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It is generally taken for granted that the rules of the Old-Age and Survivors Insurance (OASI) offered by the U.S. social security system--unlike many defined-benefit (DB) schemes, especially those based on the last earnings rule--achieve a progressive redistribution, given that lower accrual rates are applied to higher income brackets. In fact, debate on the OASI redistributions has generally focused on the interaction of its basic progressive design with other, possibly regressive and more or less transparent, sources of redistribution embedded within the system. Liebman (2002) has shown that most of the redistribution generated by OASI comes from factors other than lifetime income and that the marked dispersion of the individual internal rates of return (IRRs) for a given lifetime income can be attributed to differences in life expectancies, family status, and career length. Similar results have been reached, by Caldwell etal. (1999), Gustman and Steinmeier (2001), and Coronado et al. (2002). The underlying idea of this strand of the literature is that, once the correlation between income and life expectancy is taken into account, the intra-cohort redistribution generated by the U.S. pension rules is much less progressive than its defenders seem to believe.

The aim of this paper is to draw some policy implications by exploring the extent to which the seemingly progressive pension formula of the U.S. program is actually able to avoid the regressive redistributions that, typically, arise within DB schemes according to differences in career patterns. (1) In order to do so, we limit our analysis to the Old-Age Insurance (OAI) of the entire U.S. social security system, thus disregarding the redistributions generated by its Survivor (SI) and Disability (DI) programs. The distributions in favor of single-earner couples, associated with spouse benefits, will also be disregarded.

The benchmark against which we measure possible deviations of the OAI scheme from its aim to redistribute from high- to low-income earners is the award formula and adjustment rule of the new variety of old-age pension system, started in the early 1990s in Italy and in Sweden, that "implants" the non-redistributive formula of defined-contribution (DC) systems into pay-as-you-go (PAYG) financing. Since then, other countries have followed suit and important institutions, like the World Bank, are sponsoring the adoption of what is now known as the "Notional" or "Non-financial" Defined Contribution (NDC) scheme (see Holzmann and Palmer 2006; Holzmann, Palmer, and Robalino 2012). (2) If properly managed, the NDC scheme can ensure a substantial degree of financial stability under varying economic and demographic conditions, for any given contribution rate. A second, crucial, property of the scheme is its transparency. It is, in particular, the actual crediting of a uniform rate of return on all individual notional accounts that allows the NDC scheme to eliminate all hidden, and mostly regressive, redistributions typical of the DB, earnings-related schemes. We will, for our purpose, use the term "fairness" to refer to this property of the NDC scheme to reward individual compulsory savings evenly, though we are perfectly aware that it should not be burdened with any reference to equity. Moreover, especially when, as is the case in Sweden, individuals are informed year by year of the development of their notional accounts and the rate of return earned on their pension wealth, the new scheme allows social security contributions to be perceived as compulsory savings rather than as taxes, as it is in funded DC systems. (3)

This is why the NDC scheme should be considered a further policy option in the debate about U.S. public pensions that has, so far, focused on whether the ongoing worsening of the demographic scenario should be addressed by shifting from PAYG financing to a funded, private retirement system (e.g., Feldstein 1998), as an alternative to parametric reforms. Advocates of funding generally argue that PAYG schemes:

* tend to remunerate contributions with a lower rate of return than funded programs, and discourage national savings;

* unfairly redistribute among participants and imply distortions in the labor market since workers tend to perceive social security contributions as taxes (4);

* force individuals to run the political risk implied by policy makers facing the alternative between levying ever-increasing taxes on "young" cohorts or curbing the "once-promised" pension benefits to keep current expenditure in line with the current contribution revenue.

In turn, authors who are skeptical about the presumed curative power of funding (e.g., Barr and Diamond 2006; Orszag and Stiglitz 2001) argue that shifting towards a funded private scheme "would produce few if any demonstrable benefits" (Aaron 2011, 387). Moreover, it should be taken into account that such a transition

* does not remunerate contributions with higher rates of returns if we take into account administrative costs, risk premiums, and the costs of transition to be distributed among generations;

* exposes all retirees to the risk of heavy losses of their lifetime savings;

* cannot curb the increasing dependency ratio, that is the simple fact that (as a whole) the product generated by active workers must be sufficient to match the consumption of an increasing number of retirees. (5)

The implications of the analysis presented in this paper are that most of the problems emphasized by advocates of funding could be addressed also without embarking in the complex transition from a PAYG to a funded system.

The analytical structure and the simulations replicate the methodology used in Nistico (2013a) and Nistico and Bevilacqua (2013) to show how NDC pensions and replacement rates respond to a variety of income and retirement patterns as well as to changes in the economic and demographic scenarios, while ensuring uniformity of IRRs and financial sustainability. The focus on benefit-tax ratios of some predefined, hypothetical earnings histories enables assessment of the redistributions that the OAI program is likely to operate for some general career profiles distinguished according to retirement age, initial wage, individual/average wage ratio, earnings growth rates, and their distribution within the working life. Although we anchor our career patterns to the specific earnings histories generated by the Office of the Chief Actuary starting from actual records of the Social Security Administration (Clingman and Kyle 2016), it is worth emphasizing that our conclusions cannot be considered representative of the actual redistributive effects of the OAI program on the U.S. insured population as a whole. In fact, assessing the actual impact of alternative policy regimes in terms of poverty reduction targets or financial stability entails running complex microsimulations (e.g., Holmer 2009) on a large enough sample of insured workers, which is well beyond the scope of the present work.

The paper is structured as follows. Section II summarizes the main theoretical features of NDC schemes and their fundamental difference with respect to traditional defined-benefit arrangements. In particular, it is pointed out that the very raison d'etre of NDC schemes lies in their capability to offset the inherent unfairness of the traditional DB programs, wherein implicit individual returns on contributions tend to differ greatly with differences in career patterns. On the other hand, NDC schemes can also be designed to achieve long-term financial stability for any freely chosen contribution rate. Section III analyses the essential properties of the OAI award formula and illustrates the IRRs that it offers to some typical workers. Section IV compares the achievements of the U.S. public retirement plan with those of a hypothetical NDC scheme. Section V concludes by suggesting that a shift towards a neutral and transparent NDC arrangement--with a higher contribution rate capable of ensuring both the sustainability of the promises matured within the old system as well as the adequacy and the financial sustainability of future pension provisions--should be included among the reform options for the U.S. pension system.


