Assessing Vertical Equity in Defined Benefit Pension Plans: An Application to Switzerland

Abstract

This paper establishes a theoretical link between actuarial neutrality and the Oaxaca–Blinder decomposition to empirically assess vertical equity in public defined-benefit schemes. We demonstrate how this approach can be generalized to non-linear functions, point systems, and notional accounts. We use an aligned dynamic microsimulation model to apply this method to the first pillar of the Swiss pension system and highlight the following three key effects: (1) the impact of the accrual rate on vertical equity; (2) the assessment of actuarial neutrality through the comparison of migrants with the non-migrant population; and (3) vertical equity across marital statuses. Our findings indicate that changing societal trends, such as increased migration, female labor participation, and the rise in non-marital unions, may alter the extent of vertical equity. This has significant implications for actuarial risk management, as a higher degree of vertical equity is associated with increased pension expenses, thereby raising the financial sustainability risk of the pension system. Future research should explore these dynamics to ensure that pension systems remain both equitable and financially sustainable in the face of evolving societal trends.

1. Introduction

Collective pension systems are characterized by horizontal equity (solidarity in risk) and vertical equity (redistribution from high to low incomes). Horizontal equity means that individuals in similar situations should receive similar treatment, while vertical equity involves treating individuals according to their specific needs (McDaniel and Repetti 1992). Essentially, horizontal equity does not require redistribution between individuals with different levels of wealth, whereas vertical equity may necessitate such redistribution (Clements et al. 2014). To empirically assess vertical equity in pension schemes, we generalize the concept of actuarial neutrality, allowing us to determine the equal—or neutral—treatment of individuals. We link this concept to the Oaxaca–Blinder (OB) decomposition method, developed to assess discrimination—an example of non-neutral treatment. By establishing a theoretical link between actuarial neutrality and the OB decomposition method, this article provides a framework for empirically assessing vertical equity in pension schemes.

An empirical assessment of vertical equity is important from an actuarial perspective, as a higher degree of vertical equity is associated with increased pension expenses, which raises the financial risk of the pension system (Ponds and Van Riel 2009). As public earnings-related pension plans—such as defined benefit (DB) schemes, points, and notional accounts—aim to cover subsistence needs, they are characterized by a higher degree of vertical solidarity compared to funded, defined contribution (DC) schemes. Since the financial risk in public earnings-related schemes is borne by the scheme sponsor, which is the community of current and future taxpayers, the assessment of vertical equity becomes a matter of public interest (Broadbent et al. 2006). This issue gains further importance in the context of low fertility rates, continuous aging, and the debt crisis, which challenge vertical redistribution and have driven governments to reduce the level of vertical redistribution within public earnings-related schemes (e.g., Lindbeck and Persson 2003; Klos et al. 2022; Jarner et al. 2024). However, changing demographic and labor market patterns can have opposing effects on the extent and need for vertical equity. Increased labor market participation by women may reduce the importance of vertical equity, as women build up higher pension entitlements. Conversely, the growing number of non-marital unions diminishes the significance of intrafamilial solidarity (e.g., income splitting) and solidarity with widowed persons. Additionally, migration, characterized by fragmented and shorter contribution periods, may also influence the degree of vertical redistribution. Therefore, changing societal trends may alter the extent of vertical equity and have significant implications for actuarial risk management.

Although the evaluation of the redistributive effects in pension systems has garnered significant attention among scholars, the recent literature on DB schemes is sparse. Several studies have focused on vertical equity in collective DC schemes, where the empirical determination of vertical equity is straightforward. In the scientific literature, the Internal Rate of Return (IRR) and Net Present Value (NPV) are commonly used to assess whether individuals with different income levels receive equitable returns on their contributions (e.g., Bonenkamp 2009; Gustman et al. 2013; Platanakis and Sutcliffe 2016). This approach links vertical redistribution to the concept of actuarial fairness. However, the concept of actuarial fairness is not applicable to DB schemes, as they are often partially financed by general government revenues rather than solely by individual contributions, weakening the link between contributions and benefits (Queisser and Whitehouse 2006).

To assess vertical equity in DB schemes, previous studies have proposed utility-based measures (e.g., Borsch-Supan and Reil-Held 2001; Lefèbvre and Pestieau 2006; Auerbach and Lee 2011; Frassi et al. 2019). However, these theoretical measures are difficult to apply due to their restrictive assumptions. To avoid the discussion of the calibration of the utility function, Wolf and Caridad López del Río (2024) used financial option positions, allowing for the consideration of a variety of participants’ risks without relying on the utility function. Klos et al. (2022) proposed determining vertical equity based on progressivity measures. Since these measures were based on Lorenz curves, the question of how re-ranking effects could be accounted for when comparing two groups or scenarios remained. This issue was addressed by Bonnet et al. (2022) and Cordova et al. (2022), who applied OB-based RIF decomposition methods. However, they did not provide a conceptual link to actuarial concepts, making the interpretation of the econometric results challenging.

We contribute to the existing literature in three significant ways. First, we establish a conceptual link between actuarial neutrality and the decomposition method. By elucidating the parallels between actuarial neutrality and the Oaxaca–Blinder (OB) decomposition, we facilitate a straightforward interpretation of the econometric results. Additionally, we highlight the similarities between defined benefit (DB) schemes, point systems, and notional accounts, demonstrating the generalizability of the econometric approach to these systems. Second, we address the empirical limitation of the OB decomposition, which is restricted to linear functions, and explore how the nonparametric framework introduced by Firpo et al. (2009, 2018) can be employed to perform a decomposition on any distributional statistic. This approach utilizes the recentered influence function (RIF) to assess the extent of vertical equity in non-linear functions. Third, we apply this measure to the Swiss public pension system (Old Age and Survivors’ Insurance, OASI). We focus on the following three effects, corresponding to our three hypotheses: (a) We showcase how the pension reform titled “For a Better Life in Old Age”, set to be implemented in 2026, impacts solely the accrual rates, as demonstrated by the effects of coefficients (H1). (b) We demonstrate how actuarial neutrality holds when comparing the pension incomes of migrants and non-migrants, as shown in the effects of characteristics (H2). (c) We illustrate the differing extent of solidarity across marital statuses and the complex interplay between the effects of characteristics and coefficients (H3). By addressing migration in earnings-related pension systems, we also contribute to the relatively scant literature related to this (e.g., Fenge and Peglow 2018).

The focus of this study is on the first pillar of the Swiss pension scheme (Old Age and Survivors’ Insurance, OASI), which operates as a DB scheme. Within the framework of this article, we demonstrate how findings from DB schemes can be generalized to point systems and notional accounts, as all these systems utilize individual earnings over various years of one’s career as input variables to calculate pension entitlements. As these systems aim to cover subsistence needs, various elements of vertical equity—such as minimum pensions and care credits—are incorporated into the pension scheme, resulting in a complex and non-linear relationship between individual earnings and pension entitlements. Considering this non-linearity, we illustrate how the non-parametric framework introduced by Firpo et al. (2009, 2018) can be employed to perform a decomposition on any distributional statistic to assess the extent of vertical equity.

To assess the extent of vertical equity of the reform proposal and across different socio-economic groups, we apply a dynamic microsimulation model (DMSM) of the Swiss pension system. This approach allows us to simulate counterfactual income distributions, such as those resulting from the implementation of the reform proposal, which would otherwise not be available. To ensure the reliability of the results, we use an aligned DMSM called MIDAS_CH (Kirn and Dekkers 2023). MIDAS_CH is part of the MIDAS group of DMSMs, which also includes models in Belgium, Luxembourg, Portugal, and Hungary. These models are well documented and typically used to project, for example, future gender pension gaps or the impact of care activities on pension income (Dekkers et al. 2022; Van den Bosch et al. 2024).

In this article, we apply an OB-based RIF decomposition method, as utilized by Bonnet et al. (2022) and Cordova et al. (2022). However, their studies lacked a conceptual link to actuarial concepts, complicating the interpretation of the econometric results. Additionally, while Bonnet et al. (2022) and Cordova et al. (2022) focused on the French and German point systems, this study provides valuable insights into a DB scheme. This distinction is particularly noteworthy, as the German and French systems have been classified as continental Bismarckian welfare states, which adhere to the principle of insurance (OECD 2019). In contrast, the Swiss public pension system has been classified as a Beveridgean pension system, with a stronger emphasis on solidarity (Fuest et al. 2010).

The remainder of this study is organized as follows. Section 2 defines the concept of actuarial neutrality and presents a generalized neutrality condition. Section 3 establishes a conceptual link between actuarial neutrality and the OB decomposition to empirically assess vertical equity. Furthermore, we discuss the RIF-based decomposition method. To illustrate this approach, it is applied to the first pillar of the Swiss pension system, as described in Section 4. Section 5 introduces the microsimulation model used to simulate counterfactual pension income. Section 6 presents the results, and Section 7 concludes the article.

2. Actuarial Concepts: Actuarial Fairness and Actuarial Neutrality

Pension systems can be characterized by two actuarial concepts—actuarial fairness and actuarial neutrality. Actuarial fairness ensures that the present value of an individual’s lifetime contributions equals the present value of benefits, meaning that there is no vertical redistribution. Thus, the concept of actuarial fairness can only be applied if the pension is financed solely by contributions, such as funded DC schemes.

Schemes partly financed through general government revenues or schemes that do not have an earmarked “pension” contribution, such as public earnings-related pension plans, cannot be assessed for actuarial fairness. Therefore, in the context of this article, we refer to the concept of actuarial neutrality. As earning-related pension plans, such as DB schemes, point systems, and notional accounts, are closely related, the concept presented here could be applied without a loss of generality to those three kinds of public schemes.1

According to Queisser and Whitehouse (2006), actuarial neutrality requires that if an individual retires one year earlier, their pension benefit should be reduced by the amount they would have accrued during that year, along with an adjustment for the extended duration over which the pension will be paid. In earnings-related pension schemes, the relevant income $X_{t|t}$ measured at time t, conditional on drawing the pension at time t, is calculated as the pension entitlement $Y_t$ at time t multiplied by the annuity factor $A_t$ (the derivation of the annuity factor can be found in Appendix A).

$$X_{t|t}=Y_t{A}_t,X_{t+1|t+1}=Y_{t+1}{A}_{t+1}$$

(1)

As the annuity rate $a_t$ is the inverse of the annuity factor (Equation (A4), Appendix A), Equation (1) is equal to the following:

$$Y_t=X_{t|t}{a}_t,Y_{t+1}=X_{t+1|t+1}{a}_{t+1}$$

(2)

Therefore, the condition for actuarial neutrality is as follows:

$$X_{t|t+1}=X_{t+1|t+1}PVP{F}_{t+1}/PVP{F}_t$$

(3)

where $PVPF$ is the present value of the flow of pension payments (Queisser and Whitehouse 2006). The definition of $PVPF$ can be found in Equation (A3), Appendix A.

