2.1 Bismarckian contributory pension scheme

The problem of policymakers under such a regime is to equalize the actuarial fairness ratio (afr) of lifetime pension benefits to lifetime pension contributions.Footnote ⁹ Abstracting from education length differences, career breaks, wage growth, demographic changes or discount and indexation rates, and considering that retirement age is uniform, that actuarial fairness ratio writes,

(1)$$\eqalign{& afr_{i, j}( {{\boldsymbol ra}} ) = S_{i, j}( {{\boldsymbol ra}} ) \displaystyle{{( {l_{i, j}-{\boldsymbol ra}} ) \;\delta \;w_{i, j}} \over {{\boldsymbol ra} \;\eta \;w_{i, j}}} = S_{i, j}( {{\boldsymbol ra}} ) \displaystyle{{( {l_{i, j}-{\boldsymbol ra}} ) } \over {{\boldsymbol ra}}}\theta , \;\cr & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\rm where}\quad \theta \equiv \displaystyle{\delta \over \eta }.} $$

Centrally defined reference retirement age is ra and lifetime benefits are equal to the time spent in retirement times the annuity δw _i,j where δ is the replacement rate and w _i,j is the individual level of earnings. Note that people can die before reaching retirement age. So we have the dummy variable S _i,j(ra) = 0 if l _i,j ≤ ra and S _i,j(ra) = 1 otherwise. By definition of a Bismarckian system, lifetime contribution corresponds to the duration of the career (here, the retirement age) that multiplies the annual contributions at a rate η. We define θ ≡ (δ/η) as the (uniform) rate of replacement for each euro of contribution.Footnote ¹⁰

An alternative way of expressing reference retirement age is ra ≡ α l where l is the unique reference longevity and 0 ≤ α ≤ 1 is the share of life the reference person is supposed to spend working. Reference retirement age and reference longevity are thus isomorphic formulations in our setting.

(2)$$afr_{i, j}( {{\boldsymbol ra}} ) = S_{i, j}( {{\boldsymbol ra}} ) \displaystyle{{( {l_{i, j}-\alpha {\boldsymbol l}} ) } \over {\alpha {\boldsymbol l}}}\theta .$$

The equalization of lifetime ratios across individuals ($\forall i, \;j\;\colon \;afr_{i, j} = afr$) can be achieved via the full individualization of the retirement age, or, equivalently, via the use of each individual's longevity l _i,j when defining the retirement age.Footnote ¹¹ For any value of α, if the retirement age is fully individualized (i.e., $ra_{i, j}\equiv \alpha \; l _{i, j}, \;\;\forall i, \;j$), we verify

(3)$$afr = \displaystyle{{( {1-\alpha } ) } \over \alpha }\theta .$$

Note that, by definition, if the retirement age is fully individualized, and if α < 1,Footnote ¹² S _i,j() is always equal to 1. In other words, nobody dies before reaching his fully individualized retirement age.

Using a uniform reference retirement age (i.e.,ra ≡ α l) introduces a gap between the fair ratio and the actual one

(4)$$gap_{i, j}( {{\boldsymbol ra}} ) \equiv afr-afr_{i, j}( {{\boldsymbol ra}} ) = \theta \left[{\displaystyle{{1-\alpha } \over \alpha }-S_{i, j}( {{\boldsymbol ra}} ) \displaystyle{{l_{i, j}-{\boldsymbol ra}} \over {{\boldsymbol ra}}}} \right]$$

or equivalently when the reference retirement age is differentiated by SES category (i.e.,ra_j ≡ α l_j), the (presumably smaller) gap is

(5)$$gap_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) \equiv afr-afr_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) = \theta \left[{\displaystyle{{1-\alpha } \over \alpha }-S_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) \displaystyle{{l_{i, j}-{\boldsymbol r}{\boldsymbol a}\,_j} \over {{\boldsymbol r}{\boldsymbol a}\,_j}}} \right].$$

A graphical representation of what happens under uniform vs. differentiated retirement age appears on Figure 1. The lower part of the graph represents the distribution of realized longevity, while the upper part depicts the (lifetime) actuarial fairness ratio (afr _ij). The first-best situation amounts to ensuring that every person gets the same ratio corresponding to the horizontal dashed line. The achieved degree of actuarial fairness corresponds to the solid and doted rising lines. Under uniform retirement age (solid rising line), only those whose longevity coincides with reference longevity (l) get the first best. All the others get less or more than what lifetime equality commands. Those who reach that retirement age but die before the reference longevity l get less than what they should. And those whose longevity exceeds l get too much. Under differentiated retirement age (two different retirement ages are depicted in Figure 1 and correspond to the dotted rising lines), the number of people who get exactly what equity commands (i.e., those whose longevity corresponds to the intersection between the dashed and doted lines) rises. Also the integral of the distances between the doted lines and the dashed horizontal line become lower, reflecting a reduction of the inequity gaps in Equation (3). However, situations synonymous with afr _i,j ‘undershooting’ or ‘overshooting’ still abound.

