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Trang 5adverse 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
Trang 6158 • 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;
Trang 7Index • 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
Trang 8160 • 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
Trang 9Index • 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;
Trang 10162 • 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