When considering retirement at the statutory retirement age (SRA), actuarial neutrality ensures that individuals $m,n$ with equal relevant income $X_{t|t}$ at time t, as well as equal life expectancies, receive similar pension benefits $Y_{t|m,n}$. Given the expression $Y_t=X_{t|t}{a}_t$ from Equation (2), the condition for actuarial neutrality in earnings-related pension schemes is expressed as follows:

$$Y_{tm}=Y_{tn}+(X_{tm|t}-X_{tn|t})a_t$$

(4)

As earnings-related pension schemes are characterized by lower benefit limits $\left(a_0\right)$ and income dependent accrual rates $\left(a_t\left(X_{t|t}\right)\right)$, the neutrality condition (4) can be generalized as follows:

$$Y_{tm}=Y_{tn}+(a_{0m}-a_{0n})+X_{tn|t}(a_{tm}-a_{tn})+(X_{tm|t}-X_{tn|t})a_{tn}$$

(5)

This condition holds if all individuals receive the same minimum benefit $(a_{0m}=a_{0n})$ and have the same pension accrual rate $(X_{tn|t}(a_{tm}-a_{tn})=0)$, assuming that equal pension wealth is acquired $(X_{tm|t}=X_{tn|t})$.

3. Econometric Approach

Earnings-related pension plans can be estimated based on (2) by separate linear pension regressions for individuals $m,n$ in groups A and B (see Section 4.2 for a generalisation of this formula), as expressed in the following equation:

$$\begin{array}{c}\hfill Y_{m,n}=X_{m,n}^{\prime }{\alpha }_{m,n}+{ϵ}_{m,n},E\left({ϵ}_l\right)=0,m\in A,n\in B.\end{array}$$

As we aim to decompose the impact of the various income components, which are summed to determine the relevant income $X_{t|t}$, X represents a vector of covariates related to pension income, such as labor income under pension insurance, care credits, and potential split income between spouses along with a constant term.2 The vector $\alpha$ contains the slope parameters (corresponding to the accrual rate a) and the intercept, while $ϵ$ represents the error term.

Therefore, the mean outcome difference in pension income between the two groups can be expressed as the difference in the linear prediction at the group-specific means of the regressors. This relationship is given by the following:

$$E\left(Y_A\right)-E\left(Y_B\right)=E{\left(X_A\right)}^{\prime }{\alpha }_A-E{\left(X_B\right)}^{\prime }{\alpha }_B$$

(6)

Considering the expected values, we have the following equation:

$$E\left(Y_{m,n}\right)=E(X_{m,n}^{\prime }{\alpha }_{m,n}+{ϵ}_{m,n})=E\left(X_{m,n}^{\prime }{\alpha }_{m,n}\right)+E\left({ϵ}_{m,n}\right)=E{\left(X_{m,n}\right)}^{\prime }{\alpha }_{m,n}$$

(7)

where $E\left({\alpha }_{m,n}\right)={\alpha }_{m,n}$ and $E\left({ϵ}_{m,n}\right)=0$.

To identify the contribution of group differences in predictors to the overall outcome difference, Equation (6) could be reformulated as follows to decompose the difference in the means of the pension incomes ${\overline{Y}}_l$ between two groups ($A,B$) by the OB decomposition (Oaxaca 1973; Blinder 1973):

$${\overline{Y}}_A-{\overline{Y}}_B={({\overline{X}}_A-{\overline{X}}_B)}^{\prime }{\widehat{\alpha }}_A+{\overline{X}}_B^{\prime }({\widehat{\alpha }}_A-{\widehat{\alpha }}_B)$$

(8)

where ${\widehat{\alpha }}_A$ and ${\widehat{\alpha }}_B$ are the least squares estimates for ${\alpha }_A$ and ${\alpha }_B$, obtained separately from the two group specific samples. Furthermore, the group means ${\overline{X}}_A$ and ${\overline{X}}_B$ are used as estimates for $E\left(X_A\right),E\left(X_B\right)$.3

Since the OB-based decomposition approach (8) aligns with the generalized condition for actuarial neutrality in earnings-related pension plans (5), it can be applied to decompose pension differentials between two groups. The first summand of Equation (8) is as follows:

$$\widehat{Q}={({\overline{X}}_A-{\overline{X}}_B)}^{\prime }{\widehat{\alpha }}_A$$

(9)

It represents the portion of the outcome differential explained by group differences in the predictors (effects of characteristics). The second component (effect of coefficients) of (8) is expressed as follows:

$$\widehat{U}={\overline{X}}_B^{\prime }({\widehat{\alpha }}_A-{\widehat{\alpha }}_B)$$

(10)

It is typically attributed to discrimination, but in the context of this article, it reflects the differences in pension incomes due to variations in the accrual rates.

The detailed contributions of individual predictors are often of greater interest than the total decomposition alone. For example, one might wish to assess how much of the gender differences in pension income are attributable to differences in income between the sexes versus differences in the contribution years. Similarly, it could be insightful to identify how much of the unexplained gap is due to varying returns (accrual rates) to incomes and how much is due to differing accrual rates to contribution years. Given that the total component is the sum of individual contributions, the effects of characteristics (9) can be represented as follows:

$${({\overline{X}}_A-{\overline{X}}_B)}^{\prime }{\widehat{\alpha }}_A={({\overline{X}}_{1A}-{\overline{X}}_{1B})}^{\prime }{\widehat{\alpha }}_{1A}+{({\overline{X}}_{2A}-{\overline{X}}_{2B})}^{\prime }{\widehat{\alpha }}_{2A}+\dots$$

(11)

where ${\overline{X}}_1,{\overline{X}}_2,\dots$ are the means of the individual regressors, and ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2,\dots$ are the corresponding coefficients. Consequently, the individual contributions to the effects of coefficients (10) are the summands in the following equation:

$${\overline{X}}_B^{\prime }({\widehat{\alpha }}_A-{\widehat{\alpha }}_B)={\overline{X}}_{1B}^{\prime }({\widehat{\alpha }}_{1A}-{\widehat{\alpha }}_{1B})+{\overline{X}}_{2B}^{\prime }({\widehat{\alpha }}_{2A}-{\widehat{\alpha }}_{2B})+\dots$$

(12)

However, since the OB decomposition is not applicable to non-linear functions, we employ the decomposition method proposed by Firpo et al. (2009, 2018). Fortunately, this method allows us not only to apply it to nonlinear functions, such as those related to pensions, but also to extend the methodology to various distributional moments beyond the mean. Firpo et al. (2009) proposed a two-stage procedure that first decomposed distributional changes into an effect of coefficients and an effect of characteristics using a reweighting method. In the second stage, the two components were decomposed into each explanatory variable using the Recentered Influence Function (RIF), just as in a standard OB decomposition.

In doing so, the RIF regression would provide a linear approximation to a non-linear functional form of the distribution. If we consider the dependent variable Y (here, the logarithm of the pension income) with the distribution $F_y$ and marginal density $f_y(·)$, and let $IF(y;v)$ be the empirical distribution for an observed y for the statistic $\upsilon \left(F_y\right)$, the RIF can be defined as the sum of the distributional statistic and the influence function, as in the following equation:

$$RIF(y;\upsilon )=\upsilon \left(F_y\right)+IF(y;\upsilon )$$

(13)

For a given quantile $Q_{\tau }$, the influence function $IF(y;Q_{\tau },F)$ is given by the following:

$$IF(y;Q_{\tau },F)={\displaystyle \frac{\tau -\Pi (y<Q_{\tau })}{f_y\left(Q_{\tau }\right)}}$$

(14)

Thus, the RIF of the quantile $\tau$ is as follows:

$$\begin{array}{cc}\hfill RIF(y;Q_{\tau },F)& =Q_{\tau }+IF(y;Q_{\tau },F)=Q_{\tau }+{\displaystyle \frac{\tau -\Pi (y<Q_{\tau })}{f_y\left(Q_{\tau }\right)}}\hfill \\ \hfill RIF(y;Q_{\tau },F)& ={\alpha }_0+{\alpha }_j{X}_i+{\epsilon }_i\hfill \end{array}$$

(15)

where $Q_{\tau }$ aligns with ${\alpha }_0$ (Firpo et al. 2009).

Given its similarities to the standard OB decomposition, the effects of characteristics can be distinguished for each component and interpreted as a policy effect of changing one covariate from one level to another, maintaining the differences in the composition of the observable characteristics (Firpo et al. 2018). The effects of characteristics in the case of quantiles then yield a partial effect of a small location shift in the distribution of covariates on the distributional statistic of interest. This is also known as the Unconditional Quantile Partial Effect (UQPE) (Firpo et al. 2018).

Although the effect of coefficients is also very similar to the standard OB decomposition, there is one important difference. Because the RIF function depends on the distribution of Y, changing the distribution of X changes the distribution of Y and thus the value of $RIF(Y;\upsilon ,F)$ for a given value of Y, which also affects the coefficients in a regression of RIF. However, this problem can be addressed by using a reweighting procedure. Since the effect of coefficients reflects the differences attributed to the relationships between the dependent variable (log of pension income) and the independent variable (including the log of incomes subject to OASI), it reflects how contributions are valued in determining pension income and therefore refers to interpersonal solidarity.4

4. Institutional Background

We focus on the Swiss Old Age and Survivors’ Insurance (OASI) system, which is an element of the three-pillared pension system. The first pillar comprises the Old Age and Survivors’ Insurance (OASI) and the disability insurance, in conjunction with supplementary benefits, forming a pay-as-you-go system that is mandatory for all residents of Switzerland, regardless of their employment status. Therefore, it can be classified as a Beveridge system (Ignacio Conde-Ruiz and Profeta 2007). The second pillar is occupational pension provision, which is capital-funded (DC scheme), and the third pillar comprises tax-advantaged private savings (Kirn and Dekkers 2023).

