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Welfare effects of life annuities: Some clarifications
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Economics Letters 99 (2008) 177 – 180 www.elsevier.com/locate/econbase
Welfare effects of life annuities: Some clarifications Hans Fehr ⁎, Christian Habermann University of Würzburg, Germany Received 10 April 2006; received in revised form 15 June 2007; accepted 21 June 2007 Available online 30 June 2007
Abstract The present paper clarifies the intergenerational welfare effects of life annuities. While present young cohorts experience significant welfare gains, future generations are hurt by lower bequest. For sufficiently high interest rates long-run welfare will decrease although aggregate efficiency rises. © 2007 Elsevier B.V. All rights reserved. Keywords: Annuitization; Intergenerational income effects; Unintended bequest JEL classsification: D6; D9; E2
1. Introduction Since the seminal work of Yaari (1965) it is a standard result in the literature that a life-cycle consumer with no bequest motives will always choose to annuitize fully, provided that the market for annuities is actuarially fair. This result rests on the fact that the actuarial bonds pay the market interest rate plus a “mortality premium” during lifetime. For surviving members of a cohort this welfare gain could be split up into an (intragenerational) income and substitution effect. Since Kotlikoff and Spivak (1981) various simulation studies have shown that these welfare gains from annuitization could be quite significant. While the short-run effects of annuitization seem to be very favorable, the respective long-run implications are not so clear. As Auerbach et al. (2001) have documented, the dramatic increase in the degree of annuitization of older Americans due to their receipt of social security and health transfers reduces bequest flows substantially. The latter induces a negative (intergenerational) income effect for future generations. Consequently, it is not clear a priori, whether future generations will benefit or lose from annuitization. Kingston and Piggott (1999) indicate that the opening up of an annuities market always ⁎ Corresponding author. University of Würzburg, Sanderring 2, D-97070 Würzburg, Germany. Tel.: +49 931 31 2972; fax: +49 931 888 7129. E-mail address: [email protected] (H. Fehr). 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.06.022
increases long-run welfare. However, their result rests on the assumed equality of the interest rate and the population growth rate. In this special case the intragenerational and the intergenerational income effects cancel out and the introduction of life annuities generates a welfare gain due to the substitution effect. We will show that long-run welfare will decrease due to the elimination of unintended bequests if the interest rate is sufficiently higher than population growth rate. Some rough calculations indicate that this situation is highly relevant empirically. 2. The short-run perspective: welfare gains from annuitization As Kingston and Piggott (1999) we consider a two-period model with fixed factor prices where each cohort of individuals survives after the first period of life with probability p and dies after the second period with certainty. Individuals have no bequest motive. We assume that agents maximize an expected time-separable, homothetic utility function over consumption when young and old, given by puðc2 Þ ð1Þ 1þd c1g where δ denotes the rate of time preference and uðcÞ ¼ 1g with γ ≠ 1 as the relative risk aversion. Without a market for actuarially fair life annuities, the savings s1 of those who have died
U ð c 1 ; c 2 Þ ¼ uð c 1 Þ þ
178
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
Table 1 Short-run welfare gains from annuitization (ϕ1 − 1) a γ
p
0.5
0.5 0.6 0.7 0.5 0.6 0.7
2.0
a
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
20.0 17.6 14.1 29.5 23.0 16.7
27.1 23.3 18.1 25.5 20.1 14.7
35.6 29.5 22.3 21.9 17.4 12.9
44.9 36.0 26.4 18.8 15.0 11.2
wN A ¼ c1 þ
1
b ¼ ð1 pÞð1 þ rÞs1 :
ð2Þ
Consequently, the budget constraint of a young agent in the first b ¼ c1 þ s1, where w defines period is given by wN A ¼ w þ 1þn labor income, n the population growth rate and wN A defines aggregate wealth. Those who survive consume their resources in the second year, i.e. c2 = (1 +r)s1. Therefore, upon substitution they end up with the aggregate budget constraint c2 : 1þr
ð3Þ
Maximization of utility (1) subject to the budget constraint (3) implies that consumption when old is proportional to consumption when young: with h ¼
1þd : 1þr
ð4Þ
Substituting the first-order condition (4) back into the budget constraint (3) we get wN A and c∗1 ¼ jN A g1 h 1 : ¼ þ p 1þr
c∗2 ¼ jN A
g1 N A h w p jN A
with
Plugging the above demand equations back into the direct utility function (1), we arrive at the indirect utility function for the no-annuity case with initial wealth: V
NA
NA
w
ð5Þ
c 1 ¼ hg c 2
at the end of the first period are transferred with interest r to the next generation as bequests, i.e.