In (hypothetical) DB schemes that maintain pure PAYG financing through time, that is that year by year levy a tax rate equal to the ratio between pension expenditure and wage bill, paid-in contributions tend to be implicitly remunerated with a rate of return equal to the growth rate of the wage bill itself. We should not forget that this well-known property of PAYG schemes, known as the Samuelson-Aaron theorem, holds only in a semi-steady state in which population growth as well as mortality rates are constant through time. Moreover, the return referred to in the theorem is a "generational" one, that is it implicitly rewards the indistinct aggregate of contributions paid by a cohort as a whole, while the "individual" returns on the contributions paid by each cohort member differ significantly with differences in career patterns. (6)

As in DC systems, the NDC scheme endows each member with an interest-bearing personal account credited with the contributions paid in the system. At retirement, the first pension benefit and the adjustment rule that drive its evolution through time can be determined so as to ensure the following deposit-exhaustion constraint, namely that the money that will be withdrawn according to ex ante life expectancy at retirement equals the money deposited including the interests computed at a yearly uniform rate for all individuals:

(1) [mathematical expression not reproducible]

where a denotes the (constant) contribution rate, [[w.sub.i] individual earnings in year i, n the duration of working life, p the first pension annuity, [[pi].sub.j] and [o.sub.j] the values taken on in year j by the interest rate credited to all personal accounts and the benefit-adjustment rate, respectively; m is life expectancy at retirement. The first member of Equation (1) is the worker's pension wealth at retirement, that is, the total value of contribution payments including returns matured, while the second is the present value at retirement of the "expected" pension benefits the worker will be paid during her m years of retirement. If the system's rules ensure that Equation (1) is satisfied for all individuals, in each year the system rewards the pension wealth of all individuals with the same rate of interest even if its liabilities are not "funded" (7); moreover, if the value of [pi] were constant through time, ex ante actuarial fairness, in the sense of uniformity of all longitudinal individual IRRs, would also obtain. By solving Equation (1) for p, it follows:

(2) [mathematical expression not reproducible]

where the numerator is the account balance at retirement.

As we know, the practice with the DC system is to calculate the first pension through division of the account balance (at retirement) by a number, which hereafter will be indicated as d, reflecting life expectancy (at retirement). Equation (2) makes it clear that if it matters to satisfy Equation (1) for all members according to their ex ante life expectancy, we should impose the following condition on the divisor:

(3) [mathematical expression not reproducible]

The right-hand side of Equation (3) is a function of 2 * m - 1 variables, m, [[sigma].sub.j] and [[pi].sub.j] (Vj), of which only the life expectancy (m) of the retiring worker's cohort can be computed at the beginning of year n + 1 on the basis of the mortality tables available at that time. Conversely, the other 2 * m - 2 variables--namely the values of future pensions' adjustment rates and rates of return to be credited on the retiree's account balance in the following m - 1 years--remain unknown unless they are pre-set regardless of the yet to be devised, fundamental economic indicators. (8) On the other hand, choosing a value of d for any given m (9) amounts to attributing an arbitrary, constant value to the "deviation rate," hereafter denoted as 5, between the rates [[pi].sub.j] and a, by imposing

(4) [mathematical expression not reproducible]

so Equation (3) can be written as:

(5) [mathematical expression not reproducible]

Equation (5) makes it clear that attributing an arbitrary value to 8 gets us round the problem of ignorance about the future values of it and [[sigma].sub.j] , so that we can calculate d at retirement. Such a rate is generally referred to as "technical" or "assumed" interest rate. However, attributing an arbitrary value to [delta] "presets" the ratio that, according to Equation (4), will have to be established each year between the rate of return and the rate of pension adjustment if the aim is to satisfy Equation (1). This is why any DC system that aspires to be free to anchor the rate of return (to be credited each year on workers', as well as on retirees' notional accounts) to some variable economic indicators in order to ensure sustainability, should adopt the following pension adjustment rule ensuring that Equation (4) is satisfied:

(6) [mathematical expression not reproducible]

Calculating the first annual benefit on the basis of formulas (2), (3), and (5), and adjusting it according to rule (6), ensures that each year j all virtual accounts of both workers and pensioners are evenly remunerated according to the value that it, takes in that year. In this narrow sense, DC schemes guarantee "fairness," in the sense of neutrality or absence of redistributions.

It is worth noting that in order to achieve fairness, as defined above, the NDC "rules" yield a somewhat marked differentiation of replacement rates according to individual career patterns. In particular, they tend to produce lower replacement rates for "convex" careers than they do for "flat" or concave careers. This is easily seen if we divide by the last earnings ([w.sub.n]) both sides of Equation (2), which, taking into account Equation (3), becomes:

(7) [mathematical expression not reproducible]

Expression (7) shows that, ceteris paribus, the replacement rate granted by the NDC rules is higher for the flat-career workers whose end-of-career earnings ([w.sub.n]) is "closer in value" to all previous values of [w.sub.i] earned during the active period and lower for the convex-career workers whose [w.sub.n] significantly exceeds the previous values of [w.sub.i]. As pointed out in Nistico and Bevilacqua (2013), in the hypothetical, "benchmark case" in which individual earnings growth coincides with the rate of return credited on the account balance, the expression of the replacement rate simplifies to the following linear function of the contribution rate:

(8) [mathematical expression not reproducible]

the multiplier being approximately equal to the ratio between the working and the retirement periods. Replacement rates will exceed or fall short of Equation (8) for convex and concave careers, respectively. Close examination of Equation (8) shows that individual replacement rates are highly sensitive to retirement age in that postponing retirement determines both an increase of n and a reduction of d due to lower life expectancy. On the other hand, NDC replacement rates are generally insensitive to changing economic scenarios, in that they tend to affect earnings and benefits in the same proportion.