4.1. The Swiss OASI System

The Swiss OASI system is organized as a defined benefit (DB) plan. The pension amount is determined by the “relevant average annual income” and the years of contribution. The “relevant average annual income” includes income from dependent and self-employed work, contributions from non-employed individuals, split income (income divided between spouses), and credits for childcare and caregiving. Since non-employed persons are also subject to compulsory insurance, missing contribution years are avoided, preventing a reduction in pension income. Furthermore, the coverage of non-employed individuals ensures universal coverage—a hallmark of Beveridge systems (Disney 2004).

The contribution rate for employees is 8.7% of the gross salary for employees and 8.1% for the self-employed. Non-employed individuals must also pay the minimum contribution of CHF 503 per year. The obligation to contribute begins at the age of 20 and ends once the reference age (formerly known as retirement age) is reached. About 20% of OASI’s expenses are covered by tax revenues. These funds come from general tax revenues, as well as special levies such as tobacco and alcohol taxes.

4.1.1. The Pension Formula

Figure 1 illustrates how the average sum of income subject to the OASI is converted into pension income. The solid line reflects the current state (OASI (year 2025), left y-axis), and the replacement rates (gray lines, right y-axis) are measured as gross pension income.

The OASI is designed to “cover subsistence needs,” which is another characteristic of Beveridge systems. However, only the maximum pension corresponds to the subsistence level (Figure 1, dotted line, left y-axis). The income range for which a flat-rate minimum pension (CHF 15,120) is paid is comparatively small. The range in which the income-dependent component applies covers the incomes up to just above the gross median wage (CHF 84,792 in 2022; FSO 2023).

For incomes below the minimum pension, the replacement rate is at least 100%, as indicated by the gray solid line. This demonstrates that the redistribution effect is most pronounced for incomes below the minimum pension. Due to the constraints of minimum and maximum pension benefits, the replacement rates in the OASI (gray lines, right y-axis, Figure 1) exhibit non-linearity. This nonlinearity reflects the varying accrual rates across different income levels $\left(a_t\left(X_{t|t}\right)\right)$ (Equation (5)), embodying vertical equity (interpersonal solidarity).

Figure 1 compares the current legal framework (solid lines) with the changes proposed under the reform titled “For a Better Life in Old Age”, set to be implemented in 2026, represented by dashed lines. The proposal suggests transitioning to disbursing the first pillar pension 13 times annually, effectively increasing the annual pension income by 13/12, or 8.33%. This uniform increase results in a vertical shift in the minimum and maximum pension and a rotation of the curve (Figure 1 black dashed line). Consequently, the amount of pension accrued per unit of OASI-liable income will increase within the income range between the minimum and maximum pension, leading to a higher degree of vertical equity, which is analyzed in Section 6.2.

A reform generally begins as part of a popular initiative, an instrument of direct democracy in Switzerland involving several steps (e.g., Stadelmann-Steffen and Leemann 2024). The first step requires a collection of signatures to launch the initiative, which, in this context, was initiated by the Swiss Trade Union Federation. In the next step, the Federal Council5 would formulate a counterproposal to the initiative (which they did not in this case) and a message to the Federal Assembly.6 In this case, the Federal Council rejected the initiative. They argued that the additional benefits would further impair the financial stability of the OASI. Furthermore, the proposed 13th OASI pension would lead to inequities, as old-age pensioners would benefit both in terms of the annual pension amount and supplementary benefits, while the benefits for recipients of disability or survivors’ pensions would be calculated at lower rates. Instead of an expensive expansion of benefits for everyone, the Federal Council typically prefers targeted improvements in old-age provisions for insured persons with lower incomes (Swiss Federal Chancellery 2025). The next step is the parliamentary deliberation; however, the National Council and the Council of States rejected the initiative. The final step is typically the popular vote. This was approved by 58.2% of the eligible voters and accepted by 14 2/2 cantons (SwissVotes 2025).7 When a popular initiative in Switzerland is accepted by the electorate, it must be implemented by the Federal Council, which is scheduled for 2026 in this case.

The referendum for a 13th OASI pension was primarily accepted due to the escalating financial pressures faced by pensioners in Switzerland. Proponents argued that the rising cost of living, including rent and health insurance, rendered the existing pension schemes inadequate. They emphasized that recent reforms had bolstered the financial stability of the OASI, thereby making the introduction of the 13th pension payment feasible (Swiss Federal Council 2024). The initiative received widespread support from various political groups, including the Social Democratic Party, Green Party, and Alternative Left (SwissVotes 2025). Notably, it also garnered backing from the right-wing conservative Swiss People’s Party (SVP), which is atypical for social expansion initiatives. This support was driven by a combination of economic concerns and the desire to improve the quality of life of retirees. Additionally, an argument that resonated strongly within conservative circles, particularly in German-speaking Switzerland, suggested that if Swiss authorities were capable of allocating substantial financial resources towards development aid and the accommodation of refugees, then it would be equally justifiable to allocate similar resources to support pensioners. Consequently, an initiative originating from the left and the unions has never before garnered such significant support among right-wing voters (Stadelmann-Steffen and Leemann 2024).

4.1.2. Additional Elements of Vertical Equity

When interpreting Figure 1, it is important to note that the replacement rates are based on a constant income and an uninterrupted employment history of 44 years. Fragmented employment histories lead to lower replacement rates. To mitigate the impact of fragmented employment due to unpaid care, solidarity mechanisms are implemented for caregivers in the form of care credits. The notional income of these care credits is linked to three times the minimum annual pension at the time of pension entitlement, equating to a notional income of CHF 45,360 per year of care activity. This amount represents approximately 60% of the median gross income for women, which was CHF 76,796 per year in 2022 (FSO 2023). Consequently, individuals with incomes below the notional income threshold benefit more from care credits than those with higher incomes, resulting in a regressive compensating effect of care credits (Van den Bosch et al. 2024).

A second important redistributive element in the OASI system is spousal solidarity. Contributions made during marriage are split equally between partners when the last of both partners retires or upon divorce. In the case of retirement, the combined pension of the married and newly retired couple is capped at 150% of the maximum individual pension, resulting in a proportional reduction in each spouse’s pension income. The higher the combined pension income of the spouses, the greater the absolute reduction, leading to progressive redistribution. Additionally, the contribution years of both spouses are equalized using a weighted average, with the partner having the longer career receiving a higher weight in this calculation. This equalization process benefits the spouse with fewer contribution years.

Furthermore, we include the inactivity contributions in our analysis. These minimum contributions are made during spells of economic inactivity to prevent gaps in contribution years. If single persons are not salaried, they must pay these contributions themselves. As these minimum contributions ensure that a minimum pension can be achieved, they are a form of interpersonal solidarity.

Another element of solidarity within the OASI pension system is solidarity with widowed persons, which is implemented through survivors’ pensions, designed to prevent financial hardship for survivors (spouses, children) upon the death of a spouse or parent. If the surviving spouse retires, his or her pension will be the higher of the following: (i) their own pension increased by a survivors’ supplement of 20%, or (ii) 80% of the pension benefit of the deceased partner.8

The additional income components (such as care credits, split income, etc.) influence the extent of vertical equity through two primary channels. First, as the amount of those income contributions affect the total relevant income X (see Figure 2), it contributes to the effects of characteristics ${({\overline{X}}_{iA}-{\overline{X}}_{iB})}^{\prime }{\widehat{\alpha }}_{iA}$ (Equation (11)). Second, as the pension accrual rate is income-dependent $\left(a_t\left(X_{t|t}\right)\right)$, those income contributions also contribute to the effect of coefficients ${\overline{X}}_{iB}^{\prime }({\widehat{\alpha }}_{iA}-{\widehat{\alpha }}_{iB})$ (Equation (12)). For example, given the regressivity of care credits, the effect of coefficients may become negative in the upper income deciles but remains positive in the lower income deciles.

The economic impact of the solidarity instruments is shown by the average annual retirement pension based on marital status (

Table 1. Average annual first pillar retirement income by marital status (2022, in Swiss francs).

	Single	Married		Divorced	Widowed	Overall
		One Spouse Retired	Both Retired
Women	22,932	18,444	20,184	23,364	26,316	22,608
Men	22,572	24,072	20,808	23,904	26,844	22,344
$\Delta$ pp	1.6%	−23.4%	−3.0%	−2.3%	−2.0%	1.2%
in % of maximal first pillar pension
Women	80.0%	64.3%	70.4%	81.5%	91.8%	78.8%
Men	78.7%	83.9%	72.6%	83.3%	93.6%	77.9%

Source: BSV, 2023. Note: In 2022, the maximum first pillar pension is CHF 28,680 per year. $\Delta$ pp denotes the difference in pension income in percentage points ((pension income of women/pension income of men) − 1).

). For single individuals (column 2), the pensions for women and men are similar, based on their own contributions and credits. For married individuals with a single-pension income (column 3), the women’s pensions (CHF 18,444 per year) are about 23% lower than that of the men. This disparity likely arises because spousal solidarity instruments, like income splitting and career equalization, are not applied in the “1st insurance case”. The differences are mainly due to the lower contributions by married women. When both spouses receive an old-age pension (2nd insurance case, column 4), compensatory splitting and capping reduce pension differences to 3%. In divorce cases (column 5), where income splitting is applied but the cap is not, pension levels for both genders are higher, with smaller differences. Higher pensions result from the absence of a cap and increased labor market participation of divorced women, leading to higher OASI contributions and reduced gender differences. Widowed persons’ pensions (column 6) are similar to those of married and divorced individuals due to interfamilial solidarity.

4.1.3. Determination of Pension Income

Figure 2 illustrates how incomes subject to the OASI (labor income, unemployment benefits, disability and survivors’ pensions) and variables that correspond to the instruments of solidarity (such as inactivity contributions, care credits, and income splitting) are combined to determine the total relevant income (step 1) in a particular year $x_j$ (indexed j).

