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80 (cid:127) Chapter 10
< V∗ 2
particular, the individual typically has some influence on the outcome. Thus, the probability q, which was taken as given, may be regarded, to some extent at least, as influenced by individual decisions that involve costs and efforts. The potential conflict that this type of moral hazard raises between social welfare and individual interests is very clear in this context. Since V∗ , an increase in q decreases the first-best expected 1 ˆV1 > ˆV2, and utility. On the other hand, in a competitive equilibrium, hence an increase in q may be desirable.
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C H A P T E R 9
Pooling Equilibrium and Adverse Selection
9.1 Introduction
For a competitive annuity market with long-term annuities to be efficient, it must be assumed that individuals can be identified by their risk classes. We now wish to explore the existence of an equilibrium in which the individuals’ risk classes are unknown and cannot be revealed by their actions. This is called a pooling equilibrium.
Annuities are offered in a pooling equilibrium at the same price to all individuals (assuming that nonlinear prices, which require exclusivity, as in Rothschild and Stiglitz (1979), are not feasible). Consequently, the equilibrium price of annuities is equal to the average longevity of the annuitants, weighted by the equilibrium amounts purchased by different risk classes. This result has two important implications. One, the amount of annuities purchased by individuals with high longevity is larger than in a separating, efficient equilibrium, and the opposite holds for individuals with low longevities. This is termed adverse selection. Two, adverse selection causes the prices of annuities to exceed the present values of expected average actuarial payouts. The empirical
importance of adverse selection is widely debated (see, for example, Chiapori and Salanie (2000), though its presence is visible. For example, from the data in Brown et al. (2001), one can derive survival rates for males and females born in 1935, distinguish- ing between the overall population average rates and the rates appli- cable to annuitants, that is, those who purchase private annuities. As figures 9.1(a) and (b) clearly display, at all ages annuitants, whether males or females, have higher survival rates than the population average rates (table 9A.1 in the appendix provides the underlying data). Adverse selection seems somewhat smaller among females, perhaps because of the smaller variance in female survival rates across different occupations and educational groups.
Adverse selection may be reflected not only in the amounts of annuities purchased by different risk classes but also in the selection of different insurance instruments, such as different types of annuities. We explore this important issue in chapter 11.
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(a)
Z
Figure 9.1(a). Male survival functions (1935 cohort). (Source: Brown et al. 2001, table 1.1).
(b)
Z
Figure 9.1(b). Female survival functions (1935 cohort). (Source: Brown et al. 2001, table 1.1).
68
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Pooling Equilibrium (cid:127) 69
9.2 General Model
(9.1)
We continue to denote the flow of returns on long-term annuities purchased prior to age M by r (z), M ≤ z ≤ T. The dynamic budget constraint of a risk-class-i individual, i = 1, 2, is now
˙ai (z) = r p(z)ai (z) + w(z) − ci (z) + r (z)a(M), M ≤ z ≤ T,
z
x
z
M
M
M
(9.2)
where ˙ai (z) are annuities purchased or sold (with ai (M) = 0) and r p(z) is the rate of return in the (pooled) annuity market for age-z individuals, M ≤ z ≤ T. For any consumption path, the demand for annuities is, by (9.1), (cid:1) (cid:3) (cid:3) (cid:4)(cid:2) (cid:2) (cid:1)(cid:2) − exp ai (z) = exp r p(x) dx r p(h) dh
i = 1, 2. (cid:5) ×(w(x) − ci (x) + r (x)a(M)) dx ,
Maximization of expected utility,
T
(9.3)
(cid:2)
M
i = 1, 2, Fi (z)u(ci (z)) dz,
z
subject to (9.1) yields optimum consumption, denoted ˆci (z), (cid:1)(cid:2)
M
(9.4) (where σ is evaluated at ˆci (x)). It is seen that ˆci (z) increases or decreases with age depending on the sign of r p(z) − ri (z). Optimum consumption at age M, ci (M), is found from (9.2), setting ai (T) = 0,
, M ≤ z ≤ T, i = 1, 2 ˆci (z) = ˆci (M) exp (cid:3) 1 σ (r p(x) − ri (x)) dx
T
x
M
M
(9.5)
(cid:1) (cid:3) (cid:2) (cid:2) − exp i = 1, 2. r p(h) dh (w(x) − ˆci (x) + r (x)a(M)) dx = 0,
(9.6)
T M exp (cid:6) T M exp
Substituting for ˆci (x), from (9.4), (cid:7) (cid:6) (w(x) + r (x)a(M)) dx , (cid:6) − (cid:7)(cid:6) i = 1, 2. ˆci (M) = dx (cid:8) σ ((1 − σ )r p(h) − ri (h)) dh 1 (cid:8) x Mr p(h) dh x M
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70 (cid:127) Chapter 9
Since r1(z) < r2(z) for all z, M ≤ z ≤ T, it follows from (9.6) that ˆc1(M) < ˆc2(M). Inserting optimum consumption ˆci (x) into (9.2), we obtain the optimum demand for annuities, ˆai (z). Since ˆai (M) = 0, it is seen from (9.1) that ˙ˆa1(M) > ˙ˆa2(M). In fact, it can be shown (see appendix) that ˆa1(z) > ˆa2(z) for all M < z < T.