Figure 1. Bismarckian pension under uniform vs. diff. ret age.

Note that Figure 1 also reveals what would happen if, instead of differentiating the age or retirement, policy makers were to differentiate by SES either the contribution rate η or the replacement rate δ. That would amount to differentiating θ ≡ (δ/η), thus the slope of solid line, while keeping the retirement age unchanged (i.e., uniform). Instead of relying on ra_j to (try) to reflect realized longevity differences (i.e., any value on the horizontal axis), the planner would use θ _j. More specifically, while here we explore what happens when ra_j becomes a positive function of expected longevity, in an alternative (but fairly equivalent approach) one could examine what happens were θ _j to become a negative function of expected longevity. And echoing the point we raised in the introduction, it is likely that the outcome of that type of differentiation would not fundamentally change. That is because, at its core, the problem of pension differentiation is an imperfect information problem. How to fully individualize (and thus generate an almost infinite number of pension regimes), using only a limited number of values for the pension parameters (ra, θ, δ, η) mirroring the equally limited number of expected longevities (one for each SES j = 1…k) the policy maker knows? And combining several parameters of differentiation (i.e., simultaneously differentiating ra and θ for instance) would not bring much of change either, because the information used to implement that ‘hybrid’ approach of differentiation would still boil down to exploiting a mere correlation between SES category k and realized longevity.

Following Baurin (Reference Baurin2021), the overall reduction in the propensity of retirement policy to deviate from the full individualized first best can be expressed as an indexFootnote ¹³ ∈[0, 1] where the numerator aggregates the (absolute) values of the individual equity gaps under retirement age differentiation by SES j = 1…k, and the denominator does the same when there is no differentiation. Note that the index being a ratio, constant θ cancels out. In the numerical simulations underpinning results of Section 4.2 the differentiated reference retirement ages/longevities correspond to averages by SES category j.

(6)$$\eqalign{& I^{Bismarck}\in [ {0, \;1} ] = \displaystyle{{\sum _{\,j = 1}^k \sum _{i = 1}^{n_j} \theta \vert {( {( {1-\alpha } ) /\alpha } ) -S_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) ( {( {l_{i, j}-{\boldsymbol r}{\boldsymbol a}_j} ) /{\boldsymbol r}{\boldsymbol a}_j} ) } \vert } \over {\sum _{\,j = 1}^k \sum _{i = 1}^{n_j} \theta \vert {( {( {1-\alpha } ) /\alpha } ) -S_{i, j}( {{\boldsymbol ra}} ) ( {( {l_{i, j}-{\boldsymbol ra}} ) /{\boldsymbol ra}} ) } \vert }} \cr & \quad \quad \quad {\rm with}\quad {\boldsymbol ra}\equiv \alpha {\boldsymbol l} ; \;{\boldsymbol r}{\boldsymbol a}_j\equiv \alpha {\boldsymbol l}_j.} $$

Note finally that retirement age differentiation can generate budgetary savings (thus a surplus) in comparison with uniform retirement. This is because the overall (or average) time spent in retirement may go down. What is more, with a Bismarckian regime, the money not spent on those who live longer is likely to outweigh the cost of financing early retirement to the benefit of the short-lived individuals. In what follows we will assume that the planner absorbs the net budgetary surplus generated by the move from uniform to differentiated retirement age. An alternative is to enforce cohortal budget equivalence, i.e., to use the surplus to increase the replacement rate θ in the numerator of our (in)equity index Equation (6). We explore that alternative in detail in Appendix A2. But the key result is that the equity gains achievable via retirement age differentiation are likely to be lower. This is because the time component of the equity gaps (Equation (5)) – that inevitably still exists under retirement age differentiation – is priced at a higher rate.Footnote ¹⁴ Say differently, in the equity indexFootnote ¹⁵, the time component of the gaps in the numerator is now multiplied by a higher θ than in the denominator.Footnote ¹⁶ Thus the equity gains achieved in terms of years spent in retirement are eroded by the (remaining) year gaps weighing more in monetary terms.

