3.1. Unbalanced systems without pension floor and assistance benefit

To highlight the relationship between rules and results more directly, we consider in this section stylized PAYG structures in which two components of the system that generate non-trivial discontinuities on results are excluded, namely: (i) the minimum amount for the retirement benefit and (ii) the guarantee of assistance benefits for the elderly poor.

In the DB system in force in Brazil, the benefit amount is calculated based on a replacement rate that is the sum of two components: $\alpha \left( g \right) + \beta .n$, where the first, which varies by gender, does not depend on the number of contributions made until retirement ( $n$) and can be seen as a compensation for groups that have lower contributory densities such as women and informal or self-employed workers. To isolate the effect of these components, we consider four alternative scenarios for computing the retirement benefit amount. Specifically, using the notation from Section 2.3 (without the subscripts), the simulated systems areFootnote ²⁸:

(28)\begin{equation}{\text{na}}1){\text{ DB}}\_{\text{NA}}:\alpha \left( g \right)sm + \beta.n.sm;\end{equation}

(29)\begin{equation}{\text{na}}2){\text{ DB}}\_{\text{NA}}0:{ }\beta .n.sm;\end{equation}

(30)\begin{equation}{\text{na}}3){\text{ DC}}\_{\text{NA}}:{ }x/ve;\end{equation}

(31)\begin{equation}{\text{na}}4){\text{ DM}}\_{\text{NA}}:\,\alpha \left( g \right)sm + x/ve.\end{equation}

The DB_NA system corresponds to the DB system in use in Brazil – i.e., expression (15) without the benefit floor. The DB_NA0 scenario is a variant of DB_NA in which the compensation that is independent of the contributory density of the agent is eliminated. The DC_NA system corresponds to the benefit of the notional DC system – i.e., expression (16) without the benefit floor – in which the agent receives the capitalization of what he/she contributed ( $x$) divided by the expected life ( $ve$). And, finally, the DM_NA system is a new mixed system that adds to the benefit of the DC system, the compensation that is independent of the contributory density of the DB system.Footnote ²⁹ Systems DB_NA0 and DC_NA contain only the component that depends on the contributory density and differ from each other by computing the effect of this density differently. Similarly, systems DB_NA and DM_NA incorporate the compensation component independent of the contributory density and differ from each other for the same reason as in the previous pair of cases.

In Figure 1, graphs 1.a, 1.b, and 1.c present the simulation results of these four systems respectively for the aggregate deficit ( ${A_t}$), the effect on permanent income inequality ( $\Delta {\psi _c}$) and the rate of return for the representative agent ( ${\rho _c}$). Graphs 1.d to 1.g (1.h to 1.k) show the rates of return of men (women) for different educational groups ( ${\rho _{c,e,g}}$). Due to the similarity of the results for some educational groups, to simplify the presentation, the rates of return of the groups ( $e = 1,\,e = 2$) and of ( $e = 3,\,e = 4$) were aggregated using as weight the proportion of agents with these educational levels in each cohort over time. In all graphs of this and other sections, $t = 2016,\ldots.,2080$ and $c = 1980,\ldots,2040$.

Figure 1.

Simulation results for unbalanced systems without pension floor and assistance benefit.

Note: The graphs are: 1.a: Deficit per year, ${A_t}$; 1.b: Effect on permanent income inequality between groups by cohort, $\Delta {\psi _c}$; 1.c: Aggregate rate of return by cohort, ${\rho _c}$. The other graphs refer to the rate of return by cohort, educational group, and gender, ${\rho _{c,e,g}}$, as indicated in the figure. The lines correspond to the benefit rules detailed in the text, where: DB_NA: defined benefit system (DB); DB_NA0: the restricted variant of DB_NA; DC_NA: notional DC system; and md0: mixed system that combines components from the DC and DB systems.

Source: Simulations based on data from Pesquisa Nacional por Amostra de Domicílios (PNAD).

