Working paper No 220 2009_tcm46-221307

8/14/2019 Working paper No 220 2009_tcm46-221307

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Stabilizing pay-a

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and falling ferti lit

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Stabilizing pay-as-you-go pension schemes in the f

longevity and falling fertility: an application to the

W.L. Heeringa and A.L. Bovenberg *

* Views expressed are those of the authors and do not necessarily refle

positions of De Nederlandsche Bank.


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Stabilizing pay-as-you-go pension schemes

face of rising longevity and falling fertilit

application to the Netherlands

W.L. Heeringay and A.L. Bovenbergz

August 13, 2009

Abstract

Rising longevity and falling fertility threaten the sustainability of as-you-go pension schemes. This paper shows that maintaining the ingenerational balance in the Dutch pay-as-you-go pension scheme inface of increased longevity since the introduction of the scheme in would have required a gradual increase of the retirement age to at 68 years for the generation born in 1945. Furthermore, we show that jected increases in labour-force participation rates do not generate cient additional tax revenues to subsitute for the dearth of human cacaused by falling fertility rates.

J.E.L Classication: H5Keywords: public pension; pay-as-you-go system

We wish to thank Fanny Janssen en Coen van Duin for comments and suggesvan Kerkho provided useful research assistance. Views expressed are those of and do not necessarily reect ocial positions of De Nederlandsche Bank.

yDe Nederlandsche Bank and Netspar. Contact information: w.l.heeringa@Nederlandsche Bank, Economics & Research Division, P.O. Box 98, 1000 AB, AThe Netherlands, tel: +31 (0) 20 524 3437, fax: +31 (0) 20 524 2506.

z

Tilburg University and Netspar.


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1 Introduction

Rising longevity and falling fertility threaten the sustainability of pago (PAYG) pension schemes. Sinn (2000) advocates a partial shiftfunded pension scheme in order to absorb lower fertility rates. Generawant to consume without working when they are old, have to eithraise children in order to provide for the resources in old age. Accordin

(2000), generations that feature lower fertility should substitute nancfor human capital of children as a source of pension income. In particuinvestment in human capital as a result of falling fertility rates reduceson e.g. feeding, clothing and educating children. The released resourbe converted into additional savings for future pensions. In additiofewer children saves time. The additional time could be used to workthe additional labor income would be another source of savings. Cincreased longevity, Shoven and Goda (2008) argue that living longean increase in the age at which one becomes eligible for public pen

"retirement age") to correct for "age ination". They argue that a faijust the retirement age for mortality improvements would substantiallthe fraction of the population eligible to receive PAYG pension bene

We explore adjustments along the lines proposed by Sinn (2000) anand Goda (2008) for the PAYG pension scheme of the Netherlandscalled AOW pension scheme established in 1957. The methodologicstone of our approach is a benchmark scenario featuring constant liftility, mortality and labour-participation proles since the introductDutch PAYG pension scheme in 1957. Actual fertility, mortality an

participation proles observed since 1957, however, dier from theproles in the benchmark scenario. Given these implied demographinomic shocks, we compute adjustments in the public pension systemrestore the initial intergenerational balances (i.e. the present value of ence between taxes paid and benets received per generation) in the bscenario. In particular, we consider how the initial intergenerationacan be restored by adjusting the retirement age in response to longeviby levying non-fertility taxes in response to negative fertility shocimposing non-participation taxes in case of negative labour-force par

shocks.The rest of this paper is structured as follows. Section 2 present

model of a PAYG pension scheme, while section 3 discusses the meof the paper. Sections 4, 5 and 6 analyze the longevity, fertility anparticipation shocks occurred since 1957 as well as the adjustmentabove that could have restored the initial intergenerational balance7 explores whether positive (female) labour-participation shocks hav


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2 Model

2.1 Population and economy

Before introducing a PAYG pension scheme, we start with a simple demand economic model. Let us denote the remaining number of peoptime i aged t i at time t as ni;t with t i = f0; 1; 2;:::;Dg for Dxed maximum age an individual can reach. Accordingly, we can enumber of newborn babies at time i as ni;i.

Let us dene the fertility prole of the generation born at time i (hdenoted as generation i) as the age-specic fertility rates of that generits lifetime. More specically, suppose that at time t, each female bo

j gives birth to fFj;t female and fMj;t male babies.

1 The initial size of gi can then be dened as

ni;i =i1

Xj=iD fFj;i + f

Mj;i

nFj;i:

Let us dene the longevity prole of generation i as the age-spvival probabilities of that generation over its lifetime. The remainpopulation at time t of the generation born at time i (nFi;t) equals

nFi;t =

1 pFi;i+1

1 pFi;i+2

::

1 pFi;t

nFi;i;

with pFi;t being the mortality rates at time t from age (t i) 1 to a

the female generation i. Given the supposed maximum age D, pFi;i+denition. For reasons of simplicity, migration is not taken into accomodel.2 Equations (1) and (2) describe the dynamics of a population

Suppose individuals can earn labour income by supplying labour. age wage earned per active full-time equivalent at time t (wt) can be as

wt =YFt + Y

MtPi=t1

i=tD Fi;t

Fi;tn

Fi;t +

Pi=t1i=tD

Mi;t

Mi;tn

Mi;t

;

withYFt : aggregate female labour income at time tFi;t: labour participation of n

Fi;t (fraction of n

Fi;t being employed)

Fi;t: part-time factor of nFi;t (number of hours worked per person

as a fraction of the maximum number of hours).

1 We denote the female version of each variable with the superscript F and the mith th i t M I t b l id th d iti f l


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For later use, we dene the labour-participation prole of generthe age-specic labour-participation rates of that generation over itEquivalently, we dene the part-time prole of generation i as the apart-time factor of that generation over its lifetime.

2.2 PAYG pension scheme

Now we introduce a PAYG pension scheme providing retired peoppension benet nanced by taxes paid by non-retired people.3 Taxeproportional to average labour income. The annual revenues of the Psion scheme at time t (Tt) are then equal to the sum of the taxes pliving generations contributing to the PAYG pension scheme

Tt =t1X

i=tD

tai;t

nFi;tyFi;t + n

Mi;ty

Mi;t

;

witht: tax rate at time t:ai;t: tax dummy (equal to 1 for taxable, non-retired generations

non-taxable, retired generations at time t).yFi;t: average per capita labour income earned at time t by the fem

ation i.The linearity of the tax system implies that we do not need to wo

intragenerational heterogeneity when computing total tax revenues. Ifully abstract from intragenerational heterogeneity. We look only a

within a generation and assume that the marginal person is equal to thperson when we consider changes in the size of a particular generatioSuppose that the PAYG pension benet is proportional to the ave

rate in the economy (wt); implying that the pension benet is not reldividual earnings. Indeed, every individual is entitled to PAYG pensioafter reaching the retirement age, irrespective of lifetime earnings. Texpenditures of a PAYG pension scheme at time t (Bt) are then eqsum of the benets of all living generations beneting from the PAYscheme

Bt =t1X

i=tD

twtbi;t

nFi;t + nMi;t

;

witht: replacement rate at time t (PAYG pension benet as fractionbi;t: benet dummy (equal to 1 for retired generations and 0 othe


