Pension system design: roles and interdependencies of tax-financed and funded pensions

Abstract

The purpose of this paper is to give an overview of the roles, objectives, and trade-offs in a two-pillar pension system consisting of tax-financed, public pensions and defined contribution, individual pensions. A pension system has many moving parts and our aim is to provide the reader with an understanding of how the parts interact and work together, a theme rarely addressed in the literature. In the first part of this paper, we give a qualitative overview of market failures, behavioural aspects, and distributional issues that form the background for a multi-pillar pension system design with mandatory components. In the second part of this paper, we present three thematic, quantitative analyses that illustrate fundamental relationship concerning wealth, inequality, insurance, and demographic changes. This paper also contains a detailed description of the agent-based model used for the analyses. The model is calibrated to Danish data, but the insights drawn from the model are of general validity.

Keywords:

• Pension system design
• two-pillar pension system
• agent-based model
• income inequality
• replacement rates
• demographic changes
• sustainability

1. Introduction

A real-world pension system has many objectives, many decision makers, and many components. The system is complex, and the design requires a stand to be taken on many trade-offs, e.g. if tax-financed public pensions contribute to reduce income inequality and also decrease the incentive to voluntary saving, how large should public pensions be?

The answer to such questions is political, but it is important that the design options and trade-offs are clear to decision makers. The academic literature is sparse on work taking an overarching perspective on pension system design and tends to focus on posing and solving well-defined problems. Many papers exist on optimal investment strategies, reserve calculations, optimal bonus rules, mortality models, etc. In comparison, very few papers are concerned with how the pieces fit together, the big picture.

The aim of this paper is to consider the entire pension system, and to discuss the roles and interdependencies of the tax-financed and funded components. We do this, not by solving a specific problem, but by building a simple yet sufficiently detailed model of the pension system to illustrate fundamental dependencies and trade-offs in a two-pillar pension system. Concepts like inequality, poverty, wealth, (re)distribution, insurance, and personal and aggregate risks are important for understanding the purpose and design of pension systems. We hope that the current paper can illuminate these concepts and their dependencies. We also hope that this paper can provide a basis for appreciating how the many specialized contributions in the literature fit together and add insights to the bigger picture.

This paper has two parts. In the first, qualitative part, we give an overview of issues regarding risks, market failures, behavioural aspects, and (re)distributional goals that a pension system must take into account. In combination, these considerations imply that a certain level of intervention is warranted. For the sake of this paper, we refer to any (pension) arrangement not driven by market forces as an intervention.

The many objectives, including consumption smoothing, distribution, and insurance, imply that most real-world pension systems include elements of at least two archetypical components: a public, tax-financed, pay-as-you-go (PAYG) component offering a universal pension to all retirees, and a mandated, defined contribution (DC), funded component with benefits depending on individual, accumulated funds. The two components are referred to as pillar 1 and pillar 2, respectively, and the system as a whole as a two-pillar system, cf. World Bank (1994). As many before us, we conclude that to balance the many objectives and considerations a well-designed system is a combination of (at least) these two pillars or variations hereof. Real-world pension systems also include voluntary, private savings (pillar 3), which enable individuals to build additional buffers to cover, e.g. old-age healthcare costs and other risks. However, to narrow the focus, we consider only pillars 1 and 2 in this paper.

In the second, quantitative part, we develop an agent-based model of a society featuring a two-pillar pension system, a financial market, a labour market, and a model for fertility and mortality. The level of detail reflects the themes we want to address. Many of the trade-offs of interest only occur as a result of interacting, heterogeneous agents subject to both personal and aggregate risks, and where the two pillars have different roles. Capturing these interdependencies necessitates a model of a certain complexity. It is our hope, though, that the model is still accessible to the majority of readers.

We use the model to carry out a series of analyses under the headings:

Wealth and income inequality	Where we investigate the effects on (disposable) income and its distribution when varying the weight of the two pillars.
Insurance and replacement rates	Where we characterize the groups that benefit and contribute, respectively, to pillar 1, and find the replacement rates for these groups.
Demographic changes and stability	Where we illustrate the impact of changing mortality and fertility on a two-pillar pension system.

In summary, the analyses support the qualitative conclusion that the complementary features of pillars 1 and 2 are needed in a balanced pension system. More precisely, the distributional, insurance, and risk mitigating properties of pillar 1 imply that a (perhaps surprisingly) large group of people, and society as a whole, benefit from the existence of tax-financed basic pensions and, more generally, social benefits, while the funded part is important for ensuring adequate replacement rates.

2. Retirement saving and pension systems

The rationale for retirement saving or pensions is basic, namely, to provide a financial basis for consumption post retirement from the labour market. This fundamental role is captured by the textbook life-cycle model determining the optimal consumption and savings for a household given the income earned during working years, see, e.g. Attanasio (1999). Out of self-interest, the household saves to provide for the desired consumption possibilities when retired.¹ Funds are thus accumulated during working years and decumulated during retirement. The basic aim of the household is to smooth the consumption profile relative to the income profile, and the optimal consumption path depends on preferences (including the time preference) and the market return to savings.

While the simple life-cycle model is straightforward, matters get more complicated when leaving the stylized textbook setting by introducing risk, market failures, behavioural aspects, and distributional issues.

2.1. Risk and market failures

The real rate of return on saving is risky, and the market offers a trade-off between the expected return and risk. The life-cycle model must be amended by the portfolio problem of how best to invest the saving, possibly including a risk appetite declining with age, see, e.g. Gomes et al. (2021). This is an important extension, but it does not as such change the basic insights of the life-cycle model.

Matters get more complicated when introducing other sources of risks because the scope individuals have for risk diversification via financial markets or various insurance mechanisms may be constrained. Insurance markets differ qualitatively from most other markets since information and incentive issues – even under idealized conditions – may cause non-existence of equilibrium (no insurance), incomplete insurance, or premiums/returns deviating from actuarial fair pricing, see, e.g. Rothschild & Stiglitz (1976) and Mimra & Wambach (2014).

2.1.1. Income risk

The income profile over the life cycle is affected by both idiosyncratic and aggregate risks for a number of reasons including unemployment, promotions, skills becoming obsolete, health problems, declining work capabilities, business cycles, and structural changes. It is well established that it is very difficult to hedge labour income risks in financial markets due to moral hazard problems.² A particular issue for the adequacy of retirement saving is the ability to work at higher ages where a depreciation or loss of work capability may cause a fall in income or involuntary early retirement. The scope for insuring work capability in private insurance markets is very limited. Missing insurance possibilities give risk averse households a precautionary savings motive to create buffers to handle shocks to, e.g. wage income. Such self-insurance is incomplete for a number of reasons including that it is difficult to accumulate sufficient buffers especially for the young and low-income households, and it is inefficient since risk is not diversified across individuals. Incomplete insurance may cause a drop in consumption or forced postponed retirement despite serious impairment of work capabilities. As discussed below, mandated pension schemes or public pensions imply some implicit insurance potentially overcoming market failures.

2.1.2. Health and old-age care

Related to the above is consumption needs. The simple version of the life-cycle model assumes age-independent preferences specified over a generic consumption bundle. But consumption needs may change over the life cycle and be age dependent. A particular issue is the need for health and old-age care and how much to save for such needs. While there is some scope for health insurance on market terms, this is also a market affected by moral hazard and adverse selection problems causing market failures and non-actuarial fair pricing, see, e.g. Brown & Finkelstein (2007). However, depending on welfare arrangements the individual may have access to tax-financed health and old-age care, which may be interpreted as implicit insurance reducing the need for private (precautionary) savings, see discussion below.

2.1.3. Uncertain life span

Uncertainty in length of life is fundamental and raises the basic question of how to be sure that enough has been saved to support consumption throughout life avoiding the risk of becoming old but poor.³ A seminal contribution by Yaari (1965) – and later generalized by Davidoff et al. (2005) – showed that all life-cycle savings should be in life annuities providing payments conditional on survival. ⁴ This allows diversification of the individual survival risk and removes it from the household decision problem. Despite this clear theoretical result, it is a fact that only a small share of voluntary life-cycle savings is in annuities – the so-called annuity paradox – see Modigliani (1986) and Brown (2001) – which may be attributed to market failures (unfair pricing), behavioural factors, or insufficient financial literacy, see below. This is also an area where intervention may overcome market failures, see below.

2.2. Behavioural aspects

The life-cycle model assumes that the household rationally plan for the entire life. Both experimental studies and empirical evidence contest this model and a large behavioural economics literature has explored various alternatives to the standard ‘rational’ models. Following DellaVigna (2009), this work can be grouped in three approaches: (i) non-standard preferences (present bias), (ii) non-standard beliefs (e.g. overconfidence, projection biases), and (iii) non-standard decision making (e.g. limited attention). For recent surveys and introductions to this literature, see, e.g. Bernheim et al. (2018).

2.2.1. Non-standard preferences

Non-standard preferences displaying so-called present bias are most relevant in the present context since they explain procrastination and hence why households tend to undersave for retirement. The simplest version is so-called myopia where insufficient attention or weight is attached to future consumption possibilities which in turn imply insufficient savings which is regretted when reaching retirement ages, see, e.g. Andersen & Bhattacharya (2011).⁵ Other explanations of undersaving include quasi-hyperbolic preferences causing time-inconsistency, self-control problems, and reference dependent behaviours.

2.2.2. Non-standard beliefs

Non-standard beliefs move beyond perfect foresight or rational expectations allowing the foresight of households to be distorted, biased, or incomplete. A large literature points to individuals underestimating longevity with obvious implications for retirement savings, see Heimer et al. (2019) and O'Dea & Sturrock (2023) for analyses of subjective mortality rates.

It is well documented that financial literacy is low for a significant part of the population with the implication that non-optimal decisions may be made, see, e.g. Gomes et al. (2021) for a survey and references. Clearly, not all individuals need to be financial specialists and they can consult expert advice. However, individual acquisition of relevant information on how to invest the funds is costly, and the skills to process such information may be inadequate. Behavioural factors may also imply that advice is not demanded (‘not knowing what you do not know’) or making individuals vulnerable to biased advice or paying too high fees (raising regulatory issues), etc. Fixed costs are a barrier for both information acquisition and risk diversification, in particular for individuals with low income/wealth, see, e.g. Lusardi & Mitchell (2014).⁶

2.2.3. Non-standard decision making

Non-standard decision making implies that framing and nudging may have large effects on behaviour. Default options in choice situations have been documented to play a crucial role in schemes where individuals have to decide whether pension contributions are automatically made or require a discrete decision, see, e.g. Brown et al. (2008) and Karlan et al. (2016). Such effects may be attributed to attentional constraints (non-standard decision making). The ‘rational inattention model’ assumes a finite capacity to process information because attention is costly, see, e.g. Gabaix (2014) and Caplin et al. (2019). Attention is therefore allocated rationally to those choice options having the lowest attention costs. This may explain why pension issues attract little attention among younger households, and why default options are important for savings, see, e.g. Chetty et al. (2014). When attention is limited, more salient features of the choice environment can capture the attention of the decision maker. As a consequence, small differences in framing highlighting different aspects of the choice situation may serve as an intended or unintended nudge that influences choice, even for given economic incentives, see Thaler & Sunstein (2008). This may explain why some responses are relatively inelastic to changes in prices and income (see, e.g. Gabaix 2014, Chetty et al. 2014), and overly responsive to changes in seemingly irrelevant factors.

2.3. Distribution and other reasons for public intervention

The classical and also historical reason for intervention in pension saving and the establishment of tax-financed pension schemes is distributional. Poverty among old is a particular problem since they are in an irreversible situation (insufficient saving cannot be undone) with few degrees of freedom, especially if the work capability is significantly reduced or lost. Distributional arguments remain crucial for contemporary pension system design, including both the living standard for the older members of society (avoiding poverty) but also the objective that all should be able to enjoy a retirement period of reasonable duration (the third life phase).