The NDC scheme is also intended to ensure sustainability, that is to adjust automatically to economic and demographic shocks in accordance with the DC risk-allocation mechanism, that is without adjusting the contribution rate. It can be proved that, for any given contribution rate, sustainability is substantially guaranteed by setting the rate of return credited in each year j on all account balances equal to the growth of the system's resources according to the following equation:

(9) [mathematical expression not reproducible]

where a and [lambda] denote wages and employment growth rates, so that the right-hand side of Equation (9) is the growth rate of taxable earnings. In other words, it can be proved that a return equal to the growth rate of the contribution base keeps annual spending in line with annual contributions, whatever the contribution rate. Note that the theorem holds under the assumption that A, is constant while it is not necessary that [alpha] be also constant. (10) In Gronchi and Nistico (2008) proof is given for a four-overlapping-generations economy allowing for heterogeneous agents. On the basis of the theorem we can designate the growth rate of the contribution base as the "sustainable return." (11)

The circumstance that financial solvency of the NDC scheme does not depend on the contribution rate raises the issue of what is the appropriate level of the contribution rate to a compulsory, public pension scheme--an issue that touches upon a political dimension well beyond the scope of this work. On the other hand, the contribution rates for the existing PAYG plans in Europe, even for those that have adopted the NDC scheme, depend on the "history" of their DB liabilities. Such is the case of Italy, whose compulsory contribution rate to the public NDC pillar has reached the unparalleled value of 33%, exhausting all forms of retirement savings and leaving almost no room for occupational or private plans. By contrast, the contribution rate to the Swedish NDC pillar amounts to 16% while an additional contribution of 2.5% is due to a compulsory, public, funded pillar. The German contribution rate to the first PAYG pillar is now (2016) at 18.7% much in line with that of Sweden and of the other major European countries.


The OAI program stands out among the traditional DB schemes essentially for its progres-siveness. The pension is "earnings related" but the average of the annual earnings (12) taken into account to determine the pension benefit to which an individual is entitled ([p.sup. (E])) is split into three brackets and each bracket is associated with a specific accrual rate according to the formula:

(10) [mathematical expression not reproducible]

where [k.sub.t], [k.sub.2], and [k.sub.3] are the three accrual rates, AE is the average of the highest 35 annual earnings adjusted according to the wage index, and [B.sub.1] and [B.sub.2] ([B.sub.1] <[B.sub.2]) are the "bend points" defining the three earnings brackets. The alleged progres-siveness of the system derives from the values of the accrual rates. These are [k.sub.1] = 0.9, [k.sub.2] = 0.32, and [k.sub.3] =0.15, while the values of the two bend points, annually recomputed according to the average wage growth, were [B.sub.1] =$10,272 and [B.sub.2] = $61,884 in 2016.

The above mechanism tends to generate striking differences in replacement rates, similarly to those generated by the NDC rules. These can vary from around 90% for individuals whose average taxable earnings at retirement are lower than $10,272 to around 30% for those whose average taxable earnings at retirement amount to around $100,000 or more; the actual level of the replacement rate clearly depending not only on the individual's average earnings but also on her earnings' profile. Since 1975, social security benefits have increased in accordance with the Consumer Price Index for Urban Wage Earners and Clerical Workers (CPI-W). The cost-of-living adjustment is normally referred to as COLA.

A. Flexible Retirement Age

The actual pension benefit (p) awarded to any individual can differ markedly from the pension to which she is entitled depending on the possible deviation of the chosen retirement age from the "normal" one. More specifically,

[mathematical expression not reproducible]

where RA and RN denote the chosen and the normal retirement age, respectively, and

[mathematical expression not reproducible]

is the correction parameter for each year of difference between the normal and the chosen retirement age.

B. Financing Old-Age Benefits

To finance its expenditure, the Old-Age, Survivors, and Disability Insurance (OASDI) collects an overall contribution rate amounting to 12.4% of the payroll, the burden of which is equally shared between the employer and the employee. Since the year 2000, each employee and respective employer have paid 0.9% of the gross wage to finance the DI program while the rest of the tax, amounting to 5.3% (around 85% of the total OASDI tax rate), finances the OASI benefits, for a total of 10.6% of the payroll. As to this latter, there is no formal division between the financing quotas of the OAI and the SI programs. Spouses and survivors of U.S. workers who pay the OASDI overall contribution rate may be entitled to receive a life-long annuity. Survivor's and spouse programs are inherently redistributive, in that the personal choices of some workers could prevent them from being even potential beneficiaries of those benefits. Given that the aim of this paper is to assess the redistributive properties of the U.S. Old-Age program against differences in career patterns alone, we purged the OASI of the redistributions implied by the existence of spouse and survivor benefits by assuming that U.S. old-age benefits will be paid only to the entitled worker up to her death. Accordingly, in order to run the simulations reported in the next section, the contribution rate net of the survivor's cost has been calculated by deducting from the whole OASI rate (10.6%) the ratio between the amount of survivor benefits and the contribution base (the total earnings of insured workers). The resulting estimate of the contribution rate covering the costs of the OAI alone range from 6.4% in 1975 to 8.7% in 2015 (see the Appendix). (13)


Much of the political discussion on the U.S. social security focuses on the adequacy of such a rate, together with the value of the Trust fund, to finance current obligations, and on the possible impact on the U.S. economy of an increase in the payroll tax to ensure the solvency of the OASDI programs. Within a pure DB setting, raising the contribution rate implies, ceteris paribus, a fall in the actual and perceived IRRs to an undefined, and possibly negative level, for all those required to pay the new, higher, contribution rate for a period of time that for some of them may prove relatively long, in exchange for the same stream of pension benefits. Young active workers will experience even greater hardship if reform of the system also includes benefit cuts. On the other hand, raising the contribution rate while switching to the NDC scheme implies that the new higher contributions are transparently rewarded with the explicit, and sustainable, rate of return credited on all pension accounts, though such a rate will probably fall short of the sustainable one enjoyed, under wholly different historical circumstances, by past cohorts of retirees. The extent to which those higher contributions could also imply higher pensions under the NDC scheme will crucially depend on how the new, average yearly rate of return credited on the pension accounts compares with the IRR granted by the old rules.

