1The demographic factor is central to the debate on the necessary changes to old-age pension systems. In this article, Gustavo De Santis compares different theoretical Pay-As-You-Go pension systems, examines their respective merits and proposes a new system which has the potential to preserve the equitable character of the PAYG principle in a changing demographic context. Based on the relative standards of living of the different age groups, the system involves accumulating reserves when conditions are favourable and depleting them or going into debt in less favourable periods. The demographic structure evolves over time, and in the long run the reserves compensate the losses. The exposition is organized primarily around the demographic aspects of the different theoretical systems and is illustrated by simulations that clarify the demonstration. It offers a perspective detached from the discussions that inform the national debates in which as citizens we all have an interest.

2

“‘Would you tell me please which way I ought to go from here?’ Alice asked. ‘That depends a good deal on where you want to get to’ said the Cat. ‘I don’t care much where…’ said Alice. ‘Then it doesn’t matter which way you go’ said the Cat. ‘… so long as I get somewhere’ Alice added as an explanation.”

3Forms of economic support for the elderly have always existed. In pre-industrial times, this was primarily a private matter. Under many forms, and frequently with limited success, caring for the aged was the responsibility of families and households (see Bengtsson and Fridlizius, 1994; Reher, 1998). With the industrial revolution and the urbanisation process that ensued, proximity and therefore also solidarity between family members declined, and alternative systems of a collective nature developed gradually for the economic protection of the elderly (Conrad, 1991; Ritter, 1991). These arrangements were originally fully funded but in most cases, at least in Europe, and for a variety of reasons [1], turned later to Pay-As-You-Go (or PAYG). The difference is that, with funding, individuals are forced to save when they are adults and working, and deplete their own capital progressively in their old age when their labour income shrinks or disappears; with PAYG, current pensions are paid out of current contributions taken from younger cohorts, and there is no accumulation of capital [2].

4In recent times, PAYG pension systems have experienced serious financial difficulties that most observers tend to impute to their basic intrinsic weakness: with no collateral (i.e. savings), there is no guarantee that sufficient resources will actually be available to pay the promised future pensions. The survival of the system depends on whether the relevant variables (labour force participation and productivity, number of retirees, etc.) will evolve at least as favourably as the designers of the system imagined. Unfortunately, system designers are policy makers. They tend to be short-sighted because they know they will not be in charge forever, and because they seek the support of voters who are possibly even more short-sighted than they are [3]. All this conspires to bias the system towards excessively generous promises that may prove impossible to keep, as the recent history of most PAYG systems and their frequent adjustments suggests [4]. The difficulty of foreseeing the future availability of resources, and the related tendency to promise too much, are among the reasons why several economists prefer funding to PAYG. Its role, according to some of them, should be sharply reduced, possibly to a mere safety net for those who are both elderly and poor (see Disney, 1996, 2000; Feldstein and Ranguelova, 2000). International financial institutions strongly support this view (World Bank, 1994, 1997) or, according to some observers, impose it on recalcitrant but economically weak countries (Müller, Ryll and Wagener, 1999).

5In these pages I will not discuss all the reasons that may motivate this preference for funding. They are basically economic, and involve, for instance, the stimulus to capital accumulation and, hence, economic growth, or the assumed optimal balance between the public and the private sphere in the annuity market. Instead, I will try to show that most of the alleged shortcomings (Ponzi scheme [5], intrinsic instability, extreme variability of internal rates of return, discouraging effects on the labour market, etc.) are not characteristic of PAYG per se, and, in particular, that they do not affect the ES version that I will present below.

I – Simplifications, assumptions and “philosophy”

6To simplify matters, I will keep this exposition at a very high level of abstraction, assuming that there is only one all-encompassing intergenerational transfer system, that all workers are subject to the same rules regardless of their sex, industry, birth cohort, etc. and that management costs are negligible. Other forms of social intervention (such as poverty eradication) are also disregarded here.

7I will concentrate in particular on the demographic mechanisms that may or may not drive a PAYG intergenerational transfer system out of equilibrium. The economic side of the story will be kept as simple as possible, assuming the absence of any variation: productivity is given, and so are activity rates, unemployment, prices (there is no inflation), etc. [6] Further, I will refer exclusively to average incomes (on the one hand: wages and salaries net of contributions; on the other: transfer benefits), and will avoid the issue of individual variability. For the latter aspect, see De Santis (1997 and 2002).

8Perhaps the simplest way of approaching the pension problem is to think that current production [7] is a sort of very large pie that must be apportioned between “participants” — traditionally employed individuals and retirees, but in my proposal, more extensively, the young, adults and the elderly. The basic question is: how? Several points are worth considering already at this stage. The first is that if one ignores how large the pie is or may become in the future, and still looks for a permanent solution to the pension problem, only relative (not absolute) incomes should be considered — an approach that is still far from accepted in public debate. Incidentally, this is also difficult to apply technically, if actuarial criteria are to be met, because each retiree, with his or her own history of contributions, wants a “fair deal” and claims at least as much pie (in the form of pension benefits) as he or she paid in the past in the form of contributions, plus interest.

9The second point is that (relative) “entitlements to the pie” may be allotted on a group basis or on an individual basis. Both these “direct” solutions present shortcomings, among other reasons because demographic structures vary. Imagine for instance that the rule is that 80% of the pie should go to the employed, and 20% to the retirees. This may be fair on average, but one year there may be 90 employed and 10 retirees (whose average pension income would therefore be exceptionally high: 2.25 times as high as the net labour income of each employed person); and another year a totally different situation may emerge — say, 50 employed and 50 retirees — with the opposite effect in terms of individual welfare: each retiree now receives merely 25% of what accrues to an employed person. The intergenerational transfer system performs better if one decides that each retiree should have, for instance, 60% of what each employed person is left with after paying his/her contributions. But, in the two extreme scenarios above, this translates into extremely varying contribution rates (6.25% and 37.5%, respectively) and this, too, is scarcely advisable. This example, however unrealistic its figures may be, serves to illustrate the tendencies that might emerge in the real world if demographic factors were disregarded in the design of the system. It also suggests that it might be a good idea to save part of the pie in years of plenty (90 employed, 10 retired) and use this reserve in years of scarcity (50 employed, 50 retired) to keep both contributions and relative incomes constant, or almost so. In other words, a reasonably flexible reserve capital may help in the management of an intergenerational transfer system, even if it is essentially based on the PAYG principle.