2.2 Beveridgian pension scheme

By definition, a Beveridgian pension system would rather aim at equalizing lifetime pension benefits (B _i,j).

(7)$$B_{i, j}( {{\boldsymbol ra}} ) \equiv b \;S_{i, j}( {{\boldsymbol ra}} ) ( {l_{i, j}-{\boldsymbol ra}} ) .$$

In the above expression, b is the standard uniform annual/monthly pension (which is independent of earning w _i,j and contributions) that multiplies the time spent in retirement. Again, people can die before reaching retirement age. So we have the dummy variable S _i,j(ra) = 0 if l _i,j ≤ ra and S _i,j(ra) = 1 otherwise. Equalization of lifetime benefits ($\forall i, \;j\;\colon \;B_{i, j} = B$) can only be achieved via full-individualization of retirement age, or the corresponding reference longevity (ra _ij ≡ l _i,j − κ), where κ is the reference number of years spent in retirement.

(8)$$B\equiv b( {l_{i, j}-ra_{i, j}} ) = b ( {l_{i, j}-l_{i, j} + \kappa } ) = b \kappa .$$

Note again that S _i,j() = 1 if there is perfect individualization and if κ > 0. We logically assume κ is the time spent in retirement by the person whose longevity is equal to the reference longevity (l) under a uniform retirement policy (κ = l − ra = (1 − α)l).Footnote ¹⁷

Key is that the use of a uniform retirement age/longevity reference (ra) leads to lifetime benefit gaps

(9)$$gap_{i, j}( {{\boldsymbol ra}} ) \equiv B-B_{i, j}( {{\boldsymbol ra}} ) = b[ {\kappa -S_{i, j}( {{\boldsymbol ra}} ) ( {l_{i, j}-{\boldsymbol ra}} ) } ] $$

or equivalently when the reference retirement age is differentiated by SES category (ra_j), the (presumably smaller) gap is

(10)$$gap_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) \equiv B-B_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) = b[ {\kappa -S_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) ( {l_{i, j}-{\boldsymbol r}{\boldsymbol a}_j} ) } ] .$$

Again, we can produce a graphical representation of what is at stake under uniform vs. differentiated retire. It is to be found in Appendix A1. In more analytical terms, the reduction in the overall propensity of retirement policy to deviate from the fully individualized first best can be expressed as the following index, where the constants b cancel out

(11)$$\eqalign{& I^{Beveridge}\in [ {0, \;1} ] = \displaystyle{{\sum _{\,j = 1}^k \sum _{i = 1}^{n_j} b\vert {\kappa -S_{i, j}( {{\boldsymbol r}{\boldsymbol a}_j} ) ( {l_{i, j}-{\boldsymbol r}{\boldsymbol a}_j} ) } \vert } \over {\sum _{\,j = 1}^k \sum _{i = 1}^{n_j} b\vert {\kappa -S_{i, j}( {{\boldsymbol ra}} ) ( {l_{i, j}-{\boldsymbol ra}} ) } \vert }} \cr & \quad \quad \quad {\rm with}\quad {\boldsymbol ra}\equiv {\boldsymbol l}-\kappa = \alpha {\boldsymbol l} \cr & \quad \quad \quad \quad \quad \;\;{\boldsymbol r}{\boldsymbol a}_j\equiv {\boldsymbol l}_j-\kappa .} $$

Note again that, so far, we have assumed that the planner absorbs any net surplus/deficit generated by the move from uniform to differentiated retirement. The alternative policy would be to raise(reduce) b prorata in the numerator of the (in)equity index equation (11), and enforce cohortal budget equivalence. Under a Beveridgian regime, retirement age differentiation is likely to generate a surplus(deficit) if the overall time spent in retirement goes down(up). Not absorbing that surplus(deficit) means that policy makers raise(reduce) b. This again means that year equity gaps – that still exist under retirement age differentiation – are ‘priced’ differently. Thus the equity gains achieved in terms of years spent in retirement are eroded(amplified) by the (remaining) year gaps weighted differently. This translates into different equity gains than those reported in result Section 4.2. We analyse this cohortal-budget-equivalence alternative in detail in Appendix A2. Results, in short, suggest that under a Beveridgian regime retirement age differentiation could generate a (small) surplus, allowing for a (slightly) higher b.Footnote ¹⁸ Thus, again, our main results in Section 4.2 (derived without imposing cohort budgetary equivalence) should be seen as an upper bound of the achievable equity gains.