In this section, we simulate the functioning of a hypothetical NDC scheme, conceived as operating in the United States from now on. The outcomes of the simulations as to pension amounts, replacement rates, and implicit rates of return on contributions will then be contrasted with those of the actual U.S. pension scheme for some typical career patterns within the same economic and demographic scenario as projected for the United States in the decades to come. More specifically, we use the projected average growth rate of U.S. taxable earnings as a measure of the sustainable rate of return in Equation (9) in line with the assumption to be found in the 2016 OASDI Trustees Report (Board of Trustees 2016, Table VI.G6). Given that from 2051 (i.e., from the year of retirement in our simulations), the projected average growth rate of taxable earnings is 1.6% in real terms, in order to calculate the vector of annuity divisors we attribute the same value of 1.6% to rate 8 in Equation (5) and we rely on the most recent data on U.S. life expectancy. The resulting vector of annuity divisors is contained in Table l. (14) Note that assuming 8 equal to the sustainable real rate of return ensures that, according to Equation (6), the pension adjustment rate coincides with the inflation rate, in accordance with the OAI rules. One final caveat is necessary. We compute the IRRs by assuming that the whole contribution rates to the OAI program (as estimated in the Appendix) are actually paid by workers, including those levied on the employer. This assumption is justified by the findings of the relevant literature on the actual incidence of payroll taxes. (15)

A. Internal Individual Returns, Replacement and Adjustment Rates

Let us start by presenting the simulation of pension benefits, replacement rates, and IRRs on lifetime contributions that the U.S. pension rules and NDC model offer to five typical workers distinguished by career path and retirement age, labeled as very low, low, medium, high, and very high. These are in fact the careers used in the 2016 Trustees Report (Board of Trustees 2016, Table V.C7) to simulate benefit amounts up to 2090 and in Clingman and Kyle (2016) who emphasize their not insignificant representativeness, in particular of the medium one, which is in line with actual earnings histories of around 30% of 2015 U.S. retirees. On the other hand, the low and very low careers correspond to the earnings histories of more than 40% of 2015 U.S. retirees, while the high and very high pattern mirrors actual earnings of around 30% of the present retirees. The graphs on the right-hand side in Table 2 plot the evolution of the five annual incomes expressed in 2016 U.S. dollars. It is moreover assumed that the individuals corresponding to the very low and low careers, born in 1994, retires at age 62, whereas the individuals corresponding to the remaining careers, born in 1989, retire at 67, the normal or full retirement age.

The results of the simulations are shown in Table 3. The upper side of the table refers to the U.S. rules. The basic social security benefit as results from formula (10) is called the primary insurance amount (PIA) and depends, essentially, on the value of the Average Indexed Monthly Earnings (AIME), which ranges from to 1,628$ for the very low career to 13,016$ for the very high career. The PIA is moreover corrected according to the COLA between the year of retirement and the first year of eligibility.

The bottom row of the upper part of Table 3 shows the real IRR of the five individuals. The lower part of the table shows the pension benefit, the replacement rate, and the real IRR of a hypothetical NDC scheme financed exclusively with the OAI tax rate as computed above, that is by subtracting the ratio between survivor expenditure and the contribution base from the OASI contribution rate. The account balances at retirement of the five hypothetical workers have been transformed into an annuity by using the specific divisors corresponding to the worker's retirement age (see Table 1).

The first, interesting result of this simulation is that the U.S. scheme rewards these five hypothetical careers with an IRR higher than that provided by the hypothetical, sustainable NDC scheme. Insofar as these results mirror the career patterns of the generality of U.S. workers, they show that the financial stability of the OAI program would, in fact, require a higher contribution rate, which would, in turn, reduce the IRRs of the insured workers. As for the redistributions, which constitute the main focus of this paper, the simulation shows that the U.S. rules penalize workers with high and very high careers by granting them a stream of pension annuities implying a real IRR of 3.0% and 2.2%, respectively as opposed to the 5.2% and 4.2% granted to workers with very low and low careers, respectively. This specific example seems therefore to confirm the progres-siveness of the U.S. rules and the neutrality of the NDC scheme, which rewards all careers with a uniform IRR of 1.8% in real terms. The latter, in fact, mirrors the system's long-run sustainable rate of return as surrogated by the geometric mean of the yearly growth rates of real taxable earnings from now on to 2090.

Some considerations are called for with respect to the adjustment rate after retirement. In fact, we should bear in mind that the values of the real IRRs in Table 3 reflect the assumption that the first pension benefit remains constant in real terms for the whole retirement period--as it is in the U.S. scheme that relies on COLA adjustment and as we have also "imposed" on the hypothetical NDC scheme by appropriately setting the value of the deviation rate (8 = 1.6%) in line with the projected real rate of growth of the total earnings subject to OAI after 2051. Actually, one of the properties of the NDC scheme is that the adjustment rate is pegged to the "annual" rate of return that will be credited during retirement through the value of 8 and that a fundamental trade-off exists between the value of the first pension (the replacement rate) and the adjustment rate (Gronchi and Nistico 2008, 137). In other words, lower values of [delta] would allow for a positive real adjustment without altering the "longitudinal" IRR. Therefore, lower replacement rates could be compensated with positive real adjustment rates by setting [delta] < 1.6%, at, say, 1%. Such a choice would bring about a more sensible NDC scheme, where the value of pensions does not decrease dramatically with age, relative to the average income of workers, which by contrast tends to increase in real terms over time. The impact of different values of the technical rate 8 on the annuity dynamics for a medium-career worker is summarized in Table 4. The first pension computed with [delta]=1.6% is 7% higher than the first pension computed with [delta]= 1.0%, which, in turn, is 6% higher than that computed with 8 = 0.5%, with replacement rates following suit. On the other hand, the graphs show how higher values of the deviation rate 8 cut down the dynamics of the real value of the pension annuity. The longitudinal IRR remains independent of the value chosen for [delta]. (16)