This is to be expected: At all ages, the stochastically dominant risk class, having higher longevity, holds more annuities compared to the risk class with lower longevity.
We wish to examine whether there exists an equilibrium pooling rate of return, r p(z), that satisfies the aggregate resource constraint (zero expected profits). Multiplying (9.1) by Fi (z) and integrating by parts, we obtain
T
(cid:2)
M
T
T
(9.7)
M
M
Fi (z)(r p(z) − ri (z)) ˆai (z) dz (cid:2) (cid:2) = r (z) dz, i = 1, 2. Fi (z)(w(z) − ˆci (z))dz + aM
Multiplying (9.7) by p for i = 1 and by (1 − p) for i = 2, and adding,
T
(cid:2)
M
[( pF1(z) ˆa1(z) + (1 − p)F2(z) ˆa2(z)) r p(z)
M
T
− ( pF1(z) ˆa1(z)r1(z) + (1 − p)F2(z) ˆa2(z)r2(z))] dz (cid:2)
T
M
T
(9.8)
(cid:2) = p F2(z)(w(z) − ˆc2(z)) dz F1(z)(w(z) − ˆc1(z)) dz + (1 − p) (cid:2)
M
+ a(M) ( pF1(z) + (1 − p)F2(z)) r (z) dz.
Recall that
M
r (z) = pF1(z)r1(z) + (1 − p)F2(z)r2(z) pF1(z) + (1 − p)F2(z)
is the rate of return on annuities purchased prior to age M. Hence the last term on the right hand side of (9.8) is equal to F (M)a(M) = (cid:6) 0 F (z)(w(z) − c) dz. Thus, the no-arbitrage condition in the pooled
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Pooling Equilibrium (cid:127) 71
(9.9)
market is satisfied if and only if the left hand side of (9.8) is equal to 0 for all z:
r p(z) = γ (z)r1(z) + (1 − γ (z))r2(z),
where
(9.10)
. γ (z) = pF1(z) ˆa1(z) pF1(z) ˆa1(z) + (1 − p) ˆa2(z)
The equilibrium pooling rate of return takes into account the amount of annuities purchased or sold by the two risk classes. Assuming that ˆai (z) > 0, i = 1, 2, r p(z) is seen to be a weighted average of r1(z) and r2(z): r1(z) < r p(z) < r2(z). In the appendix we discuss the conditions that ensure positive holdings of annuities by both risk classes.
Comparing (9.9) and (9.10) with (8.25) and (8.26), it is seen that r p(z) < r (z) for all z, M < z < T. The pooling rate of return on annuities, reflecting adverse selection in the purchase of annuities in equilibrium, is lower than the rate of return on annuities purchased prior to the realization of different risk classes.
Indeed, as described in the introduction to this chapter, Brown et al. (2001) compared mortality tables for annuitants to those for the general population for both males and females and found significantly higher expected lifetimes for the former.
In chapter 11 we shall explore another aspect of adverse selection, annuitants’ self-selection leading to sorting among different types of annuities according to equilibrium prices.