2.3.1. Market failures

Public intervention may also be motivated by market failures. As an example, Eckstein et al. (1985) show in a classical contribution why life-annuities, due to adverse selection problems, may either be unavailable or offer incomplete insurance when there are different risk classes (low and high survival probabilities) unobservable to insurance companies. A mandated, funded pension saving may overcome the non-existence problem and imply that all are better off compared to the non-intervention situation. This is an example of a more general finding that intervention may overcome market failures and also serve a ‘market maker’ role by making, e.g. annuitization possible for households, see the discussion in Scharfstein (2018). Such interventions may also be motivated in addressing problems arising due to lack of financial literacy and fixed information/transaction costs. Mandated, funded pension systems are effectively a ‘financial cooperative’ where, e.g. a relatively homogeneous group of workers organized in a union/pension fund face similar problems. Investments are typically collective in the sense that the funds (although credited to an individual account) under administration are invested collectively by a professional investment team subject to a risk–return profile determined by the investment policy of the fund. There is one important proviso for this argument, namely that the governance structure of the pension fund ensures that the interests of the members are served. This is not trivial precisely because the members may not pay enough attention or have the needed competencies to challenge management.

2.3.2. Behavioural aspects

Mandated retirement saving or a tax-financed pension scheme may also overcome the consequences of behavioural aspects causing undersaving, see, e.g. Andersen & Bhattacharya (2011). While such intervention may seem straightforward it raises a crowding out issue. Mandating a certain level of saving in a funded scheme may seem a simple way to ensure that all households have saved adequately for old age. However, such a mandate will crowd out voluntary savings; the individual does not see any reason to increase savings! If voluntary and mandatory savings offer the same return, the crowding out would be one-to-one and the mandated savings would have no effect. Total savings is unchanged but the balance between mandated and voluntary savings has changed. The crowding out effect presumes that it is possible to reduce voluntary savings when the mandated savings requirement increases. This is straightforward if voluntary savings is positive, but at a sufficiently high mandated savings requirement the desired voluntary savings becomes negative. Such borrowing may be impossible (due to credit constraints) or very costly (high risk premium). It is implied that the mandate is ineffective for households not facing a borrowing constraint but increases total savings for those household who are constrained. To put it differently, the mandate is inconsequential to those households who save enough, while it may increase savings for those households who undersave (the target group).

2.3.3. The free-riding problem

In a welfare state with strong distributional goals, individuals anticipate that there is always some support, also to old with low or no savings. Hence, some may free ride on the public system and purposely undersave to increase consumption prior to retirement and then rely on public support when becoming old. This free-riding problem is another argument for a mandated pension savings requirement, see, e.g. Diamond (1977).

2.4. Pension system design

Intervention raises fundamental questions on the design of the pension system. The classical discussion of this issue takes outset in a comparison between the returns in PAYG and funded pension schemes, see discussion in e.g. Feldstein (1995) and Barr & Diamond (2006). A PAYG scheme has an implicit return which is basically determined by the growth in total wage income (wage sum),⁷ whereas a funded scheme offers an explicit return given by the market rate of return. In a so-called dynamically efficient economy,⁸ the market rate of return exceeds the implicit return of a PAYG scheme, and this gives an argument in favour of funded rather than PAYG schemes. On top of this comes the distortionary effects of the taxes needed to finance public pensions.

While the return difference is fundamental, later work has shown that other factors are also important for pension system design. A straightforward reason is that distributional objectives may call for tax-financed pensions. In addition, the different exposure to shocks and thus the (implicit) insurance offered by the two prototypes differ across different types of risks. In a funded scheme, the individual is fully exposed to, e.g. individual wage income shocks, while such exposure is smaller (none in a flat rate system) in the public scheme. Oppositely, an aggregate shock to wage income (employment) affects the implicit return in a PAYG scheme.⁹ While funded schemes are directly influenced by market return risk, a tax-financed scheme is more exposed to, e.g. fertility risk.

2.4.1. Distribution and insurance

It is well established that it is difficult, or impossible, to make a clear separation between distribution and insurance. Any insurance arrangement implies ex-post redistribution from those not experiencing the event to those who do. Public pension schemes which are redistributive are therefore also typically implying some form of (implicit or social) insurance.¹⁰ As an example, means-tested public pensions target the least well-off with low pension savings but also offer implicit insurance against, e.g. variations in wage income over the life cycle. An individual who happens to be unemployed for extended periods of time and therefore to have low savings, will be entitled to a higher public pension, and vice versa. A crucial difference with respect to risk diversification between the two prototypes arises since a public scheme running via the general budget offers scope for intergenerational risk sharing, see, e.g. Gordon & Varian (1988). It is very difficult to offer such insurance on market terms.

2.4.2. Objectives and trade-offs

Any pension system, whether funded or non-funded, is characterized in terms of financing (contributions/taxes), benefits (level and profile), and the retirement period.¹¹ Along each of these three dimensions various design options are possible, but they are interrelated through the financial constraint of the system that contributions should cover outpayments. This also has the implication that changes in fundamentals, like market returns or longevity, require a change in one or more of these dimensions to ensure the financial viability of the pension system.

The preceding discussion boils down to two general observations. First, the pension system has to address many objectives including consumption smoothing, insurance, and distribution, and considering both market failures and behavioural aspects (undersaving, financial literacy, and responses to intervention). Second, many specific designs are possible spanning a broad spectrum from a non-funded PAYG scheme offering a flat rate pension to all retired financed by taxes to a mandated, defined contribution, funded scheme where benefits depend solely on individual accumulated funds. The characteristics of the two prototypes are listed in Table 1. None of the prototypes is universally dominating in relation to the many objectives. A well-designed and balanced pension system is thus a hybrid or portfolio including different pension forms to balance the many considerations and trade-offs related to pension adequacy, incentives, insurance, and financial viability.¹²

Table 1. Comparison of prototype non-funded and funded pension systems.

	Flat rate, PAYG scheme	DC, funded scheme
Financing	General taxation or a specific contribution, typically a fraction of income	Contribution, typically a fraction of income
Entitlement	No direct link between financial contribution and entitlement – typically dependent on age (statutory retirement age) and residence requirements	Entitlement determined by contributions
Return	Implicit return determined by the growth of the total wage bill	Market rate of return
Incentives	Decoupling of relation between contributions and entitlements may reduce work and savings incentives	Incentives to work and save maintained – but can be impaired if mandated savings is considered excessive
Risks	Economic developments affecting the growth of the total wage bill	Own income over the life-course
	Demographic changes affecting the implicit return	The market rate of return
	Political risk of changes in contribution rates, benefits, and retirement ages	Political risk of changes in tax rules
Longevity	Public schemes are implicit annuities (pooling the entire population) providing benefits from retirement throughout remaining life	Depends on benefit structure – can be in the form of lifetime annuities (pooling across group members)
Risk diversification	Across the entire population and across generations for non-strict PAYG schemes	Across group members
Redistribution	Inherent in the decoupling of contributions and entitlements, can be strengthened by, e.g. means-testing	Modest redistribution can arise via pooling across risk classes or due to regulation, e.g. unisex mortality tables for determination of benefit levels

3. Two-pillar pension system model

The rest of this paper is devoted to a series of thematic analyses of a two-pillar pension system based on an agent-based model of a society evolving over time. Below we give an overview of the model, and the rationale behind it. The model is intended to represent a generic two-pillar system and the insights that can be drawn from the model are valid regardless of the real-world specifics. To a smaller or greater extent, the model and its conclusions apply to all systems in OECD (2021). That being said, the model is calibrated to Danish data and the benefit structures loosely mimic the Danish pension system, see the monograph (Andersen et al. 2022b) for details. The model and its calibration are described in detail in the appendix.

Some of the themes we address have been studied before, see, e.g. Zhu et al. (2021), He et al. (2021), Alonso-García et al. (2018), Jarner & Kronborg (2016), Alonso-García & Devolder (2016) and references therein. Most of this work, however, use very simplistic, analytically tractable models to answer specific questions, often related to optimality. In contrast, our aim here is not to solve a well-defined problem, but rather to illustrate trade-offs and dependencies in real-world pension systems. This difference in aims is reflected in our choice of model.

3.1. A note on terminology

In the literature, there are different definitions of what social benefits encompass. For the purpose of this paper, we use social benefits as the common term for all state provided benefits. In particular, retirees also receive social benefits. Public pensions are synonymous with social benefits to retirees.

3.2. Rationale

The purpose of a pension system is not only to facilitate lifetime consumption smoothing for fully-employed, average-waged, average-lifetime individuals enjoying average financial returns. In fact, such individuals are rather rare. The goal is to provide adequate, sustainable pensions for the entire population of which the majority will at some point experience unemployment, low wages, health problems, or financial downturns.

It is the presence of heterogeneous agents in different socio-economic conditions that create many of the trade-offs in the design of real-world pension systems. To study the ensuing design questions, we need on the one hand a model complex enough for at least some of these effects to occur. On the other hand, we need a model simple enough for the conclusions to stand out clearly. An agent-based model centered around labour market participation fulfils both of these criteria.

3.3. Model overview

The model consists of a population of agents spanning newborns to 100-year-olds. For a given pension system design (i.e. given tax level, contribution rates, benefit profiles, etc.), we track the income, social benefits, and other socio-economic variables of the agents over a set of simulation paths. This gives us an ensemble of agents having experienced the given pension system under different financial markets, different labour histories, and different lifetimes. From this ensemble, we compute various summary statistics.

The model has four key components:

Demographics	The model evolves in annual time steps. In each step, agents are born and agents die according to given fertility and mortality schedules, respectively. All quantities are unisex. We consider both stationary demographics (base scenario), and scenarios with increasing life expectancy and fertility deviating from replacement levels.
Pensions and social benefits	Agents pay taxes and they contribute to a DC pension scheme. In return they receive means-tested social benefits, as well as DC pensions. The latter take the form of a combination of 20-year fixed term and lifetime annuities. In each period, the level of social benefits is computed such that the total expenditure on social benefits matches (part of) the total tax revenue.
Financial market	All DC accounts are invested in a financial market with stochastic interest rates, mean-reverting stock returns and stochastic (price) inflation. The accounts are invested according to an individual, life-cycle strategy; specifically, an age-dependent fraction of each account is invested in stocks, and the remaining fraction is invested in bonds. Hence, all agents are exposed to the same financial market, but with different risk exposures.
Labour income and employment	Each agent has both a labour income process conditioned on being in work and an employment process. The actual income is the product of the two. This structure allows us to separate the largest (economic) personal risk, namely, unemployment from the ‘risk’ of having a low-income job. Wages are indexed with price inflation plus a constant productivity growth.

For reason of simplicity, many other aspects of importance to a real-world pension system are deliberately left out of the model, most notably health status, housing, bequest motives, and private savings (pillar 3). To reach familiar levels of tax, we assume that only a fraction of taxes are spent on social benefits; we do not account explicitly for the remaining public spendings. To aid the comparison of different designs, we do not allow state budget deficits, nor surpluses.

The aim of the model is to illustrate basic features of tax-financed and funded pension schemes and their interdependence. The model is purely rule-based (descriptive) leaving out behavioural responses to changes in, e.g. the pension system. Introducing such responses would significantly complicate the model, and the basic effects identified here are consistent with theoretical models incorporating such responses.¹³

3.4. Initial state and build-up phase

All simulations start in 2020 and are run for at least 100 years. In the base scenario, we use stationary demographics with time-invariant fertility and mortality schedules and an initial (equilibrium) population proportional to the survival function. In other analyses, we use a best estimate mortality forecast and lower/higher fertility but the same initial population, see Appendix A.3 for details. All pension accounts are empty at the outset and it takes a century before all accounts reflect a given system. However, many characteristics of the system stabilize before that. Some analyses are performed on the 100-year horizon, while others focus on the evolution over time.

To gain intuition for the dynamics, Figure 1 illustrates wage-deflated total taxes for 100 simulations of the base scenario, as well as the ensemble average for each year. We see that the underlying, stochastic financial market and income histories create sizeable fluctuations, but in the long run the growth in taxes equals wage inflation, i.e. the thick black line is flat. Social benefits are therefore also (implicitly) wage-indexed.