g1 h c2 c1 ¼ p
pc2 : 1þr
Maximizing Eq. (1) taking into account Eq. (5) now yields the first-order condition
We assume δ = 0.0.
wN A ¼ c1 þ
Consequently, wealth wN A does not change for the short-run cohort, but due to annuitization surviving individuals of that cohort now consume c2 ¼ ð1þrp Þs1 . Consequently, the budget constraint (3) changes to
wN A =jN A ¼ 1g
# 1g " 1g h g p þ p 1þd
In our simplified model, we can easily distinguish between the short-run and long-run implications of the introduction of annuities. The short-run only lasts for one period and refers to the first cohort that is able to annuitize his/her assets when old. This cohort still receives a bequest from the previous generation when young. All subsequent cohorts are born in the models long-run equilibrium, since they don't receive a bequest any more.
ð6Þ
and the respective demand functions c∗∗ 2 ¼
wN A jA
and
g c∗∗ 1 ¼h 1
wN A jA
with
1
j A ¼ hg þ
p 1þr
from where we derive the indirect utility function for the annuity case with unchanged initial wealth N A A 1g N A 1g w =j p A g V w h þ : ¼ 1þd 1g In order to quantify the welfare effects of annuities, we follow Kotlikoff and Spivak (1981) and compute the required percentage increase in initial wealth ϕ1 which would leave a person in the absence of an annuities market as well of as with access to an annuities market (VA(wN A) = VN A(ϕ1wN A)): 2 /1 ¼
1 3 1k 1g g
7 h þ jN A 6 6 7 : 5 A 4 1g j g p |{z} hp þ 1þd N1 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} p 1þd
Wb1
Table 1 reports the percentage increase in initial wealth for alternative parameter combinations. Not surprisingly, the welfare gains decrease with higher survival probabilities. For the extreme case p = 1 we get ϕ1 − 1 = 0. Note that higher risk aversion only increases the welfare gains from access to an annuities market if, as Kotlikoff and Spivak (1981) assume, r = δ. In this case, future consumption is equal to current consumption after annuitization, i.e. risk aversion has no effect on the consumption profile. However, if r N δ future consumption is larger than current consumption, but greater risk aversion flattens the consumption profile. The latter effect dampens the utility increase from annuitization. 3. The long-run perspective: the elimination of unintended bequests Next we consider the long-run implications of annuities. The initial equilibrium without annuities does not change compared to the previous section. However, for better comparison we substitute Eq. (2) into Eq. (3) to rewrite the budget constraint w ¼ c1 þ
c2 ð1 pÞc2 : 1þr 1þn
ð7Þ
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
179
Fig. 1. Short-and long-run welfare effects of annuities.