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Individuals either pay taxes to the PAYG pension scheme or receiv(implying bi;t = 1 ai;t). The borderline is at the "retirement age", iat which people become eligible for the PAYG pension benet. In temodel, the retirement age of a generation can be shifted by changingtax rate and replacement rate at time t are the same for all generatio

A PAYG pension scheme in fact implies a balanced budget at eactime

Tt = Bt;

which can be maintained by adjusting either the tax rate t or the rerate t (or both). In the former case, the PAYG pension scheme is of benet type; in the latter case it is of a dened-contribution type. Ain appendix A, the Dutch AOW pension scheme has been a mixturtypes since its establishment in 1957. Apart from that, it deviatesPAYG pension scheme as modelled above in three repects. First statutory tax rate is capped by law at 0.179 and applied solely to th

second tax brackets. Hence, the tax rate in equation (4) which detaxes paid to the PAYG pension scheme as a fraction of total incomcomparable with the statutory tax rate for the AOW pension schemany decits of the AOW pension scheme are supplemented by other taConsequently, retirees also contribute to the PAYG pension scheme they pay other taxes. This implies that the tax dummy ai;t in equanot strictly binary in reality, but can vary between 0 and 1 for retirdepending on the supplements from the general tax system. Seconpension benets are related to the contemporaneous statutory minim

rather than the contemporaneous average (full-time) wage in the ecassumed in our model. Finally, although we do not include immigramodel for reasons of simplicity, in reality they are entitled to AOWbenets in proportion to the period they have resided in the Netherla

3 Methodology

The methodological cornerstone of our paper is a benchmark scenario

demographic and economic lifetime proles are supposed to be consthe introduction of the PAYG scheme (i.e. f

F

s , pFs ;

Fs ,

F

s ; as anda bar denoting the benchmark value of each variable at age s, whereliminated the time subscripts because these variables are assumed stant). Imposing the balanced budget constraint of a PAYG pensi(see equation 6) on the benchmark scenario, we nd a consistent set replacement rates under the assumption that the replacement rate is


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anced budget in the face of such shocks would require adjusting tax rate (in a dened-benet pension scheme) or the replacement dened-contribution pension scheme). However, these adjustments to a rather arbitrary income redistribution among generations. For ina dened-benet scheme, non-retired generations would have to pay eif retired generations experience a positive longevity shock. Furtha dened-contribution scheme, retired generations would have to acc

benets if the labour participation of non-retired generations falls. It wmore appropriate, however, if the generations "causing" the demogreconomic shocks would bear the external eects of their behavior on tof the public pension system. This principle implies that the presenthe balance of the taxes paid minus the pension benets received by eration over its lifetime (the so-called intergenerational balances) shouunchanged in the face of demographic and economic shocks. In fact, mintergenerational balances in the face of shocks implies that we do nsarily impose the requirement of a public budget that is balanced at e

in time as is the case in a pure PAYG pension scheme. Indeed, depthe shock, the government might save by acquiring assets or dissave public debt.

4 Longevity shocks

4.1 Analytical framework

An increase in the age-specic survival rates of a generation has two

eects on its intergenerational balance. First of all, aggregate taxes pageneration rise as the average period during which people pay taxelonger. Secondly, aggregate pension benets received by that generatioalso as more people survive up to the retirement age and the average peligible people receive pension benets increases.4

Let us now consider this more formally. The direct longevity eectpaid per female generation (eTFi ) can be calculated as the (expectevalue of the dierence between on the one hand taxes paid accordactual longevity (en

Fi;j) and tax dummy (ai;j) prole, and on the o

taxes paid according to the benchmark longevity (nFi;j) and tax dumprole

eTFi =i+DXj=i+1

Ri;jjyFi;j

ai;jenFi;j ai;jnFi;j ;

ith di t f t R d d


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where rj equals the interest rate in year j. Note that we keep discconstant in the face of demographic (or economic) shocks.

Equivalently, the direct longevity eects on benets received per feeration ( eBFi ) can be calculated as the (expected) present value of ence between on the one hand benets received according to the actual(enFi;j) and benet dummy prole (bi;j), and on the other hand benetaccording to the benchmark longevity (nFi;j) and benet dummy pro

eBFi =i+DXj=i+1

Ri;jjwj

bi;jenFi;j bi;jnFi;j :The implicit longevity debt of the female generation born in year

can be expressed as

gDEB

F

i =

eBFi

eTFi ;

while the total implicit longevity debt of each generation ( gDEBi) eqgDEBi = gDEBFi + gDEBMi :

The initial intergenerational balance of generation i can be rechanging the retirement age (i.e. by changing ai;j and bi;j) such that DNote that this would imply that the pension scheme is no longer baany point of time as in the benchmark scenario. In fact, the pension now supplemented by a funded scheme for longevity shocks.

4.2 Data

4.2.1 Benchmark

A natural candidate for the longevity benchmark would be the observerates in the calender year 1957. However, the actual survival rates of g1892 (being the rst generation receiving AOW pension benets afterment in 1957 for the rest of its life) dier from these 1957 survival r

implies that generation 1892 would face a longevity shock after beinThe budgetary impact of such a shock for the AOW pension schemabsorbed in dierent ways. First of all, the tax and replacement rate ogenerations could be adjusted, which would yield optimal risk sharingenerations. However, this would be at odds with our principle of unchtergenerational balances. Alternatively, the replacement rate of genercould be lowered in order to restore its intergenerational balance. How


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that is the combination of the age-specic survival rates of generaabove age 65 and the observed age-specic survival rates in 1957 belo

Figure 1: Female longevity proles

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Age

Remainingfraction

2000

1975

1950

1925

1900

Benchmark

Figure 2: Male longevity proles

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Age

Remainingfraction

2000

1975

1950

19251900

Benchmark

4.2.2 Shocks

Figures 1 and 2 show for both genders the actual longevity proles oerations born in 1900, 1925, 1950, 1975 and 2000 as well as the belongevity prole. For the younger generations that have not died oneed to rely on projected survival rates. The ocial population proStatistics Netherlands provides survival rates up to the year 2050 (s


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subsequent generations born in the 20th century has increased. Redumortality plays a major role. In addition, improved nutrition and hhave contributed to increased longevity (Cutler and Meara, 2001).

An alternative measure for changes in longevity is the evolutionpected (remaining) life expectancy of a generation at a certain agdene the remaining life expectancy at age 65 as the expected benea generation, measuring the average period a generation receives pen

ts. Equivalently, dene the life expectancy from ages 15 to 65 as thecontribution period of a generation, measuring the average period a gpays taxes. By denition, the expected contribution period of genmaximized at 50 years.