10Notice that changes in the pensioner/employed ratio may derive from two distinct causes: participation in the labour market (an economic variable), and age structure (a purely demographic variable). To my knowledge, no one has ever tried to disentangle these two elements. Below, I will try to show how important it is to separate them, so as to be able to use actual economic behaviour (participation in the labour market) coupled with an age structure of reference.

11Traditionally, adults are expected to work in the market and the elderly are not. But, who is old? Isn’t “old age” a dynamic concept that should somehow be linked to the mean length of life, e₀? Do 50 years always mean the same, regardless of whether life expectancy at birth is 40 or 85? And, if the answer is no, what link should one establish between the increase in e₀ and the evolution of the retirement age (called here β )?

12And, finally, at socially agreed age β, why should the elderly be forced to retire, either by law or by practical (e.g. fiscal) arrangements? There may be several reasons linked to the functioning of the labour market (for instance they may have become too expensive, or unfit for a particular occupation) but, from the point of view of the transfer system, this is senseless. Ceteris paribus, more workers, including more elderly workers, make a bigger pie and a bigger slice for all claimants, which is one of the objectives that intergenerational transfer systems should pursue.

II – Three theoretical types of PAYG pension systems, plus two

13Several different versions of PAYG pension systems have been proposed so far. They all share the common feature that, in each year, the mass of contributions must match the mass of benefits, or, stated as a formula:

14

15where R = retirees,b^R = their average pension benefit, W = employed, w^W= their average gross wage, and c = the payroll contribution rate. (Notice that the focus here is not on the whole of the pie — Ww^W — but on the part that goes from the employed to the retired).

16With populations that are perfectly stationary, both demographically and economically, it is easy to identify a set of values that satisfy condition [1] once and forever. But if there is some kind of variation — demographic, economic, or both — equation [1] will not always hold, unless at least one of its variables is left free to adapt, i.e. is used as dependent. Since the number of retirees R, the employed population W, and the average gross wage w (linked to labour productivity) are exogenous to the pension system, only two variables may act as dependent: the payroll contribution rate c, and the average pension benefit b^R. Depending on which one is selected for this role, three main theoretical types of PAYG emerge (Table 1) [8].

Table 1

Classification of three theoretical types of PAYG

Dependent Label Name Instrumental variable variable(s) DC Defined Contribution c bR DB Defined Benefit bR c RS Risk Sharing rRW =bR/ wW (1-c) c, b R

Classification of three theoretical types of PAYG

17The first type fixes the contribution rate c and lets pensions benefits b^Rvary accordingly, as in the first scenario of the preceding section, so that its basic equation reads

18

19The second type fixes the average pension benefit b^Rand lets the contribution rate c vary in consequence, so that its basic equation becomes

20

21Notice that this approach relies on real (absolute, not relative) pension incomes (b^R), and is therefore (in my view) inadequate in a world where everything changes unpredictably all the time.

22The third type (Risk Sharing) fixes the ratio r^RW between the retirees’ pension benefits b^R and the employed’s net wages w^W (1 – c), as in the second scenario introduced above. In this arrangement, both pension benefits

23

24and the contribution rate c

25

26depend on the value selected for r. Equations [4] and [5] introduce a principle of “risk sharing” between employed and workers and pensioned, because whatever happens (for instance, with labour productivity, inflation, or unemployment), their relative economic distance r will remain unaffected (see Musgrave, 1981; Hagemann and Nicoletti, 1990; or Gonnot, Keilman, and Prinz, 1995).

27Elsewhere (De Santis, 2002), I show that most actual PAYG pension systems, although very different from country to country, form a fourth, distinct group that works remarkably worse than the theoretical types just described. There are two main reasons for this: a) no variable is explicitly defined as dependent, and left free to vary, so as to make sure that inflows (contributions) match outflows (payments); b) individual pensions are calculated with complex and widely varying formulas that typically take into consideration several variables such as past contributions, industry, gender, age at retirement, inflation, or expected future growth of productivity. In every period, individual and therefore also average pensions constitute an independent and largely unpredictable variable that, together with several others (survival, age at retirement, employment status, productivity, etc.), makes the system hardly manageable, scarcely transparent, and potentially unviable in the long run.

28It is precisely this type of PAYG system that is currently in financial and theoretical difficulty. But the current debate on how to reform it, especially in Italy, often reminds me of Alice’s dilemma (see epigraph): patches are frequently proposed, and sometimes implemented, in order to face the most urgent crises, but a clear vision of the (indefinitely sustainable) pension model that should be set up seems to be missing. This article sets out to present one called ES, Equitable and Stable (the fifth, in my classification). First it discusses some of its theoretical properties, and later it shows that the model performs better in simulations than all available alternatives [9].

29In the discussion and simulations that follow, I will avoid the intricacies of the economic sphere as much as possible, not because I think the system is weak in this respect (see De Santis, 2002), but because understanding its demography is already complicated enough and because its most original distinctive feature (the reference age structure) is demographic. I will not discuss either the issue of how to go over from the current system to PAYG-ES or, for that matter, of what happens when any PAYG system is modified, or matures. In all cases I will present matters as if each of the PAYG systems under scrutiny were already in full maturity.

III – The rationale of PAYG-ES

30I think that the main weakness of any long-term financial arrangement is that conditions can vary unexpectedly, thereby deceiving the hopes of, and not keeping the promises made to, those who decided, or were forced, to participate. Within limits, this is acceptable, because everybody knows that the future is uncertain; beyond certain limits, however, expectations will be frustrated, the confidence of the general public will be shaken, and the whole system may crumble, with potentially dire consequences.