Plugging the first-order condition (4) into the new budget constraint (7) we derive the demand functions g1 w h w ∗ ∗ c 2 ¼ N A and c 1 ¼ with p j˜ j˜ N A g1 h 1 1p j˜ N A ¼ þ p 1þr 1þn and the (adjusted) indirect utility function V˜
# 1g " 1g w=j˜ N A h g p ðwÞ ¼ þ : p 1þd 1g
NA
Since annuities eliminate unintended bequests, the long-run budget constraint with annuities changes to pc2 : ð8Þ wA ¼ w ¼ c1 þ 1þr Substituting the first-order condition (6) into the budget constraint (8) yields the long-run demand equations 1 w w c∗∗∗ ¼ A and c∗∗∗ ¼ hg A : 2 1 j j Following Kingston and Piggott (1999) we first consider the special situation r = n. It is easy to see that the budget constraints (7) and (8) are equivalent in this case. Consequently, the introduction of life annuities does not change the resources of agents living in the long run. Since both equilibria (c1⁎, c2⁎) and (c1⁎⁎⁎, c2⁎⁎⁎) are on the same budget constraint we must have NU c∗1 ; c∗2 ð9Þ ; c∗∗∗ U c∗∗∗ 1 2 simple due to revealed preferences. The economic intuition is that future generations experience no aggregate income effect since the elimination of unintended bequests completely neutralizes the (intragenerational) income gains from the increased returns on savings. This is shown in the left part of Fig. 1 where the initial equilibrium without annuities is A. The
introduction of annuities steepens the slope of the budget line and induces young generations (who still receive bequests from the elderly) to move to the new equilibrium in B. The resulting welfare increase could be decomposed into a substitution effect from A to C and an (intragenerational) income effect from C to B. In the long run, unintended bequests are eliminated and the budget line is shifted to the left. If r = n, the new budget line includes the initial equilibrium A so that C also reflects the new long-run equilibrium. Since we assume in Fig. 1 δ = r, households will consume equal amounts when young and old in the short and long run after the reform. In order to analyze the welfare changes when r N n, we compute the adjusted necessary wealth change from VA (w) = V˜ N A (ϕ2w), i.e. /2 ¼
j˜ N A W: jA
ð10Þ
Table 2 reports the relative wealth change for alternative parameter combinations. It should be clear that the first column shows positive values, since this reflects the situation in the left part of Fig. 1. However, a higher interest rate increases the losses from the elimination of unintended bequests which eventually results in a welfare decrease. This is shown in the right part of Fig. 1. The economic intuition behind this result is Table 2 Long-run welfare effects from annuitization ( ϕ2 − 1) a γ
p
0.5
2.0
a
0.5 0.6 0.7 0.5 0.6 0.7
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
8.0 5.2 2.8 2.7 1.5 0.8
1.5 − 2.4 − 4.2 − 8.8 − 7.7 − 6.1
− 17.7 − 21.1 − 19.8 − 21.5 − 18.1 − 14.0
− 61.3 − 59.8 − 49.7 − 35.7 − 30.0 − 23.2
We assume again δ = n = 0.0.
180
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
Table 3 LSRA welfare effects from annuitization (ϕ3 − 1) a γ
p
0.5
2.0
a
0.5 0.6 0.7 0.5 0.6 0.7
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
– – – – – –
14.7 9.0 4.7 3.7 2.2 1.1
25.3 14.5 7.1 5.0 3.0 1.5
40.4 21.6 9.9 6.7 4.0 2.1
We assume again δ = n = 0.0.
straightforward. If the interest rate is above the population growth rate, the negative (intergenerational) income effect dominates the positive (intragenerational) income effect. For small differences between r and n, the substitution effect might still ensure an overall long-run welfare gain. However, for larger differences long-run welfare decreases due to the elimination of unintended bequests. Table 2 clearly indicates that a long-run welfare reduction is quite realistic and substantial. 4. Combining the short- and long-run perspective: efficiency effects As shown before, annuitization increases short-run welfare and reduces long-run welfare for realistic assumptions about r and n. In order to assess the aggregate efficiency consequences, we introduce a compensation mechanism in the spirit of the Lump-Sum Redistribution Authority (LSRA) of Auerbach and Kotlikoff (1987, 65f.). The LSRA pays a lump-sum transfer (or levies a lump-sum tax) to each living household in the first period of the transition to bring their utility level back to the level of the initial equilibrium. The old generation in period 1 is not affected by the introduction of the annuities market. On the other hand, the lump-sum tax for the young generation in the first period is computed from VA(wN A − T1) = VN A(wN A). It is straightforward to show that T1 ¼ //1 1 wN A . T1 is always 1 positive due to the short-run welfare gains shown above. Hence, the wealth of the LSRA after period 1 is T1 and since the new steady-state is already reached in the next period, the interest income of the LSRA is transferred to future generations, i.e. Tj = rT1, j N 1. All generations after the first period have now the resources w + rT1 so that we can compute the aggregate efficiency gain w þ rT1 " w !# /1 1 1p ¼ /2 1 þ r 1þ NA /1 j˜ ð1 þ nÞ
/3 ¼ /2
ð11Þ
from V A ðw þ rT1 Þ ¼V˜ ð/3 wÞ where we have substituted the definitions for the lump-sum tax T1, bequests (2) and the NA