Figure 1. Total tax revenue in 100 simulations of the base scenario from 2020 to 2120. Taxes are wage-deflated and indexed to 1 in 2020. The ensemble average is shown as the thick black line.

The initial increase in (wage-deflated) tax revenues is due to the build-up of the DC pension system. DC pensions are taxed when paid out, but contributions are tax-exempt when paid into the system. Thus in the build-up phase taxes are effectively postponed. Only when retirees with full contribution histories emerge does the tax revenue begin to stabilize; in fact, it settles only after these retirees have used up their fixed term annuities.

4. Three thematic analyses

In the following, we discuss and quantify trade-offs and interdependencies in a two-pillar pension system. In the real world, arriving at clear-cut conclusions is complicated by the fact that there are many moving parts, not least changes in the underlying demographics, and most analytical contributions focus on a subset of aspects. To separate fundamental features from effects due to changing demographic, the analyses in Sections 4.2 and 4.3 on wealth versus inequality and contributors versus beneficiaries, respectively, are performed under stationary demographics, while analyses of stability issues due to non-stationary demographics are deferred to Section 4.4.

The structure of each thematic analysis is the same. First, there is a brief description of the construction of the data being analysed. This is followed by two separate, but connected (sub)analyses. Last, there are a few concluding remarks. The three analyses are essentially independent of each other, and they can be read in any order.

4.1. Getting acquainted with data

Before presenting the analyses, we first discuss how the underlying income distribution arises. Agents enter the labour market at age 24 and retire at the statutory retirement age 67 (base scenario). During working ages, they earn a wage (if employed) of which a fraction is paid to an individual DC scheme. Once retired, the agents receive a combination of 20-year fixed term and lifetime annuities funded by the DC scheme. The gross income is the wage net DC contributions (workers), or the sum of DC benefits (retirees).

Depending on their age and gross income, agents might receive social benefits (also known as public pensions for retirees). Personal income (i.e. the sum of gross income and social benefits) is taxed at a flat rate of the part that exceeds a basic allowance. The disposable income is the personal income net income tax.

In a system with a $45\mathrm{\%}$ tax rate and a $12.5\mathrm{\%}$ DC rate, Figure 2 shows an example of the gross and disposal incomes in 2120 in a simulation with 2000 agent types. The incomes shown are deflated with the accumulated wage inflation over the period (basis year 2020).

Figure 2. A simulation of gross (black circles) and disposable (green dots) income distributions in 2120, projected from the model starting in 2020 with 2000 agents. The horizontal blue line marks the median disposable income. The vertical dashed lines separate the working population from the non-working population. The orange line displays the theoretical average earned income conditionally on being employed. For better visualisation, the vertical axis is truncated at 1M (16 agents have a gross income exceeding this level).

The plot illustrates several features. First, we note how the age-profile of social benefits manifests itself as increasing, horizontal segments of green dots. Social benefits are means-tested and the full amount (obtainable at a given age) is given only to agents with zero gross income. For the working population, this corresponds to being unemployed in the given year. For retirees, however, a gross income of zero only occurs if they have been unemployed throughout their entire life. Since lifetime unemployment is rare there is no discernible horizontal pattern of green dots for retirees. Second, we see how both gross and disposable income decline after 20 years in retirement, when the fixed term annuity ends. However, while both decrease, disposable income decreases far less than gross income because the drop in gross income is partially compensated by higher public pensions. Third, we see that almost all employed members of the working population are net contributors, i.e. their disposable income is below their gross income, but there are also many net contributors among the younger part of the retired population. In contrast, young adults, unemployed, and the oldest part of the retired population are net beneficiaries.

Figure 2 gives a cross-sectional view of a population consisting of agents with individual labour histories. The agents experience persistent income dynamics interrupted by periods of unemployment, possibly permanent. The employment dynamics imply that the risk of being unemployed increases with age. These longitudinal aspects of the model are illustrated in Figures A6 and A7, respectively, in the appendix.

4.2. Wealth and income inequality

From the individual's point of view, the primary objective of the pension system is to provide a substitute for labour income in retirement. From society's point of view, there is also the objective that all individuals should have a level of income at or above a politically determined minimum – and possibly a separate objective of limiting income inequality among retirees, or the population at large. In this section, we analyse how the size of DC and tax-financed pensions affects the distribution of disposable incomes and thereby the trade-off between these objectives.

4.2.1. Setup

The analysis is carried out under stationary demographics (base scenario). Data is constructed by simulating a population with 2000 agent types and computing the distribution of (wage-deflated) disposable income in 2120, cf. Figure 2. From this distribution, we compute the average disposable income as a measure of permanent income (wealth) and the Gini coefficient as a measure of inequality.¹⁴ We compute wealth and inequality both for workers and retirees combined ($\mathrm{age}\ge 24$), and retirees only ($\mathrm{age}\ge 67$). The simulation is repeated 1000 times and the ensemble average of (average) income and Gini coefficient is formed. We compute and illustrate these ensemble averages for various pension systems, i.e. various tax and DC rates.

4.2.2. General population

Figure 3 shows the Gini coefficient for disposable income against average disposable income for adults of working age and above.¹⁵ We start by noting that income inequality is decreasing (lowers the Gini coefficient) in the tax rate. However, due to the assumption of exogenous expenses, tax also decreases the average disposable income. Only half the tax revenue is redistributed as social benefits, the rest is spent on other public expenses. Clearly, in the real world, the population benefits from these expenses via, e.g. health care, education, infrastructure, etc., but seen in isolation there is essentially a ‘tax on taxes’ that makes conversion of taxes to social benefits costly. In other words, achieving a high degree of income equality via taxes comes at the expense of reducing the average disposable income.

Figure 3. Disposable income inequality (Gini) against average disposable income for DC rates of 10% (leftmost), 20%, 30%, 40%, and 50% (rightmost), and varying tax rates for adults in 2120.

For a given tax rate, the effect of varying the DC rate forms a U-shaped pattern. Except at extreme tax levels (not shown), without a DC system retirees are poorer than workers and inequality in the adult population is high. Increasing the DC rate effectively transfers income from workers to retirees thereby decreasing inequality. Eventually, however, the wealth transfer becomes so large that the inequality starts to rise again, with the roles of workers and retirees reversed. Minimal inequality is achieved for DC rates around 30%.

Increasing the DC rate increases the average disposable income. This is due to the financial returns on the underlying funding base of the DC system. Capital gains give rise to higher pensions (on average) than the contributions paid into the system. In addition, the DC system itself is taxed and this contributes to a higher tax revenue and thereby higher social benefits. As mentioned in Section 2.4, higher returns on DC than PAYG systems is the classical argument in favour of the former. Indeed, Figure 3 shows that the highest average disposable income occurs in a system with no tax and highest possible DC rate. We also see, however, that to meet distributional goals we need a tax-financed element as well.

4.2.3. Retired population

The relation between average income and income inequality is different among retirees, than in the general population, cf. Figure 4. Taxes still decrease inequality and decrease average income, and higher DC rates still lead to higher average incomes. However, in contrast to the U-shaped pattern for the general population (Figure 3), there is a monotone relationship between DC rates and income inequality for retirees. Higher DC rates lead to higher income inequality among retirees.

Figure 4. Disposable income inequality (Gini) against average disposable income for DC rates of 10% (leftmost), 20%, 30%, 40%, and 50% (rightmost), and varying tax rates for retirees in 2120.

The explanation for this apparent paradox is as follows. At the personal level, a DC system (or any other savings vehicle) increases income equality between working years and years in retirement, up to a point. The same mechanism is in effect at the aggregate level. In other words, the DC system creates equality by making retirees resemble workers in terms of income. However, this implies that the income inequality of workers gets mirrored by retirees. The higher the DC rate, the more the income distribution of retirees resemble that of workers and the higher the inequality among retirees.

This dual effect of the DC system, creating equality between workers and retirees while creating inequality among retirees, illustrates that even for the same objective, i.e. income equality, trade-offs can occur when considering different subpopulations.

4.2.4. Concluding remarks

There is a fundamental trade-off between average income and income equality in the design of pension systems. Although a DC system can simultaneously improve average income and income equality at the level of the general population, it can do so only up to a point. Tax-financed social benefits are needed both to ensure a minimal income level for retirees and to reach higher levels of income equality. Neither for the general population, nor for retirees only, is it possible to achieve the highest levels of average income and income equality simultaneously, i.e. the lower right corners of Figures 3 and 4 are empty.

The analyses of this section are based on a fully phased-in DC system. The funding base of this system generates financial returns making it more cost-effective than a PAYG system. It is worth noting, however, that the establishment of the funding base comes at the expense of lower tax revenues in a long initial build-up phase, cf. Figure 1. This causes intergenerational economic inequality although, methodologically, it is less obvious how to quantify this effect. Of course, once established, all generations benefit from the financial returns generated by the funding base.

One might argue that we exaggerate the cost of providing social benefits by including exogenous expenses in the model. To measure disposable incomes, it is important to include the entire tax burden falling on households. In the model set-up, a tax increase finances both social benefits and other expenses, and hence it may be argued that we exaggerate the marginal effects of a tax increase to finance social benefits. Oppositely, we do not include the disincentive effects of taxes on work, and this tends to exaggerate the effect of tax increases on tax revenues. We acknowledge that there are a number of such non-modelled effect modifiers, some of which partially offset each other, but we do not attempt to quantify any of these.

4.3. Insurance and replacement rates

The contributions to DC pensions are typically set as a fraction of one's salary. By construction, this creates a strong (positive) link between lifetime earnings and the size of DC benefits. Tax-financed, means-tested pensions, on the other hand, are largest for people with low savings and public pensions can therefore be seen as providing insurance against low lifetime earnings. In this section, we illustrate the extent of this insurance mechanism and the effect on replacement ratios from combining the two types of pension.

4.3.1. Setup

The analysis is carried out under stationary demographics (base scenario) for a system with a tax rate of 45% and a DC rate of 12.5%. An initial population with 2000 agent types is projected for 100 years in 1000 stochastic scenarios, resulting in a pool of approximately 350,000 retirees in 2120. For each retired agent, i, in each scenario, j, the wage-deflated, net public pension is computed, $Z_{i,j}=[B_{i,j}^S(T)-R_{i,j}^{\mathrm{tax}}(T)]/{\tilde{I}}_{T,j}$, where T = 2120, $B_{i,j}^S$, and $R_{i,j}^{\mathrm{tax}}$ are the public pension and income tax, respectively, and $\tilde{I}$ is the wage-inflation index.¹⁶

We divide the pool of retirees into three groups of equal size according to their value of Z. This gives us a group of ‘Contributors’ with negative Z, a middle group of ‘Low receivers’ with mainly small, positive values of Z, and a group of ‘High receivers’ with the largest, positive values of Z. Clearly, individuals also contribute during their working years via tax payments to the pension of previous cohorts, but we focus here on the group of retirees to illustrate both the role of means-tested public pensions, DC-pensions (and thus taxable income) and the heterogeneity in the group of retirees.

4.3.2. Contributors and beneficiaries

Figure 5 gives a characterisation of the three groups of retirees in terms of their socio-economic profiles. Regarding the age distribution, Figure 5(A) shows that almost everyone of age 87 and above is a high receiver (the fixed term annuity ends at age 86). Regarding employment, Figure 5(B) shows that almost everyone with very long periods of unemployment is a high receiver, while Figures 5(C,D) show that of the three groups, high receivers have the lowest average income and the lowest average return on their savings.

Figure 5. Age distribution (Panel A), employment distribution (Panel B), labour income distribution (Panel C), and returns on pension accounts (Panel D) for three groups of retirees in 2120. The groups consist of contributors, low receivers, and high receivers of public pensions net taxes, respectively.

These observations confirm our preconception of who benefits from pillar 1: The very old, the unemployed, the low income groups, and those who have suffered financial losses. The insight of interest, however, is not that these groups benefit from pillar 1 but that almost all socio-economic characteristics are represented in the group of high receivers.

In fact, although there are distinct differences between the three groups with contributors generally being younger, more employed, with higher income, and earning higher returns than both low and high receivers, the most striking feature of Figure 5 is the considerable variability and overlap in characteristics between the groups.