9.3 Example
x
(9.11)
M
M
Assume that u(c) = ln c, F (z) = e−αz, 0 ≤ z ≤ M, Fi (z) = e−α Me−αi (z−M), M ≤ z ≤ ∞, i = 1, 2, w(z) = w constant and let retirement age, R, be independent of risk class.1 Under these assumptions, (9.6) becomes (cid:1) (cid:3) (cid:2) (cid:2) ∞ − exp (w(x) + r (x)a(M)) dx, ˆci (M) = αi r p(h) dh
1 Individuals have an inelastic infinite labor disutility at R and zero disutility at ages
z < R.
where w(x) = w for M ≤ x ≤ R and w(x) = 0 for x > R.
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72 (cid:127) Chapter 9
M
Figure 9.2. Demand for annuities in a pooling equilibrium.
z
x
x
M
M
M
(cid:1) (cid:3) Demand for annuities, (9.2), is now (cid:2) (cid:3) (cid:2) (cid:1)(cid:2) − exp (w(x) + r (x)a(M)) dx ˆai (z) = exp r p(x) dx r p(h) dh
x
M
M
(9.12)
(cid:1) (cid:3) (cid:2) (cid:2) ∞ (cid:8) (cid:7) − − 1 − e−αi (z−M) exp (w(x) + r (x)a(M)) dx. r p(h) dh
Clearly, ai (M) = ai (∞) = 0, i = 1, 2, and since α1 < α2, it follows
R
x
· ˆai (M) = w
M
M (cid:1)
that ˆa1(z) > ˆa2(z) for all z > M. From (9.1), (cid:3) (cid:1) (cid:1) (cid:3) (cid:2) (cid:2) − dx
x
M
M
(9.13)
(cid:3) + a(M) (cid:3) 1 − αi (cid:1) r p(h) dh (cid:2) exp (cid:2) ∞ − , × r (x) dx i = 1, 2. exp r (M) − αi r p(h) dh
Since r (x) decreases in x, (8.29), it is seen that if r p(x) > α1, then for i = 1, both terms in (9.13) are positive, and hence ˙ˆa1(M) > 0. From (9.12) it can then be inferred that ˆa1(z) > 0 with the shape in figure 9.2.
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Pooling Equilibrium (cid:127) 73
M
Figure 9.3. Return on annuities in a pooling equilibrium.
Additional conditions are required to ensure that ˙ˆa2(M) > 0, from which it follows that ˆa2(z) > 0, z ≥ M. Thus, the existence of a pool- ing equilibrium depends on parameter configuration. When ˆa2(z) > 0 (figure 9.2), then r (z) = δ(z)α1 + (1 − δ)α2 > r p(z) = γ (z)α1 + (1 − γ (z))α2 because when ˆa1(z) > ˆa2(z), then (figure 9.3)
δ(z) = pe−α1(z−M) pe−α1(z−M) + (1 − p)e−α2(z−M)
> = γ (z). pe−α1(z−M) ˆa1(z) pe−α1(z−M) ˆa1(z) + (1 − p)e−α2(z−M) ˆa2(z)
What remains to be determined is the optimum ˆa(M),
M
0
M
ˆa(M) = ((w − ˆc)/α)(eα M − 1), or, equivalently, consumption prior to age M, ˆc = w−α ˆa(M)/(eα M − 1). By (9.11), ˆci (M), i = 1, 2, depend directly on ˆa(M). Maximizing expected utility (disregarding labor disutility), (cid:2) (cid:2) ∞ V = e−αz ln c dz + pe−α M e−α1(z−M) ln ˆc1(z) dz
(9.14)
M
(cid:2) ∞ +(1 − p)e−α M e−α2(z−M) ln ˆc2(z) dz,
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74 (cid:127) Chapter 9
in
life:
(cid:3) (cid:2) −
(cid:3) (cid:2) −
A =
dx > 1
early (cid:5) .
Figure 9.4. Amount (cid:1) (cid:2) ∞ M exp
purchased long-term annuities of (cid:4) (cid:4) (cid:2) x R x r (x) dx/ Mr p(h) dh Mexp Mr p(h) dh
x M r p(h) dh)r (x) dx
x
M
(9.15)
with respect to a(M), using (9.11), yields the first-order condition for an interior solution that can be written, after some manipulations as (cid:1) (cid:3) = eα M − 1 w(eα M − 1) − αa(M) p α1 + p α2 (cid:6) (cid:3) × (cid:6) ∞ M exp(− (cid:1) (cid:2) − (w(x) + c(x)a(M)) dx r p(h) dh (cid:6) ∞ M exp
The left-hand side of (9.15) increases with a(M), while the right hand side decreases with a(M) (figure 9.4).
