Sample chapter for the economic theory of annuities_8 potx

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August 18, 2007 Time: 11:35am index.tex
Index
adverse selection, xiii, 4; bequests and, 82;
bundling and, 131; defined, 67;
empirical importance of, 67; general
model for, 69–71; proof of, 76; rate of
return and, 71; refundable annuities and,
144–49; risk class and, 67–76
after-tax retirement annuities (ATRA),
81n25
age, 6; coefficient of relative risk aversion
and, 29; consumption and, 12–14; life

expectancy and, 15, 18; longevity and,
4–5; no-arbitrage condition and, 24–25;
normal retirement age (NRA) and,
37–38; optimum retirement, 12–14,
21–22, 27–28, 32–40; partial
annuitization and, 33–34; payout
schemes and, 3; present/future self utility
and, 48–50; profile mismatch and, 2–3;
risk class and, 56–66; steady-state age
density distributions and, 97–108;
survival functions and, 15–18; survival
probabilities and, 1–2, 45–50; timing of
purchases and, 135–37
age composition effect, 97
aggregate resource constraint, 21–22, 32
Angeletos, G., 45
annua,9
annuities: bundling and, 131–34;
competitive equilibrium and, 23–24,
51–53, 59, 61, 65, 78–79, 135–37;
deferred, 26; defined, 1; differentiated,
81–96; equivalent level of assets and,
41–42; first best equilibrium and, 6,
21–23, 56–58, 64, 77–78, 141; full
annuitization and, 23–24; growth of,
9–10; holding, 1n1; information effects
and, 3–4; long-term, 6, 26, 59, 62–66;
low returns and, 35; money’s worth of,
10; moral hazard and, 51–55, 79–80;
no-arbitrage condition and, 5, 24–25,

60–62, 70–71; optimum taxation of,
125–28; options and, xv, 9; partial
annuitization and, 33–35; payout terms
and, 59; period-certain, 7, 81, 87–89;
pooling equilibrium and, 103–4 (see also
pooling equilibrium); predetermined
flow of returns and, 63, 147–48; profile
mismatch and, 2–3; protected, 81;
refundable, 8, 136, 144–49; regular, 81,
84, 87–89; revaluation of, 64–65;
sequential market equilibrium and,
137–40; short market history of, 9–10;
short-term, 6, 26, 33–34, 60–62; stable
flows of, 4n3; survival probabilities and,
1–2, 45–50 (see also survival
probabilities); timing of purchases and,
135–37; unintended bequests and, 1–2,
98, 104–5; welfare value and, 41–42
annuity puzzle, 2–3
Arrow, Kenneth, xv, 111
asymmetric information, 6, 118–22
Atkinson, A., xiv
Austen, Jane, 9
Averting the Old Age Crisis (World
Bank), xv
Baldwin, B. G., 11
banks, 4
behavioral models, xiv, 3, 45–47, 136–37
Benartzi, S., 136, 148
bequests, 38n15; adverse selection and, 82;

ergodic long-term distribution of, 104–5;
information effects and, 3–4; insurance
and, 81–83; motive for, 1n1; partial
annuitization and, 33–34; period-certain
annuities and, 87–89; pooling
equilibrium and, 82; private information
and, 82; renewal process and, 105;
separating equilibrium and, 82;
steady-state age density distributions
and, 104–5; unintended, 1–2, 98, 104–5
Bernheim, D. B., 1n1, 27
Bible, 1
Biggs,J.H.,11
Bodie, Z., 132, 135
bounded rationality, xiv, 3
Bowers, N. L., 11
Brown, J. R., 1–3, 13, 33, 67, 81, 140n49
August 18, 2007 Time: 11:35am index.tex
158
•
Index
Brugiavini, A., 82n26, 136, 142n51
bundling, xiii, 8; adverse selection and,
131; cost of, 131–32; pooling
equilibrium and, 133; price and, 131–34;
sustainability and, 131
Cannon, E., 10, 35
certainty equivalence, 88n30
Chiappori, A., 67
Choi, J., 45

coefficient of relative risk aversion, 29
competitive equilibrium: first best
equilibrium and, 51–53; full
annuitization and, 23–24; income
uncertainty and, 78–79; separating
equilibrium and, 59, 61; short-term
annuities and, 61; survival functions and
65; timing of annuity purchases and,
135–37
competitive separating equilibrium, 59, 61
consumption: after retirement, 12–14,
27–28; aggregate savings and, 97–102;
coefficient of relative risk aversion and,
29; competitive equilibrium and, 51–53;
elasticity and, 13–14, 31; equations for,
12–14; expected, 13; exponential
survival functions and, 42–43; first best
equilibrium and, 56–58, 83–84;
individual savings and, 98; life cycle
model and, 21–26; longevity increase
and, 30–32, 44; long-term annuities and,
63–66; moral hazard and, 51–55; no late
transactions and, 138–40; optimum
retirement age and, 22, 38–40; optimum
trajectory for, 34; pooling equilibrium
and, 67–76; present/future self utility
and, 48–50; price and, 109–17;
refundable annuities and, 8–9, 144–49;
resource constraint and, 57; short-term
annuities and, 60–62; utility of, 21,