B. More on the Alleged Progressiveness of the U.S. Social Security Scheme

The figures in Table 4 show that replacement rates are, per se, a poor indicator of the generosity of a pension scheme while they corroborate our choice to use comparison of individual IRRs as a yardstick to assess the fairness of alternative schemes. It is worth recalling that the marked, regressive disparities in individual IRRs generated by the goal of the "last-earnings rule" to level the replacement rates for all career profiles constituted the very reason behind the wide political consensus on the Italian and Swedish NDC reforms of the 1990s. (17) In fact, as emerges clearly from Equation (7) above, granting uniform IRRs to all individuals generates disparities in replacement rates. Conversely, the same equation shows that leveling replacement rates for different career patterns generates a regressive redistribution in that higher individual IRRs would be granted to fast-rising careers and vice versa. Actually, the simulations of the previous section confirm that both the NDC scheme and the U.S. rules--with three different accrual rates, lower for higher income brackets--produce the desired effect of avoiding regressive redistributions. The U.S. rules seem to do more than that, since for the five types of career patterns simulated above they produce progressive redistributions, whereas "actuarial fairness," in the sense of uniformity of longitudinal individual IRRs, obtains for the NDC rules. In order to check how robust are the progressiveness of the U.S. rules and the fairness of the NDC scheme against different career patterns, in this section we provide three other simulations. The new simulations differ from those of the previous section in that the three careers, though sharing both "starting point" and retirement age (67), part company only a few years after the job starts; the annual income of the "medium" career coincides with that of the medium career reported in Table 2 while the other two exceed the annual growth rate of the medium one by 2% and 3%, respectively, though remaining within the limits of maximum taxable earnings. The dynamics of the three careers in real terms are represented in Figure 1.

The results of the simulations, summarized in Table 5, confirm both the progressiveness of the U.S. rules and the neutrality of the NDC scheme. However, the differences in individual IRRs, and hence the progressiveness, generated by the U.S. rules are not as striking as in the previous simulations. On the other hand, the new simulations confirm that both actuarial fairness and progressiveness require a marked differentiation of replacement rates in favor of flat careers.

A fundamental difference between the U.S.-DB rules and the NDC scheme is worth emphasizing. In fact, the latter grants all its members a real return on each dollar paid in the system, whose magnitude depends almost exclusively on the average economic and demographic conditions prevailing as long as the individual remains within the system. A small variance of the individual IRRs could be observed according to the timing of the contributions paid in the system relative to the possible oscillations of the yearly rates of return credited, explicitly and transparently, on all account balances. By contrast, the rate of return on each dollar paid within the U.S. system is not only hidden but also highly sensitive to the vagaries of both the individual career and the contribution rate charged during the working period. The radical uncertainty about the individual rate of return granted by the U.S. rules clearly emerges from Figure 2, which plots the values of the individual IRRs projected for three different contribution rates against increasing values of AIME depending upon: (case i) differences in the initial earnings with the same income growth rate (in line with the average wage growth); (case ii) different growth rates of the individual income with the same initial earnings (amounting to 36.252$ in 2016 as in the medium career represented by the bold line in Figure 1).

The central, bold line reflects the actual OAI contribution rate of 8.7%. The dotted lines above and below assume a contribution rate of 6% and 20%, respectively, throughout the working period. Note that a contribution rate of 20%, in line with that prevailing in most European countries, would imply a negative IRR either for careers starting with an initial yearly income above 85,000$ in 2016 U.S.$ (case i) or for careers with an "average" initial income growing above 6% in real terms, both type of careers implying an AIME of around 10,000$ in 2016 U.S.$.

Finally, we provide the simulations taking three different career profiles of equal length (starting at 27 in 2016 and retiring at 67 in 2056) whose relative dynamics can be summarized as follows:

* The "flat" income pattern exceeds the other two from the age of 27 (2016) up to the age of 30 (2019) while remaining in-between there onwards;

* The "parabolic" pattern starts "low" but then lies above the other two from the age of 30 up to the age of 50 when it again drops below the other two;

* The "convex" pattern starts from an intermediate position, while soon dropping below the other two where it remains up to the age of 50, when it starts to outdo the other careers to reach the maximum taxable income just before retirement.

Though clearly dissimilar, the three career patterns plotted in Figure 3 generate the same account balance at retirement, and hence the same NDC pension, given that the three hypothetical individuals retire at the same age and belong to the same birth cohort (thus sharing the same annuity divisor). Moreover, as shown in Table 6, the NDC scheme also grants the same real, individual, longitudinal IRR of 1.8% to these latter careers.

On the other hand, the figures of Table 6 show that the redistributions operated by the U.S. rules can be somewhat regressive, as is the case with the traditional "last-earnings" DB schemes. In fact, the U.S. rules reward each dollar paid in the system by the individual following the exponential career and with the highest average lifetime earnings, at a real rate of return of 3.5%, which is significantly higher than the real IRRs of 3.2% granted to both the parabolic and flat careers. All three IRRs are almost twice as much as the sustainable one of 1.8%, granted by the NDC rules.

The inverse relationship between the (longitudinal) IRR and the AIME of each individual that emerged in the first two simulations (see also Figure 2) does not hold for simulations among careers of different shapes, as is the case for the three careers presented in Figure 3, one of which is convex, another flat, and the third non-monotonic, being a typical pattern for self-employed workers. On the other hand, also this last simulation confirms the neutrality of the NDC scheme, which would leave all possible redistributions to other programs of the social security.

C. Flexible Retirement Age

Traditional DB schemes, whether PAYG or funded, generally tackle the ongoing increase in life expectancy by raising the legal retirement age (or the contribution rate). DC schemes, on the other hand, allow individuals to choose their retirement pattern (within a predefined range of options), since annuity divisors (Equation (3)), decreasing for each cohort with retirement age and for each retirement age with the cohort's year of birth (see Table 1), ensure that constraint (1) holds whatever the individual choice of each cohort member (18); continuous updating of the divisors is needed for the flexibility in retirement age to jeopardize neither the financial sustainability nor the neutrality of the scheme in terms of individual IRRs for different career lengths. As seen in Section III above, the U.S. scheme mimics the functioning of the DC arrangement in that it allows retirement within the age range 62-70, with actuarial adjustments relative to the pension benefit the individual is entitled to if she retires at the "normal" age of 67. Actuarial adjustments that will be in force for the birth cohorts 1960 and later range from minus 25% to plus 32% for those who choose to retire at 62 or 70, respectively. It is important to assess the extent to which the actuarial adjustments of the U.S. scheme preserve uniformity of individual IRRs, for different individuals who, ceteris paribus, retire at different ages.

In what follows, we simulate the effects of three alternative retirement ages, 62, 67, and 70, for an individual belonging to the cohort born in 1989, who starts work at 27 in 2016 and whose wage grows in line with the medium career of previous simulations (see the bold line in Figure 1).