Graphs 1.a, 1.b., and 1.c show that, while the pure DC system DC_NA has the smallest deficit, the smallest effect (in absolute terms) on permanent income inequality and the smallest rate of return, the pure DB system DB_NA exhibits the opposite pattern. In particular, it should be noted that the DB rate of return is more than three times higher than that of the DC and well above productivity growth, which is an indicator of future deficits in the system.Footnote ³⁰ The differences between the rates of return of the DB and DC systems are accentuated for the groups with lower education (graphs 1.d to 1.k), especially in the case of women, whose term of the replacement rate that does not depend on the contribution density is higher than that of men.Footnote ³¹ These differences between the two systems were expected as the DC system is, by design, more neutral, while the DB tends to be more generous, especially to the less skilled and to women.

Graphs 1.a to 1.c also show that the mixed system DM_NA, as well as the DB_NA0 system, present intermediate results between the DB and DC systems for deficit, permanent income inequality and rate of return. This intermediate pattern of the DM_NA and DB_NA0 systems is also observed for the returns of the educational groups, both for men (graphs 1.d–1.g) and for women (graphs 1.h–1.k). However, the DM_NA system tends to present higher returns for groups with less education, especially for women, which is explained by the presence of the term that is independent of contributory density in the calculation of the benefits of the DM_NA system. It is also interesting to note that the rates of return are higher for women than for men for almost all systems, the exception being the DC structure DC_NA, which displays a neutral return across the board.

Except for the case of the DC structure DC_NA, it is interesting to note that the deficits of all other systems tend to increase in the long run with positive values even in the context without pension floor and assistance benefit. This results from the combination of some operating forces that were discussed before. Clearly, the rise in the dependency ratio operates as a deficit enhancer. But the systematic increase in the probability of contributions to social security and the rise in effective tax rates paid by workers – due to the shifts to the right of the distributions of wages and education – are deficit reducers. There are also the opposing effects associated with the presence of the fixed wage contribution cap: on one hand, it limits the total value of contributions but, on the other, it also creates a cap on benefit payment. The results in graph 1.a show that the deficit reducers were not strong enough to counterbalance the effects of the deficit enhancers.

As for the effects on inequality of permanent income and the rates of return (graphs 1.b and 1.c, respectively), we see that they tend to be relatively stable across almost all cohorts (except for the first ones) for all systems. However, this is not observed for the rate of return of the DB system DB_NA for the more educated groups workers (especially males), which indicates that the rise in the effective contribution rate tends to depress the return for these groups in that system.

It is also possible to observe in graphs 1.a to 1.c that the aggregated results of the DM_NA and DB_NA0 systems are similar. Though subject to the parameterization used, this similarity of results implicitly confirms the generosity of the DB system. The mixed system DM_NA can be seen as a DC system in which a compensation is introduced for groups with the lowest propensity to contribute – specifically, the term that is independent of contributory density in the DB system. The incorporation of this component in the DM_NA system makes its results similar to those of the DB_NA0 system, which, like the DC system, does not have this compensation component.

These results motivate the evaluation of the functioning of the mixed system DM_NA, which reduces the deficit compared to the DB system, has a greater inequality-reducing effect than the DC system and presents an intermediate rate of return between the DB and the DC, including for the different educational groups and genders. However, the abrupt change from the initial DB system to the DM_NA mixed system has the inconvenience of generating a discontinuity in the value of the benefit. For this reason, we propose an adjustment of the rule in which the benefit starts from the DB system and converges in a certain number of years to the mixed system DM_NA. More specifically, the benefit formula in this system (which we denominate DM) is given by:

(32)\begin{equation}{\text{s}}5){\text{ DM}}:{ }\alpha \left( g \right)\,sm\, + \,\left[ {\gamma \left( {\beta *n} \right)sm + \,\left( {1 - \,\gamma } \right)\,x/ve} \right],\end{equation}

where $\gamma = {\text{max}}\left\{ {0,1 - \Delta \gamma \,\left( {t - t0} \right)} \right\}$ is the function that regulates the transition phase. Unless stated otherwise, in the simulations of this system, we set $\Delta \gamma = 0.025$ and $t0 = 2016$.Footnote ³²

3.2. Unbalanced systems with pension floor and assistance benefit

In this section, we evaluate the functioning of the DB, DC, and the new mixed DM system presented in the previous subsection in a more realistic environment where the existence of the minimum benefit amount and the guarantee of the assistance benefit for the elderly poor are incorporated.

wa1) DC_WA: DC system with benefit calculated by expression (15);

wa2) DM_WA: DM system with benefit calculated by expression (32);

wa3) DM_WA_IT: DM system with instantaneous transition (i.e., with $\gamma = 0$ in expression (32));

wa4) DB_WA: DB system with benefit calculated by expression (14).