27–28
cost effects, 119–20
cross-subsidization, 5, 13
Cutler, D. M., 32n13, 35
Davidoff, T., 3, 33, 81
Deaton, A., 97
demand elasticity, 89
Diamond, P., xiv–xv, 3, 7, 14, 33, 48, 81,
109n39, 118
Dickens, Charles, 9
disability benefits, 56
Ditchik, Seth, xv
Dixit, Avinash, xv
Duncan, R. M., 11
elasticity: consumption and, 13–14;
demand, 89; optimum taxation and,
124, 127–28; price and, 113–14;
steady-state age density distributions
and, 101
envelope theorem, 31, 139
equations: aggregate resource constraint,
42–43, 47, 102–3, 120; aggregate
savings, 99–102, 106–8; behavioral
models, 46; bequests, 104–5; bundling,
132–33; competitive equilibrium, 23–24;
consumption, 12–14; differentiated
annuities, 83–86, 88–96; expected
lifetime utility, 21; expected utility of
consumption, 27; first best equilibrium,
57–58, 83, 137–38, 213; income

uncertainty, 77–79; individual savings,
98; life expectancy, 15; longevity
changes, 18–20, 30–31, 44; long-term
annuities, 63–66; moral hazard, 51–53,
55; optimum retirement age, 27–28,
35–40; optimum taxation, 122–28;
partial annuitization, 34–35;
period-certain annuities, 88; pooling
equilibrium, 69–74, 84–86, 89–93, 95,
103–4, 119–22, 129–30; positive time
preference, 32–33; present/future self
utility, 49–50; pricing, 109–17;
refundable annuities, 145–47, 150–51;
separating equilibrium, 141–43;
sequential market equilibrium, 137–40;
short-term annuities, 60–61; survival
functions, 15–18, 25–30, 57–58, 60–61,
63–66; wage increase, 29; wealth level,
12; welfare costs, xiv, 41
equilibrium, xiii; bundling and, 131–34;
competitive, 23–24, 51–53, 59, 61, 65,
78–79, 135–37; cost effects and,
119–20; efficiency of, 62–66; first best,
6, 21–23, 56–58, 64, 77–78, 141; full
annuitization and, 23–24; general
equilibrium effect and, 119; income
uncertainty and, 77–80; life cycle model
and, 21–26; long-term annuities and,
62–66; moral hazard and, 51–55;
optimum commodity taxation and, 7–8;

August 18, 2007 Time: 11:35am index.tex
Index
•
159
Pareto optimality and, 8; partial
annuitization and, 33–34; pooling,
xiii–xiv, 4 (see also pooling equilibrium);
refundable annuities and, 8–9, 144–49;
risk-class pricing and, 59; second best,
8–9; separating, xiii–xiv (see also
separating equilibrium); sequential,
137–44; short-term annuities and,
60–62; stability and, 129; timing of
purchases and, 135–37; uniqueness
and, 129
expected lifetime utility, 21
fertility rate, 99n35
Finance Act, 10
Finkelstein, A., 10, 51, 82, 93
Fisher, H. F., 81, 136
France, 38
full annuitization, 23–24
game theory, 45
Gan,L.,6,45
general equilibrium effect, 119
Genesis, Bible Book of, 1
Gerber, H., 11
Germany, 38
Gompertz-Makeham function, 11
government, xiv, 144n52

Great Depression, 9
Green, Jerry, xv, 147
hazard rate, 5, 64n22; individual savings
and, 98; longevity changes and, 18–20,
30–32, 44; survival functions and, 16–18
Hebrew University, xiii
Heifetz, Zeev, xv
HMOs, 132
Hurd, M. D., 6, 27, 45
hyperbolic discounters, 148–49
income: separating equilibrium and,
140–44; sequential market equilibrium
and, 141–44; uncertainty and, 77–80,
140–44
information, 3–4, 26; asymmetric, 6,
118–22; bequests and, 82; cost effects
and, 119–20; full, 82, 87–88, 112;
medical care and, 54; moral hazard and,
51–55; pooling equilibrium and,
119–22; private, 52–53, 82, 120; risk
class pricing and, 59; separating
equilibrium and, 82; symmetric, 54
insurance, xiii–xv, 10; annuity sales and, 1;
asymmetric information and, 119–20;
bequests and, 81–83, 87–89; bundling
and, 131–34; differentiated annuities
and, 81–96; different instruments for, 67;
disability benefits and, 56; ex ante, 57;
first best equilibrium and, 56–58, 83–84;
full information and, 87–88; income