Figure 3: Evolution expected contribution and benet period per

14

15

16

17

18

19

20

21

22

23

24

25

26

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

Years

38

39

40

41

42

43

44

45

46

47

48

49

50

Female benefit period, perfectforesight

Female benefit period, myopic

Male benefit period, perfectforesight

Male benefit period, myopic

Female contribution period(RHS)

Male contribution period (RHS)

Figure 3 shows for both genders the evolution of the expected tion and benet period, the latter both for myopic expectations (on the survival rates in a calender year) and perfect foresight (i.e. survival rates of a generation). Compared to generation 1892, the apected contribution period for men and women will have increased byfor generation 1945 and by 3.7 years for generation 2000. At the scompared to generation 1892, the average expected benet period usi

foresight (myopic) expectations will have increased by 4.7 (4.3) years ftion 1945 and by 6.9 (7.3) years for generation 2000. Hence, the rst conclusion that can be drawn is that the increase in the expected benexceeds the increase in the expected contribution period. Put dieaverage period in which benets are received has increased more thanage period in which taxes are paid. Moreover, an increase in the coperiod also implies that a larger fraction of a generation will have su


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with women over time.

4.3 Adjustment

4.3.1 Equilibrium retirement age

Using the benchmark longevity prole, we start by calculating the (implicit longevity debt for the generations born between 1892 and

general, the longevity shocks experienced since 1957 caused an implicitdebt for all generations considered. Given this longevity debt, we calibin equation (7) and bi;j in equation (9) to compensate for the longewithin each generation such that intergenerational balances are unaincreased longevity. The new retirement age is called the equilibrium age (ERA).

Figure 4 shows the equilibrium retirement age for all generationtween 1892 and 2000 for various combinations of the real interest ratwage growth after 2008. Let us rst consider the ERA with a real wa

(g) of 1% and a real interest rate (r) of 3% beyond 2008. The ERAincreases to 69.1 years for generation 1945 and to 70.9 years for generin this scenario. The dashed and dotted lines show the sensitivity offor the growth rate. Higher growth raises the ERA, as it implies morAOW pension benets. The fat lines show the sensitivity of the ERinterest rate. A higher (lower) interest rate reduces (raises) the ERApension benets are discounted more (less).7

Figure 4: Equilibrium retirement age: no participation eect

65

66

67

68

69

70

71

72

1 89 0 1 90 0 1 91 0 1 92 0 1 93 0 1 94 0 1 95 0 1 96 0 1 97 0 1 98 0 1 99 0 2 00 0

Birth year

Equilibriumr

etirementage

g=1.5,r=3

g=1.25,r=3

g=1,r=3

g=1,r=2.5

g=1,r=3.5

Figure 4 keeps age-specic labour-participation rates xed at


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benchmark prole. However, increasing the retirement age might wellspecic labour-participation for example, by raising the social normone retires from the labour force. This would reduce the required increquilibrium retirement age as higher employment rates would yield mtax revenues.8 Hence, the ERA as calculated in gure 4 must be conthe upper bound for the required increase in the retirement age as itfrom the possible impact of higher labor-force participation on interge

balances. The empirical literature on the participation eects of raisitirement age is limited. However, empirical evidence for the US (Ma2006) suggests that Americans aged 62 and above have postponedtirement by one month in response to a two-month increase of the retirement age.9 In the absence of empirical estimates for the particifect for the Netherlands, we will assume that the participation prolterms of number of people participating as well as hours worked) isout from the age of 45 until the retirement age in response to chanretirement age. More specically, by means of sensitivity analysis, w

a participation eect in response to increases in the retirement age oand 100% starting at the age of 45. For instance, in case of a 100%an increase in the retirement age by one year will increase the partic60-year olds to that of 59-year olds prior to the adjustment in the rage. This implies that the ERA required to restore the same intergebalances as before the longevity shock must be solved simultaneouslparticipation prole, requiring an iterating numerical procedure.

Figure 5: Equilibrium retirement age: with participation eec

65

66

67

68

69

70

71

72

1890 1900 1910 1920 1930 1 940 19 50 196 0 1970 1980 1990 2000

Birth year

Equilibriumr

etirementage

0%

50%

100%

Figure 5 shows the ERA10 for labour-participation eects rangin

8


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to 100%11 for r=3% and g=1% after 2008. In the case of a participaof 50%, the ERA increases to 68.1 years for generation 1945 and 6for generation 2000. In that case, on average approximately a quarrequired longevity adjustment is accounted for by participation changeby a higher retirement age. In the extreme case of a participation eecthe ERA increases to 67.3 years for the generation born in 1945 and for generation 2000, reecting a lower bound for the ERA. In tha

average almost half of the required longevity adjustment is accounparticipation changes induced by a higher retirement age. Hence,declines with the participation eect. Accordingly, depending on tassumptions and the behavioral participation eects, our calculatiothat the retirement age should gradually be raised to between approxiand 70 years for generation 1945 and between 69 and 71 years for the gborn in 2000 to absorb the impact of increased longevity on intergenbalances.

4.3.2 Alternative rules of thumb

Some authors have proposed alternative rules to determine the requirein the retirement age in the face of higher longevity. For instance, Vet al. (2006) and Shoven and Goda (2008) suggested some simpthumb. Unlike the ERAs, which are based on intergenerational balretirement ages according to these rules of thumb are determined on thdemographic elements only. We will apply these and some other rulesto the Dutch AOW pension scheme and nally compare the outcomeERAs determined above. In order to allow for a fair comparison, wperfect foresight concerning future survival probabilities when applrules and calibrate the required retirement age for generation 1892 a

Constant mortality risk at retirement The rst rule of thumbmortality risk at the retirement age constant over generations (ShGoda, 2008). For generation 1892, the probability of mortality amounted to 1.9%. According to this rule, the AOW retirement abe lowered for generations born before 1912, as their probability of

at age 65 actually increased due to the tobacco disease (Janssen et However, the probability of mortality subsequently fell sharply betweand 75, a phenomenon that is known as the compression of mortalitaccording to this rule, the AOW retirement age should be graduallyto 72.1 years for the generation 1945 and to almost 75.9 years for the g2000 (see gure 6).


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constant over generations by adjusting the retirement age (Van Da2006). For generation 1892, the RLE at age 65 amounted to 16.0 ycording to this rule, the AOW retirement age should gradually be in71.1 years for generation 1945 and 73.1 years for generation 2000. the implied increase in the retirement age exceeds the increase in thage 65 as shown in section 4.2.2. This is because the marginal rethe RLE of a generation for every year that the retirement is increas

than one year. The reason behind this is that those members of that gwith a RLE of less than one year die as the retirement age is increasyear. Put dierently, those members no longer pull down the averathe generation as a whole.

The drawback of the former rules of thumb is that they neglectments of survival rates before the retirement age. Such improvementsintergenerational balance of a generation in two ways (see appendixof all, they raise the amount of taxes contributed by a generation ovtime. Secondly, the fraction of a generation reaching the retirement

becoming eligible for retirement benets) increases. For these reasonexplore two alternative rules.

Figure 6: Required increase in retirement age: some rules of thu

63

64

65

66

67

68

69

70

71

72

73

74

75

76

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

Retirementage

Constant mortality risk

Constant benefit period

Constant ratio contribution/benefit period

Constant ratio contribution/total period

Constant generational account, noparticipation effects

Constant generational account, 50%participation effect

Constant ratio contribution over benet period The rst ruleratio of the expected contribution period over the expected benet pstant. This rule guarantees that a relative change in the period in whicare received is matched by an equally relative change in the period


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years for generation 2000.