We do not model socio-economic groups explicitly, but the lifetime average real income given employment can be regarded as proxy for socio-economic group.¹⁷ With this interpretation, Figure 5(C) shows that, except at the very highest end, members of all socio-economic groups are among net receivers of public pensions. Effectively, pillar 1 acts as an insurance against the risks of old age, unemployment, and low returns – risks that can a priori affect everyone, regardless of their socio-economic group.

4.3.3. Replacement rates

To complement the preceding analysis of who benefits from public pensions, we here look at the importance of public pensions in terms of replacement rates. The replacement rate is defined as the disposable income in the first year of retirement relative to the average disposable income in the 5-year period leading up to retirement.¹⁸ To avoid inflating the replacement rates, we look only at retirees with positive gross income throughout the 5-year period.

For the three groups of retirees previously defined, the left and right plots of Figure 6 show the (total) replacement rate and the replacement rate due to public pensions, respectively. From the right plot, we see that the average replacement rate due to public pensions ranges from around 15% from contributors to around 45% for high receivers. These differences are due to both the public pensions being higher, but also the pre-retirement wages being lower for the group of high receivers relative to the group of contributors.

Figure 6. Replacement rate distributions for (new) retirees in 2120. The groups consist of contributors, low receivers, and high receivers of public pensions net taxes, respectively. The replacement rate is either the disposable income (left plot), or after-tax public pension (right plot) divided by the average disposable income over the 5 years leading up to retirement. Only retirees with positive gross income throughout the 5-year period are included. The vertical dashed line marks a replacement level of 80 pct.; due to fewer expenses a replacement level of this size is typically sufficient to maintain pre-retirement lifestyle.

In contrast to the marked differences in replacement rates due to public pensions, the three groups have a similar distribution of (total) replacement rates, cf. left plot of Figure 6. In a sense, the two-pillar system achieves its goal of securing all groups equally. Of course, this is only on a relative scale, equal replacement rates imply differences in absolute retirement income that mirror pre-retirement income differences. It is also implied that replacement rates at retirement are a poor indicator as to the extent individuals benefit from public pensions.

In terms of relative importance, we can infer from Figure 6 that public pensions account for approximately half of the disposable income for the high receiving group, two-fifths for the low receiving group, and one-fifth for the group of contributors.¹⁹

4.3.4. Concluding remarks

To properly assess the roles and interdependencies in a two-pillar pension system, we have aimed for a model that includes most of the major personal risk factors that people are faced with in real life. Arguably the largest (economic) personal risk is unemployment risk and it has been modelled as the risk of having zero gross income in a period of time. Health (not in the model) is also a major risk but in a Danish (Nordic) context the access to tax-financed public health services reduces the implications for pension savings/income.

A key insight from the analysis is that, with a multitude of personal, non-diversifiable risks, a large proportion of the population will experience one or more adverse events that impair their DC savings to a smaller or larger extent. This in turn implies that for at least one-third of the population, public pensions are vital for securing their disposable income in retirement, despite a fully phased-in DC system. Moreover, almost all socio-economic characteristics are represented in this group.

4.4. Demographic changes and stability

The Western world is currently experiencing both increasing life expectancy and fertility rates below replacement levels. In response to increasing longevity, many European countries have implemented pension reforms linking the statutory retirement age to (remaining) life expectancy of each birth cohort or similar indexation schemes. Ceteris paribus, a higher retirement age implies more working years and therefore higher DC savings and more taxes being paid, but it also implies fewer years in retirement and therefore fewer years in which to spend the accumulated DC funds and receive public pensions. Both of these effects improve personal and public finances alike.

The effects of low fertility rates are more subtle. Leaving aside the question of shortage of workers, sub-replacement level fertility gradually changes both the age composition of the population and the absolute size of the population. This in turn limits the possibility of financing public pensions by tax from workers. However, if, at the same time, longevity and retirement age both increase, the balance between workers and retirees can be restored.

In this final analysis, we quantify the effect of changing demographics on the stability of the pension system as a whole, and pillar 1 in particular.

4.4.1. Setup

The analysis is performed for a system with a tax rate of 45% and a DC rate of 12.5%. The same initial population with 2000 agent types is projected to 2250, i.e. for 230 years, under different demographic scenarios. The long simulations are needed because demographic changes affect the system only gradually and over long horizons. For each demographic scenario, the simulation is repeated 1000 times and ensemble averages of key statistics are plotted as a function of time.

We construct a best estimate projection of future Danish (unisex) mortality, cf. Appendix A.3.2. We define a low and high fertility schedule by 75% and 125%, respectively, of the replacement level (RL) fertility schedule used in Sections 4.2 and 4.3. The low fertility schedule corresponds to the current level of fertility in many European countries. We refer to the combination of projected, improving mortality and 75% RL fertility as best estimate demographics.

Regarding retirement age (RA), we consider both a constant RA of 67 years (as in Sections 4.2 and 4.3), and a dynamic scheme in which RA increases with a fraction of life expectancy, see Appendix A.3.3 for details. With these building blocks, we construct three scenarios, in addition to the base scenario: best estimate demographics with constant RA; best estimate demographics with dynamic RA; and a scenario with improving mortality, 125% RL fertility, and constant RA.

4.4.2. Sustainability of pillar 1 pensions

For each of the four scenarios, Figure 7 shows the evolution of maximal social benefits over time.²⁰ The public sector is modelled such that the budget relation between taxes and expenditures on public pensions, and other social benefits, are taking into account, and thereby captures how demographic changes affect a PAYG pension scheme. The change in the maximal social benefit is a measure of sustainability in the sense of maintaining a given level of public pensions. If (maximal) social benefits decrease over time, the system is not able to sustain the same level of social protection for each generation.

Figure 7. The average maximal social benefit for each year from 2020 to 2250.

We first note that in all scenarios, there is an initial build-up phase of the DC system in which taxes and social benefits increase, see also Figure 1 and the description in Section 3.4. The length of the build-up phase is approximately 50 years, corresponding to the length of a full working life.

By construction, for the base scenario the level of social benefits stabilizes after the initial build-up phase. In contrast, with a constant retirement age and best estimate demographics the level of social benefits steadily declines after the initial build-up phase. From the point of view of public finances, this is the quantitative background for pension reforms. Without an increase in retirement age the level of social benefits cannot be maintained.

Figure 7 shows that, under best estimate demographics, the indexation of retirement age does indeed stabilize the level of social benefits. In this sense, retirement age indexation is sufficient to ensure the sustainability of public pensions. However, the level at which social benefits stabilize is substantially below that of the base scenario. This is due to low fertility.

Although retirement age indexation can stabilize the old-age dependency ratio (i.e. the ratio of retirees to workers), the ratio also depends on the level of fertility, see Figure A2(b) in the appendix. Low fertility skews the age composition of the population towards old ages, while high fertility has the opposite effect. From Figure 7, we can see the (marginal) effect of increasing fertility from 75% RL to 125% RL. The long-term effect of a fertility increase of this size is about three-fifths of the effect of retirement age indexation. On its own, even a fertility increase of this size is not enough to ensure sustainable public pensions. However, together with some form of retirement age indexation, increased fertility can substantially improve the level of sustainable public pensions.

4.4.3. Adequacy

Besides sustainability, adequacy is a key dimension of interest for a pension system. Figure 8 shows the average replacement rates over time for the four scenarios (the replacement rate is defined in Section 4.3.3). To illustrate the relative importance of the two pillars, Figure 8 shows both the total replacement rate, and the replacement rate attributable to public pensions. As in Figure 7, there is an initial phase-in period before the underlying patterns emerge.

Figure 8. The average replacement rate for new retirees for each year from 2050 to 2250. The replacement rate is either the disposable income (left plot), or after-tax public pension (right plot) divided by the average disposable income over the 5 years leading up to retirement. To avoid inflating the replacement rate, only retirees with positive gross income throughout the 5-year period are included.

It is clear from Figure 8 that, with improving life expectancy, retirement age indexation is needed to secure an adequate pension. In terms of the total replacement rate, an increase in fertility plays only a small role compared to an increase in the number of working years. However, in terms of the replacement rate attributable to public pensions, fertility plays a larger role than retirement age indexation, although both are declining over time.

In broad terms, retirement age indexation with low fertility leads to higher DC savings and higher total pensions on average, while constant retirement age with high fertility leads to higher public pensions but lower total pensions on average. In relative terms, pillar 2 plays a larger role in the former scenario, while pillar 1 plays a larger role in the latter scenario. The simplistic message is: increased retirement age is needed for adequacy, higher fertility is needed for equality.

4.4.4. Concluding remarks

The analysis in Section 4.3 showed that pillar 1 is effective at mitigating personal risks to which pillar 2 is vulnerable, e.g. risks of old age, unemployment, and low income. Pillar 1, on the other hand, is vulnerable to aggregate, demographic risks, e.g. longevity risk and low fertility, to which pillar 2 is only indirectly exposed (via long-term shortage of workers and increased service costs). The two analyses can therefore be seen as complementary in showing the benefits and vulnerabilities of the two pillars.

Under changing demographics, the system has to adapt to maintain sustainable and adequate pensions. The analysis focused on retirement age indexation and increased fertility. To keep the analysis simple, we deliberately looked only at these two ‘drivers’ and only at their marginal effect. The intent of the analysis was to quantify the size of the effects, and while retirement ages can be affected by changes in the statutory retirement age, it is more difficult to change fertility rates.²¹ On the contrary, in real life, the challenges faced by pension systems are not solved by a single adjustment but requires reforms and changes along many dimensions, e.g. tax incentives, expansion of the labour force, etc. Ultimately, navigating the trade-offs of these interventions is a political discipline.

5. Main takeaways

In this paper, we have taken a big picture approach to a discussion of the purpose and design of pension systems. In particular, we have tried to broaden the discussion to encompass an entire population, not only the fully-employed, average worker often focused on. From society's point of view, the pension system must balance multiple objectives including savings/wealth, economic equality, sustainability, and adequacy both in terms of preventing poverty and ensuring acceptable replacement rates.

Many of the inherent trade-offs of interest are related to socio-economic differences between agents, or demographic differences between generations of agents. We have developed an agent-based, two-pillar pension system model that incorporates most of the major risk sources and in which the resulting heterogeneities can be studied. Hopefully, the approach taken can serve as inspiration for others.

We hope to have conveyed the picture of a pension system as consisting of components (pillars) with different characteristics and different risk profiles. By combining these pillars, it is possible to achieve several objectives simultaneously as well as resilience to both personal and aggregate risk factors. A multi-pillar pension system offers outcomes better than any one pillar can provide.

We finally note that, although the model has a fair level of detail, there is still a big leap from our modelling world to the real world. Both in terms of tangible omissions, e.g. health, housing, private savings, bequests, etc., but also in terms of behavioural aspects of agents, e.g. preferences, financial literacy, endogenous goals, herding, etc. Most likely, adding some or all of these features would yield further insights, but there is also a risk of losing sight of the bigger picture by doing so. In this paper, we have strived for interpretability over detail.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors gratefully acknowledge support from the Danish Labour Market Supplementary Pension Fund (ATP) and the Rockwool Foundation under the project ‘Pension and Ageing’.

Notes

1 In the simplest set-up, the retirement age is exogenous, but it can straightforwardly be endogenized by introducing a trade-off between consumption and leisure where disutility from work may increase with age, see, e.g. Crawford & Lilien (1981).

2 There is asymmetry since savings is always possible but borrowing in periods with low income may be impossible (credit constraint) or very expensive (high risk premium). The reason is that the scope for borrowing depends critically on what types of incomes/assets the creditor can seize in case of default. While it is possible for creditors to seize tangible assets this is not so for human capital (ruling out slavery) which is the reason why it is difficult to borrow with future wage income as collateral.

3 Early death also raises an issue of protection of partner and children. This risk can be covered via life insurance, a more widely used product than life-annuities.