In Figure 2, graphs 2.a, 2.b, 2.c show several results that are worth noting. First, it is possible to observe that the incorporation of the social security floor and the guarantee of minimum benefit for the elderly poor do not qualitatively change the comparisons between the DC, DB and DM systems discussed in the previous subsection. The DM system (DM_WA and DM_WA_IT) remains an intermediary scheme between the DB (DB_WA) and the DC (DC_WA) systems for almost all variables, the only exception being the effect on inequality in the latter system (see below).

Figure 2.

Simulation results for unbalanced systems with pension floor and assistance benefit.

Notes: The graphs are: 2.a: Deficit per year, ${A_t}$; 1.b: Effect on permanent income inequality between groups by cohort, $\Delta {\psi _c}$; 1.c: Aggregate rate of return by cohort, ${\rho _c}$. The other graphs refer to the rate of return by cohort, educational group, and gender, ${\rho _{c,e,g}}$, as indicated in the figure. The lines correspond to the benefit rules detailed in the text, where: DB_WA: defined benefit system (DB); DC_WA: notional defined contribution (DC) system; DM_WA: mixed system (DM) that combines components from DC and DB systems and has a smooth transition ( $\Delta \gamma = 0.025$); and DM_WA_IT: mixed system (DM) with instantaneous transition ( $\gamma = 0$).

Source: Simulations based on data from Pesquisa Nacional por Amostra de Domicílios (PNAD).

Secondly, differently from the previous subsection, some trends clearly appear in the outcome variables for all systems. As aforementioned, it is difficult to disentangle the effects of each operating factor but it is clear that the introduction of the benefit floor and the assistance benefit is generating the trends. This is particularly noticeable for the DC_WA system, a result that shows that even a neutral structure like the notional DC can be strongly affected by the insertion of restrictions on benefit floors and assistance benefits.Footnote ³³

Compared to the context without pension floor and assistance benefit (graph 1.a), the results of graph 2.a. reveal a rise in the deficit for all simulated rules, with the DB system DB_WA displaying the largest increase, the DC system DC_WA the smallest, and the proposed mixed system DM_WA an intermediate value. Graph 2.a also reveals that the incorporation of a transition phase from the DB system to the DM system DM_WA generates only a transient impact on the deficit compared to the system without transition (DM_WA_IT). This result is interesting as it shows that it is possible to modulate the effect on the deficit of a transition from the DB system to a mixed system in which the benefit calculation includes a component that is independent of contributory density and another component entirely based on systems of defined contribution.Footnote ³⁴

The results in Graph 2.b show that the DC DC_WA system promotes a significant drop in permanent income inequality for the initial cohorts, but that this reduction gradually attenuates across the cohorts. This is because this system drastically reduces the benefits, leading to a high proportion of agents receiving the benefit floor, which substantially reduces inequality for the initial generations. The attenuation of the effect on inequality is mainly due to the shifts to the right of the distributions of wages, contributions and education, which decrease the proportion of groups restricted by the benefit floor. Graph 2.b also shows that the DB scheme DB_WA is the one that promotes the greatest reduction in inequality among cohorts. The reduction of inequality by the mixed systems DM_WA and DM_WA_IT is smaller (in module) than that of DB system, and the introduction of the transition between systems (DM_WA) has practically no effect on inequality. Unlike the trajectory of the DC scheme DC_WA, the DM_WA and DM_WA_IT systems do not present a concentration of groups that receive the benefit floor and that, as in the DC system, are particularly affected by the shift to the right of distributions of wages, contributions, and education.