consumption demand of the second period. Table 3 shows the aggregate efficiency gain after compensation. For higher interest rate values aggregate efficiency gains increase since the budget line in the case of annuities market becomes steeper. It should be also clear that our compensation mechanism only works for r N 0. For r = 0 compensation is not necessary, since welfare effects are positive in the short and long run, see Tables 1 and 2. 5. Implications for public policy The welfare implications of annuitization receive a growing interest because many countries are currently implementing pension reform strategies which force or induce households to move to individual savings accounts as a supplement for public pensions. One of the key issues in the design of these supplementary pension accounts is whether they should be mandatory annuitized or not. While various studies highlight the potential welfare gains from annuitization this paper clearly distinguishes the intergenerational income effects as well as the overall efficiency gains from annuitization. The latter are quantified in a separate experiment by neutralizing the intergenerational redistribution via lump-sum transfers. However, such compensations are hardly possible when mandatory annuities are introduced in reality. Consequently, initial generations experience a positive income effect since their surviving members receive the assets of the decreased. Of course, this reduction of unintended bequests is at the cost of subsequent generations. If this negative income effect exceeds the benefits from the insurance, future generations will experience a welfare loss. If (as is usually claimed) pension reforms are intended to improve the economic situation, especially of future generations, the mandatory annuitization of individual retirement accounts may not work as intended. Acknowledgment We would like to thank an anonymous referee for very helpful comments on an earlier version. References Auerbach, A., Kotlikoff, L.J., 1987. Dynamic Fiscal Policy. Cambridge University Press, Cambridge. Auerbach, A., Gokhale, J., Kotlikoff, L.J., Sabelhouse, J., Weil, D., 2001. The annuitization of Americans' resources: a cohort analysis. In: Kotlikoff, L.J. (Ed.), Essays on Saving, Bequests, Altruism, and Life-Cycle Planning. MIT Press, Cambridge, pp. 93–131. Kingston, G., Piggott, J., 1999. The geometry of life annuities. Manchester School 67, 187–191. Kotlikoff, L.J., Spivak, A., 1981. The family as an incomplete annuities market. Journal of Political Economy 89, 372–391. Yaari, M.E., 1965. Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies 32, 137–150.
Economics Letters 99 (2008) 177 – 180 www.elsevier.com/locate/econbase
Welfare effects of life annuities: Some clarifications Hans Fehr ⁎, Christian Habermann University of Würzburg, Germany Received 10 April 2006; received in revised form 15 June 2007; accepted 21 June 2007 Available online 30 June 2007
Abstract The present paper clarifies the intergenerational welfare effects of life annuities. While present young cohorts experience significant welfare gains, future generations are hurt by lower bequest. For sufficiently high interest rates long-run welfare will decrease although aggregate efficiency rises. © 2007 Elsevier B.V. All rights reserved. Keywords: Annuitization; Intergenerational income effects; Unintended bequest JEL classsification: D6; D9; E2
1. Introduction Since the seminal work of Yaari (1965) it is a standard result in the literature that a life-cycle consumer with no bequest motives will always choose to annuitize fully, provided that the market for annuities is actuarially fair. This result rests on the fact that the actuarial bonds pay the market interest rate plus a “mortality premium” during lifetime. For surviving members of a cohort this welfare gain could be split up into an (intragenerational) income and substitution effect. Since Kotlikoff and Spivak (1981) various simulation studies have shown that these welfare gains from annuitization could be quite significant. While the short-run effects of annuitization seem to be very favorable, the respective long-run implications are not so clear. As Auerbach et al. (2001) have documented, the dramatic increase in the degree of annuitization of older Americans due to their receipt of social security and health transfers reduces bequest flows substantially. The latter induces a negative (intergenerational) income effect for future generations. Consequently, it is not clear a priori, whether future generations will benefit or lose from annuitization. Kingston and Piggott (1999) indicate that the opening up of an annuities market always ⁎ Corresponding author. University of Würzburg, Sanderring 2, D-97070 Würzburg, Germany. Tel.: +49 931 31 2972; fax: +49 931 888 7129. E-mail address: [email protected] (H. Fehr). 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.06.022