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Appendix
Table 9.A.1.
Population
Annuitants
Age
Male
Female
Male
Female
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
0.978503 0.955567 0.931401 0.906303 0.880455 0.853800 0.826172 0.797493 0.767666 0.736589 0.704187 0.670393 0.635149 0.598456 0.560408 0.521200 0.481108 0.440451 0.399581 0.358884 0.318805 0.279836 0.242486 0.207251 0.174563 0.144767 0.118099 0.094678 0.074510 0.057496 0.043497 0.032263 0.023472 0.016760 0.011757 0.008094 0.005462 0.003608 0.002329 0.001467
0.986735 0.972336 0.956873 0.940484 0.923244 0.905086 0.885875 0.865541 0.843998 0.821157 0.796868 0.771044 0.743735 0.715046 0.685027 0.653585 0.620632 0.586205 0.550354 0.513134 0.474641 0.435065 0.394715 0.354020 0.313509 0.273776 0.235444 0.199121 0.165364 0.134641 0.107438 0.084018 0.064413 0.048453 0.035806 0.025961 0.018442 0.012814 0.008695 0.005751
0.989007 0.977086 0.964103 0.949935 0.934490 0.917697 0.899490 0.879829 0.858678 0.835989 0.811695 0.785733 0.758039 0.728578 0.697360 0.664443 0.629934 0.593975 0.556727 0.518386 0.479222 0.439561 0.399797 0.360364 0.321725 0.284338 0.248635 0.214996 0.183735 0.155093 0.129260 0.106332 0.086313 0.069084 0.054455 0.042188 0.032040 0.023776 0.017172 0.012013
0.992983 0.985266 0.976922 0.967886 0.958116 0.947530 0.936004 0.923386 0.909496 0.894166 0.877234 0.858575 0.838109 0.815799 0.791601 0.765431 0.737205 0.706870 0.674371 0.639648 0.602670 0.563491 0.522278 0.479344 0.435214 0.390583 0.346256 0.302021 0.260889 0.222355 0.187020 0.155292 0.127382 0.103228 0.082603 0.065170 0.050582 0.038510 0.028653 0.020738
A. Survival Rates for a 1935 Birth Cohort
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76 (cid:127) Chapter 9
Table 9.A.1. Continued.
Population
Annuitants
Age
Male
Female
Male
Female
105 106 107 108 109 110 111 112 113 114 115
0.003699 0.002309 0.001394 0.000813 0.000455 0.000244 0.000125 0.000061 0.000028 0.000012 0.000000
0.008094 0.005216 0.003189 0.001830 0.000974 0.000473 0.000206 0.000078 0.000024 0.000006 0.000000
0.014519 0.009766 0.006259 0.003784 0.002131 0.001100 0.000510 0.000206 0.000068 0.000017 0.000000
0.000901 0.000538 0.000311 0.000175 0.000094 0.000049 0.000025 0.000012 0.000005 0.000002 0.000000 Source: Brown et al. (2001, table 1.1)
· ˆc(z) >
B. Proof of Adverse Selection
We first prove that ˆa1(z) > ˆa2(z) for all z, M ≤ z ≤ T. From (9.5), it is seen that ˆc1(z) and ˆc2(z) must intersect at least once over M < z < T. Let · z0 be an age at which ˆc1(z0) > ˆc2(z0). By (9.4), the sign of ˆc(z) at z0 is equal to the sign of r2(z0) > r1(z0). Hence, the intersection point is · · ˆc2(z) (cid:1) 0 as z (cid:1) z0. It follows now from (9.2) that ˆc1(z) − unique, and ˆa1(z) > ˆa2(z) for all M < z < T.