To sum up, minimizing both the Bismarckian and Beveridgian inequity indices (Equations (6) and (11)) depends on the social planner being able to match the full distribution of longevity across individuals l _i,j, i.e., the different values of the horizontal axis forming the longevity distribution on the lower part of Figures 1 and A1. If she can only go for tagging (Akerlof, Reference Akerlof1978), i.e., use j = 1…k < N proxies l_j that are simply correlated to realized longevity l _i,j to differentiate treatment, and if unaccounted longevity differences are important and matter, then both policies should translate into values of our indices that are relatively close to 1. In Section 4.2, we will show simulation results illustrating this, using US data on longevity heterogeneity.

4.1 Data construction

The data used to analyse partial de-annuitization vs. retirement age differentiation are from the US. They consist of a simulation of the full distribution of longevity across a cohort of N individuals with different socio-demographic background (l _i,j; i = 1…n _j; j = 1…k) who have survived until prime age. As its core, the simulation rests on the (unavailable to us) mortality rates assembled by Chetty et al. (Reference Chetty, Stepner, Abraham, Lin, Scuderi, Turner, Bergeron and Cutler2016).

The underlying micro data comprise a sample of 1.4 billion observations from anonymized tax records, covering the years 1999–2014. Mortality data start at age 40 and are available either by gender, US state of residence and income quartile; or by gender and income percentile. We retain the gender × income version of the Chetty data. More precisely, we use the (publicly available) parameters of Gompertz functions they provide for each gender × income cell j, alongside the number of people in the US population belonging to these cells (n _j, j = i…k).Footnote ²² The parameters of the Gompertz function capture the expected differences of mortality between categories j. Whereas the predicted values delivered by each Gompertz function j provides the ‘within’ category distribution of mortality rates for each (potential) age of death.Footnote ²³ These mortality rates by age can then be multiplied by the number of individuals forming each cell j to know the number of individuals whose longevity is equal to 40, 41, …, 120.

In Figure 2 we display the Gompertz-generated distribution of longevity for men belonging to the lowest income percentile of the US male population vs. the equivalent distribution for women forming the highest income percentile of the female population. Expected/average longevity (corresponding to the dashed vertical lines) varies between the two groups: the expected longevity gap is larger than 16 years (88.7 vs. 71.9 years). Still, the solid curves capturing the full distribution of realized longevities within each group reveal that quite many women forming the upper income percentile die before the average age of 88.7 (and would deserve to retire early), while some for low-income men live beyond that age (and to not deserve to early retire).Footnote ²⁴

Figure 2. USA-[M]ale lowest income percentile and [F]emale highest percentile; expected/average longevity difference between the two groups vs. full distribution of realized longevities within these groups. Source: Chetty et al. (Reference Chetty, Stepner, Abraham, Lin, Scuderi, Turner, Bergeron and Cutler2016), based on mortality data beyond the age of 40.

4.2 Numerical simulation results

We start by assuming that our reference longevity (i.e., l) underpinning the uniform retirement policy is the average longevity in the Chetty data. Without loss of generality, and for the sake of clarity, the value of the share of life spent in employment α is chosen so that the corresponding uniform retirement age is equal to 65. Here after, the results for the retirement age policy are centred on that age of 65 (ra = 65).

Given the Chetty data documenting longevity differences only past the age of 40, the minimal retirement age under retirement age differentiation is 40. And with de-annuitization, the lump-sum payment is also assumed to intervene at 40. By definition, it consists of paying a lump-sum to all whose longevity is considered, and that is only feasible before the first individual dies, thus here at 40.Footnote ²⁵ Also, given the data used, the two policies examined here (equally) ignore the problem of the very short-lived, i.e., those who don't survive up to the age of 40. Note that the hypothetical use of a data set documenting longevity only from the age of 50 or 55 would simply inflate the number of individuals who are de facto not compensated, but without affecting our key analytical results markedly. Remember that our prime interest is to assess de-annuitization needed to match whatever can be achieved via retirement age differentiation, and by that we also mean the extent to which that policy ignores some short-lived. Remember also that the major factor driving our results is the (in)ability of the planner to match the full distribution of longevity using a few proxies. Whether it is the post 40, 50 or 55 distribution does not matter much.