This income is uprated with the mixed index u to take wage and price increases into account, and then divided by the number of contribution years to determine the average relevant income (step 2). The third step uses a conversion table to convert this average relevant income into pension income by an accrual rate a for each year of service. This is the so-called first insurance case. Finally, step 4 adjusts the resulting pension income according to personal circumstances (such as divorce, widowhood, or the capping if both spouses are retired) to implement rules for solidarity with widowed persons and solidarity between the spouses. Thus, the pension benefit in DB schemes $Y_{DB}$ (before the adjustments according to personal circumstances take place (step 4)) can be written as follows:

$$Y_{DB}=\sum _{j=0}^R{x}_{j}a{(1+u)}^{R-j}$$

(16)

where R is the year of retirement; $x_j$ total relevant income in a particular year; and $a_j$ the accrual rate for each year of service (Queisser and Whitehouse 2006).

4.2. Relevance

The findings from the DB schemes can be generalized to point systems and notional accounts, as all these systems use individual earnings over various years of an individual’s career as input variables to calculate pension entitlements.

In point systems, the pension benefit $\left(Y_{PP}\right)$ depends on the value of a point $\left(v\right)$ and is divided by the cost of the pension point $\left(k\right)$ in a given year, with $\left(p\right)$ representing the uprating factor as follows:

$$Y_{PP}=\sum _{j=0}^R{\displaystyle \frac{x_j{v}_j}{k_j}}{(1+p)}^{R-j}$$

(17)

For notional accounts, the pension benefit $\left(Y_{NA}\right)$ is based on the inflow, which is the product of wages and the contribution rate $\left(x_{j}c\right)$. The notional capital increases each year by the notional interest rate $\left(\eta \right)$. Upon retirement, the accumulated notional capital is divided by a notional annuity factor A as follows:

$$Y_{NA}=\sum _{j=0}^R{\displaystyle \frac{x_{j}c}{A}}{(1+\eta )}^{R-j}$$

(18)

Queisser and Whitehouse (2006) concluded that if the uprating procedures for the DB schemes, pension points, and notional interest rates were the same $(u=p=\eta )$, the structure of these three equations would be very similar. In this case, the accrual rate of the DB $a_j$ is equivalent to the ratio of the value of the pension point to its cost $(v_j/k_j)$ and the ratio of the notional account contribution rate to the annuity factor $(c/A)$.

Given the similarities between the DB schemes, points systems, and notional accounts, the functional relationship of the neutrality condition (4) and the generalized neutrality condition (5) is applicable to all those schemes, and the decomposition approach can be utilized. Then, depending on the system analyzed, the estimated regression coefficient $\alpha$ corresponds to the DB accrual rate $a_j$ in the case of DB schemes, the ratio of the pension point value corresponds to its cost $(v_j/k_j)$ if point systems are analyzed, and the ratio of the contribution rate corresponds to the annuity factor $(c/A)$ for notional accounts.

5. Simulation and Calibration

5.1. Model Description

As this study aims to decompose the differences in pension income into those arising from earnings-based contributions and the cushioning elements of vertical equity, as well as to explore how changes in women’s behavior in the labor market impact the significance of redistributive mechanisms within the OASI, we employ a microsimulation model that allows us to simulate counterfactual life trajectories. By projecting diverse labor market scenarios, we can simulate the future pension incomes for both men and women and decompose the gender disparities in pension income to assess the significance of vertical equity.

We use MIDAS_CH (Microsimulation for the Development of Adequacy and Sustainability) (Dekkers 2013; Kirn and Dekkers 2023), which is a dynamic microsimulation model of the discrete-time, longitudinal-aging type (see Li et al. 2014). In this model, the simulation takes place at the levels of the individual and the household. This approach allows for the inclusion of all non-linearities and redistributive elements in the pension system. Furthermore, it is dynamic in that, starting from the Swiss SILC dataset of 2018, the model creates a longitudinal dataset by simulating the demographic and labor market histories of all individuals in the dataset, up to the simulation horizon in 2070. This is mainly achieved using individual transition probabilities that combine behavioral equations in the form of logistic regressions with “alignment by sorting” techniques (Dekkers et al. 2012; Li and O’Donoghue 2013). In that sense, even though the model is new, it aligns with the traditional models, including SESIM (Sweden), Mosart (Norway), T-DYMM (Italy), MIDAS_BE (Belgium), MIDAS_LU (Luxembourg), Destinie (France), and CBOLT (US).

Similar to most datasets, the SILC data for Switzerland include weights to address the over- or under-sampling of specific cases, disproportionate stratification, and survey non-response (Verma et al. 2006). Consequently, frequency weights must be applied in DMSM to achieve unbiased results. One straightforward method to establish an accurate DMSM is to expand the initial dataset using the frequency weights and then sample from the expanded dataset. However, since the weighted objects need to be aligned with external constraints, such as projected demographic developments, the interaction between alignment and weighting procedures must be considered to avoid simulation errors. Therefore, we follow the approach proposed by Dekkers and Cumpston (2012), combining a shared weights approach with the alignment of external constraints and a carrying forward algorithm to address any misalignment. Through this approach, the simulation results are adjusted to match the official demographic projections by the FSO (e.g., FSO 2017, 2020b). Additionally, other relevant official projections on household formation and dissolution, labor market participation, unemployment, caregiving rates, part-time work rates, earnings growth rates, and more are used to align the simulation results of MIDAS_CH.

Table A1. Data sources utilized for alignment procedures.

Alignment	Source	Data	Dimensions	Age Groups	Scenario Variants
Mortality	FSO	BEVNAT, ESPOP, STATPOP	gender	year	Scenario A-00-2020
Births	FSO	BEVNAT, ESPOP, STATPOP	female	year	Scenario A-00-2020
Population	FSO	BEVNAT, ESPOP, STATPOP	gender	5-year brackets	Scenario A-00-2020
Fertility	FSO	BEVNAT, ESPOP, STATPOP	female	year	Scenario A-00-2020
Marriage	FSO	BEVNAT	gender, civil state	5-year brackets
Divorce	FSO	BEVNAT	gender	5-year brackets
Disability	FSO	IV	gender	5-year brackets
Care	FSO	SGB	gender	year
Care giving	FSO	SGB	gender	year
Early retirement	FSO	NRS	gender	year
Labor market participation	FSO	SAKE	gender	year
Part-time	FSO	SAKE	gender	5-year brackets	3 categories of PT work
Unemployment	FSO	SAKE	gender	5-year brackets
Immigration	FSO	STATPOP	gender, nationality	Year	Scenario A-00-2020
Emigration	FSO	STATPOP	gender, nationality	Year	Scenario A-00-2020

Notes: BEVNAT = Statistics of Natural Population Movement (Statistik der natürlichen Bevölkerungsbewegung); ESPOP = Statistics of the Annual Population Status (Statistik des jährlichen Bevölkerungsstandes); STATPOP = Statistics of Population and Households (Statistik der Bevölkerung und der Haushalte); SAKE = Swiss Labour Force Survey (Schweizerische Arbeitskräfteerhebung); IV = Disability Insurance Statistics (IV-Statistik); NRS = New Pensions Statistics (Neurentenstatistik); SGB = Swiss Health Survey (Schweizerische Gesundheitsbefragung).

in Appendix B provides an overview of the data sources utilized for the alignment procedures.

5.2. Simulation and Alignment

The objective of a dynamic microsimulation model (DMSM) such as MIDAS_CH is to produce long-term pension projections. Given that pensions are influenced by factors like historical earnings, labor market participation, and residency status, it is essential to simulate the intertemporal evolution of various characteristics, including demographic aging, household size, marital status, and activity rates. This is typically accomplished using discrete-dependent variable models, such as logistic regressions. However, if these models are considered sub-optimal or if researchers prefer to leverage existing projections to ensure consistency and avoid redundancy, alignment techniques are often employed. Various alignment methods exist, but Li and O’Donoghue (2014) and Li et al. (2014) found that for simulation models focusing on distributional impacts, the sidewalk hybrid and SBDL methods9 yielded the best results. Due to the low computational efficiency of sidewalk hybrid methods, the “alignment by sorting” approach of the SBDL method has been widely used in complex microsimulation models, including MIDAS_CH.

To ensure consistency with the exogenous macroeconomic forecasts, the earnings are indexed and the macroeconomic variables are adjusted. This process aligns simulated micro-level earnings with macroeconomic wage growth projections. Furthermore, the thresholds for the pension of the first pillar are adjusted using the arithmetic mean of the nominal wage index and the consumer price index, also known as the mixed index.

The forecast for the short-term wage–price index relies on SECO’s 2022 projection. Due to the lack of official long-term estimates for the wage and price index, it is presumed that the future growth rates of nominal wages will align with the average annual growth rates observed over the past two decades, as reflected by the nominal wage index (FSO 2022a). Furthermore, it is assumed that real wage growth will match the inflation rate, ensuring complete inflation compensation. Figure 3 presents the nominal and real wage indices based on official statistics (with 1993 as the base year), along with the projected nominal and real wage indices, the consumer price index, and the mixed index.

5.3. Simulation of Population Projections and Labor Market Dynamics

In our population simulation, we utilize the population projections from the official statistics. The Federal Statistical Office (FSO) in Switzerland bases its projections on various hypotheses regarding the future trends in fertility, mortality, and migration. The projection method, known as the component method or population balance, determines the annual population size by accounting for natural demographic events (births and deaths) and spatial movements (immigration and emigration) (FSO 2020b). The reference scenario (A-00-2020) continues the trends observed in recent years and covers the period up to 2050 (FSO 2020b). For our simulation, we adopt the assumptions of the reference scenario and extend the projections to the year 2070. In the following sections, we provide a detailed description of the components of the population balance, focusing on demographic events (Section 5.3.1) and spatial movements (Section 5.3.2). Given that demographic changes also impact the working population, the simulation of labor market participation—an essential determinant of future pension income—is detailed in Section 5.3.3.

5.3.1. Fertility and Mortality

In Switzerland, the trend in fertility has shifted toward later childbearing over the past three decades. The number of young mothers has decreased significantly, while the proportion of mothers aged 30–34 increased steadily from the early 1970s until 2001. The share of mothers aged 35 and older continues to rise. According to the FSO (2020b) reference scenario (A-00-2020), this trend is expected to persist, with the probability of giving birth peaking in the early 30s and remaining relatively high until the mid-30s. The projections indicate that total fertility rates (TFRs) will remain stable at around 1.5 children per woman, with a continued shift towards older maternal ages (FSO 2020b).