Constant ratio contribution over total period Although intupealing, the former rule still does not explicitly account for the frageneration that reaches the retirement age which matters for the intional balance, as mentioned before. Therefore, a better rule would bthe ratio of the expected contribution period over the remaining life e

at age 15 (henceforth denoted as the expected total period) constangeneration.12 In fact, under this rule, both the expected contributiopected benet period are determined at the age of 15. Adhering towould require a gradual increase of the AOW retirement age to 69.8generation 1945 and to 72.2 years for generation 2000. Hence, the results in a higher AOW retirement age than the former, as it explicinto account the fact that a larger fraction of a generation reaches the rage.

The latter rule does not account for the fact that longevity changes

early in the life cycle outweigh longevity changes late in the lifecycle time-discounting. It is mainly for this reason that the ERAs as calculaare lower than retirement ages implied by the second alternative rule 6). Indeed, the ERA takes into account not only changes in the expectthe contribution over the benet ratio, but also the fraction of a generreaches the retirement age and the time discount factor. Hence, thwhy we consider the ERA superior to the considered alternative rules

Figure 7: Public debt due to longevity shocks

-3

-2

-1

0

1

2

3

4

5

1950 1975 2000 2025 2050

Calender year

Fractionoftotalincome

Retirement age 65

ERA

12 This is a modied version of the rule presented in Van Dalen et al. (2006),


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4.3.3 Public debt due to longevity shocks

Figure 7 shows public debt13 of the AOW pension scheme before the change in the retirement age14 . With a constant retirement agexploding, due to the longevity shocks that have occurred since 19retirement ages equal to the ERAs calculated before, the governmenassets to cover future longevity increases. Hence, gure 7 shows that the retirement age to ERA levels will restore the sustainability of

pension scheme in the face of longevity shocks.

5 Fertility shocks


5.1.1 Shocks

The rst-order demographic eect of a decrease in age-specic fertilian immediate fall in the number of babies born. However, it is importathat there are also higher-order eects. As future generations will bfewer babies today implies also fewer babies in the future, even if tage-specic fertility rates would be temporary. Hence, the full impla fertility shock can only be determined in a recursive fashion by ccomplete demographic projections after and before (temporary) fertili

Let us now analyze this formally. Suppose that the number of femgenerated by female generation i at time t changes from age-specic b

value fFti to bfFi;t in year t, but remains at the age-specic benchmafF

ti after year t. Suppose also that the fertility proles of all othgenerations stay at the benchmark from year t on. Denote the projecthe arbitrary female generation l at time j before the fertility shock

and after the fertility shock as bnFl;jji;t. The marginal change in the sdue to this temporary fertility shock at time t caused by the female gi (nF

l;jji;t) then equals

nF

l;jji;t =bnFl;jji;t nFl;jji;t;for l = [t;1] and j = [l; l + D].

In terms of intergenerational balances, a negative fertility shock ima loss in taxes paid and a fall in future pension benets received by fuerations. The present value at time t of the change in future taxes coby all generations born after time t caused by the fertility shock at bF


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bTFi;t =1Xl=t

l+DXj=l

Rt;jjal;jyFl;jn

Fl;jji;t:

Equivalently, the present value at time t of the change in futurbenets received by all generations born after time t caused by thshock at time t attributable to the generation born at time i (

bBFi;t)

bBFi;t =1Xl=t

l+DXj=l

Rt;jjbl;jwjnFl;jji;t:

Hence, the implicit non-fertility debt arising due to the fertility

time t attributable to generation i ( dDEBFi;t) can be expressed asdDEBFi;t = bBFi;t bTFi;t;

where it can be shown that dDEBFi;t will converge to a nite value in cally ecient economy.15

The total implicit non-fertility debt of generation i ( dDEBi;t) caufertility shock at time t can now be expressed as

dDEBi;t = dDEBFi;t + dDEBMi;t:5.1.2 Adjustment

The initial intergenerational balance can be restored in two alternativrst possibility would be to levy an additional "non-fertility" tax on gewhen this generation is raising children. As generation i saves time andthat would otherwise be spent on raising children in the period follnegative fertility shock, it would be logical to levy such a non-fertilthis period. In that case, the government in fact saves on behalf of gewhich would not have to save privately to absorb the negative consethe fertility shock for its consumption at old ages. We will denote thas the "non-fertility tax young" (NFTY) variant, as additional taxes a

young ages. Suppose the child-rearing period equals S years analogtime it takes to raise children until they enter the labour market at adene the income growth rate of generation i at time t + m as gi;t+mcan spread the implicit non-fertility debt over S years by convertingannuity dened as

d SX (1 + gi t) (1 + gi t+1) (1 + gi t+ )


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with non-fertility taxes xi;t+m = (1 + gi;t) (1 + gi;t+1) ::: (1 + gi;t+m)a constant fraction of the income of generation i at time t + m.

A second possibility to restore the intergenerational imbalance canegative fertility shock, would be to lower the replacement rates of thbenets generation i will receive. In fact, this would be equivalent(gross) pension benets. Therefore, we denote this variant as the "notax old" (NFTO) variant. Let us now consider the NFTO-variant mor

Suppose the present value of all non-fertility debts incurred by geduring its life can be expressed as

dDEBi = i+DXt=i

Ri;t dDEBi;t:The average non-fertility tax rate on pension benets required to s

non-fertility debt (i) in the NFTO-variant can now be expressed as

i = dDEBiBE Ni ;with BE Ni being the present value at birth of the gross pension benetby generation i. The net replacement rate of the pension benet regeneration i at time t (bi;t) then equals

bi;t = (1 i) t:Accordingly, the net replacement rates at time t (

bi;t) now dier amo

ations depending on the fertility shocks that can be attributed to the

In fact, by levying a non-fertility tax, in both variants the PAYscheme is supplemented with a funded scheme for fertility shocks. with the NFTY-variant, in the NFTO-variant non-fertility taxes wilin a later stage of the life cycle, implying that there will be less publover time. However, in the NFTO-variant, forward-looking individuat consumption smoothing will compensate the future fall in pensiofollowing a negative fertility shock by additional savings when they aHence, in terms of national saving, the increase in private savings in tvariant will compensate for the lower level of additional public savi

is in fact the scenario that Sinn (2000) had in mind; he advocatedfunded system to supplement a smaller public PAYG system.

5.2 Data

5.2.1 Benchmark


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that ultimately generates a constant population. A necessary conditioa fertility prole is that so-called completed fertility (reecting the nbabies a woman gives birth to over her lifetime) amounts to 2.1. Accto arrive at a benchmark fertility prole, we proportionally decreasespecic fertility rates registered in the calender year 1957 in such acompleted fertility equals 2.1.