uncertainty and, 77–80; low returns and,
35; medical testing and, 3n2; optimum
annuity taxation and, 125–28;
period-certain annuities and, 81, 87–89;
pooling equilibrium and, 84–95; profile
mismatch and, 2–3; reinsurance and, 10;
reverse life, 2; self-selection and, 82–83,
90–93; separating equilibrium and, 84;
survival probabilities and, 1–2;
unintended bequests and, 1–2
investment, xiv; annuity sales and, 1;
competitive equilibrium and, 51–53; low
returns and, 35; moral hazard and,
51–55; optimum retirement age without
annuities, 38–40; risk class and, 56–66;
short-sightedness and, 48
Israel, 38
James, E., 10
Kinugasa, T., 97
Köszegi, B., 48
Kotlikoff, L., 40
labor utility, 22, 44, 56, 71n23, 73
Laffont, J J., 121n47
Lagrange multiplier, 123
Laibson, D., 6, 45, 48, 136, 148
law of large numbers, 21–22
Lee, R., 97
Levy, H., 16n5
life cycle model: competitive equilibrium
and, 23–25; expected lifetime utility

and, 21; exponential survival functions
and, 25–26; first best equilibrium and,
21–23
life expectancy, 15, 18; aggregate savings
and, 97–102; expected lifetime utility
and, 21; longevity changes and, 18–20;
optimum taxation and, 122–30; pricing
issues and, 109–17; sequential market
equilibrium and, 137–40
liquidity, 35, 136
August 18, 2007 Time: 11:35am index.tex
160
•
Index
longevity, 4–6; age composition effect and,
97; aggregate savings and, 97–108;
average, 82; bundling and, 131; changes
in, 18–20, 30–32, 35–38, 44; elasticity
and, 31; first best equilibrium and,
56–58, 83–84; income uncertainty and,
77–80; individual savings and, 98–100;
life cycle model and, 21–26;
period-certain annuities and, 7;
refundable annuities and, 144–49; risk
class and, 56–66, 70; steady-state age
density distributions and, 97–108;
survival functions and, 101 (see also
survival functions); timing of purchases
and, 135–37; unintended bequests and,
98, 104–5

Madeson, Avital, xv
Manski, Charles F., 6, 45
markets, xiii–xv, 5; bundling and, 131–34;
competitive annuity, 112; compulsory
purchase, 10; differentiated annuities
and, 81–96; full information and, 112;
Great Depression and, 9; income
uncertainty and, 77–80; information
effects and, 3–4; insurance, 82 (see also
insurance); low returns and, 35; medical
prices and, 54; no late transactions and,
138–40; partial annuitization and,
33–34; replacement rates and, 2–3; risk
class and, 56–66; sequential equilibrium
and, 8, 137–44; short history of, 9–10;
survival frequencies and, 6; voluntary
purchase, 10; welfare value and, 41–42
Martimort, D., 121n47
Mason, A., 97
McFadden, D., 6, 45
McGarry, K., 45
medical care, 3n2, 54, 132
Merrill, A., 45
Merton, R., xiv
Middle Ages, 9
Miles, D., 97
Milevsky, M. A., 11, 81
Miller, T., 97
Mirrlees, J. A., xiv–xv, 7, 78n24, 111, 118
Mitchell, Olivia S., 148

mixed pooling equilibrium, 90–93
modified Ramsey-Boiteux conditions, 124
money’s worth, 10
moral hazard, 6; competitive equilibrium
and, 51–53; defined, 51; income
uncertainty and, 79–80; medical care
and, 54; no-arbitrage condition and, 53;
private information and, 52–53
Murtaugh, M., 131–32
Myles, G., xiv
national defined contribution systems,
13n4
New Financial Order, The (Shiller), xv
no-arbitrage condition, 5, 24; long-term
annuities and, 62; moral hazard and, 53;
pooling equilibrium and, 70–71;
short-term annuities and, 60–61;
survival functions and, 25
normal retirement age (NRA), 37–38
optimum commodity taxation, 7–8;
pooling equilibrium and, 122–30
optimum income tax model, 78n24
optimum retirement age, 37–40; adverse
selection and, 67–76; competitive
equilibrium and, 51–53; first best
equilibrium and, 56–58; income
uncertainty, 77–80; life cycle model and,
21–22, 27–28; long-term annuities and,
63–64; partial annuitization and, 33–35;
positive time preference and, 32–36;