The pooling rate of return is a weighted average of the two risk-class rates of return, r1(z) < r p(z) < r2(z), provided ˆai (z) > 0, i = 1, 2. From (9.2) and (9.5), a sufficient condition for this is that w(z)+r (z)a(M)− ˆci (z) strictly decreases in z, i = 1, 2. By (9.5), this ensures that there exists some z0, M < z0 < T, such that w(z) + r (z)a(M) − ci (z) (cid:1) 0 as z (cid:1) z0. By (9.2), this implies that ˆai (z) > 0 for all z, M < z < T. Assuming that r p(z) − r1(z) > 0, a sufficient condition for ˆa1(z) > 0 is that w(z) + r (z)a(M) is nonincreasing in z. Assuming further that r p(z) − r2(z) < 0, a more stringent condition is needed to ensure that ˆa2(z) > 0 for all M < z < T. Thus, the existence of a pooling equilibrium depends on parameter configuration.
Since ˆc1(z) − ˆc2(z) (cid:2) 0 as z (cid:2) z0 (where z0 satisfies ˆc1(z0) − ˆc2(z0) = 0). ˆRi , satisfies ˆR1 (cid:1) ˆR2 as ˆRi (cid:1) z0, Accordingly, optimum retirement age, i = 1, 2.
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C H A P T E R 8
Uncertain Future Survival Functions
8.1 First Best
So far we have assumed that all individuals have the same survival functions. We would now like to examine a heterogeneous population with respect to its survival functions.
A group of individuals who share a common survival function will be called a risk class. We shall consider a population that, at later stages in life, consists of a number of risk classes. Uncertainty about future risk-class realizations creates a demand by risk-averse individuals for insurance against this uncertainty. The goal of disability benefits programs, private or public, is to provide such insurance (usually, because of verification difficulties, only against extreme outcomes). Our goal is to examine whether annuities can provide such insurance.
In order to isolate the effects of heterogeneity in longevity from other differences among individuals, it is assumed that in all other respects— wages, utility of consumption, and disutility of labor—individuals are is to analyze the first-best resource allocation and alike. Our goal alternative competitive annuity pricing equilibria under heterogeneity in longevity.
It is difficult to predict early in life the relevant survival probabilities at later ages, as these depend on many factors (such as health and family circumstances) that unfold over time. For simplicity, we assume that up to a certain age, denoted M, well before the age of retirement, individuals have the same survival function, F (z). At age M, there is a probability p, 0 < p < 1, that the survival function becomes F1(z) (state of nature 1) and 1 − p that the survival function becomes F2(z) (state of nature 2). Survival probabilities are continuous and hence F (M) = F1(M) = F2(M). It is assumed that F1(z) stochastically dominates F2(z) at all ages M ≤ z ≤ T.
Let c(z) be consumption at age z, 0 ≤ z ≤ M, and ci (z) be consumption at age z, M ≤ z ≤ T, of a risk-class-i (state-i) individual, i = 1, 2. Similarly, Ri is the age of retirement in state i, i = 1, 2.
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Uncertain Future Survival Functions
(cid:127) 57
M
T
R1
(cid:3) (cid:2)(cid:1) (cid:1) An economy with a large number of individuals has a resource con- straint that equates total expected wages to total expected consumption: (cid:1)
M
0
M
F (z)(w(z) − c(z)) dz + p F1(z)w(z) dz − F1(z)c1(z) dz
T
R2
(8.1)
(cid:3) (cid:2)(cid:1) (cid:1)
M
M
+(1 − p) = 0. F2(z)w(z) dz − F2(z)c2(z) dz
M
T
R1
Expected lifetime utility is (cid:3) (cid:1) (cid:1) (cid:2)(cid:1)
0
M
0
V = F (z)u(c(z)) dz + p F1(z)u(c1(z)) dz − F1(z)e(z) dz
T
R2
(8.2)
0
M
(cid:2)(cid:1) (cid:1) (cid:3) . +(1 − p) F2(z)u(c2(z)) dz − F2(z)e(z) dz
1(z), R∗
2). It can easily be shown that c∗(z) = c∗
1
1
∗
, c∗ Denote the solution to the maximization of (8.2) subject to (8.1) by 1(z) = = R∗. The solution = R∗ 2 (c∗(z), c∗ 2(z), R∗ 2(z) = c∗ for all 0 ≤ z ≤ T and that R∗ c∗ (c∗, R∗) satisfies
(8.3)
∗ = c
∗ (R
∗
(8.4)
(cid:3) u
, ) = β W1(R∗) + (1 − β) c z1 W2(R∗) z2
∗ )w(R
∗ )) = e(R
∗ (R (cid:4)
M
T
0 F (z) dz +
M
M Fi (z) dz is life expectancy, Wi (R) = M Fi (z)w(z) dz are expected wages until retirement in
(c ), (cid:4) (cid:4)
where zi = (cid:4) R 0 F (z)w(z) dz + state i, i = 1, 2, and
β = , 0 ≤ β ≤ 1. pz1 pz1 + (1 − p)z2
This is an important result: In the first best, optimum consumption
and age of retirement are independent of the state of nature. to full This is equivalent, as we shall demonstrate,
insurance against longevity risk and against risk-class classification. When infor- mation on longevity (survival functions) is unknown early in life, individuals have an interest in insuring themselves against alternative risk-class classifications, and the first-best solution reflects such (ex ante) insurance.