The results of the simulations, summarized in Table 7, show that the longitudinal IRR (3.6% in real terms) is insensitive to retirement age, thus proving the neutrality of the U.S. actuarial adjustments. As expected, the same neutrality is guaranteed by the annuity divisors in the NDC scheme. Consistently, also the sensitivity of replacement rates to retirement age shows no difference in the two systems, with the replacement rate at 70 being more than twice as much as the replacement rate at 62. (19)


With the current demographic transition in progress in the United States and the low growth rates of the payroll expected in the coming decades, continuing the U.S. pension system with the present computation and adjustment rules calls for a significant increase in the payroll tax rate. According to the Board of Trustees (2016), covering the costs of the OASDI programs in the next 50 years requires the payroll tax to increase, gradually, by roughly 4.5 percentage points, that is from the present level of 12.4% to around 17% (under the intermediate assumptions). According to Feldstein and Liebman (2002, 4), an even higher increase in the payroll tax, by around 7 percentage points, will be needed to avoid cutting benefits. Such a shift would imply an overall tax of around 20% of the payroll, much in line with the European standards, but almost unconceivable for the U.S. policy makers. Feldstein and Liebman belong to the group of those, mentioned in the introduction, that foresee a more efficient alternative in accepting the heavy, short-run cost of the transition towards an investment-based system of personal accounts to be added to the existing PAYG plan. A deposit of 3% of the payroll would do the job (2002, 4).

U.S. citizens consider their social security a costly but effective tool to redistribute in favor of low-income classes, at least upon retirement. This is why one of the main arguments in favor of diverting financial resources from the old PAYGDB system towards a new, funded pillar based on personal accounts lies in the poor redistributive properties of the OAI program once the lower life expectancy of the income classes that the pension formula should sustain is taken into account. Our paper has given some further support to this latter thesis by showing that, despite its appearance and regardless of differences in life expectancies, the OAI formula has poor redistributive properties vis-a-vis some, not infrequent, career patterns. We are aware that by focusing on a few hypothetical earnings histories we cannot evaluate to what extent our analysis is relevant for the actual U.S. workers protected by the OAI program, nor can we take into account real life events such as job changes or jobs that do not offer pensions under different scenarios. Nevertheless, it is our conviction that understanding the "general" redistributive properties of the OAI program as opposed to the "neutral" NDC arrangement is a necessary, preliminary step in view of discussion of the alternative reform options. In fact, our use of the NDC benchmark to assess the fairness of the U.S. scheme has brought out the fact that a shift towards a personal accounts model does not entail the costly transition from PAYG to funding. A clear-cut separation between an OAI plan, designed to be both neutral and sustainable, and the other redistributive programs of the social security with set-apart financing sources would do the job of ensuring, transparently, the appropriate redistributions. Moreover, a shift towards a neutral and transparent NDC arrangement--with a higher contribution rate capable of ensuring at the same time the sustainability of the promises matured within the old system as well as the adequacy and the financial sustainability of future pension provisions--would allow, at no transition costs, the higher pension contributions not to be perceived as a tax. On the contrary, the "new" contributions to the social security would be, correctly, perceived as compulsory savings, necessary to provide U.S. workers for their own retirement income in the new economic and demographic scenario. It is, of course, possible that such a choice might fall short of the opportunities offered by the financial returns of a hypothetical investment-based system. Whether or not this is true, we will see in the long run. In the meantime, we should not forget that there is no such thing as a free lunch and that the trade-off between net income during the active period and income during our ever-increasing retirement horizon is unavoidable, regardless of the pension design or, indeed, of the very existence of a public, compulsory OAI system.

Estimate of the Contribution Rate Covering the Costs of the OAI Program

          Earnings Total in
          Covered Employment          Total Annual Benefits Paid from
          (in Millionsof Dollars)      OASI Tri
                     Reported Taxable           Survivors  Benefits
                                                           In % Taxable

Year      (1)        (2)               (3)      (4)        (4)/(2)
1970        531,600    415,600          28,796    7,428    1.79
1971        559,700    426,960          33,413    8,602    2.01
1972        617,900    484,110          37,122    9,481    1.96
1973        686,700    561,850          45,741   12,356    2.20
1974        746,700    636,760          51,618   13,843    2.17
1975        787,600    664,660          58,509   15,544    2.34
1976        874,700    737,700          65,699   17,257    2.34
1977        960,100    816,550          73,113   19,070    2.34
1978      1,092,600    915,600          80,352   20,706    2.26
1979      1,222,200  1,067,000          90,556   23,140    2.17
1980      1,328,800  1,180,700         105,074   26,654    2.26
1981      1,450,900  1,294,100         123,795   30,874    2.39
1982      1,516,600  1,365,300         138,800   33,612    2.46
1983      1,615,200  1,454,100         149,502   35,163    2.42
1984      1,800,800  1,608,800         157,862   36,627    2.28
1985      1,936,800  1,722,600         167,360   38,617    2.24
1986      2,081,800  1,844,400         176,845   40,693    2.21
1987      2,237,000  1,960.000         183,644   42,111    2.15
1988      2,432,800  2,088,400         195,522   44,788    2.14
1989      2,578,700  2,239,500         207,977   47,419    2.12
1990      2,703,800  2,358,000         222,993   50,745    2.15
1991      2,760,500  2,422,500         240,436   54,689    2.26
1992      2,917,800  2,532,900         254,939   58,049    2.29
1993      3,022,900  2,636,100         267,804   61,226    2.32
1994      3,197,000  2.785,200         279,118   64,002    2.30
1995      3,401,800  2,919,100         291,682   67,083    2.30
1996      3,587,600  3,073,500         302,914   69,759    2.27
1997      3,858,721  3.285,000         316,311   72,505    2.21
1998      4,172,641  3,524,900         326,817   73,940    2.10
1999      4,467,110  3,749,600         334,437   75,336    2.01
2000      4,819,870  4.008.500         352,706   77,848    1.94
2001      4,919,536  4,167,900         372,370   81,359    1.95
2002      4,938,294  4,250,100         388,170   83,973    1.98
2003      5,068,917  4,355,000         399,892   85,634    1.97
2004      5,370,545  4,554,500         415,082   87,737    1.93
2005      5,668,730  4,766.000         435,373   90,073    1.89
2006      6,049,719  5,043,400         460,457   93,300    1.85
2007      6,381,306  5,268,200         485,881   96,555    1.83
2008      6,496,180  5,432,800         509,056   99,348    1.83
2009      6,184,514  5,271,200         557,160  105,380     2.0
2010      6,318,485  5,307,700         577,448  105,740     2.0
2011      6,594,469  5,486,200         596,212  106,310     1.9
2012      6,898,266  5,712,312         637,948  110,346     1.9
2013 (a)  7,146,829  5,913,015         672,175  112,032     1.9