The shift in the fertility patterns and trends across different age groups is illustrated by Figure 4. It illustrates the likelihood that a woman of a specific age will give birth within a given year. This “age-specific fertility rate” (ASFR) is calculated as the number of births to women of a specific age group per 1000 women in that age group per year.

The Total Fertility Rate (TFR) for Swiss and foreign national mothers born in Switzerland is similar, but it is higher for women born abroad (FSO 2020a; Rojas et al. 2018). Variations in the fertility rates are affected by levels of educational attainment, as prolonged education postpones the start of families (Van Hek et al. 2015) and raises the opportunity costs linked to career breaks (Gustafsson 2001). While the majority of children are still born within marriages, the incidence of non-marital births has surged more than fivefold since 1970. These non-marital births predominantly occur among single women, most of whom live in consensual partnerships (FSO 2020a). To capture these fertility differences, the SDBL alignment method simulates the a priori probability of giving birth using a logistic regression model that incorporates cohabitation and marriage status, educational attainment level, age, and the number of siblings as variables. The probability of giving birth by age (see Figure 4) is used to align the a priori probability with projected fertility rates.

Regarding mortality, we proceed similarly as we do for life expectancy. Switzerland boasts one of the highest life expectancies at birth globally. In 2019, the average life expectancy at birth was 81.9 years for men and 85.9 years for women. This upward trend is also evident in the remaining life expectancy at age 65, which has seen substantial growth (FSO 2017). Due to the partially observed mortality at older ages, it can be assumed that the next generations will also live significantly longer. The average remaining lifespan of women and men born in 1952, who turned 65 in 2017, is expected to be a little over 21 years and just under 25 years, respectively. Men and women born in 2017 are likely to live an average of 28 and 30 years, respectively, after their 65th birthday (FSO 2022b).

To account for this increase in life expectancy, the projected probabilities of mortality by age for both women and men have been incorporated into the modeling of mortality risks. Figure 5 illustrates the probability of death for men and women by age in different decades. This indicator shows the likelihood that a man or woman of a certain age will die before reaching a specific future age under the current mortality conditions.

The risk of death for both genders increases rapidly with age; between 20 and 30 years, the increase is exponential. For men, just before the age of 60, the risk is 5 per mille, and for women around the age of 55, it is 2 per mille, ten times higher than for 30-year-olds (not shown in the graph). Between 60 and 80, mortality rates increase tenfold again, reaching 50 per mille for 80-year-old men and just over 20 per mille for women. For women between 80 and 95 and men between 80 and 105, the risk increases tenfold once more, to 200 per mille for 95-year-old women and 500 per mille for 105-year-old men. The probability of death for 105-year-old women is the same as for men of the same age (FSO 2022b).

Due to flu epidemics and heatwaves, the development of life expectancy in Switzerland has been more irregular in recent years than in the past. Although the increase in life expectancy in Switzerland is slowing down, there are many indications that it could increase significantly again in the coming decades. According to the reference hypothesis, the increase in life expectancy is being slowed. As gender differences in health behavior continue to decrease, the difference in life expectancy between women and men is also narrowing (FSO 2020b).

5.3.2. Immigration and Emigration

Switzerland is a country characterized by significant migration, necessitating the MIDAS_CH model to account for both immigration and emigration. Annually, approximately 145,000 people immigrate to Switzerland, while around 126,000 emigrate. In 2019, Switzerland, with a population of approximately 8 million, experienced around 17 immigrations and 15 emigrations per 1000 residents. This migration rate surpasses that of other similarly sized Western European nations, including Austria (12/8 per 1000), Belgium (13/9 per 1000), and the Netherlands (12/6 per 1000). In 2018, immigrants made up 29.7% of Switzerland’s population, one of the highest percentages in Europe, surpassed only by Luxembourg, excluding micro-states like Andorra and Liechtenstein.

The immigrant profile in Switzerland shows some differences compared to the EU countries. As indicated by the Swiss Mobility Migration Survey, professional reasons are the primary driver for a majority of the immigrations to Switzerland (Steiner and Wanner 2019). Approximately 73% of immigrants benefit from the ‘Agreement on the Free Movement of Persons’ (AFMP) with the EU. Other categories of migrants, including labor migrants from non-EU countries (1.7%), accompanying family members (17%), and humanitarian migrants (5.5%) (OECD 2020), play a less significant role.

Two main challenges emerge when modeling immigration. Firstly, the characteristics of new immigrants, such as the panel history and weighting information required for the shared weights approach (Schonlau et al. 2013), are unknown. This issue can be addressed using the donor approach (Duleep and Dowhan 2008), where the existing individuals in a dataset are used to represent potential immigrants. The donor sample is then duplicated and incorporated into the dataset. Secondly, immigrants often arrive as households, whereas alignment tables are typically expressed in individual numbers. To address this, we follow the method outlined by Chénard (2000).

Given that the sociodemographic characteristics of immigrants vary by their country of origin (FSO 2019), we categorize individuals in the model into three groups based on their birthplace, namely Switzerland (returning emigrants), the European Economic Area (EEA), and other countries. The alignment process follows the projections of the FSO (2020b) (Scenario A-00-2020), which are designed for these three categories. Figure 6 shows the age distribution of immigrants from these regions. The graph demonstrates that the simulated numbers of immigrants are in line with the projections by the FSO.

The age distribution of immigrants from the three regions of origin exhibits slight variations, with the majority falling within the 25–29 age bracket (see Figure 6). Since the obligation to contribute to the OASI begins at age 20, later migration results in fewer contribution years, leading to lower pension incomes. However, as these reductions are actuarially fair, we hypothesize that the effect on coefficients should be non-significant. The differences in lifetime incomes due to shorter contribution periods should be reflected in the effects of individual characteristics.

5.3.3. Development of Working Population

Modeling labor market participation is a crucial component of MIDAS_CH, as reductions and interruptions in employment significantly impact pension income after retirement. Therefore, the employment and part-time work rates, particularly for women, are of special importance. Although the overall employment rate of women in Switzerland is high, it is often at a reduced level of employment.

To capture the heterogeneity of work intensity, we model it in three steps. First, we determine whether an individual works by combining alignment information with the results of a logistic regression model, which includes variables such as age, civil status, and previous work status (employed, unemployed, others). Next, for those who are employed, we use another logistic regression model (considering gender, number of children under the ages of 11 and 15, civil status, age, and education level) along with alignment information to simulate whether the person works part-time or full-time. In the third and final step, we model the extent of part-time work for individuals chosen to work part-time in the initial step. We categorize them into the following three groups: (i) less than 50% of full-time duration, (ii) between 50% and 69% of full-time duration, and (iii) between 70% and 89% of full-time duration. These categories correspond with the projections from FSO (2020b), which we utilize in our alignment tables (see

Table 2. Projections labor market participation.

	2020	2025	2030	2035	2040	2045	2050	2070
Perm. resident population (mill.)	8.6	9.1	9.4	9.8	10.0	10.2	10.4	11.2
Annual growth in %	0.8	0.8	0.8	0.6	0.5	0.4	0.4	0.4
Due to migration balance in %	0.6	0.6	0.6	0.5	0.4	0.3	0.3	0.3
Due to birth surplus in %	0.3	0.2	0.2	0.2	0.1	0.1	0.1	0.1
Foreigners’ share in %	25.5	26.6	27.9	29.0	29.8	30.5	31.1	33.5
Share of under 15-year-olds in %	15.1	15.1	15.0	14.8	14.6	14.4	14.2	13.8
Share of +65-year-olds and in %	18.9	20.3	22.1	23.6	24.4	25.0	25.6	28.6
Share of +80-Year-Olds in %	5.4	6.2	7.0	7.6	8.5	9.8	10.7	13.7
Age quotient in %	30.9	34.0	38.3	41.8	43.6	44.9	46.5	51.5
Youth quotient in %	32.6	33.7	34.6	35.1	35.1	35.0	35.0	34.5
Overall employment rate in %	58.0	57.0	56.0	55.2	54.7	54.2	53.7	52.4
Employment Rate of 15–64 in %	84.4	84.3	84.3	84.6	84.7	84.6	84.5	84.5
% of +65 y per 20–64-y workers	35.6	39.2	44.0	47.9	49.9	51.5	53.3	57.1

Source: Projections by FSO (2020b) according to reference scenario (A-00-2020) up to 2050, and own projections for years 2050–2070.

).

Table 2 presents the projection of the working population. The development of the working population is influenced not only by demographic changes but also by the employment rate. According to the reference scenario (A-00-2020), the working population aged 15 and older will continuously increase, reaching 5.6 million by the end of 2050 (+11.2% compared to 2020). As a similar trend is expected for both men and women, the proportion of women in the workforce remains unchanged at 46.7%. However, the picture changes when considering the development of the workforce in full-time equivalents; the female working population will grow more significantly than the male working population during the period under study (+17.1% to 2.0 million compared to +7.0% to 2.7 million). This development is attributed to a trend towards higher employment rates among women and increased part-time employment among men. Consequently, the difference in the employment rate between men and women will decrease from 10.9 percentage points in 2020 to 9.2 percentage points in 2050.

According to the reference scenario (A-00-2020), the employment rate of those aged 15 and older will decrease by 5.6 percentage points between 2020 and 2050, reaching 62.7% in 2050. This decline is closely related to the aging population, with a significant increase in the population of retirement age expected during the study period. However, the employment rate of those aged 15 to 64 will remain relatively stable (2020: 84.4%; 2050: 84.5%).

6. Results

Using three empirical examples, we demonstrate the application of the decomposition method and its linkage to the concept of actuarial neutrality. In the first example, we analyze the introduction of a thirteenth monthly pension. Our first hypothesis is that this reform, which only changes the accrual rate, will be reflected solely in the effect of coefficient (Hypothesis 1).

In the second application, we decompose the differences in the pension incomes of immigrants and non-immigrants. Here, we expect that, due to their comparatively shorter contribution periods, immigrants will show a significant effect of characteristics on their pension incomes. Meanwhile, the effect of coefficients will not be significant (Hypothesis 2).