31PAYG-ES is basically a set of rules that is meant to guarantee that an intergenerational transfer system is viable, i.e. can exist forever with almost unchanging characteristics, independently of how the demographic and the economic situations evolve. It is always possible to introduce changes, obviously, but the important point is that, up to a limit, modifications will not prove necessary. In other words, the system can last forever, at least in principle. In addition to this, the system possesses a few other desirable features, in particular:

1. almost perfect intergenerational, or actuarial, equity is assured [10]: generations pay in contributions as much as they receive in benefits;
2. the contribution load can remain relatively constant over time;
3. relative incomes (employed vs. retired; young vs. adults vs. elderly) can remain constant over time;
4. everybody can keep on working for as long as he/she pleases: pension benefits will, in all cases, start to be paid exactly at the predefined age P (e.g. 60 or 65), regardless of the employment status of the elderly;
5. dynamic threshold ages (for instance, a retirement age that grows over time as life expectancy increases, according to pre-defined criteria) can be introduced in a way that is consistent with the general framework;
6. child benefits may be incorporated in the intergenerational transfer system, an extension over traditional pension systems that I find theoretically defensible and technically useful (cf. below).

32Let us now see how the proposed system works in practice. Already in equation [1] above, two main policy decisions emerge as crucial: on the normal age at retirement β, and on whether pensions payments should be more or less generous. Indeed, given w^W(average wage, exogenous to the pensions system), a lower retirement age β increases the retired-to-employed ratio R/W; and a higher pension transfer b^Rtranslates into a higher equilibrium payroll contribution rate c.

33The picture is more complex than this, however. Let us consider an enlarged version of equation [1], that is

34

35or

36

37where R = retirees; E = elderly, W = employed, A = adults: b^E = b^R(R/E) is therefore the average pension income of each elderly person; while w^A = w^W(W/A) is the average net labour income of each adult. The interest of using equation [7] instead of equation [1] is that it highlights that the two socio-economic categories customarily used in this case, W and R, i.e. employed and pensioned, derive from a compositional effect (how large the three basic demographic groups are: Y, A, and E), and from two behavioural effects: participation in the labour market, W/A, and in the pension system, R/E. Besides, with equation [7] the policy choice of setting ages α and β (beginning and end of adulthood) emerges explicitly as crucial.

38On average, each adult pays (c w^A) in contributions. Equation [7] implies that all those who work pay contributions, regardless of their age [12]; on the other hand, only those who are in a convenient age bracket may receive an intergenerational transfer benefit b: each young person receives b^Y(that may be zero, and in this case the system reduces to a simple pension system), whereas each elderly receives on average b^E. Elderly workers intervene in both kinds of transfers: they pay a contribution cw^Aout of their salary, but they also receive an average benefit b^E.

39Currently, no existing pension system incorporates child benefits among its transfers (i.e. b^Y = 0). There are good historical reasons for this, but, in my opinion, including the young among the recipients of these benefits should not be excluded a priori, on several grounds. Theoretically speaking, if one accepts the principle that an intergenerational transfer system should protect those who are not of working age, children (below a conventionally defined age α, like 15 years, for instance) are as entitled as the elderly to such a transfer. Besides, pension systems may exert a depressing effect on fertility (see e.g. Harrod, 1950; Nugent, 1985; Cigno, 1991; Cigno and Rosati, 1992), which, in the long run, causes population aging and decline, and therefore undermines the basis of the system itself, so that a counteracting mechanism, like explicit child benefits, may well be needed.

40Finally, there is also a practical reason. As the simulations of Section IX show, including child benefits among the provisions of an intergenerational transfer system exerts a non-trivial stabilization effect, because, very frequently, the relative weights of the two age groups move in opposite directions, and this limits the variability of the total transfer needs within each period.

41But the main element of innovation in the PAYG-ES transfer system is the notion of reference age structure, which serves two main purposes. On the one hand, it constitutes a standard of reference for appreciating the peaks and troughs of the current age structure. In particular, I will speak of “favourable” demographic phases when there are relatively more adults in the current than in the reference age structure, and “unfavourable” demographic phases when the reverse is true (more young and/or elderly in the current than in the reference age structure). One important implication of the use of the reference (instead of the current) age structure is that, in periods when the current age structure is favourable, the PAYG-ES system tends to accumulate resources (or a reserve capital) that it depletes in periods when the age structure is unfavourable. The topic is discussed more in depth in Section IV.

42On the other hand, the reference age structure constitutes a point of attraction, or a pivotal line, around which, by definition, the current age structure oscillates. Figure 1, representing a case of favourable age structure (proportionally more adults in the actual than in the reference population), gives an idea of how the principle works [13].

Figure 1

Actual and reference age structure for Italy in 2001

Actual and reference age structure for Italy in 2001

43In Section V I will argue that, for all practical purposes, one can use as “reference” the age structure of the stationary population associated with the current life table, but, for the time being, let us just assume that this reference age structure is given. As Figure 1 shows, adding a policy decision on the threshold ages α and β (beginning and end of adulthood), produces Y, A and E, i.e. the proportions young, adult and elderly in the reference age structure. For instance, in the case of Figure 1 (e₀ = 82; α = 15; β = 65), one finds that Y = 18.2%, A = 59.5%, and E = 22.3%.

44In this hypothetical population, equation [7] becomes

45

46and the average adult net [14] wage transforms into w^A(1 – c). Now, let us apply a modified version of Risk Sharing, and let us assume that a general policy agreement has been reached, for the present and for the future, on the most convenient ratio r between average transfers b and adults’ average net wages, so that transfer benefits for the young and for the elderly, are

47

48For instance, excluding child benefits results in r^Y = 0 (and therefore also b^Y = 0); at the other extreme of the age scale, virtually every arrangement is possible, but, just to fix ideas, let us imagine that pension benefits should average 60% of (net average) wages, so that r^E = 0.6. With equation [9] in mind, equation [8] can be re-written as

49

50After re-organisation, one obtains the equilibrium contribution rate

51

52For instance, with the assumptions set forth above (e₀ = 82; α = 15; β = 65; r^Y = 0, r^E = 0.6), the resulting equilibrium contribution rate is c= 18.4%.