Importantly, the first-best allocation, (8.3) and (8.4), involves trans- fers across states of nature. Let S denote expected savings up to
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58 (cid:127) Chapter 8
M
∗
(8.5)
age M, defined as the difference between expected wages and optimum consumption: (cid:1)
0
F (z)(w(z) − c ) dz. S =
Define optimum transfers to risk class i, denoted T∗ i
R∗
T
∗
∗
(8.6)
, as the excess of expected consumption over expected wages from age M to T less expected savings during ages 0 to M: (cid:1) (cid:1)
∗ zi + Wi (R
∗ T i
M
M
= c F (z)w(z) dz − S = c ). Fi (z) dz −
∗ T 1
(8.7)
By (8.6), (cid:2) (cid:3) , = z1(1 − β)
∗ T 2
(cid:2) − W1(R∗) z1 (cid:3) . = z2β W2(R∗) z2 − W2(R∗) z2 W1(R∗) z1
< 0. We have assumed that wages, w(z), are nonincreasing with z.1 Under this assumption transfers to the stochastically dominant group with higher life expectancy are positive, T∗ > 0, and transfers to the 1 dominated group with shorter life expectancy are negative, T∗ 2 Since F1(z) stochastically dominates F2(z),
M
M
R∗ M F2(z) dz
R∗ M F1(z) dzz1 F
0 F (z) dz + z2
≥ w(zc) W2(R∗) z2 (cid:6) − W1(R∗) z1 (cid:5) (cid:4) (cid:4) (cid:4) (cid:4) 0 F (z) + × −
M
M
T M F2(z) dz
T M F1(z) dz
(cid:6) (cid:4) (cid:4) (cid:4) (cid:5) (cid:4)
0 F (z) dz + z2
0 F (z) + z1
(8.8)
− > = 0,
1 Recall that w(cid:3)(z) ≤ 0, 0 ≤ z ≤ T, is a sufficient condition for the unique determination
of optimum consumption and retirement.
= 0. where zc is the age at which the two functions Fi (z)/zi , i = 1, 2, intersect. The resource constraint (8.1) means that total expected transfers are 0: pT∗ 1 + (1 − p)T∗ 2
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8.2 Competitive Separating Equilibrium (Risk-class Pricing)
Consider a competitive market in which individuals who purchase or sell annuities are identified by their risk classes. Identification is either exogenous or due to actions of individuals that reveal their risk classes.2 As above, during ages 0 to M, all individuals are assumed to belong to the same risk class. At ages beyond M, individuals belong either to risk class 1 or to risk class 2 and, accordingly, their trading of annuities is at the respective risk-class returns.
Whether a competitive annuity market can or cannot attain the first-best allocation depends on the terms of the annuities’ payouts. We distinguish between short-term and long-term annuities. A short- term annuity pays an instantaneous return and is redeemed for cash by a surviving holder of the annuity.3 A long-term annuity pays a flow of returns, specified in advance, over a certain period of time or indefinitely. When the short-run returns of annuities’ depend only on age according to a known survival function, the purchase or sale of a long-term annuity is equivalent to a sequence of purchases or sales of short-term annuities. However, upon the arrival of information on and the differentiation between risk classes, this equivalence dis- appears. Once information on an individual’s risk class is revealed, the terms of newly purchased or sold annuities become risk-class- to zero specific. The no-arbitrage condition, which is equivalent expected profits, now applies separately to each risk class. On the other hand, long-term annuities purchased prior to the arrival of risk-class information yield a predetermined flow of returns which, in equilibrium, reflect the expected relative weight of different risk classes in the population. Because of their predetermined terms,
long-term annuities pro- insurance against risk-class classification. This will be demon- the efficiency of competitive annuity for vide, strated to be crucial markets. We shall first show that if annuities are only short-term,
the availability of
2 This is further discussed in chapter 9. 3 In practice, of course, “instantaneous” typically means “annual,” that is, a 1-year
annuity.