savings and, 12–14
optimum transfers, 58
Pareto optimality, xiv, 8
Paxson, C., 97
pay-as-you-go systems, xiv
pension funds, 4
Pindyck, R., 131
pooling equilibrium, xiii–xiv, 4, 6; annuity
demand and, 69, 70; asymmetric
information and, 119–22; bequests and,
82; bundling and, 133; cost effects and,
119–20; couples and, 40–41; defined, 67;
general model for, 69–71; insurance and,
84–93, 95; mixed, 90–93; no annuities
and, 103–4; no-arbitrage condition
and, 70–71; optimum commodity
taxation and, 122–30; period-certain
annuities and, 88–89; Ramsey-Boiteux
conditions and, 118, 124–25, 127; rate
of return and, 71; stability and, 129;
uniqueness and, 129; welfare function
and, 120
August 18, 2007 Time: 11:35am index.tex
Index
•
161
population theory: aggregate savings and,
97–102; fertility rate and, 99n35;
steady-state age density distributions
and, 97–108

portfolios, 4, 145; refundable annuities
and, 8–9, 144–49
positive constant rate of interest, 32–36
positive time preference, 32–36
Poterba, J., 10, 51, 82, 93
poverty rates, 14
present/future self utility, 48–50
price, xiii; actuarially fair, 109; bundling
and, 131–34; competitive annuity
market and, 112; elasticity and, 89n33,
113–14; equilibrium and, 4 (see also
equilibrium); first-best, 109–12, 115–16;
full information and, 82, 112; hyperbolic
discounters and, 148–49; medical care
and, 54; no late transactions and, 140;
numeraire, 120–21, 125–26; optimum
commodity taxation and, 122–30;
period-certain annuities and, 89;
predetermined, 8–9, 147–48; refundable
annuities and, 8–9, 144–49; risk-class,
59; second-best optimum, 113–15;
sequential market equilibrium and,
137–44; stability and, 129; time
preference and, 109, 125; utilitarianism
and, 109; welfare function and, 110–11;
zero-profits conditions and, 122
Princeton University, xiii
private information, 4
Prudential, 10
Ramsey, F. P., 7, 118

Ramsey-Boiteux conditions, 118, 124–25,
127
rational individuals, 3, 5–6; predetermined
flow of returns and, 63; present/future
selves and, 48–50; risk pooling by
couples and, 40–41; self selection and,
82–83, 90–93; survival probabilities and,
45–50; timing of purchases and, 135–37
reform, 4–5
refundable annuities, 8, 136, 144–49
reinsurance, 10
renewal process, 105
replacement rates, 2–3
retirement, xiv; aggregate savings and,
97–102; benchmark calculations for,
12–14; coefficient of relative risk
aversion and, 29; consumption after,
12–14, 27–28; cross-subsidization and,
13; exponential survival functions and,
42–43; first best equilibrium and, 56–58;
full annuitization and, 23–24; longevity
increase and, 35–38; no-arbitrage
condition and, 24–25; normal retirement
age (NRA) and, 37–38; optimum age for,
12–14, 21–22, 27–28, 32–40; optimum
transfers and, 58; partial annuitization
and, 33–34; pooling equilibrium and,
67–76; present/future self utility and,
48–50; survival probabilities and, 45–50;
wage increase effects and, 29

retirement-consumption puzzle, 27–28
reverse life insurance, 2
risk, xiii, xv, 33; coefficient of relative risk
aversion and, 29; competitive separating
equilibrium and, 59; differentiated
annuities and, 81–96; disability benefits
and, 56; income uncertainty and, 77–80;
life cycle model and, 21–26; longevity
and, 4–6, 21–26 (see also longevity); low
returns and, 35; pooling by couples and,
40–41; pricing and, 59; self-selection
and, 82–83, 90–93; survival functions
and, 15–18, 59
risk class, 4, 7–8, 109; competitive
equilibrium and, 65; defined, 56;
exponential survival functions and,
65–66; first best equilibrium and, 56–58;
long-term annuities and, 62–66; lower
longevity and, 70; optimum taxation
and, 125–28; optimum transfers and, 58;
pooling equilibrium and, 67–76; pricing
and, 59; separating equilibrium and, 61;
short-term annuities and, 60–62;
stochastically dominant, 70; weighted
equilibrium and, 67
Rohwedder, S., 27
Romans, 9
Rothschild, M., 67
Rubinfeld, D., 131
Salanie, B., xv, 67, 118, 119n44, 121n47