          Tax Rates as a Percent
          of Taxable Earnings
          OASI  Estimate of OAI

Year      %     %
1970      7.3   5.5
1971      S.I   6.1
1972      S.I   6.1
1973      8.6   6.4
1974      s.s   6.6
1975      s.s   6.4
1976      8.8   6.4
1977      8.8   6.4
1978      S.6   6.3
1979      8.7   6.5
1980      9.0   6.8
1981      9.4   7.0
1982      9.2   6.7
1983      9.6   7.1
1984      10.4  8.1
1985      10.4  8.2
1986      10.4  8.2
1987      10.4  8.3
1988      11.1  8.9
1989      11.1  8.9
1990      11.2  8.0
1991      11.2  8.9
1992      11.2  8.9
1993      11.2  8.9
1994      10.5  8.2
1995      10.5  8.2
1996      10.5  8.3
1997      10.7  8.5
1998      10.7  8.6
1999      10.7  8.7
2000      10.6  8.7
2001      10.6  8.6
2002      10.6  8.6
2003      10.6  8.6
2004      10.6  8.7
2005      10.6  8.7
2006      10.6  8.8
2007      10.6  8.8
2008      10.6  8.8
2009      10.6  8.6
2010      10.6  8.6
2011      10.6  8.7
2012      10.6  8.7
2013 (a)  10.6  8.7
2014            8.7
2015            8.7

(a) Last available.
Source: Authors' elaboration based on Statistics Annual Statistical
Supplement 2014.


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Nistico: Professor, Department of Economics and Law, University of Cassino and Southern Lazio, Cassino 03043, Italy. Phone 39-339-5962800, Fax 39-06-48903 253, E-mail

Bevilacqua: Researcher. CREAM--Creativity and Motivations Economic Center, University of Cassino and Southern Lazio, Cassino 03043, Italy. Phone 39-338-4277442, Fax 39-0776-937447, E-mail

(1.) Although Liebman (20021 recognizes that the timing of the contributions paid in the system (for a given income and life expectancy) could account for the dispersion of the individual IRRs, his work does not address this issue.

(2.) For a reconstruction of the political and scientific background behind the emergence of the Italian and Swedish reforms, see Gronchi and Nistico (2006, 508, note 6).

(3.) The fact that this transparency can also be achieved with the NDC scheme is acknowledged by Feldstein (2002, 7), one of the most influential advocates of funding.

(4.) For discussion of how the system design affects the individual perception of pension contributions as a tax, with its distortive and redistributive properties, or as compulsory savings, see Valdes-Prieto (2005), Lindbeck and Persson (2003, 2006), Barr and Diamond (2006), Disney (2004).

(5.) A detailed proposal for reforming the U.S. public pension system balancing revenue increases and benefit reductions is in Diamond and Orszag (2004). On the other hand, Kotlikoff and Sachs (1997), Feldstein (1998) as well as Kotlikoff and Burns (2004) argue that the pension crisis awaiting the United States can hardly be avoided with "small" adjustments in the generosity of the system and in payroll taxes, so the shift towards funded, private pension plans will be unavoidable. Although the defenders of privatization recognize the heavy transition costs entailed, according to Shaviro (2000, 156-7) there is no reason why the old-rules pension levels should be entirely guaranteed, a possible reduction of existing benefits being a feasible strategy to finance the transition towards funded, individual accounts.

(6.) Proof that the constancy of the wage growth is not a necessary condition for the theorem to hold is provided in Gronchi and Nistico (2008) within a clear-cut earnings-related framework that, realistically, allows for different retirement ages and career patterns. The first proofs of the theorem for a fully steady state can be found in Samuelson (1958), Aaron (1966), Buchanan (1968) and, in Italian, in De Finetti (1956). Note that in Samuelson (1958) the theorem is not explicitly proved in a defined-benefit context and that in Aaron (1966) the award formula is not related to individual earnings, in that the first annual benefit is equal to the average wage of active workers.

(7.) The NDC scheme cannot preclude those negligible differences in individual (longitudinal) IRRs due to the different impact of the variability of the yearly rate of return on individual accounts according to specific contribution "histories." See Gronchi and Nistico (2008, 139, footnote 16).

(8.) Although, in principle, there is no reason why both the adjustment and the conventional rate of interest should not be preset, say the first at 2% and the second at 3%, such a choice would configure a pension system, wherein both intra-and inter-generational actuarial fairness would be strictly guaranteed, but financial sustainability would not, with the pension system bearing the financial risk arising from possible discrepancies between the dynamics of the main economic and demographic indicators and the preset rates.

(9.) In fact, once 5 has been set, according to Equation (5) the value of d increases with m, i.e., the expected payout duration, which decreases when retirement age increases. In turn, given the present decreasing trend in mortality rates, the payout duration increases with the year of birth. Therefore, the definition and updating of the values generated by Equation (5) should determine lower divisors for higher retirement ages within the same birth cohort and, for each given retirement age, higher divisors for younger cohorts.

(10.) Actually, the theorem "inverts" and extends to PAYG DC schemes, the Samuelson-Aaron theorem, which was conceived for DB schemes whose financial stability is ensured by the contribution rate. Valdes-Prieto (2000, 407-8) erroneously includes the variability of a among the causes of temporary unbalances. Constancy of the population age structure is also a necessary condition for the theorem to hold.

(11.) The Swedish NDC scheme has chosen the average wage growth as the standard rate of return, while activating a sort of brake in the dynamics of liabilities, namely an "automatic balance mechanism," which is triggered whenever liabilities exceed assets computed on the basis of the current contribution flow. For an explanation of how assets can be computed in a PAYG system and of how the automatic adjustments of the rate of return can ensure long-term solvency also when mortality and employment growth rates are not constant through time, see Settergren and Mikula 2006. Skeptical appraisals of the possibility to enhance automatic adjustment mechanisms capable to insulate public PAYG systems from political risk can be found in Valdes-Prieto (2006).