then a competitive annuity market cannot attain the first best. Subse- long-term quently we shall demonstrate that annuities enables the competitive annuity market to attain the first best.
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8.3 Equilibrium with Short-term Annuities
(8.9)
During the first phase of life, individuals have the same survival functions and the purchase or sale of annuities is governed by
˙a(z) = r (z)a(z) + w(z) − c(z), 0 ≤ z ≤ M,
z
z
x
0
0
(8.10)
0 0 ≤ z ≤ M.
or, since a(0) = 0, (cid:7) (cid:8) (cid:7)(cid:1) (cid:8) (cid:2)(cid:1) (cid:1) − , a(z) = exp r (x) dx exp r (h) dh (cid:3) (w(x) − c(x)) dx
M
(8.11)
Applying the no-arbitrage condition, r (z) = f (z)/F (z), (8.10) can be rewritten as (cid:1)
0
F (z)(w(z) − c(z)) dz. F (M)a(M) =
Maximization of expected utility for 0 ≤ z ≤ M yields constant consumption, denoted c, whose level depends, of course, on the expected level of annuities held at age M, F (M)a(M). This level of annuities, (8.11), is equal to expected total savings up to age M.
(8.12)
Since all annuities are short-term, the stock a(M) is converted into new annuities by individuals alive at age M. The dynamics after age M are governed by the relevant risk-class rate of return. Consider an individual who belongs to risk class i, i = 1, 2. Denote the annuities held by this individual by ai (z). The purchase and sale of annuities are governed by
˙ai (z) = ri (z)ai (z) + w(z) − ci (z), M ≤ z ≤ T,
z
z
z
M
M
M
or (cid:7) (cid:8) (cid:7)(cid:1) (cid:8)(cid:5)(cid:1) (cid:1) − exp ai (z) = exp ri (x)dx ri (h)dh
(8.13)
(cid:6)
, M ≤ z ≤ T, ×(w(x) − ci (x)) dx + a(M)
where ri (z) is the rate of return on annuities held by risk-class-i individuals. At age M the individual holds ai (M) = a(M), having converted savings into risk-class-i annuities. The no arbitrage condition applies to each risk class separately, ri = fi (z)/Fi (z), i = 1, 2. Taking, in
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T
(8.14)
(8.13), z = T and ai (T) = 0, we obtain (cid:1)
M
Fi (z)(w(z) − ci (z)) dz + F (M)a(M) = 0.
M
T
(8.15)
Maximization of expected utility for M ≤ z ≤ T, conditional on being in state i, yields constant optimum consumption, denoted ci . From (8.11) and (8.14), c and ci are related by the condition (cid:1) (cid:1)
0
M
F (z)(w(z) − c) dz + i = 1, 2 Fi (z)(w(z) − ci ) = 0,
(8.16)
(cid:3) u
(cid:3) (ˆc) = pu
(cid:3) (ˆc1) + (1 − p)u
(with w(z) = 0 for Ri ≤ z ≤ T). Maximization of expected utility, (8.2), with respect to c, taking into account relation (8.15), yields
(ˆc2).