In terms of ranking between systems, the rate of return of the representative agent presented in graph 2.c shows qualitatively similar results to those of section 3.1, that is, the DB presents a higher rate than that of the DM which, in turn, exhibits a higher rate than that of the DC. It can also be noted in graph 2.c that the difference in return between the mixed systems with and without transition (DM_WA and DM_WA_IT, respectively) is eliminated along the initial cohorts, which shows that with the transition the return converges smoothly to its long-term trend.

The sets of graphs 2.d to 2.g and 2.h to 2.k respectively show decreasing trends in the intergenerational return for men and women, especially for groups with lower education. The comparison of these graphs with the ones in the previous subsection confirms, for most systems and educational groups by gender, the downward effect of introducing additional restrictions due to the benefit floor and the assistance benefit. It is also notable that the introduction of these restrictions increases the rate of return for the lowest education groups, especially for females. In fact, as before, the rates of return for women are higher than those for men for almost all systems, except for the DC structure DC_WA for the most educated group, which displays equal return between the genders. As in the aggregate case, it is also noted that the difference in return between the mixed systems with and without transition (DM_WA and DM_WA_IT) disappears for initial cohorts of more educated groups – and it does not even show up for the least educated ones.

All simulations presented until this section assume that the deficit is financed by an external source that does not impact the agents’ returns. This makes the simulation results not strictly comparable. For example, as we have seen, the DB system generates greater returns than the mixed DM system, but its deficit is much higher. The next section introduces rules for adjusting the contributions of active and inactive groups to make all systems financially sustainable and, in that sense, more comparable.

3.3. Balanced systems with pension floor and assistance benefit

In a financially balanced pension system, revenues and expenditures are equal over the years and, to ensure this equality within the systems, active and inactive generations contribute to a fund that finances imbalances. The additional contribution of the active is a proportion of the contribution wage ( ${\tau _t}$) and the contribution of the inactive is a rate ( $\tau {b_t}$) of the benefit received. It is assumed that active and inactive agents who respectively receive more than $L$ and $Lb$ multiples of the minimum wage contribute to this fund, that is, agents with lower income are excluded from the adjustment contribution. The rates of active and inactive groups are determined per year according to two classes of rules inspired in Auerbach and Lee (Reference Auerbach and Lee2011), namely:

$A(\alpha )$: defines rates ${\tau _t}$ and $\tau {b_t}$ so that the proportion of the deficit paid by the inactive generations is equal to $\alpha $; and,

$X\left( {\beta ,\gamma } \right)$: defines the contribution rate of inactive generations, $\tau {b_t}$, according to expression (36) below and calculates the contribution rate of active generations, ${\tau _t}$, in a residual way to cancel the deficit (expression (37)).

The first class of rules distributes the adjustment burden between active and inactive groups according to a single policy parameter $\left( \alpha \right)$ and has the advantage of controlling a relevant dimension, which is the relative burden between the two groups. The second class of rules is a stylized form of the system in use in Germany, where the contribution of the inactive is adjusted both by the variation of the ratio between the size of the inactive and the active population and by the variation in gross wages. In this rule, the inactive can be seen as a partner of the social security system, contributing more if the elderly population proportionally increases and contributing less (may be even with negative rates) when the wages increase. That is, the inactive generations have the system’s burden and bonus, and the active generations pay what is necessary to eliminate the deficit. Using the notation from section 2.3, the following equations define aggregate variables that are used in the two classes of adjustment rules. Let:

(33)\begin{equation}f{a_t} = \,\mathop {{\sum _{c,e,g}}}\limits_{} \mathop {{\sum _{i \in I\left( {c,t} \right)}}}\limits_{} I(s{c_{c,e,g,i}},L*smi{n_t})*{z_{c,e,g,i}}*po{p_{c,i,g}}\,{\text{and}}\end{equation}