increases long-run welfare. However, their result rests on the assumed equality of the interest rate and the population growth rate. In this special case the intragenerational and the intergenerational income effects cancel out and the introduction of life annuities generates a welfare gain due to the substitution effect. We will show that long-run welfare will decrease due to the elimination of unintended bequests if the interest rate is sufficiently higher than population growth rate. Some rough calculations indicate that this situation is highly relevant empirically. 2. The short-run perspective: welfare gains from annuitization As Kingston and Piggott (1999) we consider a two-period model with fixed factor prices where each cohort of individuals survives after the first period of life with probability p and dies after the second period with certainty. Individuals have no bequest motive. We assume that agents maximize an expected time-separable, homothetic utility function over consumption when young and old, given by puðc2 Þ ð1Þ 1þd c1g where δ denotes the rate of time preference and uðcÞ ¼ 1g with γ ≠ 1 as the relative risk aversion. Without a market for actuarially fair life annuities, the savings s1 of those who have died
U ð c 1 ; c 2 Þ ¼ uð c 1 Þ þ
178
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
Table 1 Short-run welfare gains from annuitization (ϕ1 − 1) a γ
p
0.5
0.5 0.6 0.7 0.5 0.6 0.7
2.0
a
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
20.0 17.6 14.1 29.5 23.0 16.7
27.1 23.3 18.1 25.5 20.1 14.7
35.6 29.5 22.3 21.9 17.4 12.9
44.9 36.0 26.4 18.8 15.0 11.2
wN A ¼ c1 þ
1
b ¼ ð1 pÞð1 þ rÞs1 :
ð2Þ
Consequently, the budget constraint of a young agent in the first b ¼ c1 þ s1, where w defines period is given by wN A ¼ w þ 1þn labor income, n the population growth rate and wN A defines aggregate wealth. Those who survive consume their resources in the second year, i.e. c2 = (1 +r)s1. Therefore, upon substitution they end up with the aggregate budget constraint c2 : 1þr
ð3Þ
Maximization of utility (1) subject to the budget constraint (3) implies that consumption when old is proportional to consumption when young: with h ¼
1þd : 1þr
ð4Þ
Substituting the first-order condition (4) back into the budget constraint (3) we get wN A and c∗1 ¼ jN A g1 h 1 : ¼ þ p 1þr
c∗2 ¼ jN A
g1 N A h w p jN A
with
Plugging the above demand equations back into the direct utility function (1), we arrive at the indirect utility function for the no-annuity case with initial wealth: V
NA
NA
w
ð5Þ
c 1 ¼ hg c 2
at the end of the first period are transferred with interest r to the next generation as bequests, i.e.
g1 h c2 c1 ¼ p
pc2 : 1þr
Maximizing Eq. (1) taking into account Eq. (5) now yields the first-order condition
We assume δ = 0.0.
wN A ¼ c1 þ
Consequently, wealth wN A does not change for the short-run cohort, but due to annuitization surviving individuals of that cohort now consume c2 ¼ ð1þrp Þs1 . Consequently, the budget constraint (3) changes to
wN A =jN A ¼ 1g
# 1g " 1g h g p þ p 1þd
In our simplified model, we can easily distinguish between the short-run and long-run implications of the introduction of annuities. The short-run only lasts for one period and refers to the first cohort that is able to annuitize his/her assets when old. This cohort still receives a bequest from the previous generation when young. All subsequent cohorts are born in the models long-run equilibrium, since they don't receive a bequest any more.
ð6Þ
and the respective demand functions c∗∗ 2 ¼
wN A jA
and
g c∗∗ 1 ¼h 1
wN A jA
with
1
j A ¼ hg þ
p 1þr
from where we derive the indirect utility function for the annuity case with unchanged initial wealth N A A 1g N A 1g w =j p A g V w h þ : ¼ 1þd 1g In order to quantify the welfare effects of annuities, we follow Kotlikoff and Spivak (1981) and compute the required percentage increase in initial wealth ϕ1 which would leave a person in the absence of an annuities market as well of as with access to an annuities market (VA(wN A) = VN A(ϕ1wN A)): 2 /1 ¼
1 3 1k 1g g
7 h þ jN A 6 6 7 : 5 A 4 1g j g p |{z} hp þ 1þd N1 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} p 1þd
Wb1
Table 1 reports the percentage increase in initial wealth for alternative parameter combinations. Not surprisingly, the welfare gains decrease with higher survival probabilities. For the extreme case p = 1 we get ϕ1 − 1 = 0. Note that higher risk aversion only increases the welfare gains from access to an annuities market if, as Kotlikoff and Spivak (1981) assume, r = δ. In this case, future consumption is equal to current consumption after annuitization, i.e. risk aversion has no effect on the consumption profile. However, if r N δ future consumption is larger than current consumption, but greater risk aversion flattens the consumption profile. The latter effect dampens the utility increase from annuitization. 3. The long-run perspective: the elimination of unintended bequests Next we consider the long-run implications of annuities. The initial equilibrium without annuities does not change compared to the previous section. However, for better comparison we substitute Eq. (2) into Eq. (3) to rewrite the budget constraint w ¼ c1 þ
c2 ð1 pÞc2 : 1þr 1þn
ð7Þ
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
179
Fig. 1. Short-and long-run welfare effects of annuities.