M 0 F (z) dz
Optimum consumption during early ages, 0 ≤ z ≤ M, is a weighted average of optimum consumption of the two risk classes after age M.4 Rewriting (8.15), (cid:4)
(8.17)
T M Fi (z) dz
, (cid:4) i = 1, 2. ˆci = Wi (Ri ) − ˆc
(8.18)
Equations (8.16) and (8.17) determine the optimum ˆc and ˆci , i = 1, 2. Optimum retirement age in state i, ˆRi , is determined by the familiar condition
(cid:3) u
i = 1, 2. (ˆci )w( ˆRi ) = e( ˆRi ),
Can the solution to (8.16)–(8.18) be the first-best allocation? To see that this is not possible, suppose that c∗ = ˆc = ˆc1 = ˆc2 and ˆR1 = ˆR2 = R∗. Then (8.17) implies that W1(R∗)/z1 = W2(R∗)/z2. It has been assumed, however, that F1(z) stochastically dominates F2(z), and therefore, as shown above, W1(R∗)/z1 < W2(R∗)/z2. This proves that the first-best solution is impossible. Specifically, stochastic dominance of F1(z) over F2(z) implies that ˆc1 < ˆc < ˆc2 for any given R. Hence, by (8.18), ˆR1 > ˆR2 (figure 8.1).
(cid:1)
M
4 The optimum amount of annuities at age M,
F (z)(w(z) − (cid:2)c) dz, may be
1 F (M)
0
negative, which means that a surviving individual undertakes at age M a contingent debt equal to this amount.
We summarize: When there are only short-term annuities, a separating competitive equilibrium is not first best. Competitive equilibrium leads to consumption and retirement ages that differ by risk class.
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62 (cid:127) Chapter 8
Figure 8.1. Optimum retirement ages by risk class.
The reason for this result is straightforward: The first best requires insurance against risk-class classification that entails transfers across states of nature. These transfers cannot be implemented with short-term annuities. We shall now demonstrate that with long-term annuities the competitive equilibrium is first best.
8.4 The Efficiency of Equilibrium with Long-term Annuities
Suppose that annuities can be held by individuals for any length of time and that their future stream of returns is fully specified at the time of purchase or sale. We continue to denote the annuities held by individuals during their early ages by a(z), 0 ≤ z ≤ M. The rate of return on these annuities at age z is denoted, as before, by r (z). Competitive trading in these annuities satisfies the no-arbitrage condition, r (z) = f (z)/F (z), 0 < z ≤ M.
Under full information about the identity of annuity purchasers and sellers, trades in annuities by individuals older than M are performed at risk-class-specific rates of return. Thus, an individual of age z > M
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T
M Fi (z)r (z) dz.
M
(8.19)
(cid:4) who belongs to risk class i, trades annuities at the rate of return ri (z) = fi (z)/Fi (z), i = 1, 2. After age M, the stock of long-term annuities held at age M, a(M), continues to provide, contingent on survival, a predetermined flow of returns, r (z). The individual may sell (when a(M) > 0) or repay a contingent debt (when a(M) < 0) at risk-class- specific prices that reflect the expected returns of these annuities to this individual, a(M) The dynamics of the individual’s budget up to age M are the same as in (8.7), and hence (8.11) holds. With constant optimum consumption, c, (cid:1)
0
F (z)(w(z) − c) dz. F (M)a(M) =
The purchase or sale of annuities by a risk-class-i individual is governed by
(8.20) where ri (z) = fi (z)/Fi (z) and ai (M) = 0. Multiplying both sides of (8.20) by Fi (z) and integrating by parts, we obtain
i = 1, 2, ˙ai (z) = ri (z)ai (z) + w(z) − ci (z) + r (z)a(M), M ≤ z ≤ T,
T
T
(8.21)
(cid:1) (cid:1)
M
M
i = 1, 2, Fi (z)(w(z) − ci ) dz + a(M) Fi (z)r (z) dz = 0,
M
T
T
or, by (8.19), (cid:1) (cid:1)
M
M
0
(cid:1) F (z)(w(z) − c) dz Fi (z)(w(z) − ci ) dz − 1 F (M)
∗
(8.23)
, is determined by Fi (z)r (z) dz = 0 (8.22) (with w(z) = 0 for Ri ≤ z ≤ T). The optimum age of retirement in state i, R∗ i
(cid:3) u
∗ (ci )w(R
i ) = e(R
i ),
i = 1, 2.
Multiplying (8.22) by p for i = 1 and by 1 − p for i = 2, and adding, we obtain
T
T
(cid:1) (cid:1)
M
M
p F1(z)(w(z) − c1) dz + (1 − p) F2(z)(w(z) − c2) dz
M
T
0
M
(8.24)
(cid:1) = 1 (cid:1) F (z)(w(z) − c) dz ( pF1(z) + (1 − p)F2(z))r (z) dz = 0. F (M)