53The basic idea behind PAYG-ES is simply the following:

54The details of PAYG-ES will be discussed shortly, but some of its merits should emerge already at this stage. By making use of a reference age structure, and of three demographic groups instead of two social categories, PAYG-ES never matures: all of its variables in eq. [11] must be defined as if the system were, and were to remain indefinitely, on an equilibrium path.

55No variable in PAYG-ES is, or needs to be, forecast or projected: accepting that the reference age structure is the age structure of the current stationary population (see Section V), everything (including the economic dimension, not discussed here) is based on observed (current or past) variables only. In other words, no forms of distortion can be introduced in the system through over-optimistic, or simply inaccurate, forecasts, although instrumental variables may obviously always be adapted if collective preferences (on the policy variables α, β, r^Y, or r^E) change.

56This is of the greatest importance. Take survival conditions, for instance: changes in mortality are virtually sure to take place, but they are difficult to predict (see for example Wilmoth, 2001; Oeppen and Vaupel, 2002). On the other hand, they are precisely and timely measured in all the developed countries. With ES, it is possible to decide beforehand how to cope with variations in survival, for instance adapting the basic contribution rate c, or the threshold ages α and β : an example is given in the simulations of Section IX. What matters is that the necessary parametric adjustments in the system can be made automatically, and need not become discretionary, i.e. create discontinuities with the past, pass through public debates (parliamentary decisions, government bills), with the delays and social contrasts that this generates, and the generational inequities it introduces (Auerbach, Gokhale, and Kotlikoff, 1994).

57On the other hand, since everything revolves around the reference age structure, variations in fertility and migration play only a relatively minor role. In other words, the system is relatively protected from demographic shocks (but see next section and the simulations).

58We may note in passing that economic shocks also affect PAYG-ES much less than any other PAYG transfer system [15], although this point is not developed here (see De Santis, 2002). For instance, lower labour productivity (i.e. lower w^W) and/or lower activity rates (i.e. lower W/A, for instance, because of unemployment, expanding underground economy, etc.) translate into a lower average adult gross wage w^A, and this, given r and c, automatically lowers the value of the average transfer payment b. Changing signs, the same happens for higher labour productivity, or an increase in the employment ratio (the ratio W/A), or even inflation (causing an increase in the nominal, but not in the real value of wages and benefits). All of these events share two common features: they affect the standard of living of the three basic demographic classes of the community in the same way (this is guaranteed by the constant proportionality factor r), but they do not affect the transfer system itself, which keeps working as before.

IV – PAYG-ES in detail

59With PAYG-ES three sets of variables must be distinguished: instrumental (or policy); exogenous, and dependent.

60One of the policy variables is r in equation [9], i.e. the ratio between average pension transfers and average net wages. Everything else given, once r has been decided, the value of cfollows automatically from equation [11]. In other words, one cannot have both generous pension allocations relative to net wages (high r) and low payroll rates (c) — not even in phases with a favourable age structure, because here it is the reference and not the current age structure that counts.

61Two more policy variables are the threshold ages α and β that separate the young from the adults (α) and the adults from the elderly (β). It is the choice of these thresholds α and β that, together with exogenously imposed survival conditions, determine the shares Y, A, and E (i.e. the proportions of young, adult and elderly in the reference population). Note that α and β need not be constant over time: actually, as shown in Section IX, it is possible (and, to me at least, preferable) to adapt them to the evolution of mortality, for instance in such a way that Y, A, and E remain constant as mortality risks change.

62Another policy variable must be added to keep the reserve capital at an acceptable level that is close to its target level K, which may be zero or (preferably) greater than zero. In each year, actual contributions Aw^Ac do not necessarily match actual transfer payments Eb^E + Yb^Y. Contributions exceed payments when the demographic conditions are favourable, and fall short of them in the opposite case. On top of this the existing capital (or deficit) K yields (or must be financed at) a market rate of interest i, producing a flow iK [16]. The algebraic sum of all of these flows gives the yearly variation in the reserve capital ΔK

63

64Ideally, the system should be designed in such a way that K oscillates only moderately around its reference value K, so that imbalances ΔK are never large, alternate in sign, and no ad-hoc adjustment is ever necessary. However, in particular circumstances, and notably when the demographic structure is distorted, ΔK may remain constantly above or below zero for some years, and this may cause the actual reserve capital to deviate too much from its target value K. Several options are available to contain this drift. One of these is to add a variable component to the reference payroll contribution rate c, c^v, for instance according to a (growing) function k

65

66so that the overall contribution rate becomes

67

68The policy variable is the function k in eq. [13], giving the reactivity of the payroll contribution rate to imbalances in the reserves of the system. Obviously, the steeper k is, the quicker the reserves of the system get back to equilibrium, but, on the other hand, the greater the deviation of the actual payroll contribution rate c is from its target value c, the less the advantage of having a reference age structure in attenuating the effects of the demographic fluctuations. The variability of c around its target value cis admittedly a disadvantage of the proposed system. However, this variability proves relatively moderate in computer simulations; it may be constrained within pre-defined limits by ad hoc measures, and, most importantly, it is compensated by the fact that it permits other relevant variables of the system to remain constant (see Sections VII to IX for an example) [17].

69So, to summarize, there are several policy (or instrumental) variables: r (ratio between transfers and net wages), α and β (threshold ages: these, together with e₀ — exogenous — determine the proportions Y, A, and E), and k (effect on the payroll contribution rate of a gap between the optimal K and the actual K reserve level).

70At the same time, there are also several exogenous variables: survival conditions (determining the age structure of the reference population); other demographic variables (determining the age structure of the actual population); labour market conditions, and notably employment status and productivity (determining gross salaries w and, together with c, individual and average contributions C); and the market rate of interest i.

71Two sets of independent variables determine the dependent ones: the proportions Y, A, and E in the actual and Y, A, and E in the reference population; the various payroll contribution rates (c, c^v, c), net wages w(1 – c), and net transfers b.