(12.) Contrary to our choice to refer to annual earnings and benefits, the rules of the U.S. Social Security refer to the monthly pension benefit.

(13.) It is worth recalling that in the NDC scheme, as implemented in Sweden, there is no survivor insurance, abolished in 1990, and the benefits to needy survivors are thus financed by the redistributive programs of social security.

(14.) We calculated the divisors on the basis of the projected probabilities of death by calendar year to be found for the "intermediate assuptions" in the 2016 Trustee Report, which shows the expected trend of U.S. life expectancies up to 2090 (Table V.A.4). In particular for the two 1989 and 1994 birth cohhorts we used the 2050 and 2055 mortality tables, respectively. On the U.S. projected Life Tables, see also Bell and Miller (2005b).

(15.) For a detailed survey of the literature, see Gronchi and Nistic6 (2008, 141-3). For a graphical analysis of the incidence of pension contributions in a simple two-sectors model see Nistico (2013b).

(16.) A small variance of the IRRs would show up if the growth rate of taxable earnings were assumed to vary through time after 2014.

(17.) For a reconstruction of the political and theoretical debate preceding the Italian and Swedish NDC reforms, see Gronchi and Nistico (2008, 132 footnote 3).

(18.) In Sweden, a lower limit to the retirement age has been set at 61. For employed workers, postponing retirement beyond 67 requires the employer's approval. In Italy, the lower limit was initially set at 57 and is now being raised to face the ongoing increase in pension expenditure due to the retiring baby-boom cohorts.

(19.) Both in the U.S. scheme and in the NDC scheme the overall effect on pension benefits of postponing (bringing forward) retirement can be decomposed into an economic and a demographic effect (Nistico and Bevilacqua 2013, 15-6). The economic effect arises because of the extra pension rights earned by postponing retirement, whereas the demographic effect measures, ceteris paribus, the positive impact on benefits due to lower life-expectancy at retirement. The reverse applies when individuals opt for early retirement. The relative weight of the economic effect clearly differs according to the career pattern.
AIME: Average Indexed Monthly Earnings
COLA: Cost-of-Living Adjustment
CPI-W: Consumer Price Index for Urban Wage Earners and Clerical Workers
DB: Defined-Benefit
DC: Defined-Contribution
DI: Disability Insurance
IRRs: Internal Rates of Return
NDC: "Notional" or "Non-financial" Defined Contribution
OAI: Old-Age Insurance
OASDI: Old-Age, Survivors, and Disability Insurance
OASI: Old-Age and Survivors Insurance
PAYG: Pay-As-You-Go
PIA: Primary Insurance Amount
SI: Survivor Insurance

Results of the Simulations (Contribution Record 40 Years, Amounts in

Indicators              Very Low   Low      Medium   High     Very High

Average earnings        15,846     28,511    63,436  101,497  155,744
whole career
U.S. Social Security
Pension System
AIME                     1,628      2,929     5,411    8,658   13,016
PIA                      1.355      1,772     2,754    3,649    4,392
Actuarial adjustment    -30%       -30%      -        -         -
First pension (annual)  11,385     14,882    33,053   43,790   52,709
Replacement rate        68.4%      49.7%     59.9%    49.6%    26.9%
Real IRR                 5.2%       4.2%      3.6%     3.0%     2.2%
NDC Pension System
Account balance         75,975     136,702  313,868  502,189  766,105
Divisor                 19.799     19.799    16.722   16.722   16.722
First pension (annual)   3,837      6,905    18,770   30,032   45,815
Replacement rate        23.1%      23.1%     34.0%    34.0%    23.4%
Real IRR                 1.8%       1.8%      1.8%     1.8%     1.8%

Source: Authors' elaboration.

Results of the Simulations for Working Careers Plotted in Figure 1

                           U.S. Social Security
                   Medium  Steep   Very
                   Career  Career  Steep Career

Average earnings   63,436  97,572  121,163
First pension ($)  33,053  43,280   47,522
                   59.9    36.2     27.7
Real IRR (%)        3.6     3.3      3.0

                   Medium  Steep   Very
                   Career  Career  Steep Career

Average earnings   63,436  97,572  122,576
First pension ($)  18.770  27,720   34.160
                   34.0    23.2     19.5
Real IRR (%)        1.8     1.8      1.8

Source: Authors' elaboration.

Results of Simulations for the Working Careers Plotted in Figure 3

                               U.S. Social Security
                  Exponential                Parabolic
                  Pattern      Flat Pattern  Pattern

Average earnings  94,699       87,857        85,371
Account Balance   -            -             -
First pension     42,518       41,361        41,969
                  22.4         46.3          86.8
Real IRR (%)       3.5          3.2           3.2

                  Exponential                Parabolic
                  Pattern      Flat Pattern  Pattern

Average earnings   94,699       87,857        85,371
Account Balance   438,264      438,264       438,264
First pension      26,209       26,209        26,209
                   13.8         29.4          54.2
Real IRR (%)        1.8          1.8           1.8

Note: Retirement age: 66 years, contribution record: 40years (U.S.
dollar, 2016 chained prices).
Source: Authors' elaboration.

Effects of Different Retirement Age

                      Case 1    Case 2    Case 3

Age at retirement     62 years  67 years  70 years
Year of retirement    2051      2056      2059
U.S. Social Security
Pension System
First pension ($)     23,071    33,053    40,986
Replacement rate (%)  36.8      59.9      80.2
Real IRR (%)           3.6       3.6       3.6
NDC Pension System
First pension ($)     13,489    18,770    22,889
Replacement rate (%)  21.5      34.0      44.8
Real IRR (%)           1.8       1.8       1.S

Note: The U.S. Social Security against the NDC Scheme
(U.S. dollar, 2016 chained prices).
Source: Authors' elaboration.
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Author:Nistico, Sergio; Bevilacqua, Mirko
Publication:Contemporary Economic Policy
Article Type:Report
Geographic Code:1USA
Date:Jul 1, 2018

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