In the third application, we decompose the differences in the pension incomes across marital status. Here, we expect that, due to the complex interactions between various elements of solidarity and the pension formula, the effects of characteristics, as well as the effects of coefficients, will be significant (Hypothesis 3).

6.1. Descriptive Statistics

To decompose the pension income, we measure the pension income and all the other characteristics at the individual level. Therefore, our dependent variable is the individual first pillar pension. As independent variables, we include all the components that are crucial for determining the relevant income used to derive the OASI pension income. We differentiate these components as follows:

• Work History Variables—Aggregated incomes subject to OASI, including labor income and contributions from non-employed persons (e.g., unemployment benefits, disability, and survivors’ pensions);
• Inactivity Contributions—Included as a form of interpersonal solidarity for single persons and spousal solidarity for married persons;
• Sum of Childcare and Care Credits—Used as an instrument of caregiver solidarity;
• Partner’s Pension Assessment Base—Includes incomes subject to OASI, inactivity contributions, childcare, and care credits. The bases of both partners are summed and split for married or divorced partners, serving as an instrument of spousal solidarity;
• Partner’s Contribution Years—Included as they are equalized when determining the pension for married partners with two pensions, serving as an instrument of spousal solidarity.

Table 3. Simulated avg. annual first pillar retirement income by marital status (2070, in Swiss francs).

	Single	Married		Divorced	Widowed	Overall
		One Spouse Retired	Both Retired
Women	34,621	32,206	25,991	32,992	30,431	30,616
Men	36,786	38,990	26,035	35,486	32,197	32,285
$\Delta$ pp	−5.9%	−17.4%	−0.2%	−7.0%	−5.5%	−5.2%
in % of maximal first pillar pension
Women	79.8%	74.2%	59.9%	76.0%	70.1%	70.6%
Men	84.8%	89.9%	60.0%	81.8%	74.2%	74.4%
Observations	17,179	1241	12,166	7862	20,583	59,031

Source: Projected annual pension incomes (first pillar) for the year 2070. The reported scenario is the base scenario. The simulated maximum first pillar pension in 2070 is CHF 43,392 per year. $\Delta$ pp denotes the difference in pension income in percentage points ((pension income of women/pension income of men) − 1).

presents the simulated average annual first pillar pension incomes by marital status for the base scenario in 2070. The simulation results align well with the descriptive statistics in Table 1, indicating that our model provides a reasonable fit.

A comparison of the simulated future pension incomes (Table 3) with the current pension levels (Table 1) reveals that the largest disparities are found among married couples when one spouse is retired. These pensions do not undergo splitting, and married women exhibit lower labor market participation compared to unmarried or divorced women, resulting in comparatively lower pension incomes, even in future projections. However, as female labor market participation is expected to increase, gender disparities in pension income are anticipated to decline. An additional noteworthy finding is the gender disparity in pension income among singles. Contrary to the observed results (Table 1), the pension income difference for single women is negative (−5.9% compared to the observed +1.6%). This discrepancy is attributed to a higher proportion of unmarried mothers (who have comparatively lower labor market participation than women without children) compared to the status quo in the observed data.

6.2. Application 1: Assessing the Impact of the Accrual Rate on Vertical Equity

To demonstrate the policy relevance of the decomposition analysis, we examine the reform initiative “For a Better Life in Old Age,” which will come into effect in 2026. This reform proposes distributing the first pillar pension 13 times per year (hereafter referred to as “13 × OASI 2026”), thereby increasing the annual pension income by 8.33%. This increase is illustrated by the “shift” in the pension formula (see Figure 1, black dashed line).

Since this reform only affects the accrual rate a, we hypothesize that the effect is captured by the effect of coefficients $\left({\overline{X}}_{iB}^{\prime }({\widehat{\alpha }}_{iA}-{\widehat{\alpha }}_{iB})\right)$, according to the generalized neutrality condition (Equation (5)). Here, ${\widehat{\alpha }}_{iA}$ denotes the pension accrual rate under the reform option, and ${\widehat{\alpha }}_{iB}$ represents the rate under the status quo. Thus, we hypothesize that the estimated differential between both accrual rates corresponds to the 8.33% increase in annual pension income.

To analyze the impact of the reform “13 × OASI 2026” on vertical equity, we compare the pension incomes of women and men under the status quo (OASI 2026) and the reform proposal.

Table 4. Decomposition of pension differences at median (singles, in percentage points).

	Decomposition						In Percent
	Coeff.	Robust SE	z	P > |z|	[95% conf. int.]		%	[95% conf. int.]
Men
$\Delta$ pp	0.0800	0.0025	32.30	0.000	0.0752	0.0849
Eff. of char.	0.0000	0.0018	0.00	1.000	−0.0036	0.0036	0	−4.5	4.5
Eff. of coeff.	0.0800	0.0017	47.52	0.000	0.0767	0.0833	100	95.6	106.3
Women
$\Delta$ pp	0.0800	0.0036	22.43	0.000	0.0730	0.0870
Eff. of char.	0.0000	0.0026	0.00	1.000	−0.0050	0.0050	0	−6.3	6.3
Eff. of coeff.	0.0800	0.0025	32.41	0.000	0.0752	0.0849	100	93.7	106.3

Source: Projected annual pension incomes (first pillar) of the year 2070. The pension incomes of singles are reported. Notes: 95% confidence intervals in brackets. The coefficients of the reform scenario (13 × OASI 2026) are used as a reference. $\Delta$ pp denotes the difference in pension income in percentage points ((13 × OASI 26)-OASI 2026).

and

Table 5. Decomposition of pension differences at centiles of income distribution (13 × OASI 2026 – OASI 2026, singles, in percentage points).

	q10		q25		q50		q75		q90
Overall
Men (13 × OASI 2026)	10.361 ***	(0.005)	10.512 ***	(0.003)	10.627 ***	(0.002)	10.706 ***	(0.001)	10.755 ***	(0.001)
Men (OASI 2026)	10.281 ***	(0.005)	10.432 ***	(0.003)	10.547 ***	(0.002)	10.625 ***	(0.001)	10.675 ***	(0.001)
$\Delta$ pp (13 × OASI 26- OASI 26)	0.080 ***	(0.007)	0.080 ***	(0.004)	0.080 ***	(0.002)	0.080 ***	(0.002)	0.080 ***	(0.002)
Effects of characteristics	0.000	(0.005)	0.000	(0.004)	0.000	(0.002)	0.000	(0.001)	0.000	(0.001)
Effects of coefficients	0.080 ***	(0.005)	0.080 ***	(0.003)	0.080 ***	(0.002)	0.080 ***	(0.002)	0.080 ***	(0.001)
Effect of coefficients
Incomes subject to OASI	0.000	(0.405)	0.000	(0.235)	−0.000	(0.162)	−0.120	(0.125)	−0.329 ***	(0.098)
Inactivity contributions	0.000	(0.067)	−0.000	(0.037)	0.000	(0.021)	0.004	(0.015)	0.004	(0.011)
Child care credits	0.000	(0.002)	−0.000	(0.001)	−0.000	(0.001)	−0.000	(0.001)	−0.000	(0.001)
Care credits	0.000	(0.010)	0.000	(0.006)	−0.000	(0.003)	−0.001	(0.002)	−0.003	(0.002)
Contribution years	0.000	(0.138)	0.000	(0.068)	−0.000	(0.046)	−0.019	(0.032)	−0.026	(0.020)
Constant	0.080	(0.358)	0.080	(0.192)	0.080	(0.130)	0.216 *	(0.107)	0.434 ***	(0.092)
Overall
Women (13 × OASI 2026)	10.237 ***	(0.006)	10.411 ***	(0.004)	10.567 ***	(0.003)	10.666 ***	(0.002)	10.724 ***	(0.002)
Women (OASI 2026)	10.157 ***	(0.006)	10.331 ***	(0.004)	10.487 ***	(0.003)	10.586 ***	(0.002)	10.644 ***	(0.002)
$\Delta$ pp (13 × OASI 26- OASI 26)	0.080 ***	(0.009)	0.080 ***	(0.006)	0.080 ***	(0.004)	0.080 ***	(0.003)	0.080 ***	(0.002)
Effect of characteristics	0.000	(0.006)	0.000	(0.005)	0.000	(0.003)	0.000	(0.002)	0.000	(0.001)
Effect of coefficients	0.080 ***	(0.006)	0.080 ***	(0.004)	0.080 ***	(0.002)	0.080 ***	(0.002)	0.080 ***	(0.002)
Effect of coefficients
Incomes subject to OASI	0.000	(0.357)	−0.000	(0.221)	0.000	(0.152)	−0.008	(0.114)	−0.183	(0.102)
Inactivity contributions	−0.000	(0.004)	0.000	(0.002)	0.000	(0.002)	−0.000	(0.001)	−0.002	(0.002)
Child care credits	0.000	(0.004)	−0.000	(0.002)	0.000	(0.002)	−0.000	(0.002)	−0.002	(0.002)
Care credits	0.000	(0.017)	−0.000	(0.009)	0.000	(0.005)	−0.000	(0.003)	−0.002	(0.003)
Contribution years	0.000	(0.139)	−0.000	(0.081)	0.000	(0.053)	0.001	(0.034)	−0.002	(0.025)
Constant	0.080	(0.320)	0.080	(0.195)	0.080	(0.134)	0.087	(0.102)	0.272 **	(0.094)

Source: Projected pension incomes (first pillar) of the year 2070. The pension incomes of singles are reported. Notes: SE in parentheses. * $p<0.05$, ** $p<0.01$, *** $p<0.001$. The coefficients of the reform scenario (13 × OASI 2026) are used as a reference. $\Delta$ pp denotes the difference in pension income in percentage points.

present the results of the RIF decomposition of the simulated pension income at the median and across the income distributions. The dependent variable is the (log) first pillar pension income at the individual level. As independent variables, we consider (the log of) all the variables that affect pension income. The log-log estimation reports changes in the percentage points, with $\Delta$ pp denoting the difference in pension income $X_A-X_B$ in percentage points.10

The computation of the decomposition components is relatively straightforward; however, the derivation of standard errors for these components presents significant challenges. Standard errors are crucial because they provide essential information about the reliability and robustness of the findings, making them indispensable measures of statistical precision. Although the bootstrap technique offers a solution for estimating standard errors, it is computationally intensive and slow. Previous estimators, such as those proposed by Oaxaca and Ransom (1998) and Greene (2003), often produce biased results due to the assumption of fixed regressors. To address these issues, Jann (2008) introduced new, unbiased variance estimators to estimate standard errors in the decomposition analysis.