V – The reference age structure

72In this Section I will not be able to prove, but will try to show, that the current stationary population (i.e. the stationary population associated with the current life table, with L_x years lived at each age, and life expectancy e₀ = ΣL_x) can be used as a proxy for the reference age structure that PAYG-ES requires. This section relies heavily on the notion of long run — meaning possibly thousands of years. This does not imply that the proposed system reaches its equilibrium values only in the long run. On the contrary, PAYG-ES is constantly in, or very close to, equilibrium, thanks to the adjustment process of equations [13] and [14] or to similar mechanisms. But proving that the demographic structure converges towards the assumed reference curve guarantees that the adjustments of equations [13] and [14] will only be needed to fine-tune the system, not to fix structural imbalances, and will operate only for relatively short periods. “Short” in this context may mean 100 years or more, but once again, this barely affects the system. In the rather extreme simulations of Sections VIII and IX, for instance, the actual payroll contribution rate exceeds its reference value for over a century, but the difference (within 3 percentage points, in that case) always falls within the margins that, by assumption, had been collectively agreed upon in advance, while everything else functions normally.

73Let us initially assume that migration is absent, and mortality constant. In the very long run (thousands of years), the rate of demographic growth must be virtually identical to zero, otherwise the population would explode, or disappear [18]. With no migration and constant mortality, this can be achieved only through fluctuations in fertility, such that the long-run reproduction rate approaches unity. In this scenario, each cohort N_t of newborn in year t can be expressed as

74

75where N is the (unknown) average of N_t, while n_t is its deviation from average, which I am assuming is distributed with average equal to 0, and a given, although possibly unknown variance. In any year t, the population age structure is given by

76

77and the expected value for this quantity is the stationary population.

78With changing mortality (and therefore also changing reference population structure) I cannot prove any convergence, but even with rather extreme variations in mortality, all the simulations I ever attempted gave excellent results when the reference population was based on the corresponding period life table. This probably depends on the fact that year-to-year variations in life expectancy are modest in relative terms (i.e. Δe₀/e₀), even when they are high by historical standards. Besides, their sole effect on PAYG-ES comes through a variation in the proportions of the three basic reference groups, Y, A, and E, and these proportions change only modestly, anyway.

79The same holds for migration: it affects the actual age structure, for sure, but should it also be included in the reference age structures, at least those cases when there is a consolidated tradition of in- or out-migration? My answer is negative: in my simulations I have invariably verified that using the “simple” reference age structure (i.e. that derived from the current life table) gives excellent results. This probably depends on several factors. In the first place, it is only age-specific net migration that matters, i.e. that may change the age structure of the actual with respect to the reference population. Secondly, net migration could in principle prove perfectly neutral if its age structure were identical to that of the reference population. In practice, this is never the case since migration takes place at relatively young ages but, and here comes the final reason, even when they are high by historical standards, age-specific migration flows tend to be relatively modest with respect to a reasonably large population, or at least short-lived, and therefore their long-term impact on the age structure proves modest, or even negligible.

VI – A theoretical comparison between forms of PAYG pension systems

80Table 2 presents a synthesis of the differences between the four types of PAYG pension systems I listed above.

Table 2

A synoptic view of four variants of PAYG

">Characteristics Name (Label) Defined Contribution DC Defined Benefit DB Risk Sharing RS Equitable and Stable ES Theoretical/Empirical Theoretical Theoretical Theoretical Theoretical Status in literature Known Known Known New(a) Main relevant variables cbWRwW ? cbWRwW ? crRWWRwW ? EA??? c cve0r YAEYAEKwA Budget constraint C=B Yearly Yearly Yearly Over several years Age structure used Current Current Current Reference Do variations in any of the following affect (impair) the working of the pension system (regardless of other impacts they may have, e.g. on the standards of living)? Employment ratio Yes Yes Yes No Labour productivity No {Yes}(b) No No Inflation No {Yes}(c) No No Fertility and migration Yes Yes Yes {No} Main constant term(s) c B rRW rEA, c Main variable term(s) b c C c {, }(d) What if mortality decreases? b decreases C increases c increases; b decreases *c ; b   and  (a) But see De Santis (1995, 1997, 2002). (b) Unless the rate of growth of pension benefits equals the rate of growth of wages. (c) Unless the rate of growth of pension benefits incorporates inflation. (d) In three different variants: see next row. Legend c = payroll contribution rate r = ratio between pensions and taxable wages b = average pension amount C = total contributions of one year A = adults W = employed E = elderly R = retired B = total pensions of one year Y = young w = gross wages  = threshold age at which old age conventionally starts (retirement age)  = threshold age at which adulthood conventionally starts {} = approximately, or depending on circumstances

A synoptic view of four variants of PAYG

81The use of r (ratio between transfers and net wages) makes PAYG-Equitable and Stable similar to Risk Sharing. But ES constitutes an improvement over RS in several ways, because of its flexible budget constraint (K tends towards K, but does not need to coincide with it all the time) and its reserve capital (K can be greater than zero), because it uses a reference age structure, and because it refers to age classes (Y, A, E), and therefore incorporates changes in the employment ratio (W/A) through the variable w^A = w^W(W/A). Ceteris paribus, lower participation in the labour force or growing unemployment indicate lower per capita production, and this automatically lowers the average transfer benefits b in ES.

82Let us briefly consider the effects of a few possible exogenous shocks.

VII – Computer simulations: background

90In the preceding section, I tried to prove that ES is preferable to other forms of PAYG by referring to a few of its theoretical properties: independence of changes in the current age structure (and therefore, also, of fertility and migration), and independence of all kinds of economic variations, like inflation, activity, labour productivity, etc. In this section I will try to prove that it also performs better “empirically”, i.e. in simulations.

91As mentioned, in PAYG-ES some variables are given exogenously, some are chosen instrumentally, and finally, some result as consequences. The simulations of Sections VIII and IX, for instance, are based on the assumptions in Table 3.