4 If there is a bequest motive, the desired bequest should be saved in standard assets.

5 Diamond & Spinnewijn (2011) argue that the present-bias problem is particularly prevalent for low- and middle-income groups (present bias is correlated with income), adding a distributional dimension to the undersavings problem.

6 Empirical evidence documents that higher wealth is generally associated with higher returns, also when compared for similar risk levels, see, e.g. Deuflhard et al. (2019) and Fagereng et al. (2020).

7 If each cohort pays a given contribution rate on their wage income to finance a pension to the current old, the implicit return to the individual is determined by the ratio between the pension received as old and the contribution made when young which is equal to the ratio of total wage income when old to young which is one plus the growth rate of total wage income (the implicit return).

8 Refers to a situation where there is no scope for making a Pareto efficient allocation benefitting some cohorts without harming others. It can also be interpreted as a situation where there is no overaccumulation of capital, see, e.g. Blanchard & Fischer (1989).

9 In general equilibrium aggregate shocks affecting wage income may also affect the market return, and hence aggregate shocks tend to affect both systems, but in different ways.

10 Hence, if there is a political objective of distribution, a funded scheme would not be sufficient, and hence the implicit insurance is a side benefit.

11 The relation between the contribution period and the benefit period, where the latter is typically determined by the difference between longevity and the statutory retirement period.

12 This was also the key point in the well-known (World Bank 1994) on pensions. For a more formal analysis, see, e.g. Matsen & Thøgersen (2004).

13 A particular issue is the possible crowding out of voluntary savings to mandatory savings in occupational pension schemes. Empirical evidence generally find this crowding out to be modest, for evidence for Denmark, see, e.g. Chetty et al. (2014). The model of the public sector ensures balance between changes in tax revenue and expenditures on social benefits. For a small open economy, it is also a reasonable assumption to assume factor prices (wages, rates of returns) to be exogenous.

14 The Gini coefficient is the standard measure of inequality in a population, see Gini (1936). In a population where everyone has the same income the Gini coefficient is 0, and in an (infinite) population where all income is concentrated at one person the Gini coefficient is 1.

15 For reference, the Danish Gini coefficient of equivalized disposable income was 27.7% in December 2022 (Eurostat). In Figure 2, the Gini coefficient for disposable income for workers and retirees is 25.4%.

16 See Appendices A.2 and A.4 for details. The explicit scenario index, j, is omitted in the appendix.

17 This is not the income actually received but the (counterfactual) income had the agent been fully employed in all working years. It is computed as the average of the Υ-process, cf. Appendix A.5.

18 All incomes are adjusted with wage-inflation before averaging and forming the ratio.

19 For each of the three groups, the number is computed as the ratio of the median replacement rate in the right plot of Figure 6 to the median replacement rate in the left plot of Figure 6.

20 The maximal social benefit is the benefit that a retiree with no other income receives. Since all other social benefits are set relative to this level, it is also a measure of the general level of social benefits.

21 As noted elsewhere, the model is rule-based and the agents are assumed to follow the rules. Of course, regarding fertility in particular it is hard to imagine the real-world intervention needed to dramatically increase fertility overnight. The high fertility scenario is chosen, not for realism, but to get an effect of comparable size to RA indexation, elucidating the relative importance of fertility and mortality changes.

22 For the analyses in the paper, we use $\zeta =3\mathrm{\%}$.

23 Age ‘67’ should be read as the current retirement age, which can change over time depending on the underlying demographics and rules for indexation (hence the profile s will also depend on time, but for ease of presentation we leave that detail out of the notation).

24 In a two-sex population, NRR is typically defined as the average number of daughters that a woman gives birth to.

25 The old-age dependency ratio is computed as the size of the retired population divided by the population of age 24 to (retirement age − 1).

26 Strictly speaking, it is the uncertainty of pension contributions that matter. In our case, however, the DC scheme is mandatory and uncertain conversion of labour income to contributions is not an issue.

Appendix. An agent-based pension system model

In the proceeding sections, we present in detail the agent-based pension system model used throughout the second part of the paper. Table A1 provides an overview of the most important notations for ease of reference.

Table A1. Overview of notations (and their default values).

Symbol		Description
τ   $(45\mathrm{\%})$		Tax rate
ζ   $(3\mathrm{\%})$		Technical rate used to evaluate the price of unit liabilities
${\gamma }^{\mathrm{DC}}$  $(12.5\mathrm{\%})$		Fraction of labour income contributed to the DC system
${\gamma }^L$ $(55\mathrm{\%})$		Fraction of DC contribution paid to life annuities
${\gamma }^M$  $(1/3)$		Fraction of income used to determine social benefit tapering in the means-testing mechanism
${\gamma }^{\mathrm{EE}}$  $(1/2)$		Fraction of total tax spent on exogenous (non-modelled) expenses
${\gamma }^R$  $(2/9)$		Fraction of expected total lifetime spent in retirement.
$W_i$		Labour income for individual i
$Y_i,Y_i^{\mathrm{disp}}$		Gross and disposable income for individual i
$C_i^{\mathrm{DC}},C_i^F,C_i^L$		DC-, fixed term annuity-, and lifetime annuity contribution for individual i
$B_i^F,B_i^L,B_i^S$		Fixed term annuity-, lifetime annuity-, and social benefit for individual i
$u^F,u^L$		The price of unit liabilities for a fixed term annuity and a lifetime annuity
$V^F,V^L$		The fixed term and lifetime annuity reserve
$I_t,{\tilde{I}}_t$		Price and wage inflation processes
$N(x)$		Number of individuals alive at age x at time $t_0$
$N_B(t)$		Number of individuals born at time t
$\mu (t,x)$		Force of mortality at time t age x
${\xi }_{t+1}$		Financial return from time t to t + 1
$R(t)$		Total revenue from taxes at time t
$L(t,x)$		Maximal social benefit receivable at time t age x
$x^P(t)$		Pension age in effect at time t
$G(x)$		Average real wage of a working individual aged x
${\mathrm{{\rm Y}}}_i(t)$		Real labour income for individual i conditioned on being at work
$E_i(t)$		Employment process for individual i

A.1. Agents

We model a society of individuals (agents) evolving over time in annual time steps. Given a (real) population with $N(x)$ individuals of age x at time $t_0$, we generate a representative set of agents, $A_1,\dots ,A_M$, in the ratio $\mathrm{\Delta }:1$, where Δ is a scale factor. More precisely, we generate agents such that $\mathrm{\#}\{i:\text{agent}{A}_i\text{is of age}x\text{at time}{t}_0\}=\lfloor N(x)/\mathrm{\Delta }\rceil \text{for all}x$, where $\lfloor y\rceil$ denotes integer rounding. For example, starting with the Danish population for ages $x=0,\dots ,100$ in year $t_0=2020$ and $\mathrm{\Delta }=200$, we would get a set of $M=\mathrm{29,112}$ agents with an age distribution closely mimicking that of the Danish population.

The agents are generated with idiosyncratic socio-economic factors. Specifically, the agents differ by their level of mortality relative to base mortality rates, and their annual labour income during working years. These socio-economic factors are fixed throughout the lifetime of the agent. In addition, the agents hold a number of financial data, e.g. income, tax paid, pension contributions, pension savings, social benefits, etc. These data are updated in each step of the projection.

A.1.1. Newborns, survivors, and deceased

We take as given a projection of base mortality rates spanning the relevant ages and future years. We denote by $\mu (t,x)$ the force of mortality at age x at time t. An agent, $A_i$, alive and of age x at time t survives to time t + 1 with probability $\exp (-z_i\mu (t,x))$, where $z_i$ is the agent's level of frailty. Thus agents with high values of z have a lower survival probability than agents with low values of z. The z's thereby introduce life expectancy heterogeneity among the agents.

In addition to mortality rates, we also take as given a projection of fertility rates spanning the relevant reproductive ages and future years. We denote by $F(t,x)$ the unisex, fertility rate at age x at time t. The expected number of newborns (conceived at time t, but delivered at time t + 1) is given by (A1) ${\lambda }_B(t)=\sum _{i\in \mathcal{I}(t)}F(t,x_i),$(A1) where $x_i$ is the age of agent $A_i$ at time t, and $\mathcal{I}(t)$ denotes the set of (indices of) agents alive at time t. In practice, fertility rates are zero above age 50, say, but conceptually every agent can give rise to a new agent as expressed by Equation (A1(A1) ${\lambda }_B(t)=\sum _{i\in \mathcal{I}(t)}F(t,x_i),$(A1) ).

Box 1: The number of newborns

Arguably, since operating in an agent-based model, the actual number of newborns, $N_B(t)$, should be drawn from a Poisson-distribution with mean ${\lambda }_B(t)$ reflecting the intrinsic stochasticity of births in a finite population. On the other hand, thinking of the agents as representatives of the larger population used at initialization we would like the fluctuations of the two populations to be of similar (relative) magnitude. We therefore demand that the standard deviation of the number of births divided by the population size is the same in the two populations. We achieve this by letting $N_B(t)={\lambda }_{\det }(t)+\text{Poisson}({\lambda }_B(t)-{\lambda }_{\det }(t))$, where ${\lambda }_{\det }(t)=\lfloor {\lambda }_B(t)(\mathrm{\Delta }-1)/\mathrm{\Delta }\rfloor$. It is easy to see that $\text{std}(N_B(t))/M\approx \sqrt{\mathrm{\Delta }{\lambda }_B(t)}/(\mathrm{\Delta M})$, which is (approximately) equal to the standard deviation of the number of births to the population size in the original population.

As explained in Box 1, the actual number of newborns, $N_B(t)$, is taken as a (convex) combination of ${\lambda }_B(t)$ and a Poisson variate. The newborns are added to the set of agents as agents of age 0 at time t + 1, i.e. $M(t+1)=M(t)+N_B(t)$, where $M(t)$ is the number of agents that have been seen up to time t, and $M(t_0)=M$ is the initial number of agents. Hence, after the first step in the simulation, the full set of agents consists of $A_1,\dots ,A_{M+N_B(t_0)}$; after two steps the full set of agents consists of $A_1,\dots ,A_{M+N_B(t_0)+N_B(t_0+1)}$, and so on.

The full set of agents conveys the idea of an ever-expanding set of uniquely identifiable agents. The simulation, however, is centered around the transitions from one point in time to another. We have (A2) $\mathcal{I}(t+1)=(\mathcal{I}(t)\setminus \mathcal{D}(t))\cup \mathcal{B}(t),$(A2) where $\mathcal{D}(t)$ and $\mathcal{B}(t)$ denote the set of (indices of) agents dying during $(t,t+1]$ and being born at t + 1, respectively. Considering the sizes of the sets occurring in Equation (A2(A2) $\mathcal{I}(t+1)=(\mathcal{I}(t)\setminus \mathcal{D}(t))\cup \mathcal{B}(t),$(A2) ), we get the well-known balancing equation for a closed population: $|\mathcal{I}(t+1)|=|\mathcal{I}(t)|-|\mathcal{D}(t)|+|\mathcal{B}(t)|$, stating that, without migration, the size of a population changes only as a consequence of deaths and births. For later use, we also introduce the notation $\mathcal{S}(t)$ for the agents alive at t that survive to t + 1, i.e. $\mathcal{I}(t)=\mathcal{S}(t)\cup \mathcal{D}(t)$.

Figure A1 illustrates a projected age distribution after 80 years, based on agent representations of varying size. Although the variability depends on the granularity of the representation, all the distributions follow the underlying mean projection closely. This is due to the regularization of the number of newborns. In principle, the number of deaths could be regularized in a similar manner, but since most deaths occur at ages with relatively high mortality rates the death distribution has (comparatively) low variability and the need for regularization is much smaller.

A.2. Pensions and social benefits

All agents enter the labour market at age 24. In general, the retirement age of the agent depends on the life expectancy evolution, cf. Appendix A.3.3. For ease of presentation, we here assume a fixed retirement age of 67 for everyone. The labour income of agent $A_i$ for the period from t to t + 1 is denoted $W_i(t)$, assuming $24\le x_i<67$ where $x_i$ is the age of the agent at time t. If the agent is unemployed the labour income is zero, but the formulas below are still valid. The agent-specific labour income process is described in Appendix A.5. Agents below age 24, or in retirement, have no labour income, but retired agents may have a defined contribution (DC) pension consisting of a combination of 20-year fixed term and lifetime annuities.