(34)\begin{equation}f{b_t} = \,\mathop {{\sum _{e,g,c}}\ {\sum _{i \gt A\left( {c,e,g} \right)\mathop \cap \nolimits^ i \in I\left( {c,t} \right)}}}\limits_{} I({b_{c,e,g}},Lb*smi{n_t})*{z_{c,e,g,i}}*po{p_{c,i,g}},\end{equation}

where $I\left( {x,y} \right)\, = \,x$ if $x \geqslant y$ and $0$ otherwise, $sc$ is the contribution wage, $b$ is the benefit, $\left( {L,Lb} \right)$ are the thresholds below which active and inactive do not contribute to balance the systems, and $smin$ is the minimum wage. Quantities $f{a_t}$ and $f{b_t}$ represent, respectively, the total funds arising from wages of the active and benefits of the inactive that are subject to the contribution to financially balance the system in period $t$.

Adjustment rule $A(\alpha )$:

(35)\begin{equation}{\tau _t} = \left( {1 - \alpha } \right)\!.\, x_{t}{\text{ and }}\tau {b_t} = \alpha .\,x_t\end{equation}

where ${x_t} = \left[ {{G_t} - {R_t}} \right]/\left[ {\left( {1 - \alpha } \right)\!.\,f{a_t} + \alpha .\,f{b_t}} \right]$.

Adjustment rule $X\left( {\beta ,\gamma } \right)$:

(36)\begin{equation}\tau {b_t} = 1 + (\tau {b_{t - 1}} - 1)*\left[ {\gamma \Delta s{l_t} - \beta \Delta r{d_t}} \right],\end{equation}

where:

$\Delta r{d_t} = r{d_t} - r{d_{t - 1}}$: rate of increase in the ratio between the number of inactive and active (old-age dependency ratio);

$\Delta s{l_t} = s{l_t} - s{l_{t - 1}}$: rate of increase in average gross wage,

and

(37)\begin{equation}{\tau _t} = \left[ {{G_t} - {R_t} - \left( {\tau {b_t}*f{b_t}} \right)} \right]/f{a_t}.\end{equation}

Applying these rules, contributions paid and benefits received by the agent $\left( {c,e,g} \right)$ who contributes to the adjustment at age $i$ are expressed in a cash flow manner by:

(38)\begin{equation}{f_{c,e,g,i}} = \left\{ {\begin{array}{*{20}{c}} { - v{c_{c,e,g,i}}{ - \,_{c + i}}*I\left( {s{c_{c,e,g,i}},L*smi{n_{c + i}}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{if}}\,i \lt {A_{c,e,g}}\,} \\ {\left[ {{b_{c,e,g}}\, - \,{b_{c + i}}*I\left( {{b_{c,e,g}},Lb*smi{n_{c + i}}} \right)} \right]*p{s_{c,g,i}}{\text{ if}}\,i \geqslant {A_{c,e,g}}\,} \end{array}} \right..\end{equation}

where $I\left( {x,y} \right)\, = \,x$ if $x \geqslant y$ and $0$ otherwise, $vc$ is the contribution value based on the contribution wage, $b$ is the benefit, $\left( {\tau ,\tau b} \right)$ are the additional rates of contribution to balance the systems from the active and inactive, respectively $\left( {L,Lb} \right)$, are the thresholds below which active and inactive do not contribute to balance the systems, $smin$ is the minimum wage, and $ps$ is the survival probability.

Simulations were chosen following the strategy of comparing DB and DM systems in different situations. For rule $A\left( \alpha \right)$, we define a set of simulations assuming that active generations finance the entire deficit (i.e., $\alpha \, = 0$) or that they share the burden in equal proportion with the inactive generation (i.e., $\alpha \, = 0.5$). For the $X\left( {\beta ,\gamma } \right)$ rule, we use the parameterization $\beta = 0.5$ and $\gamma = 1$, which implies that the inactive receive the entire wage gains and contribute half of the increase in the dependency ratio. In all cases, parameters $\left( {L,Lb} \right)$ are fixed in $\left( {3,1.15} \right)$.Footnote ³⁵