Table 3

Scenario no. 1 – description

3(a) – Exogenous variables Variable Symbol Value Total fertility rate TFR 2.11.6 Life expectancy at birth e0 77 years Average labour productivity ( = gross wage) w 1 Employment ratio (Employed/Adults) W/A 0.6 Retirees/Elderly ratio R/E 0.6 3(b) – Instrumental or policy variables Variable Symbol Value Lower limit for adulthood  15 years Upper limit for adulthood ( = “retirement" age)  65 years Relative(a) child benefit [ = bY / wA (1-c)] rY 0% Relativea pension benefit [ = bE / wA (1-c)] rE 75.4% Reference level of reserve capital K 1 year of pension benefits Sensitivity of cv k cv = 3% (K –K)/BYBE(b) (a) Relative to an adult’s average net labour earnings. (b) But subject to the condition that |cv| 3%. 3(c) – Resulting values for the dependent variables Variable Symbol Comes from Value Reference payroll contribution rate c eq. [11] 18.7% Variable payroll contribution rate cv eq. [13] –3% < cv <3% Overall payroll contribution rate ( = c+cv) c eq. [14] 15.7% <c <21.7% Share of the young in the reference population Y (e0; 19.2% Share of adults in the reference population A (e0;;) 61.9% Share of the elderly in the reference population E (e0;) 18.9% Adults’ gross labour earnings wA eq. [6]–[7] 0.600 Adults’ net labour earnings wA(1–c) eq. [9] 0.488 Average pension benefit for the elderly bE eq. [9] 0.368 Average pension benefit for retirees bR eq. [4] 0.613

Scenario no. 1 – description

92Starting from these assumptions, let us consider two different demographic evolutions that may represent the case of the majority of the developed countries [20]:

• in scenario No. 1, everything else being equal, fertility declines at first (the TFR, total fertility rate, or number of children per woman, reaches 1.6) and resumes its equilibrium value (2.1) later;
• in scenario No. 2, the mean length of life (or life expectancy at birth e₀) increases from 77 to 85 in a hundred years, and then stabilizes at the new, higher level, while the TFR adapts, so that, eventually, the population returns to stationarity.

93In scenario No. 1, only 4 types of PAYG will be compared: Defined Contribution, Defined Benefit, Risk Sharing, Equitable and Stable. In the second scenario, on the contrary, since e₀ increases, we will need to distinguish between three sub-variants for each of them (2, 2’, and 2”), depending on what varies. In scenario 2, threshold ages do not move: it is the payroll contribution rate c that adapts (except for DC, where it is pension benefits b that decrease, c being fixed by definition).

94In scenario 2’, age at pension β increases in such a way that all economic variables remain fixed in the ES system (c, r, …): in practice, as e₀increases from 77 to 85, β changes from 65 to about 69.

95Finally, in scenario 2”, adult threshold ages a and β increase in such a way that the proportions Y, A, and Ein the reference populations do not vary, and this constancy extends therefore also to all other variables in the ES system (c r, …). In practice, with e₀increasing from 77 to 85, α reaches 16.5 (up from 15), and β 70 (up from 65).

96The criteria for comparing the performances of the various forms of transfer systems are, needless to say, a crucial issue. It is perhaps worth remembering that one of the reasons of dissatisfaction with PAYG pension systems is that they are unpredictable. Current rules may not be valid tomorrow, depending on how circumstances evolve. Hence, my criteria for excellence will be based on constancy: the less a key variable is forced to change, the better. With this in mind, let us consider four different dimensions:

• Variability over time of the yearly contribution rate c_t(ideally approaching 0). Some PAYG transfer schemes cause c to deviate from its original level: this implies that some generations are taxed differently from others, with possible undesirable economic consequences (e.g. increase in labour costs, etc.). The ideal is a scheme that leaves the contribution rate c constant at the socially preferred level.
• Variability over time of r_t(ideally approaching 0). Some PAYG transfer schemes cause variations in the yearly ratios r between pension benefits b and net wages w(1-c), thus making the young and the aged richer or poorer relatively to the adults. Since this is typically an unwanted consequence of unforeseen demographic (or — not discussed here — economic) dynamics, a solution that leaves the ratio r unaffected is preferable.
• Actuarial (inter-generational, or simply generational) equity implies that each generation pays in contributions C just as much as it receives in benefits B. A possible measure of generational equity is therefore the difference D^g = B^g - C^g, to be calculated (in real terms [21]) for each generation g. The ideal is a scheme that guarantees perfect generational equity (D^g = 0), or at least that deviates from this objective as little as possible. Absence of equity, however, can be revealed by two slightly different measures:
• average of D^gover time (ideally approaching 0). If μ(D^g) differs from zero, the system proves sensitive to demographic variations (increase or decrease of population) that tend to cause generational gains or losses in the transfer system;
• variance of D^gover time (ideally approaching 0). If var(D^g) differs from zero, the system proves sensitive to demographic variations (increase or decrease of population) and tends to distribute the resulting gains and losses inequitably across the generations considered in the simulation (with some earning or losing more than others).

VIII – Computer simulations: scenario No. 1

97In scenario No. 1, mortality does not vary (e₀ is constant at 77 years), but fertility falls to 1.6 and then gets back to reproduction level (Figure 2b). This causes the population to eventually lose about a third of its members (Figure 2c). Since the system is displacing resources towards the elderly, population reduction implies a sort of capital loss (Lee, 1994; 2000a), which will have to be redistributed somehow among the participating cohorts.

Figure 2

Computer simulation of population, scenario No. 1

Computer simulation of population, scenario No. 1

98In this example, with constant e₀, the reference age structure does not change: however, the actual age structure will be disturbed in the course of time. For instance, the proportion of the young falls from 19% to 15% before getting back to 19%; the adults (aged 15-64) expand to 64%, then shrink to 59%, and eventually resume their reference share of 62%; the elderly, finally, peak at 24%, but then fall back to 19% (Figure 2d–2f). Note, however, that Figure 2e illustrates one of the basic mechanisms behind PAYG-ES: favourable ( A > A) and unfavourable (A < A) periods alternate, and are roughly of the same order of magnitude if one chooses a proper standard of reference.