We assume that a fraction, ${\gamma }^{\mathrm{DC}}$, of the labour income is contributed to the DC system. The contributions are tax-exempt, but pension benefits are taxed when they are paid out. Assuming the agent survives the period, the gross income of the agent is given by (A3) $Y_i(t)=W_i(t)+B_i^F(t)+B_i^L(t)-C_i^{\mathrm{DC}}(t),$(A3) where $B_i^F(t)$ and $B_i^L(t)$ are the benefits from the fixed term and lifetime annuities, respectively, and $C_i^{\mathrm{DC}}(t)={\gamma }^{\mathrm{DC}}{W}_i(t)$.

Everyone pays the same flat rate, say, $\tau =45\mathrm{\%}$, on their income in excess of a basic allowance, $D(t)$, see Box 2 for further details. The taxes collected (including revenues from the DC system) are redistributed to the population in the form of social benefits. The tax revenue from personal income is (A4) $R_i^{\mathrm{tax}}(t)=\tau {(Y_i(t)+B_i^S(t)-D(t))}^{+},$(A4) where $B_i^S(t)$ denotes social benefits paid to individual i. Thus, after taxes and social benefits, each agent has a disposable income of (A5) $Y_i^{\mathrm{disp}}(t)=Y_i(t)+B_i^S(t)-R_i^{\mathrm{tax}}(t).$(A5) In the next sections, we describe the dynamics of the underlying DC pension accounts, and the computation of tax revenues and social benefits.

Box 2: Calibration of tax rates

Taken as a 10-year average (2012–2021), the total personal income tax and consumption tax (VAT) (Table OFF12, Statistics Denmark) divided by the total income (Table INDKP102, Statistics Denmark) gives a tax rate of $47\mathrm{\%}$. This number reflects the rate of taxation of income when it is either earned or spent.

The basic allowance is $D(t)={\tilde{I}}_t{D}_{\mathrm{base}}$ where ${\tilde{I}}_t$ is the wage inflation process of Appendix A.4, and the base amount is $D_{\mathrm{base}}=\mathrm{46,700}$ DKK, which is the basic allowance in Denmark in 2021. This number is chosen to match the level of the wages, which are based on Danish income data from 2021, cf. Appendix A.5.

A.2.1. Pension account dynamics and taxes

The DC contribution is split such that a fraction, ${\gamma }^L$, of the contribution enters into a lifetime annuity account and the remaining part into a fixed term annuity account, see also Box 3. The fixed term and lifetime annuities differ in the way benefits are calculated and the ownership of the accounts of the dead. Lifetime annuity accounts of deceased policyholders are transferred to surviving policyholders, giving rise to an additional return from these bequests. In contrast, fixed term annuity accounts belong to the estate of the deceased, which in our case is the state. Financial returns on either type of DC accounts are tax-exempt.

Box 3 Calibration of the annuity fraction (${\gamma }^L$)

Taken as a 10-year average (2012–2021), the split between the contribution to pension schemes that enter lifetime and fixed term annuities is 55/45 (cf. www.fogp.dk/en/statistics/statistics-on-life-insurance/). Thus we let ${\gamma }^L=55\mathrm{\%}$.

Consider an agent $A_i$ of age x at time t. The total and annuity-specific contributions of the agent during the period from t to t + 1 are given by (A6) $C_i^{\mathrm{DC}}(t)={\gamma }^{\mathrm{DC}}{W}_i(t),C_i^L(t)={\gamma }^L{C}_i^{\mathrm{DC}}(t),C_i^F(t)=(1-{\gamma }^L)C_i^{\mathrm{DC}}(t).$(A6) To facilitate computation of the benefits, we introduce the price of unit liabilities at time t for an agent of age x: (A7) $\begin{array}{rl}{u}_t^L(x)& =\sum _{y=x}^{101}(1+\zeta {)}^{-(y-x)}{S}_t(y|x)\text{for}x=67,\dots ,101,\end{array}$(A7) (A8) $\begin{array}{rl}{u}^F(x)& =\sum _{y=x}^{86}(1+\zeta {)}^{-(y-x)}\text{for}x=67,\dots ,86,\end{array}$(A8) where ζ is the technical rate²² , and $S_t(y|x)$ is the cohort-specific probability of an agent of age x at time t to reach age y. The survival probabilities are computed using base mortality levels, i.e. the same mortality surface is used for all agents.

The lifetime annuity account evolves according to the dynamics (A9) $\begin{array}{rl}{B}_i^L(t)& =\frac{V_i^L(t)}{u_t^L(x_i)}{1}_{(67\le x_i)},\end{array}$(A9) (A10) $\begin{array}{rl}{V}_i^L(t+)& =V_i^L(t)+C_i^L(t)-B_i^L(t),\end{array}$(A10) (A11) $\begin{array}{rl}{V}_i^L(t+1)& =(1+{\xi }_{t+1})\exp (z_i\mu (t,x_i))V_i^L(t+),\end{array}$(A11) where ${\xi }_{t+1}$ is the financial return on the account over the period from t to t + 1. Note that we use the actual mortality of the agent, $z_i\mu (t,x_i)$, in the mortality bequest term (rather than $\mu (t,x_i)$). This is done to avoid an implicit financing of the system. One might think of the agents as being in different pension companies with similar agents. The use of base mortality rates in $u_t^L$ implies a (slight) front-loading, or back-loading of benefits for some agents, but it does not lead to a systematic loss/profit.

The fixed term annuity account evolves according to the dynamics (A12) $\begin{array}{rl}{B}_i^F(t)& =\frac{V_i^F(t)}{u^F(x_i)}{1}_{(67\le x_i<86)},\end{array}$(A12) (A13) $\begin{array}{rl}{V}_i^F(t+)& =V_i^F(t)+C_i^F(t)-B_i^F(t),\end{array}$(A13) (A14) $\begin{array}{rl}{V}_i^F(t+1)& =(1+{\xi }_{t+1})V_i^F(t+).\end{array}$(A14) We think of the situation where everyone is alive at the beginning of the period at which point they receive labour income, make contributions and receive benefits; this is reflected in the intermediate account values, $V_i^L(t+)$ and $V_i^F(t+)$. Immediately afterwards some agents die, leaving behind their $t+$-accounts to the surviving lifetime annuity policyholders and the state, respectively. In addition, we assume that the benefits of the dead are transferred to the state. In this way, the state receives some compensation for the otherwise tax-exempt lifetime annuity system. Thus the total tax revenue from income taxes and the pension system is given by (A15) $R(t)=\sum _{i\in \mathcal{S}(t)}{R}_i^{\mathrm{tax}}(t)+\sum _{i\in \mathcal{D}(t)}[V_i^F(t+)+B_i^F(t)+B_i^L(t)].$(A15)

A.2.2. Social benefits

The purpose of social benefits is to ensure that everyone has a minimal level of net income, although not the same level. The social benefits are computed from the agent's gross income, $Y_i(t)$, and their age.

First, we introduce an age-profile, $s(x)$, which determines the relative, maximal social benefit that the different age groups can receive. We suppose that (A16) $\begin{array}{r}s(x)=\{\begin{array}{ll}0& x<18\\ 0.45& 18\le x<24,\\ 0.9& 24\le x<67,\\ 1& x\ge 67,\end{array}\end{array}$(A16) with the interpretation that young adults can receive roughly half of the (maximal) amount receivable for the retirees, while the working population can receive at most 90% of the retirees.²³ The maximal level of social benefits at time t is denoted $L(t)$, and, with slight abuse of notation, we let $L(t,x)=L(t)s(x)$.

Second, we define a means-testing mechanism such that the full benefit tapers off with a fraction, ${\gamma }^M$, of the income in excess of $L(t,x)$. Thus, assuming an agent, $A_i$, of age x at time t with a gross income of $Y_i(t)$, the social benefit receivable is given by (A17) $B_i^S(t)={(L(t,x)-{\gamma }^M[Y_i(t)-L(t,x){]}^{+})}^{+}.$(A17) The level of benefits, $L(t)$, is given implicitly by the requirement that total social benefits equals total tax revenues less exogenous expenses, (A18) $\sum _{i\in \mathcal{S}(t)}{B}_i^S(t)=(1-{\gamma }^{\mathrm{EE}})R(t),$(A18) where $R(t)$ is given in Equation (A15(A15) $R(t)=\sum _{i\in \mathcal{S}(t)}{R}_i^{\mathrm{tax}}(t)+\sum _{i\in \mathcal{D}(t)}[V_i^F(t+)+B_i^F(t)+B_i^L(t)].$(A15) ), and ${\gamma }^{\mathrm{EE}}$ is the fraction of total tax spent on non-modelled expenses, cf. Box 4.

Assuming ${\gamma }^M<1$, the social benefit, $B_i^S(t)$, ensures that the agent has a net income of at least $L(t,x)$. For the analyses in the paper, we use ${\gamma }^M=1/3$.

Box 4 Calibration of exogenous expenses $({\gamma }^{\mathrm{EE}})$

Taken as a 10-year average (2012–2021), expenditures on public pensions and other social benefits account for $22.0\mathrm{\%}$ and $12.3\mathrm{\%}$, respectively, of the Danish total tax revenue. These numbers are computed by dividing the amount spent on public pensions and social benefits (Table ESSPROS1, Statistics Denmark) with the Danish total tax revenue (Table OFF12, Statistics Denmark). To achieve realistic social benefit levels, we adjust ${\gamma }^{\mathrm{EE}}$ to reflect the fact that our model only has income tax. Income tax makes up for $70.4\mathrm{\%}$ of the total tax revenue (10-year average, 2012–2021). Whence, we let ${\gamma }^{\mathrm{EE}}=1/2(\approx \frac{34.3\mathrm{\%}}{70.4\mathrm{\%}}$).

A.3. Demographics

The impact of demographics will be illustrated by use of scenarios. Within each scenario, vital rates, i.e. mortality and fertility rates, are deterministic functions of time and age. We define a base scenario with stationary demographics, and two alternative scenarios with a best estimate projection of future Danish (unisex mortality) and varying fertility rates.

A.3.1. Stationary and stable populations

A population with an age distribution which remains constant over time is referred to as stable. A stable population of constant size is said to be stationary. Assuming constant fertility and mortality rates, and zero net migration in all ages, a stable population will arise over time, and it is independent of the initial age distribution, see, e.g. Pollard (2015), Preston et al. (2001) and Sharpe & Lotka (1911). For ease of notation, we assume a continuous-time model in this subsection. Further, since the vital rates are assumed constant we drop the usual time argument.

The age-composition of the limiting stable population is given by (A19) $c(x)=\frac{\exp (-\mathrm{rx})S(x)}{{\int }_0^{\mathrm{\infty }}\exp (-\mathrm{rx})S(x)\mathrm{d}x},$(A19) where r is the intrinsic growth rate defined as the unique solution to the equation (A20) ${\int }_0^{\mathrm{\infty }}\exp (-\mathrm{rx})S(x)F(x)\mathrm{d}x=1.$(A20) The intrinsic growth rate determines the growth of the population once the limit c has been achieved. The actual growth rate of the population differs to a smaller or larger extent from r, depending on how far the initial age distribution is from c.

The net reproduction rate (NRR) is the average number of offsprings by each member of the population.²⁴ Formally, it is defined as (A21) $\mathrm{NRR}={\int }_0^{\mathrm{\infty }}S(x)F(x)\mathrm{d}x.$(A21) Comparing (A20(A20) ${\int }_0^{\mathrm{\infty }}\exp (-\mathrm{rx})S(x)F(x)\mathrm{d}x=1.$(A20) ) and (A21(A21) $\mathrm{NRR}={\int }_0^{\mathrm{\infty }}S(x)F(x)\mathrm{d}x.$(A21) ), it follows that NRR = 1 is equivalent to r = 0, and that NRR greater (smaller) than 1 is equivalent to r greater (smaller) than 0. A level of fertility that ensures NRR = 1 for given mortality schedule is called replacement level fertility.