Figure 8: Fertility proles: generations 1935-1950

0

0.05

0.1

0.15

0.2

0.25

15 20 25 30 35 40 45 50

Age

Birthsperwoman

1935

1940

1945

1950

Benchmark

Figure 9: Fertility proles: generations 1955-1970

0

0.05

0.1

0.15

0.2

0.25

15 20 25 30 35 40 45 50

Age

Birthsperwoman

1955

1960

1965

1970

Benchmark

5.2.2 Shocks


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unexpected fertility shocks can be attributed to the generation respothe fertility shock. The area below each line represents the completeAs argued above, the benchmark fertility is characterized by completeof 2.1, which is consistent with a constant population in the long completed fertility of the generations born before 1943 exceeded 2.1that these generations caused positive fertility shocks. However, the cfertility of the generations born since 1943 has fallen rapidly, imp

these generations have caused negative fertility shocks. Finally, notespecic fertility at older ages fell for the oldest generations considereincreased again for younger generations. The former can be explainby the increased use of contraception, which has facilitated a more smpattern over the lifecycle of women. The latter can be explained byto postpone the motherhood-stage in the lifecycle, which in turn is increased female labour participation at younger ages.

5.3 Adjustment

This sub-section calculates the non-fertility taxes required to restoreerational balances in response to fertility shocks. Using the benchmarprole, we start by calculating the implicit non-fertility debt as dention 5.1.1. In general, the present value of female benets that are sava negative fertility shock exceeds the present value of the male beneas women tend to live longer. At the same time, the loss in male to a fertility shock exceeds the loss in female taxes, as men tend to ethan women. Both eects make boys more valuable to the governmethan girls. Overall, the loss in taxes due to negative fertility shocks ebenets saved, implying that lower fertility generates positive non-fertAs discussed in section 5.1.2, this implicit non-fertility debt could bby levying non-fertility taxes at young (NFTY) or old ages (NFTO).

Figure 10: Non-fertility taxes at young ages: generations 1930-1

-0.025

0

0.025

0.05

15 25 35 45 55 65

Fractionofincome

1930

1940

1950

1960


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Figure 11: Non-fertility taxes at young ages: generations 1970-2

-0.05

-0.025

0

0.025

0.05

15 25 35 45 55 65

Age

Fract

ionofincome

1970

1980

1990

2000

5.3.1 Non-fertility tax young

Recall that in the "non-fertility tax young" (NFTY) variant, the gecausing the negative fertility shock pay a non-fertility tax shortly afterhas occurred. More precisely, the implicit non-fertility debt due at a cis smoothed over 20 years when calculating non-fertility taxes due. Fand 11 show the implied non-fertility taxes due in this variant for geborn between 1930 and 2000 as fraction of their contemporaneous incnon-fertility taxes due for generation 1930 are actually negative overlife cycle, as its completed fertility (3.0) exceeds the completed ferti

benchmark (2.1). Put dierently, generation 1930 would be entitled tobonus in this variant. Generations born between 1930 and 1960 inireceive fertility bonuses as age-specic fertility remains relatively highages (see gure 8). However, as mentioned before, at older ages thedrops signicantly due to increased use of contraception. Consequentlages they have to pay non-fertility taxes. For the youngest generationother way around. In fact, non-fertility taxes due for generations b1970 show a typical hump-shaped pattern peaking around age 30. Hmentioned before, at older ages their fertility is relatively high due to p

motherhood, making these generations entitled to fertility bonuses. Ftion 2000, the non-fertility taxes due peak at age 30 at 2.5% of contemincome.

5.3.2 Non-fertility tax old

Non-fertility debts in the "non-fertility tax old" (NFTO) variant are s


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Figure 12: Non-fertility tax at old ages

0.22

0.24

0.26

0.28

0.30

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

Averagereplacem

entrate

Gross

Net, after non-fertility taxesdue

(represented by the dashed line in gure 12) for generations born bactually exceed gross replacement rates, as their completed fertilitythe benchmark. Hence, these generations would receive a bonus on thpension benets in this variant, just as they would receive a tax bothey are young in the NFTY-variant. However, the AOW pension bgenerations born after 1943 would be taxed to compensate for their no

debt. As a consequence, generation 2000s average net replacement rabe approximately 2 percentage points below its gross replacement ra

Figure 13: Public debt AOW pension scheme due to fertility sh

-1

-0.5

0

0.5

1

1950 1975 2000 2025 2050

Fractionoftota

lincome

Public debt without NFT

Public debt with NFTO

Public debt with NFTY


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5.3.3 Public debt due to fertility shocks

Figure 13 shows the impact of fertility shocks on public debt as well pact of non-fertility taxes on public debt. As a reference, the solid line public debt of the AOW pension scheme due to fertility shocks in thof any non-fertility tax. The exploding debt pattern clearly illustrateAOW pension scheme becomes unsustainable in the absence of comnon-fertility taxes. The dotted (dashed) line shows the evolution of p

when a non-fertility tax would be levied at young (old) ages. In botnon-fertility savings are initially negative, as generations with a high cfertility receive a fertility bonus. However, this is reversed when fertstart to fall. Under both variants, the AOW pension scheme will starmulate assets. However, in the NFTY-variant, more assets are generin the NFTO-variant, due to the dierence in the timing of the notax in the life cycle. Summarizing, generations born since 1943 havean implicit non-fertility debt, which might have been serviced eitheing non-fertility taxes at young ages or taxing their AOW pension be

eectively cutting net replacement rates) at old ages.

6 Labour-force participation shocks


The higher its labour participation, the more taxes a generation paAOW pension scheme. However, a labour-participation shock does notbenets a generation receives, as these benets are not related to a geearnings (see equation 5). Hence, a positive labour-participation shockthe intergenerational balance. Let us consider this now more formdirect impact of taxes paid by generation i due to a labour-force pashock (TFi ) is equal to the present value of the dierence between o

hand the taxes paid according to the actual participation prole (and on the other hand the taxes paid according to the benchmark parprole (sszs for s = j i)

TFi =

i+D

Xj=i+1

Ri;jj i;ji;jzi;j sszswjai;jnFi;j;with zi;j being the age-specic wage-factor ofn

Fi;t scaling w

Fi;t to wt (see

A). This implies that the implicit non-participation debt of generatio

DEBi = TFi T

Mi :


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"non-participation" taxes at the moment these shocks occur. These tabe levied on the generations causing the negative labour-participationorder to restore the initial intergenerational balance.

6.2 Data

Labour-participation proles (i;ji;jzi;j) consist of three componentsdata about the participating fraction (i;j) and hours worked (i;j) oation are available; these will be analyzed more closely below. Unfolong series of the age-specic wage factor (zi;j) are not available. Ionly have data for the age-wage factor in the period 1995-2005. Foculations, we calibrated zi;j in such a way that the implied develoaggregate labour income is consistent with the actual development of labour income.

6.2.1 Labour participation

We start by analyzing shocks in labour participation. As benchmark tion prole, we take the age-specic labour-participation rates in thyear 1957. Figures 14 and 15 show the (expected) labour-participatiof the generations 1935, 1950, 1965, 1980 and 2000. For the younger gewe relied on projections generated by CPB Netherlands Bureau for Policy Analysis ranging to 2050. Beyond 2050, we assumed constant alabour participation. Note that perfect foresight is not a necessaryfor labour-participation shocks (unlike longevity shocks): unexpected tion shocks can be attributed to the generation responsible for the par

shock.