99Indeed, depending on the kind of PAYG system in force, the impact of this scenario on the payroll contribution rate may prove sizeable. From its initial level of 18.7%, the payroll rate generally rises (although not in Defined Contribution, by definition), to more than 24% in the case of Defined Benefits, to 23% with Risk Sharing, and to less than 22% with the Equitable and Stable solution (Figure 3).

Figure 3

Payroll contribution rate c under scenario No. 1 (%)

Payroll contribution rate c under scenario No. 1 (%)

100In two variants of PAYG, namely Risk Sharing and Equitable and Stable, the ratio r between the elderly’s pension benefits b^Eand the adults’ net labour earnings w(1 – c) remains unaffected at 75.4%, but in the remaining two variants this ratio oscillates significantly, sometimes dropping to 58% (Defined Contribution) [22], and sometimes climbing to 81% (Defined Benefit) (Figure 4).

Figure 4

Cross-sectional “equity” under scenario No. 1 (ratio between pension benefits and net adult earnings)

Cross-sectional “equity” under scenario No. 1 (ratio between pension benefits and net adult earnings)

101Finally, a look at intergenerational equity [23] in Figure 5 shows that all the PAYG systems considered here suffer when population declines (precisely because they transfer resources forward). They react differently, however, either in terms of the absolute amount of the loss (graphically represented by the area comprised between the x axis and each curve), or in terms of how they distribute the loss among the generations. Sometimes they concentrate it in relatively few cohorts, and sometimes they spread it upon several of them.

Figure 5

Intergenerational equity under scenario No. 1: difference between the overall amount of pension benefits and the total contributions paid over life, relative to the annual adults’ net labour earnings, per cohort (in %)

Intergenerational equity under scenario No. 1: difference between the overall amount of pension benefits and the total contributions paid over life, relative to the annual adults’ net labour earnings, per cohort (in %)

102Table 4 provides a summary of these results. Although the variables refer to different, non-comparable dimensions (variability of contribution rate; variability of relative standard of living of elderly and adults; intergenerational equity), there seems to be little doubt that Defined Contribution displays the worst performance, because it introduces unacceptably high variability in transfer benefits [24]. In this simulation, for example, around year 150, the elderly become extremely poor relative to the adults (see Figure 4). Defined Benefit is systematically worse than both Risk Sharing and Equitable and Stable; these two perform similarly, but the latter proves slightly preferable in all respects.

Table 4

Synthetic results of scenario No. 1

Type Characteristic Variance (x 10,000) Average c(a) r(b) D(c) D(d) DC c=constant 0.00 298.21 7.54 –0.73% DB b= constant 28.50 27.64 9.33 – 0.87% RS r= constant 17.19 0.00 6.86 – 0.84% ES RS+reference age structure 14.61 0.00 5.84 – 0.85% Note: For all indicators zero is the best possible outcome. (a) Equilibrium payroll contribution rate (variance over 300 years  10,000). (b) Ratio between pensions and wages (variance over 300 years  10,000). (c) Difference between benefits and contributions, relative to 1 year of pay (variance over 200 cohorts  10,000). (d) Difference between benefits and contributions, relative to 1 year of pay (average over 200 cohorts).

Synthetic results of scenario No. 1

IX – Scenario No. 2, and the introduction of child benefits

103Figure 6 presents the demography of scenario No. 2, with increasing life expectancy (from 77 to 85), slightly lower fertility (so as to ensure long term stationarity), and a population that grows by about 10% (thanks to longer life). With fixed threshold ages, the population becomes considerably older: those over 65, for instance, increase from 19% to 24.5% of the total population, while the young (0-14) shrink to 17.5% and the adults to about 58%. With moving threshold ages, however, aging becomes much less important. In Figure 7, we present the proportion of the elderly under the three subscenarios (2, 2’ and 2’’) described earlier. This offers a reminder that the definitions of youth, adulthood and old age depend very much on conventions, which need not remain fixed as survival conditions change.

Figure 6

Computer simulation of population, scenario No. 2, (fixed threshold ages)

Computer simulation of population, scenario No. 2, (fixed threshold ages)

Figure 7

Proportion E of the elderly in the population under scenario No. 2, with different values of threshold age β (scenarios 2, 2’ and 2")

Proportion E of the elderly in the population under scenario No. 2, with different values of threshold age β (scenarios 2, 2’ and 2")

104Let us further complicate the picture by considering two alternative transfer systems in all four scenarios (1, 2, 2’ and 2"), one with and one without child benefits, but in all cases, such that the initial equilibrium payroll rate is 18.7%. We therefore obtain 8 cases: for each of them, the performance of the four PAYG systems is presented in Table 5.

Table 5

Synthetic results of various scenarios, with and without child benefits

Type Characteristic Variance (×10,000) Average Variance (×10,000) Average c(a) r(b) D(c) D(d) c(a) r(b) D(c) D(d) Without child benefits(e) With child benefits(f) No. 1: fluctuations in fertility DC c=constant 0.00 298.21 7.54 –0.73% 0.00 53.27 1.16 –0.31% DB b=constant 28.50 27.64 9.33 –0.87% 9.53 3.84 3.11 –0.35% RS r=constant 17.19 0.00 6.86 –0.84% 5.99 0.00 2.22 –0.34% ES RS+reference age structure 14.61 0.00 5.84 –0.85% 5.90 0.00 2.18 –0.33% No. 2: life expectancy increases, threshold ages (αβ) do not vary DC c=constant 0.00 891.77 0.17 0.14% 0.00 199.62 0.70 0.36% DB b=constant 106.33 110.67 6.15 0.97% 43.85 18.70 2.63 0.63% RS r=constant 60.96 0.00 3.91 0.77% 26.41 0.00 2.07 0.57% ES RS+reference age structure 60.09 0.00 3.60 0.75% 26.06 0.00 1.92 0.56% No. 2’: life expectancy increases, age at pension (β) adapts DC c=constant 0.00 32.95 0.90 0.32% 0.00 2.58 0.30 0.233% DB b=constant 2.18 1.91 0.99 0.42% 0.35 0.13 0.30 0.232% RS r=constant 1.42 0.00 0.93 0.40% 0.23 0.00 0.29 0.232% ES RS+reference age structure 1.28 0.00 0.84 0.39% 0.21 0.00 0.26 0.226% No. 2’’: life expectancy increases, both threshold ages (αβ) adapt DC c=constant 0.00 15.26 0.88 0.268% 0.00 2.50 0.13 0.07% DB b=constant 0.88 0.74 0.39 0.261% 0.34 0.13 0.05 0.07% RS r=constant 0.59 0.00 0.43 0.263% 0.22 0.00 0.06 0.07% ES RS+reference age structure 0.55 0.00 0.38 0.257% 0.20 0.00 0.05 0.06% Note: for all indicators zero is the best possible outcome. (a) Equilibrium payroll contribution rate (variance over 300 years, ×10,000). (b) Ratio between pensions and wages (variance over 300 years, ×10,000). (c) Difference between benefits and contributions, relative to 1 year of pay (variance over 200 cohorts, ×10,000). (d) Difference between benefits and contributions, relative to 1 year of pay (average over 200 cohorts). (e) Pensions start at about 75% of salaries. (f) Pensions start at about 50% of salaries; child benefits at 25%.