Our results consist of the simulated values of the gains generated by retirement age differentiation in terms of inequity gap indexes exposed in Equations (15) and (19), one for the Bismarckian system and one for the Beveridgian one. In both cases we estimate numerically the gains achieved by resorting to 200 different reference retirement ages/longevities (i.e., j = 1…200, corresponding to 2 genders × 100 income percentiles). The differentiated retirement ages we use are visible in Figures 3 and 4. They correspond to each of our SES categoryj's average longevity (multiplied by α with Bismarck, or minus κ with Beveridge).

Figure 3. Bismarckian differentiated pension ages.

Figure 4. Beveridgian differentiated pension ages.

The key result is the one about the equity gains achieved via differentiation. It is reported on top of Table 1. We see values of 0.963 and 0.964 for (respectively) the Bismarckian and Beveridgian schemes. These are our best estimates of how much inequity indices can be brought down via retirement age differentiation. Remember that these indices are equal to (θ ^′/θ); (b ^′/b) as stated in Equations (15) and (19). Thus 1 − 0.963 or 1 − 0.964 tell us about the % reduction of pension instalmentFootnote ²⁶ required to generate an equivalent gain in terms of pension fairness. We see it is relatively limited: less than 4% points of reduction of θ (Bismarck) or of the basic pension b (Beveridge) would be enough to generate the same equity gains as extensive retirement age differentiation across 200 gender × income categories; with retirement ages ranging from 56.5 to 69.7 (Bismarckian) or 54.2 to 71 (Beveridge) displayed in the lower part of Table 1. Also, one can get an idea of what the above results imply for the importance of the lump-sum paid upfront. Assuming that the (by 4%) reduced pension instalments are paid from the age of 65 to retirees with a 20-year life expectancy, following Equation (17), in a Beveridgian system, the lump-sum would represent approximately 0.004 × 20 × 12 = 9.6 month-equivalents of the current pension instalment. The calculus is less straightforward for Bismarckian system but results are similar.

Table 1. Numeric results: values of the equity gap index^a achieved via retirement age differentiation and the degree of de-annuitization needed to match these gains [without cohortal budget equivalence^b]

^a The simulated values of equity gap indexes exposed in Equations (15) and (19). The propensity of these indices to be below 1 tells us simultaneously about (i) the equity gains that can be achieved via retirement age differentiation and (ii) the degree of de-annuitization required to generate an equivalent gain in terms of pension fairness. We report here the gains achieved by resorting to 200 different reference retirement ages/longevities (i.e., j = 1…200, corresponding to 2 genders × 100 income percentiles). The differentiated retirement ages we use are those visible in Figures 3 and 4. In the lower part of this table we report only a sample of them.

^b Meaning that we assume that the planner ‘absorbs’ the net budgetary surplus(deficit) generated by the move from uniform to differentiated retirement age. Thus the move does not translate into higher(lower) value of θ and b to guarantee that the same total budget is spent on the cohort of pensioners.

^c I ^Bismarck ∈ [0, 1] as defined by Equation (15).

^d I ^Beveridge ∈ [0, 1] as defined by Equation (19).

4.3 Winners, losers

In this section, we explore the question of the support/opposition that partial de-annuitization might encounter. Our approach is quite simple. We consider that the support for de-annuitization depends on the share of short-lived individuals who get a higher lifetime actuarial fairness ratio (or benefits with a Beveridgian system). In Figure 5, it corresponds to all the individuals whose longevity is inferior to l*.

Figure 5. Partial de-annuitization: winners vs. losers.

Algebraically and numerically, one can show that the ‘indifferent’ pensioner is not influenced by the degree of de-annuitization. The point of indifference in Figure 5 and the corresponding longevity l* are fixed. Whatever the intensity of de-annuitization, there is always the same share of pensioners who gain from de-annuitization vs. uniform retirement age. Proof of this is in Appendix A4.Footnote ²⁷

This said, it is important to stress that the value l* varies with the reference retirement age (ra). In Figure 6 we report simulation results on the share of (relatively short-lived) pensioners who would gain from de-annuitization. It is clearly a rising function of the reference retirement age. In policy terms this means that the support for de-annuitization is likely to rise the more policy makers increase the (unique) retirement age.

Figure 6. Partial de-annuitization: share of winners and ref. retirement age.