The rules determine for each year the contribution rates of active and inactive generations alive in year $t$ and, as they fluctuate over time, cause the agents of different cohorts to be heterogeneously affected. The simulated rules are:

wab1) DB_WA_B0: DB system with adjustment $A\left( 0 \right)$ (i.e., $\tau {b_t} = 0,\forall t$);

wab2) DM_WA_B0: DM system with adjustment $A\left( 0 \right)$ (i.e., $\tau {b_t} = 0,\forall t$);

wab3) DB_WA_B5: DB system with adjustment $A\left( {0.5} \right)$ (i.e., $\tau {b_t}{ = _t}$);

wab4) DM_WA_B5: DM system with adjustment $A\left( {0.5} \right)$ (i.e., $\tau {b_t}{ = _t}$);

wab5) DB_WA_BX: DB system where the rate of the inactive group is given by $X\left( {0.5,1} \right)$ and residual balance is provided by the active group;

wab6) DM_WA_BX: DM system where the rate of the inactive group is given by $X\left( {0.5,1} \right)$ and residual balance is provided by the active group.

Figures 3 and 4 present results for two subsets of these rules in balanced systems where the deficit is zero. The graphs of these figures are similar to the previous ones, except for the pairs (3.a, 4.a) and (3.b, 4.b), which respectively present the value of the rates ( ${\tau _t},\tau {b_t}$) that are required from active and inactive groups to eliminate the deficit.

Figure 3.

Simulation results for balanced systems with pension floor and assistance benefit – a rules.

Notes: The graphs refer to: 3.a: Adjustment rate of the active group per year, ${\tau _t}$; 3.b: Adjustment rate of the inactive group per year, $\tau {b_t}$; 3.c: Effect on permanent income inequality between groups by cohort, $\Delta {\psi _c}$; 3.d: Aggregate rate of return by cohort, ${\rho _c}$. The other graphs refer to the rate of return by cohort, educational group, and gender, ${\rho _{c,e,g}}$, as indicated in the figure. The lines correspond to the benefit rules detailed in the text, where: DB_WA_B0: DB system with adjustment $A\left( 0 \right)$; DM_WA_B0: DM system with adjustment $A\left( 0 \right)$; DB_WA_B5: DB system with adjustment $A\left( {0.5} \right)$; and DM_WA_B5: DM system with adjustment $A\left( {0.5} \right)$.

Source: simulations based on data from pesquisa nacional por amostra de domicílios (PNAD).

Figure 4.

Simulation results for balanced systems with pension floor and assistance benefit – a and x rules.

Notes: The graphs refer to: 4.a: Adjustment rate for the active group per year, ${\tau _t}$; 4.b: Adjustment rate for the inactive group per year, $\tau {b_t}$; 4.c: Effect on permanent income inequality between groups by cohort, $\Delta {\psi _c}$; 4.d: Aggregate rate of return by cohort, ${\rho _c}$. The other graphs refer to the rate of return by cohort, educational group, and gender, ${\rho _{c,e,g}}$, as indicated in the figure. The lines correspond to the benefit rules detailed in the text, where: DB_WA_B5: DB system with adjustment $A\left( {0.5} \right)$; DM_WA_B5: DM system with adjustment $A\left( {0.5} \right)$; DB_WA_BX: DB system with adjustment $X\left( {0.5,1} \right)$; and DM_WA_BX: DM system with adjustment $X\left( {0.5,1} \right)$.

Source: Simulations based on data from Pesquisa Nacional por Amostra de Domicílios (PNAD).

Graph 3.a shows that DB systems DB_WA_B0 and DB_WA_B5 require higher contribution rates from the active group than DM systems DM_WA_B0 and DM_WA_B5. This difference between the systems can also be observed in graph 3.b for the rules involving the contribution of inactive generations, i.e., the DB_WA_B5 rule compared to DM_WA_B5. These results are consistent with the difference in deficits presented in sections 3.1 and 3.2 between these two types of systems.