Figure 14: Female labour-participation proles

0.0

0.2

0.4

0.6

0.8

1.0

15 25 35 45 55 65

Age

Labou

rparticipation

2000

1980

1965

1950

1935

Benchmark


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Figure 15: Male labour-participation proles

0.0

0.2

0.4

0.6

0.8

1.0

15 25 35 45 55 65

Age

Lab

ourparticipation

2000

1980

1965

1950

1935

Benchmark

shocks shown above. Secondly, labour participation at young ages has

both genders, which can be attributed to the fact that more years on education. Although this reduces the tax base in the short termens the tax base in the long run if real wages and participation ratesages are higher due to better education. Finally, male labour partichigher ages has fallen substantially in comparison with the benchmis especially the case for the oldest generations under consideration.non-monotonic development of male old-age labour participation acroFor generations born before 1915 (reected in the benchmark), older tured high participation rates. However, participation rates fell sub

for generation 1935, which reects the policy in the 1980s to withdworkers from the labour market and provide them with disability otirement benets in order to reduce youth unemployment. For the gborn since 1950, labour participation at higher ages is projected tagain in view of more actuarially fair supplementary pension benetslonger discourage people from working longer.

6.2.2 Part-time factor

Now we will analyze shocks in the part-time factor since 1957. As bpart-time prole, we take again the age-specic hours worked in thyear 1957. Figures 16 and 17 show the (expected) part-time factogenerations 1950, 1965, 1980 and 2000, where age- and gender-spetime proles for the period 2007-2100 have been assumed to be constanfrom 2007. Note that perfect foresight is not a necessary condition i th t ti f t ( lik l it h k ) t d h


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Figure 16: Female part-time proles

0.0

0.2

0.4

0.6

0.8

1.0

15 25 35 45 55 65 75

Age

P

art-timefactor 2000

1980

1965

1950

1935

Benchmark

Figure 17: Male part-time proles

0.0

0.2

0.4

0.6

0.8

1.0

15 25 35 45 55 65 75

Age

Part-timefactor 2000

1980

1965

1950

1935

Benchmark

6.3 Adjustment

6.3.1 Non-participation taxes due

Figure 18 shows the female non-participation taxes due as a fractiotemporaneous income. Actually, almost all female generations are ea participation bonus caused by the increase in female participation.participation bonuses peak for the female generation 2000 at 15 perincome. Figure 19 shows the male non-participation taxes due as of contemporaneous income. All male generations considered sho


28/43

Figure 18: Non-participation taxes due: women

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Age

Frac

tionofincome

1935

1950

1965

1980

2000

Figure 19: Non-participation taxes due: men

0.00

0.10

0.20

0.30

0.40

20 30 40 50 60

Age

Fractionofincome

1935

1950

1965

1980

2000

fully compensate for the male non-participation taxes.

6.3.2 Public debt due to labour-participation shocks

Figure 21 shows the public debt of the AOW pension scheme due tparticipation shocks. Public debt due to participation shocks is reecting the fact that increases in female participation do not fullysate for the decreases in male labour participation. By denition, pdue to participation shocks will be zero when male generations pay participation taxes due and female generations receive their participati


29/43

Figure 20: Non-participation taxes due: total

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Age

Frac

tionofincome

1935

1950

1965

1980

2000

Figure 21: Public debt AOW pension scheme due to participation

0

1

2

3

4

5

1950 1975 2000 2025 2050

Calender year

Fractionoftotalincome

7 Participation bonuses vs. non-fertility

due

So far, we have assumed that negative fertility shocks are compensanon-fertility taxes, while positive participation shocks are compensaparticipation bonuses. As noted before however, one could argue thafertility shocks are directly related to positive (female) participatioHence, it is interesting to analyze whether participation bonuses hpensated for non-fertility taxes due. More specically, we will dete


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taxes between genders would be arbitrary. Therefore, we decided tonon-fertility taxes on a fty-fty base. Then we calculate for each gand gender the balance of the participation bonuses and the fertility Finally, for every generation we calculate the present value of this baexpress it as a percentage of the present value of future AOW pensioin order to allow for a fair comparison between generations and gender22 and 23 show the resulting balance for female and male generations 1892 as a fraction of the present value of future pension benets.

Figure 22: Participation bonus versus fertility taxes due: wom

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

Fractio

nofPVb

enefits

Non-fertility tax due

Participation bonus

Balance

Figure 23: Participation bonus versus fertility taxes due: me

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

FractionofPVb

enefits


Participation bonus

Balance

The balance is positive for all female generations, as participatio


31/43

the decrease in male labour participation has added to the existinnon-fertility debt.

Figure 24: Participation bonus versus fertility taxes due: tota

-0.3

-0.2

-0.1

0

0.1

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Birth year

FractionofPVb

enefi

ts


Participation bonus

Balance

Figure 24 shows that the balance for men and women has overallative for almost all considered generations. Hence, these calculatiooverall that changes in labour participation have not fully compensatfall in fertility. Thus, the coincidence of dropped fertility, increaslabour participation and decreased male labour participation has been detrimental for the sustainability of the AOW pension scheme.

8 Summary and conclusions

This paper explores adjustments that maintain intergenerational balPAYG pension scheme in the face of fertility and longevity shocks. Gwith fewer children save time and resources, allowing them to workmore. Imposing non-fertility taxes on such generations generates therequired to restore intergenerational balances in the face of fertiliMoreover, raising retirement ages in response to longevity shocks enthe agents who live longer pay for the additional retirement benets th

over their longer expected lifetime.The methodological cornerstone of our paper is a demographic and

benchmark scenario where longevity-, fertility- and labour-participatioas well as retirement ages remain constant and in which the PAYGscheme features a balanced budget at each point in time. The benchnario allows us to determine the longevity-, fertility- and labour-for


32/43

identifying the demographic and economic shocks per gender and gsince 1957 before calculating their impact on the intergenerational ball generations born between 1892 and 2000. In general, age-speciprobabilities have increased substantially for both genders. Althoustill tend to live longer than men, the survival probabilities of men anare converging. At the same time, completed fertility for generatafter 1943 has fallen below the benchmark which is consistent with apopulation in the long run. Finally, gender-specic labour-participatihave converged since 1957 as men have reduced their labour participatwomen have increased their labour participation.

Our calculations suggest that the retirement age should be increleast 68 years for the generation born in 1945 in order to compensacreased longevity. Furthermore, we show that projected increases in laparticipation rates do not generate sucient additional tax revenuethe dearth of human capital as a result of declining fertility rates.


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References

[1] Cutler, D.M. and E. Meara (2001), "Changes in the age distrmortality over the 20th century", NBER Working Papers, 8556Bureau of Economic Research.

[2] Dalen, H.P. van, F. van Poppel and H. van Solinge (2006), "Lanlater met pensioen?" [Live longer, retire later?], Demos, vol. 22, p

[3] Janssen, F., A. Kunst and J. Mackenbach (2007), "Variations in told-age mortality decline in seven European countries, 19501999of smoking and other factors earlier in life", European Journal of Pvol. 23 (2), pp. 171-188.

[4] Mastrobuoni, G. (2006), "Labor supply eect of the recent sority benets cuts: empirical estimates using cohort discontinuitieWorking Paper no. 136.

[5] Shoven, J.B. and G.S. Goda (2008), "Adjusting government policination", NBER Working Papers, 14231, National Bureau of Research.