Samuelson, Paul, xv
savings: age composition effect and, 97;
aggregate, 4–5, 7, 97–108; benchmark
calculations for, 12–14; changes in
longevity and, 97–102; consumption
and, 12–14; cross-subsidization and, 13;
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Index
exponential survival functions and,
102–3; hyperbolic discounters and,
148–49; individual, 98–99; low returns
and, 35; no annuities and, 103–4;
optimum retirement age without
annuities and, 38–40; optimum transfers
and, 58; positive constant rate of interest
and, 32–33; poverty rates and, 14;
sequential market equilibrium and,
137–40; short-sightedness and, 48;
steady-state age density distributions
and, 97–108; timing of purchases and,
135–37; unintended bequests and, 98,
104–5
self selection, 82–83, 90–93
separating equilibrium, xiii–xiv, 21n8;
bequests and, 82; competitive, 59;
income uncertainty and, 140–44;
insurance and, 84; period-certain
annuities and, 87–89; risk class pricing

and, 59; short-term annuities and, 61;
survival functions and, 59
sequential equilibrium, 137–44
Sheshinski, E., 24n12, 103n37
Shiller, Robert, xv
short-sightedness, 48
Smith, J. P., 45
social security, xv, 2–3, 109; normal
retirement age (NRA) and, 38; reform
and, 4–5
Society of Actuaries, 11
Solow, R., xv
Song, X., 10
Spillman, B. C., 131–32
Spivak, A., 40
steady-state age density distributions:
aggregate savings and, 97–108; defined,
99; elasticity and, 101; no annuities
and, 103–4; survival functions and,
102–3; unintended bequests and,
104–5
Stiglitz, J., xiv, 67
stochastic dominance, 5, 16, 56
subjective time preference, 5
subsidization, 5, 13, 114, 144n52
survival functions, xiii, 15–18; aggregate
savings and, 97–105; competitive
separating equilibrium and, 59;
exponential, 25–26, 42–43, 47–48,
65–66, 102–3; first best equilibrium and,

56–58; income uncertainty and, 77–80;
life cycle model and, 25–26; longevity
increase and, 30–32, 44; long-term
annuities and, 62–66; no-arbitrage
condition and, 25; risk class and, 56, 59;
sequential market equilibrium and,
137–40; short-term annuities and,
60–62; steady-state age density
distributions and, 102–3
survival probabilities, 1, 6–7; aggregate
savings and, 97–102; behavioral effects
and, 45–47; game theory and, 45; life
expectancy and, 15, 18; longevity
changes and, 18–20; moral hazard and,
51–55; pooling equilibrium and, 67–76;
present/future selves and, 48–50; rates
for, 75–76; sequential market
equilibrium and, 137–40;
short-sightedness and, 48; subjective
beliefs and, 45–50
taxes, xiv, 84n28, 144n52; elasticity and,
124, 127–28; general equilibrium effect
and, 119; optimum commodity, 7–8,
122–30; optimum income tax model
and, 78n24; pooling equilibrium and,
122–30; Ramsey-Boiteux conditions
and, 118, 124–25, 127; risk class and,
125–28; subsidization and, 114;
zero-profits conditions and, 122
Thaler, 136, 148

TIAA-CREF, 81n25
time preference, 5; annuity purchases and,
135–37; optimum retirement age and,
32–36, price and, 109, 125
Tonks, I., 10, 35
tontine,9
Ulpianus, Domitius, 9
unintended bequests, 1–2, 98, 104–5
United Kingdom, 93; annuity market
history and, 9–10; bequests and, 81–82;
bundling and, 132; purchase timing and,
135–36
United States: annuity market growth in,
9–10; Great Depression and, 9; normal
retirement age (NRA) and, 38;
period-certain annuities and, 81; poverty
rates and, 14; replacement rates and,
2–3; social security and, 2–3; timing of
purchases and, 135
Utkus, Stephen, 148
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wages, 84n28; coefficient of relative risk
aversion and, 29; exponential survival
functions and, 42–43; first best
equilibrium and, 56–58; income
uncertainty and, 77–80, 140–44;
optimum retirement and, 29; optimum

transfers and, 58
Warshawsky, M. J., 131–32
welfare, xiv; longevity increase and, 30–32,
44; market value and, 41–42;
period-certain annuities and, 87–89;
pooling equilibrium and, 120; price and,
110–11
World Bank, xv
Yaari, M., 1n1, 5, 33
Young, J., 81, 136
Zissimopoulos, J. M., 45