specif-
ically, optimum retirement ages of the two groups are ‘pooled together’.

    Proposition 7: In a Pooling Equilibrium with continuous annuitization,
                                      ∗   e       ∗   e
if σ ≥ 1 for all z, 0 ≤ z ≤ T , then R1 > R1 and R2 < R2 .

   Proof. By (57), since h01 (z) > 0,
                                   R
                                   R                                       R
                                                                           R
                                       G(z)w(z)dz                                G(z)h1 (z)w(z)dz
                                   0                                       0
      e1 (R) = c01 h1 (R) =
      c                                                       h1 (R) >                                  (60)
                                   R
                                   T                                             R
                                                                                 T
                                       G(z)h1 (z)dz                                  G(z)h1 (z)dz
                                   0                                             0

   By (53) and (58)
                                       (G(z)h1 (z))0    f1 (z)
                                                     <−                                                 (62)
                                        G(z)h1 (z)      F1 (z)
                                            G(z)h1 (z)                F1 (z)
Inequality (62) implies that            R
                                        T                  and    R
                                                                  T              satisfy the ‘single-crossing’
                                            G(z)h1 (z)dz              F1 (z)dz
                                        0                         0
condition (the latter crossing ‘from below’) and hence, by (27),
                                                 
                       ZR 
                           G(z)h1 (z)        F1 (z)  
                          T              − T           dz > 0                                         (63)
                          R                R          
                       0     G(z)h1 (z)dz     F1 (z)dz
                              0                               0


                                                         27
for all R, 0 < R < T . In view of (60) and (63),

                                  R
                                  R                                  R
                                                                     R
                                      G(z)h1 (z)w(z)dz                   F1 (z)w(z)dz
                                  0                                  0
                e1 (R) >
                c                                                >                      = c∗ (R)
                                                                                           1       (64)
                                      R
                                      T                                  R
                                                                         T
                                          G(z)h1 (z)dz                       F1 (z)dz
                                      0                                  0

                                                         ∗   e
This inequality and conditions (56) and (11) imply that R1 > R1 . A similar
                                      ∗
argument can be used to show that R2 < R  e2 .   ¥

                                                 e
    Note that it has not been shown whether R1 is 0larger or smaller than
                                                                      0
e                                            (G(z)h1 (z))
R2 . Indeed, observe that, by (58) and (59), G(z)h1 (z) > (G(z)h22(z)) implying
                                                           G(z)h (z)
           G(z)hi (z)
that   R
       T                  , i = 1, 2, satisfy the ‘single-crossing’ condition. Hence, there
           G(z)hi (z)dz
       0
exists an age Rc , 0 < Rc < T , such that

                              G(R)h1 (R)                   G(R)h2 (R)
                                                   S                           , as R S Rc         (65)
                          R
                          T                            R
                                                       T
                              G(z)h1 (z)dz                 G(z)h2 (z)dz
                          0                            0

                              R
                              R
                                  G(z)w(z)dz
   Since ei (R) =
         c                    0
                              R
                              T                  hi (R), it follows from (65) that e1 (R) S e2 (R)
                                                                                   c        c
                                  G(z)hi (z)dz
                              0
    e        e                            e            e    e
and φ1 (R) T φ2 (R), as R S Rc . Thus, if R1 < Rc then R1 > R2 , while if
e             e     e
R1 > Rc then R1 < R2 .




                                                            28
   Appendix A
   Maximization of (40) subject to (42) with respect to c(z), R ≤ z ≤ T , R
                                             ˆ
and BR yields F.O.C. for the optimum, (ˆ(z), R):
                                        c
                                      ˆ
                   u0 (ˆR (z)) − v 0 (BR ) = 0,
                       c                          R≤z≤T                    (A.1)

                               ˆ          ˆ
                             a(RR ) − φR (RR ) = 0                         (A.2)

where φR (R) = u0 (ˆR (R))w(R), for all R, 0 ≤ R ≤ T . It is seen from (A.1)
                   c
that consumption after retirement is constant, cR (z) = cR (R), R ≤ z ≤ T .
                                               ˆ        ˆ

   The (Hamiltonian) dynamic optimization condition with respect to c(z),
0 ≤ z ≤ R, is
                                 "                          #
               c0R (z)
               ˆ                      ˆ
                         1 f (z) v 0 (BR (z)) − u0 (ˆR (z))
                                                      c
                       =                                           (A.3)
               cR (z)
               ˆ         σ F (z)          u0 (ˆR (z))
                                              c

Note that equation (6) in the paper is a special case of (A.3) when v0 (B) = 0.