The fund that finances the imbalances of the systems is based on the total contribution of active (inactive) groups, $f{a_t}$ ( $f{b_t}$), who receive more than $L\,\left( {Lb} \right)$ minimum wages. In general, the $f{a_t}$ portion of the total fund is substantially larger than the portion, even taking into account the population aging process. This difference explains the proximity of the rates of the pair of rules DB_WA_B5 and DB_WA_B0 and, especially, of the pair DM_WA_B5 and DM_WA_B0 (graph 3.a). One implication of this result is to reveal a potential limitation of the $A\left( \alpha \right)$ rule, since the contribution of inactive generations does not generate a significant reduction in taxation on active workers even when the distribution of imbalances is carried out in a proportional way, i.e., with $\alpha = 0.5$.

As can be seen in Figure 3.c, all rules generate a reduction in permanent income inequality. However, this graph also shows that all rules display pronounced trends that attenuate across cohorts. This pattern is essentially due to two factors acting in opposite directions. On the one hand, the increase in the contribution of active and inactive agents observed in graphs 3.a and 3.b raises the proportion of agents who receive the minimum wage and this accumulation of agents at the minimum generates a reducing effect on inequality. On the other hand, shifts to the right in the distribution of wages, contributions, and education work to reduce the concentration of agents receiving the minimum. The trends observed in graph 3.c show that this second effect more than offsets the first effect across generations.

Another point to be observed in graph 3.c is that the DB system tends to reduce inequality more than the DM system. However, both systems show similar trends across cohorts. In fact, maintaining the trends observed in graph 3.c, the effect of the two types of systems tends to disappear for the last cohorts, with the difference that this occurs for later generations in the DB system than in the DM system.

Graph 3.d reveals that all rules show a downward trend in aggregate return across cohorts, with the rate of return converging towards the unit value in most rules for the final cohorts.Footnote ³⁶ Another point to be highlighted in graph 3.d is a pronounced accentuation of the downward trends of the DB and DM systems comparable to those analyzed in section 3.2 (graph 2.c) – specifically DB_WA for the DB case and DM_WA for the DM case. In other words, the introduction of the contribution of active and inactive generations to eliminate the growing deficits of the DB and DM systems (as shown in graph 2.a.) intensifies the downward trends in the representative agent’s return already observed in the context without financial equilibrium. It is also noticeable in graph 3.d that the rules that tax the inactive in the same type of system have a systematically lower return than those that do not tax them (i.e., DB_WA_B5 compared to DB_WA_B0 or DM_WA_B5 compared to DM_WA_B0). These patterns in rates of return are also verified for almost all educational groups of both men (graphs 3.e to 3.h) and women (graphs 3.i to 3.l).Footnote ³⁷ These results show that the rate of return is strongly influenced not only by increases in the taxation of the active generations, but also by the adjustment contribution of the inactive ones. The comparison of the rates of return of males and females shows that the introduction of adjustment contributions on the active and inactive groups does not change the previous result that the latter group has higher rates than the former.

We selected rules DB_WA_B5 and DM_WA_B5 to compare with rules DB_WA_BX and DM_WA_BX, which are based respectively on a DB and a DM system whose benefits are affected by changes in both the dependency ratio and gross wages. The results are shown in Figure 4, whose graphs are arranged in the same way as in Figure 3.

The simulations in Figure 4 make it possible to compare the rules $\left( {A,\,X} \right)$ and the systems (DB, DM), that is, they allow the analysis of the partial effects of changing the adjustment rule given the system – i.e., comparing DB_WA_B5 with x2b or DM_WA_B5 with x2m – or from changing the system given the adjustment rule – i.e., comparing DB_WA_B5 with DM_WA_B5 or x2b with x2m. Graph 4.a shows that the adjustment rules DB_WA_BX and DM_WA_BX display lower contribution rates from the active group than their counterparts in rule $A$, DB_WA_B5 and DM_WA_B5 respectively. This result indicates that the class of rules that modulate benefits based on the dependency ratio and the evolution of wages lessens the additional taxation that falls on active generations. Graph 4.a also shows that system DB (DB_WA_B5 and DB_WA_BX) exhibits higher rates than their MD counterparts (DM_WA_B5 and DM_WA_BX, respectively).