It follows from Equation (A19(A19) $c(x)=\frac{\exp (-\mathrm{rx})S(x)}{{\int }_0^{\mathrm{\infty }}\exp (-\mathrm{rx})S(x)\mathrm{d}x},$(A19) ) that in a stationary distribution (r = 0), the age composition is proportional to the survival function. If, however, the population is expanding (r>0) the age distribution is shifted towards the younger ages, and, conversely, if the population is contracting (r<0) the age distribution is shifted towards the older ages.

A.3.2. Demographic scenarios

We consider two mortality projections, with and without improvements, respectively. The projection with improvements is constructed as a median projection from a Lee–Carter model of Danish, unisex mortality with Gamma-distributed frailty, see Jarner & Jallbjørn (2022) for further details on estimation and forecasting in this model. The projection without improvements uses the Danish period mortality for 2020 in all years.

We construct a (static) replacement level fertility profile from Danish, total fertility rates in 2020. More precisely, we scale the Danish fertility profile from 2020 such that, when combined with period mortality from 2020, it gives a net reproduction rate of 1. We extend this to a fertility projection by assuming constant age-specific fertility rates over time. In addition to replacement level fertility, we also consider fertility of $75\mathrm{\%}$ and $125\mathrm{\%}$ of replacement levels, respectively. In 2021, the total fertility rate in Europe was 1.53 live births per woman, and the former fertility schedule is meant to reflect this level (recalling that our fertility rate is a unisex quantity).

We consider in total three demographic scenarios: static mortality and replacement level fertility (the base scenario), and improving mortality with a fertility schedule corresponding to either 75% or 125% of replacement levels (RL). In all scenarios we use an initial population with an age distribution proportional to the period survival function in 2020. In combination with the static mortality projection and replacement level fertility, this gives a stationary population.

The age distribution of the population in 2120 under the three demographic scenarios is shown in Figure A2(a). Since the base scenario (red line) is stationary, the age distribution in 2120 for this scenario is equal to that of the initial population. Compared to this profile, the composite effect of mortality improvements and low fertility causes a particularly old population. We also note, that even after 100 years the effect of the initial population is still clearly present as ‘ripples’ caused by cohorts of the ‘wrong’ size passing through their peak reproductive ages.

A.3.3. Retirement age

Faced with increased life expectancies, many countries have introduced indexation of the (statutory) retirement age to ensure financial sustainability of the pension systems. Different schemes exist with features such as smoothing, lagging, rounding, and partial indexation. We do not aim to reproduce any real-world scheme in particular. Instead, we choose a very simple mechanism which aims at keeping the lifetime fraction spent in retirement (approximately) constant across generations.

Specifically, we compute the retirement age in effect at year t as (A22) $x^R(t)=\underset{x}{\arg \min }|\frac{e_x(t)}{e_0(t)}-{\gamma }^R|,$(A22) where $e_x(t)$ is remaining life expectancy at age x using the period mortality of year t, and ${\gamma }^R$ is the fraction of expected total lifetime spent in retirement.

Assuming improving mortality, Figure A3(a) shows the period and cohort life expectancy at birth together with the retirement age from 2020 to 2120 under two different indexation schemes; the initial retirement age, $x^R(2020)$, is 67 and 70 for ${\gamma }^R=2/9$ and ${\gamma }^R=1/5$, respectively. Over 100 years, the (period) life expectancy increases with 9.24 years, while the retirement age increases with 5 years, from 67 to 72 years for ${\gamma }^R=2/9$, and from 70 to 75 years for ${\gamma }^R=1/5$. We have used ${\gamma }^R=2/9$ for the analyses in the paper, which gives an initial retirement age equal to the current statutory retirement age of Denmark.

In Equation (A22(A22) $x^R(t)=\underset{x}{\arg \min }|\frac{e_x(t)}{e_0(t)}-{\gamma }^R|,$(A22) ), one might argue that it would be more ‘correct’ to use cohort life expectancies for the same birth cohort in the numerator and denominator of the life expectancy ratio. However, we see from Figure A3(a) that the cohort life expectancy for newborns in 2020 is in fact very close to the period life expectancy in 2090 when this generation is about to retire. Further, we see from Figure A3(b) that the period and cohort life expectancy at retirement are close. Therefore, it makes only little difference whether cohort or period life expectancies are used in Equation (A22(A22) $x^R(t)=\underset{x}{\arg \min }|\frac{e_x(t)}{e_0(t)}-{\gamma }^R|,$(A22) ).

In Figure A3(b), the period and cohort life expectancy at age 67 and at retirement age is illustrated. The expected time in retirement increases by approximately one and three quarter years over the period, which is comparable to (but a little less than) two-ninth of the increase in life expectancy at birth, as intended.

The old-age dependency ratio, i.e. the ratio of the retired to the working part of the population, is a commonly used measure to illustrate the effect of changing demographics. Figure A2(b) shows the old-age dependency ratio for the three demographic scenarios.²⁵ For the scenario with mortality improvements (and low fertility), the ratio is shown assuming both constant and indexed retirement age. With improving mortality and constant retirement age, the old ratio increases throughout the period. In contrast, with mortality improvements and retirement age indexation, the old-age dependency ratio resembles that of the static scenario. In that sense, the retirement age indexation has the effect of stabilizing the old-age dependency ratio. A high fertility schedule has roughly the same effect.

A.4. Financial market

The capital market model is described in Jarner & Preisel (2017), see also Munk et al. (2004) and Jørgensen & Slipsager (2016), but for ease of reference we restate it here. The model is well suited to generate long-term projections with plausible associations between stocks, bonds, and inflation. In particular – and in contrast to a standard Black–Scholes model – the mean-reverting equity risk premium process, if suitably calibrated, guarantees well-behaved stock prices on long horizons.

The model features five stochastic components

• A short rate ($r_t$) driving a bond market of all maturities

• An equity index ($S_t$)

• A mean-reverting equity risk premium ($x_t$)

• An inflation index ($I_t$)

• A mean-reverting expected inflation process (${\pi }_t$).

The model of Jarner & Preisel (2017) also contains break–even inflation (BEI) curves for pricing inflation-indexed bonds, and inflation swaps, but these will not be needed.

A.4.1. Short rate, inflation, and stock returns

We assume that the short (nominal) interest rate follows an Ornstein–Uhlenbeck process, (A23) $\mathrm{d}{r}_t=\kappa (\overline{r}-r_t)\mathrm{d}t+{\sigma }_r\mathrm{d}{W}_t^r,$(A23) where $\overline{r}$ is the long-run mean of the short interest rate, κ describes the degree of mean reversion, ${\sigma }_r$ is the interest rate volatility, and $W^r$ is a standard Brownian motion.

The stock index (total return index) is assumed to evolve according to the dynamics (A24) $\frac{\mathrm{d}{S}_t}{S_t}=(r_t+x_t)\mathrm{d}t+{\sigma }_S\mathrm{d}{W}_t^S,$(A24) where $r_t$ is the short rate from (A23(A23) $\mathrm{d}{r}_t=\kappa (\overline{r}-r_t)\mathrm{d}t+{\sigma }_r\mathrm{d}{W}_t^r,$(A23) ), $x_t$ is the time-varying expected excess return (or risk premium) from investing in stocks, ${\sigma }_S$ is the stock index volatility, and $W^S$ is a standard Brownian motion. We further assume that the excess return follows an Ornstein–Uhlenbeck process, (A25) $\mathrm{d}{x}_t=\alpha (\overline{x}-x_t)\mathrm{d}t-{\sigma }_x\mathrm{d}{W}_t^S,$(A25) where $\overline{x}$ denotes the long-run equity risk premium, α describes the degree of mean reversion towards this level, and ${\sigma }_x$ is the excess return volatility. Note that the stock index and the expected return processes are locally perfectly negatively correlated, i.e. a stock return above or below its expected value will ‘cause’ a change in the (future) expected return in the opposite direction. This interaction induces a mean reversion in the stock returns over time.

We also introduce a price index process, I, interpreted as the nominal price of a real consumption good. We assume that I evolves according to the dynamics (A26) $\frac{\mathrm{d}{I}_t}{I_t}={\pi }_t\mathrm{d}t+{\sigma }_I\mathrm{d}{W}_t^I,$(A26) with (A27) $\mathrm{d}{\pi }_t=\beta (\overline{\pi }-{\pi }_t)\mathrm{d}t+{\sigma }_{\pi }\mathrm{d}{W}_t^{\pi },$(A27) where ${\pi }_t$ is the expected rate of inflation, $\overline{\pi }$ is the long-run mean inflation rate, β describes the degree of mean reversion, ${\sigma }_{\pi }$ is the volatility of the expected inflation rate, ${\sigma }_I$ is the volatility of the price index, and $W^I$ and $W^{\pi }$ are standard Brownian motions. The price index is thus influenced by both expected inflation and unexpected inflation shocks, where the expected inflation forms a persistent process while the shocks are uncorrelated.

To index wages (and implicitly taxes, social benefits, etc.), we introduce in addition to the price index, I, the wage index (A28) ${\tilde{I}}_t=I_t\exp (0.015t).$(A28) That is, for simplicity and in line with the assumptions of CRE (2022), the price inflation adjusted for a constant productivity growth of $1.5\mathrm{\%}$.

A.4.2. Term structure of interest rates

To complete the model, we also need to specify the evolution of the yield curve, i.e. the term structure of interest rates. Assuming no arbitrage, the term structure is uniquely given by the short rate dynamics under P, i.e. Equation (A23(A23) $\mathrm{d}{r}_t=\kappa (\overline{r}-r_t)\mathrm{d}t+{\sigma }_r\mathrm{d}{W}_t^r,$(A23) ), and the assumed market price of interest rate risk, ${\lambda }^r={\lambda }^r(t,r_t)$, see, e.g. Chapter 21 of Björk (2009).

Assuming a linear market price of risk, it can be shown that the term structure of interest rates is of the form considered by Vasicek (1977). In particular, the price at time t of a zero-coupon (government) bond maturing at time $T\ge t$ is given by (A29) $p_t(T)=\exp \{G(\mathrm{\Delta })-H(\mathrm{\Delta })r_t\},$(A29) with $\mathrm{\Delta }=T-t$, (A30) $\begin{array}{rl}H(\mathrm{\Delta })& =\frac{1}{a}(1-\exp \{-a\mathrm{\Delta }\}),\end{array}$(A30) (A31) $\begin{array}{rl}G(\mathrm{\Delta })& =(b-\frac{{\sigma }_r^2}{2a^2})(H(\mathrm{\Delta })-\mathrm{\Delta })-\frac{{\sigma }_r^2}{4a}{H}^2(\mathrm{\Delta }),\end{array}$(A31) and where a and b are parameters controlling the slope and level of the yield curves.

A.4.3. Life-cycle investment profiles

We assume that all DC accounts, for both fixed term and lifetime annuities, are invested in a mix of stocks and bonds. Specifically, we assume that at any point in time a fraction of the account, $w_t^S$, is invested in stocks and the remaining fraction is invested in 5-year zero-coupon (government) bonds. At the end of the annual time step, the stocks and the now 4-year bonds are sold and the proceeds are reinvested in stocks and 5-year bonds, and so on. Thus, the return, ${\xi }_{t+1}$ for the period from t to t + 1 takes the form (A32) ${\xi }_{t+1}=w_t^S\frac{S_{t+1}}{S_t}+(1-w_t^S)\frac{p_{t+1}(t+5)}{p_t(t+5)}-1.$(A32) We further assume that the equity share follows a simple, piecewise linear, life-cycle investment strategy. For given agent, the strategy has a constant equity share of ${\overline{w}}^S$ until 15 years before retirement, at which point it decreases to ${\overline{w}}^S/2.5$ at retirement. This level is kept for the remaining period of the policy. We use an equity cap of ${\overline{w}}^S=50\mathrm{\%}$.