   The two differential equations (A.3) and (41) in the paper, with initial
           ˆ
condition BR (0) = B0 and the end-condition cR (R) satisfying (A.1) and
                                                ˆ
                                           ˆ
(A.2), determine the optimum path (ˆR (z), BR (z)), 0 ≤ z ≤ R.
                                   c

    The budget constraint, (42) in the paper, and (A.2) are used jointly to
          ˆ         ˆ ˆ
solve for RR and cR (RR ). Figure A.1 displays the phase diagram for the
optimum path. A higher R corresponds to a higher cR (0). The path drawn
                                                      ˆ
                                                ˆ
                     ˆ R satisfying (A.2), with BR = B0 , i.e. steady state.
in the Figure is for R

    Unlike the model with no bequest motive, optimum consumption first
increases and then decreases with age. The reason is that the marginal utility
of bequests initially exceeds the marginal utility of consumption, leading to
accelerated accumulation of assets and slowly increasing consumption. As
                                                                          ˆ
assets increase, eventually this relation is reversed. At point A (at age RR ),
                                                       RR
                             ˆ ˆ             ˆ ˆ
assets have accumulated to BR (R), where BR (RR ) = (w(z)−ˆR (z))dz+ BR .
                                                                c          ˆ
                                                         0
                                    ˆ
                                    R
                                    RR
At this point, cumulative savings                                 ˆ ˆ        ˆ
                                        (w(z)−ˆR (z))dz > 0 (i.e. BR (RR ) > BR )
                                              e
                                    0
                           ˆ ˆ                          ˆ
are annuitized to a flow of cR (RR ), reducing assets to BR .

                                         29
                     ˆ ˆ                                    ˆ
    The proof that cR (RR ) < c∗ (and hence, by (A.1), BR < B ∗ ), is the same
as the proof of Proposition 3 in the paper. The assumption that B0 = BR , i.e.ˆ
                                                                                ˆ
                                                       ˆ R ) for all z, 0 ≤ z ≤ RR ,
steady-state, is needed to establish that cR (z) > cR (R
                                          ˆ        ˆ
which is used in that proof. It may not hold for an arbitrary B0 < BR .     ˆ




                                        30
   Appendix B
   Groups 2’s survival function, F2 (z), is stochastically dominated by that of
group 1, F1 (z). First we want to show that when annuities are priced at the
pooled survival function, G(z), and the payment of benefits can be adjusted
optimally, group 2 individuals annuitize all their savings.

    Let e2 (z) be consumption at age z, R ≤ z ≤ T , from non-annuitized
resources. The zero-expected profits condition is

                   ZT                          ZT                Z2
                                                                 R

                        G(z)c(z)dz + G(R)           e2 (z)dz −        G(z)w(z)dz = 0        (B.1)
                   0                          R2                 0

   Maximization of expected utility

              Z2
              R                          ZT                                   Z2
                                                                              R

   V2 =            F2 (z)u(c2 (z))dz +        F2 (z)u(c2 (z) + e2 (z))dz −         F2 (z)a(z)dz
              0                          R2                                   0
                                                                                            (B.2)

subject to (B.1) yields F.O.C. with respect to c(z), R ≤ z ≤ T ,


                              F2 (z)u0 (c2 (z) + e(z)) − λ2 G(z) = 0                        (B.3)

where λ2 , λ2 > 0, is a constant. Hence, for any z, z > R2 ,
                                ∂V2
                                       = λ2 [G(z) − G(R2 )] < 0,                            (B.4)
                               ∂e2 (z)

which means that, at the optimum, all savings are annuitized.

   Suppose now that annuities provide a constant flow of payments, denoted
by b. The zero expected profits condition is now

         ZT                        ZT                 Z2
                                                      R

    b2        G(z)dz + G(R2 )           e2 (z)dz −         G(z)(w(z) − c2 (z))dz = 0        (B.5)
         R2                        R2                 0




                                                    31
and expected utility is

         Z2
         R                          ZT                              Z2
                                                                    R

  V2 =        F2 (z)u(c2 (z))dz +        F2 (z)u(b2 + e2 (z))dz −        F2 (z)a(z)dz = 0
         0                          R2                              0
                                                                                      (B.6)

   Maximization of V2 subject to (B.5) with respect to e2 (z) and b2 yields
F.O.C.