[6] Sinn, H.W. (2000), "Why a funded pension system Is needed andnot needed," International Tax and Public Finance, vol. 7 (4/5), pp


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A Benchmark scenario

Here we consider how we can derive a set of tax and replacement rates with the balanced-budget constraint in the benchmark scenario. Denper capita labour income earned at time t by the female generation i

yFi;t = Fi;t

Fi;tz

Fi;twt;

with zF

i;t being the age-specic wage-factor of nF

i;t scaling wF

i;t toage-specic wage-factors of generation i over its lifetime will be denrelative wage prole. Combining equations (3), (4), (5) and (23) with(6) gives the following equilibrium condition required for balanced bthe benchmark scenario

t

t=

Pt1i=tD bi;t

nFi;t + n

Mi;t

Pt1

i=tD ai;t

hnFi;t

Fi;t

F

i;tzFi;t + n

Mi;t

Mi;t

M

i;tzMi;t

i = kt;

with kt being the eective old-age dependency ratio at time t: the ranumber of PAYG pension beneciaries at time t over the eectivePAYG tax payers at time t in the benchmark scenario. Note that the dethe eective old-age dependency ratio deviates somewhat from the condenition of the old-age dependency ratio where i;t = i;t = zi;t = genders. Note also that the average wage (wt) is irrelevant for the paraand t. This is because we suppose that benets received from and taxthe PAYG pension scheme at time t are both indexed to average waequation (24), the tax rate at time t required for a balanced den

PAYG pension scheme in the benchmark scenario can now be expres

t = ktt;

for given replacement rates t. Alternatively, the replacement rate rtime t for a balanced dened-contribution PAYG pension scheme in tmark scenario can be expressed as

t =t

kt;

for given tax rates t. Figure 25 shows the development of the taxreplacement rates (t) and the eective old-age dependency ratio (all demographic and economic variables are kept at their benchmarNote that the eective old-age dependency ratio reaches its steady-s(0.57) in an oscillating movement. Using equation (5) we can derive threplacement rates since 1957 from the actual total AOW benets pa


35/43

should have been increased from 0.06 in 1957 to 0.12 in 2006 in the bscenario. Hence, the AOW pension scheme has been a mixture of Din the past. However, beyond 2006, we will treat the AOW pension scDB PAYG pension scheme in the benchmark scenario. Hence, the rerate is kept constant at 0.26. Changes in the eective old-age dependafter 2006 are now absorbed by parallel changes in the tax rate in ordthe budget balanced.

Figure 25: PAYG parameters benchmark scenario

0.0

0.1

0.2

0.3

0.4

1950 1975 2000 2025 2050 2075 2100

Calender year

0.0

0.2

0.4

0.6

0.8

k (RHS)

B Balancing longevity changes by adjusti

retirement age: a stylized model

In section 4.3.1 we derived the equilibrium retirement age by means of simulations. This appendix provides a theoretical decomposition of thchange of the equilibrium retirement age. To this end, we use a simpstylized version of the model applied in section 4.3.1. First of all, we regender dierences in the demographic and economic variables considondly, we suppose that the economy is in the steady-state, implying and stable tax rate. Thirdly, all variables are dened in continuous t

B.1 Population and economy

The remaining population at time t of the generation born at timeexpressed as


36/43

si;t: function representing the cumulated survival probabilities atthe generation born at time i.

Suppose that all working individuals of generation i earn wage wt with

wi;t = wi;i exp [(t i) g] ;

for g being the constant wage growth rate and

wi;i = w0 exp[ig] :

B.2 PAYG pension scheme

Suppose the generation born at time i pays a proportional tax rateincome. Total taxes paid by individuals of generation i at time t at afS; Rig (taxi;t) then yield

taxi;t = parti;twi;t;

with parti;t being a function representing the average labour par(hours worked as a fraction of the maximum number of hours) atgeneration i.

Suppose that parti;t is a positive function of the retirement age R

parti;t = parti;t (Ri) :

Hence, we assume that the higher the retirement age, the longer pparticipate.Suppose that the government determines the retirement age of ge

(Ri) as a function of its cumulative survival function (si;t)

Ri = R (si;t) :

Put dierently, we suppose that the longer people live, the hightheir statutory retirement age to compensate for "age ination" (ShGoda, 2008).

Finally, every individual of the generation born at time i receivbi;t at time t at ages t i = fRi; Dg

bi;t = wi;t;

ith b i th d l t t


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B.3 Intergenerational balance

The total amount of taxes a generation contributes to the PAYG pensiover its lifetime depends on several factors. First among them is the fthe generation i still alive at age S (si;i+S): the larger si;i+S, the mopays. The second factor is the average life expectancy of that generaage S to age Ri: the longer it lives on average in this period, the mit pays. Third is the labour participation of that generation: the h

labour participation, the more taxes it pays. Fourth is the wage growthat generation: the higher the growth rate, the more taxes it paythe present value of tax payments at young ages is higher than tax pahigher ages. In order to summarize the latter three factors in one nuintroduce the concept of the economically expected eective contribut(Ei [Ri S]) of the generation born at time i. Formally, we dene Eas

Ei [Ri S] = Rt=i+Rit=i+S

ni;tparti;texp[(ti)g]exp[(ti)r]dt

ni;i+S;

implying

Ei [Ri S] = f (si;t;parti;t; Ri) :

Note that in the special case of r = g and parti;t = 1 for all S < t

Ei [Ri S] =

Zt=i+Rit=i+S

si;t

si;i+Sdt;

which is exactly the demographic denition of the life expectancyS to age Ri.

The present value of total taxes paid by the generation born at tcan now be expressed as

Ti =

Zti=Riti=S

ni;ttaxi;t

exp [(t i) r]dt:

The total amount of benets that a generation receives from t

pension scheme over its lifetime depends on several factors. First amis the fraction of the generation i still alive at age Ri (si;i+Ri): the largthe more benets a generation will receive. The second factor is the life expectancy of that generation after retirement: the longer it lives othe more benets it receives. Third is the wage growth rate during its rperiod: the higher the wage growth rate, the more benets it receive


38/43

Ei [D Ri] =

Rt=i+Dt=i+Ri

ni;texp[(ti)g]exp[(ti)r]

dt

ni;i+R;

implying

Ei [D Ri] = f(si;t; Ri) :

Note that in the special case of r = g

Ei [D Ri] =

Zt=i+Dt=i+Ri

si;t

si;i+Ridt;

which is exactly the demographic denition of the (remaining) life efrom age Ri to age D.