Table 4 presents the aggregated decomposition results at the median, along with the 95% confidence intervals for the estimated coefficients. As the reform option “13 × OASI 2026” results in an increase of 8.3%, the estimated effect of coefficients is 8% for both men and women. The effects of characteristics (Equation (9)), which reflect differences in pension income due to variations in covariates, are zero because the covariates—such as income subject to the OASI or the amount of care credits—remain unaffected by the reform. The narrow confidence intervals for the effects of coefficients indicate a precise estimate of the regressor’s coefficient. When expressed as a percentage share of the difference in pension income, these differences can be fully attributed to the effects of coefficients.

Table 5 reports the decomposition results across the income distributions. When considering the detailed decomposition of the effects of coefficients (Equation (12)), a parallel shift is reflected in the constant term, which is adjusted accordingly to represent the difference in intercepts between both groups. Interestingly, in the upper income decile, a portion equal to ${\overline{X}}_{1B}^{\prime }({\widehat{\alpha }}_{1A}-{\widehat{\alpha }}_{1B})$ is transferred from the constant to the part affected by different slope coefficients. This reflects the trade-off of the OB, where the differentials are attributed to group membership components (difference in intercepts) and the part attributed to differences in slope coefficients. The conclusion is that the detailed decomposition results for the unexplained portion are only meaningful for variables where scale shifts are not permitted (Jann 2008). Another interesting result is that the number of contribution years has a negative coefficient (albeit non-significant), indicating that a higher pension is achieved with the same number of contribution years.

6.3. Application 2: Assessing Actuarial Neutrality

As most immigrants arrive within the 25–29 age bracket (see Figure 6), they contribute for fewer years to the old age insurance compared to non-immigrants. This is illustrated in the density plot of contribution years (see Figure 7a). The density plot for non-immigrants is skewed to the left, with a pronounced peak at 44 contribution years, indicating continuous contribution periods. In contrast, the density plot for immigrants is bimodal; the lower peak represents those who immigrated at a young age (e.g., as children), while the higher peak reflects the shorter contribution periods of those who immigrated as adults. Interestingly, the density plot of the average income subject to the OASI shows only a slight difference between the immigrants and non-immigrants. This suggests that average income levels are not significantly different, likely because immigration to Switzerland is primarily driven by professional opportunities. However, since immigrants contribute fewer years, their first pillar pension income is much smaller than that of non-immigrants, as illustrated by the lower peak in the density plot (see Figure 7b).

We hypothesize that the disparity in pension income between the immigrants and non-immigrants (${\overline{Y}}_A-{\overline{Y}}_B$) is primarily attributable to fewer contribution years and lower incomes subject to the OASI, as reflected in the effects of characteristics. Given that the pension accrual rate is adjusted for each year of service (Equation (16)), we anticipate that the number of contribution years will significantly enter into the effect of coefficients. However, since the reduction in pension incomes due to fewer contribution years aligns with the principle of actuarial neutrality (Equation (4)), we expect the overall effects of coefficients to be non-significant.

Table 6. Decomposition of pension differences at median (singles, in percentage points).

	Decomposition						In Percent
	Coeff.	Robust SE	z	P > |z|	[95% conf. int.]		%	[95% conf. int.]
$\Delta$ pp	−0.0604	0.0042	−14.25	0.00	−0.069	−0.052
Eff. of char.	−0.0561	0.0036	−15.73	0.00	−0.063	−0.049	92.8	84.3	101.4
Eff. of coeff.	−0.0043	0.0028	−1.54	0.12	−0.010	0.001	7.2	−1.4	15.8

Source: Projected annual pension incomes (first pillar) of the year 2070. The pension incomes of singles are reported. Notes: 95% confidence intervals in brackets. The coefficients of the immigrants (IMs) are used as a reference. $\Delta$ pp denotes the difference in pension income in percentage points (immigrants − non-immigrants).

presents the aggregated results of the RIF decomposition of the differences in pension income between the immigrants (IMs) and non-immigrants (NIMs) at the median. The overall difference in pensions is 6.04%, which is statistically significant. This difference is mainly driven by the effect of characteristics (11), which captures the impact of between-group differences in the explanatory variables $({\overline{X}}_A-{\overline{X}}_B)$, evaluated using the coefficients for immigrants ${\widehat{\alpha }}_A$. Thus, the estimated coefficients of the explanatory variables denote the percentage-point impact of each of these on the difference in pension income, then expressed as a percentage share of the difference in pension income, the effect of characteristics explains 92.8% of the overall difference. The effect of coefficients, which is not significant, further contributes to the difference −0.43%; expressed as a share of the difference in pension income, the effect explains 7.16% of the overall difference.

Table 7. Decomposition of pension differences at centiles of income distribution (Immigrants vs. non-immigrants, singles, in percentage points).

	q10		q25		q50		q75		q90
Overall
Immigrants	10.146 ***	(0.008)	10.326 ***	(0.005)	10.475 ***	(0.004)	10.585 ***	(0.003)	10.661 ***	(0.003)
Non-immigrants	10.238 ***	(0.005)	10.406 ***	(0.003)	10.535 ***	(0.002)	10.615 ***	(0.001)	10.667 ***	(0.001)
Total gap (IM − NIM)	−0.092 ***	(0.009)	−0.080 ***	(0.006)	−0.060 ***	(0.004)	−0.030 ***	(0.003)	−0.006 *	(0.003)
Effects of characteristics	−0.088 ***	(0.007)	−0.068 ***	(0.005)	−0.056 ***	(0.004)	−0.032 ***	(0.002)	−0.022 ***	(0.002)
Effects of coefficients	−0.004	(0.006)	−0.011 **	(0.003)	−0.004	(0.003)	0.002	(0.003)	0.016 ***	(0.004)
Effects of Characteristics
Incomes subject to OASI	−0.030 ***	(0.004)	−0.024 ***	(0.003)	−0.012 ***	(0.001)	−0.009 ***	(0.001)	−0.007 ***	(0.001)
Inactivity contributions	0.005 ***	(0.001)	0.000	(0.000)	−0.001 ***	(0.000)	−0.001 *	(0.000)	−0.002 ***	(0.000)
Child care credits	0.001 *	(0.001)	0.001 *	(0.000)	0.000	(0.000)	0.001 **	(0.000)	0.001 **	(0.000)
Care credits	0.000	(0.002)	−0.005 ***	(0.001)	−0.007 ***	(0.001)	−0.006 ***	(0.001)	−0.004 ***	(0.001)
Contribution years	−0.065 ***	(0.005)	−0.041 ***	(0.003)	−0.036 ***	(0.002)	−0.018 ***	(0.001)	−0.010 ***	(0.001)
Effects of Coefficients
Incomes subject to OASI	−0.207	(0.446)	0.897 ***	(0.243)	0.004	(0.182)	0.100	(0.144)	0.177	(0.126)
Inactivity contributions	−0.011	(0.011)	0.012 *	(0.006)	0.003	(0.005)	−0.018 ***	(0.004)	0.009 *	(0.004)
Child care credits	0.003	(0.004)	−0.005 **	(0.002)	−0.007 ***	(0.001)	0.003 *	(0.001)	0.003	(0.002)
Care credits	−0.047 ***	(0.014)	−0.027 ***	(0.007)	−0.000	(0.005)	0.004	(0.004)	0.006	(0.004)
Contribution years	−0.227	(0.167)	0.078	(0.083)	0.908 ***	(0.061)	0.485 ***	(0.046)	0.337 ***	(0.037)
Constant	0.486	(0.398)	−0.967 ***	(0.201)	−0.913 ***	(0.155)	−0.571 ***	(0.127)	−0.515 ***	(0.122)

Source: Projected pension incomes (first pillar) of the year 2070 according to the scenarios. The pension incomes of singles are reported. Notes: SE in parentheses. * $p<0.05$, ** $p<0.01$, *** $p<0.001$. The coefficients of immigrants are used as a reference. $\Delta$ pp denotes the difference in pension income in percentage points (immigrants - non-immigrants).

presents the detailed results of the RIF decomposition of the differences in pension income between the immigrants (IMs) and non-immigrants (NIMs). The migrants–non-migrants differential in the first pillar pension income of singles at the 10% quantile is −9.2% and declines to −0.6% at the 90% quantile. At the median, the observed differences in pension income amounting to −CHF 2190 (−6%) can be attributed to the sum of the effect of characteristics, quantified as −CHF 2044 (−5.6%), and the effect of coefficients, quantified as CHF −146 (−0.04%).11

The detailed decomposition of the effect of characteristics (Equation (9)) reveals that it is mainly driven by differences in contribution years. The coefficient of contribution years (−0.036) indicates that these differences contribute to a disparity in the pension income amounting to CHF 1314 per year at the median. If the immigrants had the same number of contribution years as the non-immigrants, while holding constant the differences in the composition of the other observable characteristics, the disparities would be much smaller (0.06 − (−0.036) = −2.4%). Additionally, lower incomes subject to the OASI and fewer (child-) care credits also contribute to the differences. The former increases the $\Delta$ pp by 1.2 pp and the latter by 0.8 pp (see Table 7).

The effect of coefficients $\left({\overline{X}}_{iB}^{\prime }({\widehat{\alpha }}_{iA}-{\widehat{\alpha }}_{iB})\right)$ is non-significant in almost all the income deciles, or if significant, the impact is rather small. A detailed decomposition of the coefficients’ effects indicates that the difference attributed to slope coefficients (e.g., contribution years) is offset by the portion attributed to the constant. Thus, the OB decomposition (8) is reduced to the following:

$${\overline{Y}}_A-{\overline{Y}}_B={({\overline{X}}_A-{\overline{X}}_B)}^{\prime }{\widehat{\alpha }}_A$$

(19)

which corresponds to the equation of actuarial neutrality (4). Thus, the results suggest that the differences in pension incomes between the immigrants and non-immigrants are solely driven by differences in characteristics, and that the adjustment of pension income due to missing contribution years follows the principle of actuarial neutrality.