Synthetic results of various scenarios, with and without child benefits

105The main results may be summarized in relatively few words:

• In all the scenarios presented here (and indeed, in all the scenarios I ever tried) the inclusion of child benefits improves the performance of every PAYG transfer system in all possible respects. Every PAYG transfer system with child benefits is characterized by lower variability in payroll rates c, in the relative incomes of the three age classes r, and in generational equity D.
• The inclusion of child benefits, however, has only a limited impact on the ranking of the systems.
• In all cases, Defined Contribution seems to introduce too much variability in the relative income of the elderly and the young relative to that of the adults, especially with fixed threshold ages (scenario No. 2), and without child benefits. This leaves Equitable and Stable as the undisputedly preferable PAYG transfer system in all demographic scenarios and in all possible respects.
• All systems perform better if threshold ages vary, especially when the variation regards both α and β, but the ES version is constantly the best (and, incidentally, it is the only one for which the variation of α and β is internally consistent, because it derives from the use of the reference population).
• One shortcoming of the Equitable and Stable version, though, is that it requires flexibility in the reserve capital. Let us consider scenario No. 1: there is almost a century of below-replacement fertility, with no compensating immigration; the population eventually declines to about two thirds of its original size; and there are generous pension transfers, but no child benefits. During the period when the actual reserve level is lower than the reference level (equivalent to one year of pension benefits), the difference between the two drops to – 280% of the annual pension benefits in the worst year; the actual reserve K then corresponds to – 180% of the yearly pension benefits, which represents the overall debt of the transfer system — something that may be in the neighbourhood of (–)27% of the GDP of the time (Figure 8). This, in my opinion, is not unacceptable, because the scenario is extreme. Even in the worst year, contributions cover at least 93% of transfer benefits [25] and, most importantly, the situation is by definition temporary. The intrinsic characteristics of the system ensure that imbalances will compensate (and debts will be repaid) in the long run, which cannot be said of most existing PAYG transfer systems. However, if such a debt is judged excessive, several measures can be taken to limit it: for instance, lower pension/wage ratios; the inclusion of child benefits; the predefinition of an accepted range of variation for the reserve capital (for instance: it shall never become negative), although, in this case, the equilibrium contribution rate c will have to increase more than indicated in these simulations, thereby reducing the advantage of ES over RS.

Figure 8

Under scenario No. 1, difference between the actual and the reference level of: (a) the reserve capital (in % of one year of pension benefits) and (b) the payroll contribution rate (in percentage points)

Under scenario No. 1, difference between the actual and the reference level of: (a) the reserve capital (in % of one year of pension benefits) and (b) the payroll contribution rate (in percentage points)

Conclusions

106Aging is probably a predicament (MacKellar, 2000), but pension systems are a problem, and one that aging aggravates. The problem is complex, especially in those very frequent instances where specific arrangements exist for different industries; where expectations have been created and rights conferred on faulty bases; and where changes in the rate of economic growth, or in the pace of demographic change, have been unexpectedly rapid.

107In this article I show that the theoretical versions of PAYG proposed so far in the literature, despite their high level of abstraction (leading in fact to practical inapplicability), when confronted with a series of possible changes in the economic or the demographic sphere, cannot keep their most basic parameters constant: contribution rate, intergenerational equity, and relative income of the young, the adults and the elderly. However, I also contend that the basic principles of PAYG may be preserved and revived in a new version, called PAYG-ES, Equitable and Stable. The proposed system uses the idea of Risk Sharing, where a constant ratio r is in all cases maintained between the incomes of pensioners and workers, so that economic shocks are evenly spread over all segments of the population. It expands it, however, in that it also tends to spread the effects of demographic shocks evenly over the various cohorts. In order to do so, it must introduce some elasticity in its accounts. Contributions and benefits must balance only in the long run, so that the reserves of the system oscillate around their long-run equilibrium value, which may or may not be equal to zero.

108On a higher level of abstraction, one could say that what is really difficult is to keep some parameters constant over time when everything else is subject to change. Without this element of stability, however, the very rationale for the existence of a pension system vanishes, and so do popular confidence and support. PAYG-ES solves the problem by keeping relative indicators constant, both on the economic side (the ratio r between average transfers and average net labour incomes) and on the demographic side (and this is the novelty of the present proposal). In the latter case, one could push the argument further and tend to fix, over time, the span of life an average individual spends in the three basic demographic phases: youth, adulthood, and old age. This ideal can be approached by letting conventional threshold ages move. This alone, together with a constant r (ratio between benefits and net wages) guarantees that everything else in the PAYG transfer system will remain practically constant.

109ES solves the technical problems posed for PAYG transfer arrangements by varying economic and demographic circumstances. But this is only a small step. Convincing the general public that only relative measures can be kept constant over time, that each worker’s pension income in t years will be a fraction of the salaries that will prevail by then, and that pension benefits will start at an age β that keeps moving from year to year (depending on survival conditions) is probably going to take a long, long time.