The present value of total benets received by the generation boi (Bi) can now be expressed as

Bi =Zti=Dti=Ri

ni;tbi;t

exp[(t i) r]dt:

Using equations (36) and (40), the intergenerational balance at timgeneration born at time i (neti;t) can be expressed as

neti;t = ni;iw0 exp[ig] ( si;i+S [Ei (Ri S)] si;i+Ri [Ei (D Ri

B.4 Longevity changes

Now consider a change in the age-specic mortality probabilities of gi (i.e. a change in si;t) and call this a longevity change. The total dof the intergenerational balance of the generation born at time i (nettime t equals:

dneti;t

dsi;t= ni;iw0 exp[ig]

8>>>>>>>>>:

h@si;i+S@si;t

[Ei (Ri S)] +

h@si;i+Ri@si;t

[Ei (D Ri)] +

+ si;i+S@Ri@si;t

@[Ei(RiS)]@Ri

@Ri@si;t

h@si;i+Ri@Ri

+ si;i+S@Ri@si;t

@parti@Ri


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@Ri

@si;t=

h@si;i+Ri@si;t

[Ei (D Ri)] + si;i+Ri@[Ei(DRi)]

@si;t

i

h@si;i+S@si;t

[Ei

si;i+S

h@[Ei(RiS)]

@Ri+

@parti;t@Ri

@[Ei(RiS)]@parti;t

i

h@si;i+Ri@Ri

[Ei (D

Using equation (43), the required change of the retirement age into the longevity shock can be disentangled in eight partial eects.

with the numerator of equation (43), which measures the eect of changes on taxes paid and benets received by a generation. The between square brackets determines the partial eect of longevity cthe benets paid to generation i. We denote this as the longevitybenets. More specically, this eect depends on the partial eect of

changes on the cumulated survival function of generation i at age Riand the partial eect of longevity changes on the expected benet

generation i (@[Ei(DRi)]@si;t

). The second term between square brack

numerator determines the partial eect of longevity changes on the tributed by generation i. More specically, this eect depends on teect of longevity changes on the cumulated survival rate of generatioS (

@si;i+S@si;t

) and the partial eect of longevity changes on the expecte

contribution period of generation i (@[Ei(RiS)]@si;t

).

We now turn to the denominator of equation (43), which measuresterbalancing eect of changes in the retirement age in response to changes on taxes paid and benets received by a generation. The reects the partial eect of changes in the retirement age on the t

tributed by generation i. We call this the retirement-age eect on taspecically, this eect depends on both the (partial) direct and indiof changes in the retirement age on the expected eective contributi

(@[Ei(RiS)]@Ri

and@parti;t@Ri

@[Ei(RiS)]@parti;t

). The second term in the denom

termines the partial eect of changes in the retirement age on the bceived by generation i: We call this the retirement-age eect on benespecically, this eect depends on the partial eect of changes in the r

age on the cumulated survival rate of generation i at age Ri (@si;i+R@si;t

partial eect of changes in the retirement age on the expected bene@[Ei(DRi)]

@Ri.

Table 1: decomposition partial eects changes in longevity for gen

via size generation via eective

Taxes Longevity @si;i+S@si;t

[Ei (Ri S)] si;i+S@[Ei

@@[Ei(R


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Table 1 summarizes the partial eects identied above. Note thpation eects require a behavioral response of the members of the ginvolved: they have to decide whether they will increase labour particresponse to changes in the retirement age.

C Data sources

C.1 Mortality dataAge- and gender-specic mortality data as from 1935 can be found onsite of Statistics Netherlands (http://statline.cbs.nl/statweb). For tbefore 1935, we relied on age- and gender-specic mortality data frFor the period 2008-2050, we have relied on mortality gures used tics Netherlands 2008 population projections. Mortality gures for t2050-2100 were generated using the LCFIT-version of the Lee-Car(http://simsoc.demog.berkeley.edu/).

C.2 Fertility data

For the female generations born between 1935 and 2000, (projectionspecic fertility data (with fertility age classes ranging from 15 untilcan be found on the website of Statistics Netherlands (http://statline.c"Vruchtbaarheidscijfers per geboortegeneratie"). For generations bor1900 and 1935, only total fertility is available. Fertility proles for erations were constructed by proportionally increasing the fertilitygeneration 1935 in such a way that the implied total fertility is consisactual total fertility.

C.3 Wages

The overall nominal wage sum since 1957 is (with some data breachesbeen interpolated) available on the website of Statistics Netherlands (hGender-specic total wage series were allotted in proportion to the laticipation of each gender (see below). Long series for the age-factor zavailable. As a proxy for the age prole in wages we used the relati

age-wage factor in the period 1995-2005. For our calculations, we zi;t in such a way that the implied development of aggregate labourconsistent with the actual development of aggregate labour income.

C.4 Interest rates and ination


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C.5 Labour participation

Overall labour participation per gender since 1957 (with some datathat have been interpolated) is available on the website of Statistics Ne(http://statline.cbs.nl/statweb) as well as a age-specic labour particive-year period. For the period 2007-2050, we applied the projectedgender-specic labour-participation proles as projected by CPB NeBureau for Economic Policy Analysis. Beyond 2050 we assumed that a

labour participation proles remain constant.

C.6 Part-time factor

Data on total hours worked since 1957 can be found in the database ogen Growth and Development Centre (http://www.ggdc.nl/), whileNetherlands provided age proles on hours worked for men and wome1995 and 2007. In order to create long series for age and gender speworked, total hours worked since 1957 were allotted to men and wom

portion to the hours worked between 1995 and 2007. Moreover, the in hours worked between 1995 and 2006 was used as a proxy to allot tworked per gender to individual ages. Finally, in order to calculate afactor, we assumed a maximum of 48 hours per week to be workeover, age- and gender-specic part-time proles for the period 2007-assumed to be constant starting from 2007.

Previous DNB Working Papers in 2009


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No. 198 Peter ter Berg, Unification of the Frchet and Weibull DistributNo. 199 Ronald Heijmans, Simulations in the Dutch interbank paym

analysis

No 200 Itai Agur, What Institutional Structure for the Lender of Last ReNo 201 Iman van Lelyveld, Franka Liedorp and Manuel Kampman, Areinsurance risk

No. 202 Kerstin Bernoth and Andreas Pick, Forecasting the fragility of sector

No. 203 Maria Demertzis, The Wisdom of the Crowds and Public PolicNo. 204 Wouter den Haan and Vincent Sterk, The comovement between

activity

No. 205 Gus Garita and Chen Zhou, Can Open Capital Markets Help ANo. 206 Frederick van der Ploeg and Steven Poelhekke, The VolatiParadox of Plenty

No. 207 M. Hashem Pesaran and Adreas Pick, Forecasting Random WalNo. 208 Zsolt Darvas, Monetary Transmission in three Central Europ

from Time-Varying Coefficient Vector AutoregressionsNo. 209 Steven Poelhekke, Human Capital and Employment Growth

Areas: New Evidence

No. 210 Vincent Sterk, Credit Frictions and the Comovement between ConsumptionNo. 211 Jan de Dreu and Jacob Bikker, Pension fund sophistication and No. 212 Jakob de Haan and David-Jan Jansen, The communication

Central Bank: An overview of the first decadeNo. 213 Itai Agur, Regulatory Competition and Bank Risk TakingNo. 214 John Lewis, Fiscal policy in Central and Eastern Europe with r

inertia and the role of EU accession

No. 215 Jacob Bikker, An extended gravity model with substitution appliNo. 216 Arie Kapteyn and Federica Teppa, Subjective Measures of Risk Portfolio Choice

No. 217 Mark Mink and Jochen Mierau, Measuring Stock Market Contto the Sub-prime Crisis

No. 218 Michael Biggs, Thomas Mayer and Andreas Pick, Credit and ecNo. 219 Chen Zhou, Dependence structure of risk factors and diversificat


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