6.4. Application 3: Assessing Vertical Equity Across Marital Statuses

Table 8. Decomposition of pension differences at median (women vs. men, in percentage points).

	Single		Married				Divorced		Widowed
			One Spouse Retired		Both Retired
Overall
Women	10.487 ***	(0.003)	10.395 ***	(0.015)	10.484 ***	(0.002)	10.480 ***	(0.003)	10.142 ***	(0.006)
Men	10.547 ***	(0.002)	10.621 ***	(0.005)	10.503 ***	(0.003)	10.536 ***	(0.003)	10.162 ***	(0.009)
$\Delta$ pp	−0.060 ***	(0.003)	−0.226 ***	(0.015)	−0.019 ***	(0.004)	−0.056 ***	(0.004)	−0.020	(0.010)
Effect of characteristics	−0.067 ***	(0.004)	−0.209 ***	(0.021)	−0.082 ***	(0.005)	−0.103 ***	(0.005)	−0.122 ***	(0.011)
Effect of coefficients	0.007	(0.004)	−0.017	(0.019)	0.062 ***	(0.005)	0.047 ***	(0.004)	0.102 ***	(0.011)
Effect of characteristics
Incomes subject to OASI	−0.072 ***	(0.003)	−0.083 ***	(0.023)	−0.085 ***	(0.003)	−0.092 ***	(0.004)	−0.129 ***	(0.007)
Inactivity contributions	−0.014 ***	(0.003)	−0.028	(0.015)	−0.010 ***	(0.001)	−0.019 ***	(0.003)	−0.018 ***	(0.004)
Child care credits	0.019 ***	(0.001)	0.007 *	(0.003)	0.000	(0.000)	0.004 ***	(0.001)	0.001 *	(0.001)
Care credits	0.006 ***	(0.001)	−0.032 ***	(0.007)	0.009 ***	(0.001)	0.001	(0.001)	0.009 ***	(0.002)
Contribution years	−0.007 ***	(0.001)	−0.072 ***	(0.013)	0.018 ***	(0.002)	−0.001	(0.001)	0.006	(0.004)
Pension assessment base (partner)			−0.009 ***	(0.002)	0.004 ***	(0.001)	0.010 ***	(0.002)
Contribution years (partner)					−0.006 ***	(0.001)
Effect of coefficients
Incomes subject to OASI	0.761 ***	(0.161)	−0.342	(0.670)	3.088 ***	(0.179)	0.232	(0.198)	1.783 ***	(0.402)
Inactivity contributions	0.135 ***	(0.015)	0.020	(0.024)	0.007 *	(0.003)	0.048 ***	(0.013)	0.049 ***	(0.010)
Child care credits	0.008 ***	(0.001)	−0.000	(0.008)	0.006 *	(0.003)	0.025 ***	(0.005)	−0.016 *	(0.007)
Care credits	0.022 ***	(0.004)	0.107 ***	(0.023)	0.039 ***	(0.006)	0.008	(0.005)	0.066 ***	(0.012)
Contribution years	−0.370 ***	(0.050)	1.150 ***	(0.227)	−1.790 ***	(0.073)	−0.497 ***	(0.063)	−1.062 ***	(0.106)
Pension assessment base (partner)			−0.206 ***	(0.061)	0.111 **	(0.034)	0.799 ***	(0.136)
Contribution years (partner)					0.118 *	(0.050)
Constant	−0.550 ***	(0.134)	−0.952	(0.518)	−1.201 ***	(0.157)	0.120	(0.170)	−1.518 ***	(0.326)
Observations (women; men)	7967	9212	357	884	9949	10,214	5255	4868	4677	1575

Source: Projected pension incomes (first pillar) of the year 2070 according to the scenarios. The pension incomes of singles are reported. Notes: SE in parentheses. * $p<0.05$, ** $p<0.01$, *** $p<0.001$. The coefficients of women are used as a reference. $\Delta$ pp denotes the difference in pension income in percentage points.

presents the results of the RIF decomposition of the simulated pension income at the 50th quantile by marital status. The results illustrate the varying extent of vertical equity across different marital statuses. In summary, unmarried women tend to have slightly higher pension incomes compared to married women, primarily due to their more intensive employment histories. However, married women benefit significantly from the solidarity mechanisms within the first pillar pension system, even in cases of divorce or widowhood. These mechanisms, which include spousal solidarity, result in marginally lower pension incomes for married men. Consequently, the gender differences in pension incomes are more pronounced among singles than among other marital statuses.

Furthermore, it illustrates how the instruments of solidarity impact the pension differences via two channels—the effects of characteristics and the effects of coefficients. The detailed analysis of the effects of characteristics indicates that, for instance, care credits affect the total relevant income X; they contribute to the effects of characteristics as follows: ${({\overline{X}}_{iA}-{\overline{X}}_{iB})}^{\prime }{\widehat{\alpha }}_{iA}$. For example, for single individuals, the effects of characteristics—which reflect differences in pension income due to differences in covariates—would result in a difference of −6.7%. The effects of characteristics are mainly driven by differences in the income subject to the OASI. The coefficient of earnings subject to the OASI (−0.072) indicates that differences in the earnings contribute to the gender differences in the pension income amounting to CHF 2741. If women had the same OASI-liable income as men, holding constant the differences in the composition of the other observable characteristics, the gender disparities would become slightly positive (0.06 − (−0.072) = +1.2%). However, the impact of the lower incomes subject to the OASI is partly offset by childcare and care credits. The former reduces the $\Delta$ pp by 1.9pp and the latter by 0.6pp. So, this measure of solidarity with caregivers reduces the $\Delta$ pp by 2.5 percentage points, as women, on average, accumulate a higher total of care credits than men.

In the year that both spouses become entitled to an old-age pension (married couples with two pension incomes), their pension income is recalculated. Pension equalization between the spouses in terms of pension amount and contribution years and the limitation of both spouses’ pensions to 150% of the maximum individual pension reduce the gender disparities significantly (−1.9% in contrast to −22.6 of married persons with one pension). This three-stage equalization of pension incomes reinforces the effect of coefficients so that the solidarity effect is second strongest (+6.2%) in comparison across all marital statuses, only surpassed by the solidarity of widows’ pensions. Thus, the extent of interpersonal solidarity can be reflected in the strong coefficient of income subject to the OASI, which significantly reduces the differences in pension income. Solidarity through pension equalization between the spouses—both in terms of pension amount and contribution years—is reflected by the opposite signs of the respective variables (income subject to OASI/pension assessment base of partner; contribution years/contribution years of partner). The results differ in cases of divorce. In this case, only income splitting is applied, but pension incomes are not equalized nor capped. As a result, the extent of solidarity, indicated by the effects of coefficients, is less strong and is equal to 4.7%. Consequently, the impact of lower incomes subject to the OASI, as estimated by the effects of characteristics, is only partially offset by interpersonal solidarity, as reflected in the effects of coefficients of teh incomes subject to the OASI and spousal solidarity, captured through the pension assessment base of the partner.

7. Discussion

This paper establishes a theoretical connection between actuarial neutrality and the Oaxaca–Blinder (OB) decomposition to empirically assess vertical equity in public earnings-related pension schemes. By clarifying the parallels between actuarial neutrality and OB decomposition, we suggest a straightforward interpretation of the econometric results. Furthermore, since the OB decomposition only applies to linear functions, we discuss how the nonparametric framework introduced by Firpo et al. (2009, 2018) can be used to perform a decomposition on any distributional statistic. This approach utilizes the recentered influence function (RIF) to determine the extent of vertical equity in non-linear functions. Third, we highlight the similarities among the DB schemes, point systems, and notional accounts, demonstrating the generalizability of the econometric approach to these systems. Finally, we demonstrate the use of dynamic microsimulation models to create the necessary data for an empirical analysis of social security systems.

To showcase this approach, we evaluated the vertical equity of the first pillar of the Swiss pension system. Our analysis focused on the following three key aspects: First, we examined the impact of the pension reform titled “For a Better Life in Old Age”, which is scheduled for implementation in 2026. This reform proposes distributing the first pillar pension 13 times per year instead of monthly, which is equivalent to a 8.3% increase. This reform affects only the accrual rate, which is reflected in the coefficients’ effects (Hypothesis 1). A closer examination of the detailed decomposition of the effects of coefficients reveals that the interpretation for variables with scale shifts is limited. Nevertheless, the total effect aligns precisely with the expected value, demonstrating the method’s applicability.

Second, we demonstrated the validity of actuarial neutrality by comparing the pension incomes of migrants and non-migrants. According to the theoretical linkage between actuarial neutrality and the OB decomposition, our empirical findings confirm the hypothesis that actuarial neutrality should be solely reflected in the effects of characteristics (Hypothesis 2). Additionally, by addressing the aspect of migration within pension systems, we contribute to the relatively scant literature on this topic.

Third, we examined the heterogeneous impact of instruments of vertical solidarity by analyzing the extent of vertical solidarity across different marital statuses. The empirical findings confirm that these instruments affect both the effects of characteristics and the effects of coefficients (Hypothesis 3). The results reveal that married women benefit more from the solidarity mechanisms within the first pillar pension system than single women, even in cases of divorce or widowhood. Given the evolving family models, characterized by the decline in the male breadwinner model and the rise in non-marital unions and the workforce participation of married women, it might be relevant to orient these mechanisms away from marital status and towards care-giving activities, such as child and care credits.

To build on these findings, future research could include non-marital unions in the analysis to provide a more comprehensive understanding of the impact of solidarity mechanisms across different family structures. Additionally, extending the analysis to point systems or NDC systems could offer valuable insights into how different pension schemes affect various demographic groups. Exploring these areas could help policymakers design more inclusive and equitable pension systems that better reflect contemporary family dynamics.

Moreover, considering the dynamics in migration and changes in female labor participation, it is important to assess the extent of vertical equity within the context of actuarial risk management. A higher degree of vertical equity is associated with increased pension expenses, which in turn increases the financial sustainability risk of the pension system. Future research should explore these aspects to ensure that pension systems remain both equitable and financially sustainable in the face of changing societal trends.