Plugging the first-order condition (4) into the new budget constraint (7) we derive the demand functions g1 w h w ∗ ∗ c 2 ¼ N A and c 1 ¼ with p j˜ j˜ N A g1 h 1 1p j˜ N A ¼ þ p 1þr 1þn and the (adjusted) indirect utility function V˜
# 1g " 1g w=j˜ N A h g p ðwÞ ¼ þ : p 1þd 1g
NA
Since annuities eliminate unintended bequests, the long-run budget constraint with annuities changes to pc2 : ð8Þ wA ¼ w ¼ c1 þ 1þr Substituting the first-order condition (6) into the budget constraint (8) yields the long-run demand equations 1 w w c∗∗∗ ¼ A and c∗∗∗ ¼ hg A : 2 1 j j Following Kingston and Piggott (1999) we first consider the special situation r = n. It is easy to see that the budget constraints (7) and (8) are equivalent in this case. Consequently, the introduction of life annuities does not change the resources of agents living in the long run. Since both equilibria (c1⁎, c2⁎) and (c1⁎⁎⁎, c2⁎⁎⁎) are on the same budget constraint we must have NU c∗1 ; c∗2 ð9Þ ; c∗∗∗ U c∗∗∗ 1 2 simple due to revealed preferences. The economic intuition is that future generations experience no aggregate income effect since the elimination of unintended bequests completely neutralizes the (intragenerational) income gains from the increased returns on savings. This is shown in the left part of Fig. 1 where the initial equilibrium without annuities is A. The
introduction of annuities steepens the slope of the budget line and induces young generations (who still receive bequests from the elderly) to move to the new equilibrium in B. The resulting welfare increase could be decomposed into a substitution effect from A to C and an (intragenerational) income effect from C to B. In the long run, unintended bequests are eliminated and the budget line is shifted to the left. If r = n, the new budget line includes the initial equilibrium A so that C also reflects the new long-run equilibrium. Since we assume in Fig. 1 δ = r, households will consume equal amounts when young and old in the short and long run after the reform. In order to analyze the welfare changes when r N n, we compute the adjusted necessary wealth change from VA (w) = V˜ N A (ϕ2w), i.e. /2 ¼
j˜ N A W: jA
ð10Þ
Table 2 reports the relative wealth change for alternative parameter combinations. It should be clear that the first column shows positive values, since this reflects the situation in the left part of Fig. 1. However, a higher interest rate increases the losses from the elimination of unintended bequests which eventually results in a welfare decrease. This is shown in the right part of Fig. 1. The economic intuition behind this result is Table 2 Long-run welfare effects from annuitization ( ϕ2 − 1) a γ
p
0.5
2.0
a
0.5 0.6 0.7 0.5 0.6 0.7
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
8.0 5.2 2.8 2.7 1.5 0.8
1.5 − 2.4 − 4.2 − 8.8 − 7.7 − 6.1
− 17.7 − 21.1 − 19.8 − 21.5 − 18.1 − 14.0
− 61.3 − 59.8 − 49.7 − 35.7 − 30.0 − 23.2
We assume again δ = n = 0.0.