Looking at the contribution rates of the inactive group, graph 4.b evinces that the DB system combined with rule $A$ (DB_WA_B5) taxes a higher rate from this group than the MD system with the same rule (DM_WA_B5). Thus, in the case of adjustment rule $A$, the DB system has a higher tax burden on both the active and the inactive generations than the MD system. By construction, the inactive group is charged in the same way by rule $X$ for both the DB (DB_WA_BX) and the MD (DM_WA_BX) systems. Interestingly, as displayed in graph 4.b, the contribution rate of the inactive for this rule has an inverted-U shape, showing that the force of the variation in wages is capable to surpass that of the dependency ratio for the later period. This shape makes the curve of the $X$ rule to cross from above the curves of the $A$ rule (DB_WA_B5 and DM_WA_B5), which implies that the burden on the inactive group to help balancing the system is only smaller in the former rule than for the latter at the end of the period of analysis.

As can be seen in graph 4.c, the effects on inequality of rule $X$ (DB_WA_BX and DM_WA_BX) are very similar to their counterparts in rule $A$ (DB_WA_B5 and DM_WA_B5, respectively). Graph 3.c also shows that rules based on the DB system (DB_WA_B5 and DB_WA_BX) have a systematically greater effect (in module) than rules based on the DM system (DM_WA_B5 and DM_WA_BX). As before (see graph 3.c), the main difference between these two systems is that relative to the DM scheme, the effect tends to cancel out for the DB system for later cohorts.

In graph 4.d, it is possible to see that rule $X$ (DB_WA_BX and DM_WA_BX) presents the same pattern of falling return for the representative agent as rule $A$ (DB_WA_B5 and DM_WA_B5). This result confirms what was mentioned earlier about the compressing effect that the introduction of contributions of active and inactive generations has on the representative agent’s return. It is observed, however, that rule $X$ exhibits greater returns than rule $A$ both for the DB system (DB_WA_BX compared to DB_WA_B5) and for the DM system (DM_WA_BX compared to DM_WA_B5). This result indicates that the rate of return of the class of rules that adjusts benefits according to the evolution of demographics and wages tends be higher than that based on the proportional distribution of the burden between active and inactive groups. Keeping the rules fixed, graph 4.d also shows that the rates of return are higher for the DB system (DB_WA_B5 and DB_WA_BX) than their counterparts in system MD (DM_WA_B5 and DM_WA_BX, respectively).

As can be seen in graphs 4.e to 4.h and 4.i to 4.l, except for the group of women with less education, the other educational groups of both sexes experience a pattern of decline in returns similar to that of the representative agent. It is also possible to see in these graphs the higher returns of the $X$ rules in relation to their corresponding $A$ rules for educational groups of both sexes. These results indicate that rule $X$ can generate greater returns to agents than rule $A$ across cohorts. Notwithstanding the downward trend in returns for both genders, as before females have higher rates of return than those of males.

Finally, to assess the impacts of the balanced systems on intergenerational equity, Table 2 presents the standard deviation of rates of return between cohorts for both the representative agent and different educational groups by gender. As seen earlier, these rates of return show downward trends, but even in this situation, this measure can capture the differences in return inequality between systems. Table 2 shows that, in general, the simulation of the DM system with rule $X$ (i.e., DM_WA_BX) is the one with the smallest dispersion, while the simulation of the DB system with rule $A\left( {0.5} \right)$ (i.e., DB_WA_B5) is the one with the greatest dispersion. The other rules/systems constitute intermediate cases, but with a tendency for those based on rule $A$ and system $DB$ to exhibit higher intergenerational inequity. It is interesting to note that the systems/rules with the lowest (highest) standard deviation tend to require the lowest (highest) taxation on the active group. One explanation for this pattern is that adjustments made earlier in the life cycle are less discounted than those that occur later (Auerbach and Lee, Reference Auerbach and Lee2011). It is also interesting to note that intergenerational inequality in returns is higher for females than for males in all education groups.

Table 2.

Standard deviation of the rate of return across cohorts by gender and education group

Notes: The systems/rules are detailed in the text and summarized in appendix C. Bold (underlined) numbers indicate the lowest (largest) standard deviation for each column in the table.

Source: Simulations based on data from Pesquisa Nacional por Amostra de Domicílios (PNAD).