                 F2 (z)u0 (b2 + e2 (z)) − λ2 G(R2 ) = 0,           R≤z≤T              (B.7)

and
                    ZT                                   ZT
                         F2 (z)u0 (b2 + e2 (z))dz − λ2        G(z)dz = 0              (B.8)
                    R2                                   R2


   Suppose that at the optimum, e(z) = 0 for all R2 ≤ z ≤ T . Then, from
(B.6) and (B.7),
                                                 
                                 F (R )                  ZT
                ∂V2              2 2            G(R2 ) 
                                                        
                           = λ2  T          − T         G(z)dz                      (B.9)
                ∂e2 |e2 =0      R             R        
                                    F2 (z)dz     G(z)dz R2
                                   R2              R2


which, by (53) and (26) in the paper, is positive. Hence, e2 (z) > 0 for some
z, R ≤ z ≤ T . Next, we want to show that b2 > 0 at the optimum. Suppose
b2 = 0 and hence (B.7) holds for all z, R2 ≤ z ≤ T . Using (B.8),
                                                          
                                                  ZT
               ∂V
                          = λ2 G(R2 )(T − R2 ) − G(z)dz              (B.10)
               ∂b2 |b2 =0
                                                              R2

which is positive. Hence, at the optimum b2 > 0.




                                              32
References
[1] Abel, A.B. (1986), “Capital Accumulation and Uncertain Lifetimes with
    Adverse Selection”, Econometrica, 54, 1079-1097.

[2] Brugiavini, A. (1993), “Uncertainty Resolution and the Timing of An-
    nuity Purchases”, Journal of Public Economics, 50, 31-62.

[3] Diamond, P. and S. Valdez-Prieto (1994), “Social Security Reforms”, in
    Bosworth, B., R. Dornbush and R. Labàn (eds.), The Chilean Economy:
    Policy Lessons and Challenges. (Brookings Institution).

[4] Eckstein, Z., M. Eichenbaum and D. Peled (1985), “Uncertain Lifetimes
    and the Welfare Enhancing Properties of Annuity Markets and Social
    Security”, Journal of Public Economics, 26, 303-326.

[5] Friedman, B.M. and M. Warshawsky (1990), “The Cost of Annuities:
    Implications for Saving Behavior and Bequest”, Quarterly Journal of
    Economics, 420, 135-154.

[6] Merton, R.C. (1981), “On the Role of Social Security as a Means for Ef-
    ficient Risk-Bearing in an Economy where Human Capital is not Trade-
    able”, N.B.E.R. Working Paper, no. 743.

[7] Mitchell, O.S., J.R. Poterba, M.J. Warshawsky and J.R. Brown (1999),
    “New Evidence on Money’s Worth of Individual Annuities”, American
    Economic Review, (forthcoming).

[8] Poterba, J.R. (1997), “The History of Annuities in the United States”,
    N.B.E.R. Working Paper No. 6001.

[9] Rothschild, M. and J. Stiglitz (1976) “Equilibrium in Competitive Insur-
    ance Markets: an Essay on the Economics of Incomplete Information”,
    Quarterly Journal of Economics, 90, 624-649

[10] Sheshinski, E. and Y. Weiss (1981), “Uncertainty and Optimal Social
     Security Systems”, Quarterly Journal of Economics, 95, 189-206.

[11] Walliser, J. (1998), “Understanding Adverse Selection in the Annuities
     Market and the Impact of Privatizing Social Security”, Congressional
     Budget Office.

                                    33
[12] Yaari, M.E. (1965), “Uncertain Lifetime, Life Insurance and the Theory
     of the Consumer”, Review of Economic Studies, 32, 137-150.




                                    34
   c




           ^
           cR(Z)



^
cR(R)




       0                      R   T   z




                   Figure 1
1              1
   =   T
z2
       ∫ F ( z )dz
       0
           2




                             F2 ( z )
                         T

                         ∫ F ( z )dz
                         0
                              2




1              1
   =   T                                             F1 ( z )
z1
       ∫ F ( z )dz
           1
                                                 T

       0                                         ∫ F ( z )dz
                                                 0
                                                      1




                     0                  zc                      T   z



                                             Figure 2
        σ >1
φ                                           a( R)




                                            ψ (R )




                                            φ R (R)



                               φ (R )

    0   R*                ^
                          RR            ^
                                        R             T   R




               Figure 3
                   ψ ( R) = φ R ( R)




e




                           φ (R )
1



    0         1                R
              α



        Figure 4
           ε



           1



                                ε




                   0                                        R   M

θ ∂φ                                            (a)
     = 1 − σε
φ ∂θ




           1
                       σ1 < 1
        1− σ1
               0
                                    σ >1                    R       M

        1−σ
                                                      (b)

                                           Figure 5
           a( R)
ψ
                      a’( R)
                                 ψ 2 ( R)   ψ 1 ( R)




    0   ^  ^
        R2 R1 1          ^ ^
                      Rc R` R`        1     R
                          1 2
              α2                      α1




               Figure 6
                                u ' (c ) = υ ' ( B )[c ' = 0]
                                                     ˆ
   ^
   c




  c*

^ R(0)
c


c ^
^ R(RR)




       0      ^
           B0=BR           B*                BR ^ R
                                             ^ (R )             B




              Figure A.1

								
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