180
H. Fehr, C. Habermann / Economics Letters 99 (2008) 177–180
Table 3 LSRA welfare effects from annuitization (ϕ3 − 1) a γ
p
0.5
2.0
a
0.5 0.6 0.7 0.5 0.6 0.7
r 1.0040 − 1
1.0140 − 1
1.0240 − 1
1.0340 − 1
– – – – – –
14.7 9.0 4.7 3.7 2.2 1.1
25.3 14.5 7.1 5.0 3.0 1.5
40.4 21.6 9.9 6.7 4.0 2.1
We assume again δ = n = 0.0.
straightforward. If the interest rate is above the population growth rate, the negative (intergenerational) income effect dominates the positive (intragenerational) income effect. For small differences between r and n, the substitution effect might still ensure an overall long-run welfare gain. However, for larger differences long-run welfare decreases due to the elimination of unintended bequests. Table 2 clearly indicates that a long-run welfare reduction is quite realistic and substantial. 4. Combining the short- and long-run perspective: efficiency effects As shown before, annuitization increases short-run welfare and reduces long-run welfare for realistic assumptions about r and n. In order to assess the aggregate efficiency consequences, we introduce a compensation mechanism in the spirit of the Lump-Sum Redistribution Authority (LSRA) of Auerbach and Kotlikoff (1987, 65f.). The LSRA pays a lump-sum transfer (or levies a lump-sum tax) to each living household in the first period of the transition to bring their utility level back to the level of the initial equilibrium. The old generation in period 1 is not affected by the introduction of the annuities market. On the other hand, the lump-sum tax for the young generation in the first period is computed from VA(wN A − T1) = VN A(wN A). It is straightforward to show that T1 ¼ //1 1 wN A . T1 is always 1 positive due to the short-run welfare gains shown above. Hence, the wealth of the LSRA after period 1 is T1 and since the new steady-state is already reached in the next period, the interest income of the LSRA is transferred to future generations, i.e. Tj = rT1, j N 1. All generations after the first period have now the resources w + rT1 so that we can compute the aggregate efficiency gain w þ rT1 " w !# /1 1 1p ¼ /2 1 þ r 1þ NA /1 j˜ ð1 þ nÞ
/3 ¼ /2
ð11Þ
from V A ðw þ rT1 Þ ¼V˜ ð/3 wÞ where we have substituted the definitions for the lump-sum tax T1, bequests (2) and the NA
consumption demand of the second period. Table 3 shows the aggregate efficiency gain after compensation. For higher interest rate values aggregate efficiency gains increase since the budget line in the case of annuities market becomes steeper. It should be also clear that our compensation mechanism only works for r N 0. For r = 0 compensation is not necessary, since welfare effects are positive in the short and long run, see Tables 1 and 2. 5. Implications for public policy The welfare implications of annuitization receive a growing interest because many countries are currently implementing pension reform strategies which force or induce households to move to individual savings accounts as a supplement for public pensions. One of the key issues in the design of these supplementary pension accounts is whether they should be mandatory annuitized or not. While various studies highlight the potential welfare gains from annuitization this paper clearly distinguishes the intergenerational income effects as well as the overall efficiency gains from annuitization. The latter are quantified in a separate experiment by neutralizing the intergenerational redistribution via lump-sum transfers. However, such compensations are hardly possible when mandatory annuities are introduced in reality. Consequently, initial generations experience a positive income effect since their surviving members receive the assets of the decreased. Of course, this reduction of unintended bequests is at the cost of subsequent generations. If this negative income effect exceeds the benefits from the insurance, future generations will experience a welfare loss. If (as is usually claimed) pension reforms are intended to improve the economic situation, especially of future generations, the mandatory annuitization of individual retirement accounts may not work as intended. Acknowledgment We would like to thank an anonymous referee for very helpful comments on an earlier version. References Auerbach, A., Kotlikoff, L.J., 1987. Dynamic Fiscal Policy. Cambridge University Press, Cambridge. Auerbach, A., Gokhale, J., Kotlikoff, L.J., Sabelhouse, J., Weil, D., 2001. The annuitization of Americans' resources: a cohort analysis. In: Kotlikoff, L.J. (Ed.), Essays on Saving, Bequests, Altruism, and Life-Cycle Planning. MIT Press, Cambridge, pp. 93–131. Kingston, G., Piggott, J., 1999. The geometry of life annuities. Manchester School 67, 187–191. Kotlikoff, L.J., Spivak, A., 1981. The family as an incomplete annuities market. Journal of Political Economy 89, 372–391. Yaari, M.E., 1965. Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies 32, 137–150.