The capital market parameters used are shown in Table A2. They originate from Jarner & Preisel (2017), but are adapted to be in line with those of CRE (2022). We use the same (average) equity return of $\overline{r}+\overline{x}=6.5\mathrm{\%}$ and inflation of $2\mathrm{\%}$, while our average return for holding 5-year bonds of $3\mathrm{\%}$ is slightly lower than that of CRE (2022). For the mean-reverting stock premium, we have used parameter set 2 at p. 27 of Jarner (2021). This implies that the annualized volatility on stocks decreases from $15\mathrm{\%}$ to $10\mathrm{\%}$ at infinite horizons, after 40 years it is approximately $12\mathrm{\%}$. With these parameters, the standard 60/40 mix of stocks and bonds has an average, (price) real return of approximately $3\mathrm{\%}$.

Table A2. Capital market parameters. The model is started in equilibrium as shown in the last row (time 0 means the initial year of the simulation, e.g. year 2020).

Rate		Stock		Inflation		Correlations
κ	0.09	α	0.06	β	0.05	${\rho }_{\mathrm{rS}}$	0
$\overline{r}$	0.025	$\overline{x}$	0.04	$\overline{\pi }$	0.02	${\rho }_{\mathrm{r\pi }}$	0.80
${\sigma }_r$	0.01	${\sigma }_x$	0.003	${\sigma }_{\pi }$	0.005	${\rho }_{\mathrm{S\pi }}$	−0.25
a	0.03	${\sigma }_S$	0.15	${\sigma }_I$	0.005
b	0.065
$r_0$	0.025	$x_0$	0.04	${\pi }_0$	0.02

A.5. Labour income

A key aim of the analysis is to quantify the benefits of a two-pillar system for a population faced with both financial and biometric risks. At the individual level, the two most important financial risk factors for the adequacy of DC pensions are uncertainty of financial returns and uncertainty of labour income.²⁶ To reflect the importance of the labour income, we model the labour income process in some level of detail. Specifically, we use a stochastic wage process with an age-dependent profile in combination with a duration model for employment status.

Assuming a labour market entry age of 24 and a retirement age of 67, the labour income for agent $A_i$ at time t takes the form (A33) $W_i(t)={\tilde{I}}_t{\mathrm{{\rm Y}}}_i(t)E_i(t)1_{(24\le x_i<67)},$(A33) where ${\tilde{I}}_t$ is the (wage) inflation process, ${\mathrm{{\rm Y}}}_i(t)$ is the real labour income conditioned on the agent being in work, and $E_i(t)$ is the agent's employment process (it is 1 if the agent is in work at time t, and 0 otherwise). The inflation process is part of the financial market of Section A.4; the two other processes are described below.

A.5.1. Real labour income conditioned on being in work

We let $G(x)$ denote the average (real) wage of a person of age x. Inspired by Andersen et al. (2022a), we model the logarithm of G as the fifth-order polynomial (A34) $\log G(x)=1.557+97.6\tilde{x}-323.9{\tilde{x}}^2+502.8{\tilde{x}}^3-324.7{\tilde{x}}^4+38.5{\tilde{x}}^5,$(A34) where $\tilde{x}=x/100$. The G-profile is calibrated to 2021-data from Statistics Denmark, see Figure A4.

Figure A1. Three simulated age distributions in 2100, projected from the Danish population in 2020 with scaling $\mathrm{\Delta }=$ 200, 1000, and 5000, respectively (corresponding to an initial number of agents of 29,112, 5823, and 1167, respectively). For ease of comparison, the size of each age cohort is multiplied by Δ, i.e. by the number that each agent represents in the underlying population. The black line is a mean projection based on the same rates for mortality and fertility.

Figure A2. Illustration of the age distribution after 100 years and the evolution of the old-age dependency ratio for the three demographic scenarios defined in Appendix A.3.2. (a) Age distribution in 2120 for three demographic scenarios and the same initial population in 2020 and (b) old-age dependency ratio, i.e. the size of the retired population to the size of the working population.

Figure A3. Period and cohort life expectancies at birth, at retirement age (PA) and at age 67 from 2020 to 2120, assuming mortality improvements. (a) The two lower lines show the retirement age computed by Equation (A22(A22) $x^R(t)=\underset{x}{\arg \min }|\frac{e_x(t)}{e_0(t)}-{\gamma }^R|,$(A22) ) with ${\gamma }^R=2/9$ and ${\gamma }^R=1/5$, respectively and (b) The retirement age is computed by Equation (A22(A22) $x^R(t)=\underset{x}{\arg \min }|\frac{e_x(t)}{e_0(t)}-{\gamma }^R|,$(A22) ) with ${\gamma }^R=2/9$.

Figure A4. Calibration of Υ; 2021-income data from Statistics Denmark (table INDKP201) shown as dots. The full line displays the calibrated model given in Equation (A36(A36) ${\mathrm{{\rm Y}}}_i(t)=G(x_i;x_{i,min},x_{i,\mathrm{ret}})\exp [\sum _{y=x_{min}}^{x_i}(y^{1/4}{\sigma }_{\mathrm{{\rm Y}}}{U}_{i,y}-y^{1/2}{\sigma }_{\mathrm{{\rm Y}}}^2/2)],$(A36) ) with G as in (A34(A34) $\log G(x)=1.557+97.6\tilde{x}-323.9{\tilde{x}}^2+502.8{\tilde{x}}^3-324.7{\tilde{x}}^4+38.5{\tilde{x}}^5,$(A34) ), estimated by regressing log-income on the scaled age variables, and income volatility, ${\sigma }_{\mathrm{{\rm Y}}}=0.03$, calibrated to match the lower- and upper income quartiles.

To handle changing retirement ages (and potentially changing labour market entry ages), we define the average wage of age x for a minimal entry age of $x_{min}$ and a retirement age of $x_{\mathrm{ret}}$ by (A35) $\log G(x;x_{min},x_{\mathrm{ret}})=G(17+(x-x_{min})\frac{66-17}{x_{\mathrm{ret}}-1-x_{min}}),$(A35) i.e. by stretching the profile in Equation (A34(A34) $\log G(x)=1.557+97.6\tilde{x}-323.9{\tilde{x}}^2+502.8{\tilde{x}}^3-324.7{\tilde{x}}^4+38.5{\tilde{x}}^5,$(A34) ) to cover the ages $x_{min},\dots ,x_{\mathrm{ret}}-1$.

Next, we define the Υ-process for agent $A_i$ by adding an idiosyncratic, exponentiated random-walk with mean one to G (A36) ${\mathrm{{\rm Y}}}_i(t)=G(x_i;x_{i,min},x_{i,\mathrm{ret}})\exp [\sum _{y=x_{min}}^{x_i}(y^{1/4}{\sigma }_{\mathrm{{\rm Y}}}{U}_{i,y}-y^{1/2}{\sigma }_{\mathrm{{\rm Y}}}^2/2)],$(A36) where the $U_{i,y}$'s are independent, standard normal variates, ${\sigma }_{\mathrm{{\rm Y}}}$ is the volatility of permanent income shocks, and $x_{i,min}$ and $x_{i,\mathrm{ret}}$ are the agent's minimal age of entry and age of retirement, respectively.

The idea behind the construction of Υ is to capture the career-dependent nature of wages as a series of permanent income shocks, as well as the typical age-profile of wages. The construction is similar to that of Zeldes (1989), Carroll (1922), and Andersen et al. (2022a), except that we leave out transitory income shocks from Υ. In our model, transitory income shocks are captured by the employment process, E.

One final point to note. To increase the heterogeneity of agents, we let $x_{min}=x_{\mathrm{work}}-6$, where $x_{\mathrm{work}}$ is the age from which the agent begins to work, i.e. all agents have a shadow labour income process that starts at age 18, 6 years prior to their actual entry to the labour market. Thus when agents receive their first salary they already differ.

Figure A5. Probability of changing employment status as a function of time spent being employed (top panel) or unemployed (bottom panel).

Figure A6. Examples of the labour income process in Equation (A33(A33) $W_i(t)={\tilde{I}}_t{\mathrm{{\rm Y}}}_i(t)E_i(t)1_{(24\le x_i<67)},$(A33) ). The processes are started 6 years prior to labour market entry at age 24. The first two vertical, dashed lines mark this 6-year period; the last dashed line marks retirement at age 67.

Figure A7. Risk of unemployment by age under the two-state duration model for different probabilities of being employed 5 years after becoming unemployed. The model is calibrated under the assumption that the 1-year risk of becoming unemployed (while currently employed) is $5\mathrm{\%}$ and the probability of being employed 1 year after becoming unemployed is $50\mathrm{\%}$, see Box 5. For the analyses in the paper, a 5-year risk of $87.5\mathrm{\%}$ is used.

Box 5: Calibration of E

The employment process, $\{E_i(t)\}$, is calibrated to match targets for the risk of becoming employed and unemployed. Specifically, for the transition employed →unemployed, β is calibrated to match a transition probability of $5\mathrm{\%}$ on a 1-year horizon. For the transition unemployed →employed, α and δ are calibrated to match transition probabilities of $50\mathrm{\%}$ and $87.5\mathrm{\%}$ on 1-year- and 5-year horizons, respectively. This results in an intensity of becoming unemployed of $\beta =0.0513$, and a reactivation intensity starting at $\alpha =0.8021$ with a decay factor of $\delta =0.2994$. Figure A5 shows the corresponding sojurn time distributions.

The 1-year targets have been determined using unemployment data from ‘Styrelsen for Arbejdsmarked og Rekruttering’, available at jobindsats.dk, and employment data from the KAS300-table of Statistics Denmark. To ensure realistic tail behaviour of unemployment frequencies, the 5-year target of $87.5\mathrm{\%}$ is determined such that the probability of being unemployed at age 60 is roughly $20\mathrm{\%}$, matching the actual employment frequency at this age as reported in the RAS200-table of Statistics Denmark, see Figure A7.

A.5.2. Employment process

We model the agent-specific employment process as a two-state duration model. The two states are encoded 0 (unemployed) and 1 (employed). For ease of presentation, we formulate the model in continuous time and discretize the dynamics afterwards.

When employed, we assume a constant intensity of becoming unemployed, ${\lambda }_{10}(t)=\beta$. However, when unemployed, we assume an exponentially decaying intensity of getting a job, ${\lambda }_{01}(t)=\mathrm{\alpha exp}(-\mathrm{\delta d})$, where d is the time (duration) since entering the state of unemployment. We assume that α, β, and δ are all strictly positive.

Decaying reactivation intensities is only one of many well-documented features of employment dynamics. We focus on this aspect because it is a simple and effective way to capture the risk of primary concern for workers, i.e. the risk of losing their job. Creating a similar level of labour force heterogeneity by the Υ-process alone would require an unrealistic high volatility of permanent shocks, ${\sigma }_{\mathrm{{\rm Y}}}$.

The sojurn time, $\overline{T}$, is defined as the time spent in a state before leaving it. In a continuous-time model, we have (A37) $\begin{array}{rl}{P}_0(\overline{T}>t)& =\exp (-{\int }_0^t\mathrm{\alpha exp}(-\mathrm{\delta s})\mathrm{d}s)=\exp (-\alpha [1-\exp (-\mathrm{\delta t})]/\delta ),\end{array}$(A37) (A38) $\begin{array}{rl}{P}_1(\overline{T}>t)& =\exp (-{\int }_0^t\beta \mathrm{d}s)=\exp (-\mathrm{\beta t}),\end{array}$(A38) where $P_x$ denotes the probability conditioned on entering state x at time 0. Note that the distribution of $\overline{T}$ under $P_0$ is defect, meaning that there is a non-zero probability (of $\exp (-\alpha /\delta )$) of not leaving state 0. This corresponds to there being a non-zero probability of not coming back to work when unemployed. Since unemployment is an absorbing state for a fraction of those entering it, the model implies an increasing probability of unemployment as a function of age.

For use in our discrete-time model, we define the integer-valued sojurn time, T, by (A39) $P_x(T=n)=P_x(\overline{T}>n-1)-P_x(\overline{T}>n)\text{for}n=1,2,\dots .$(A39) Thus, in discrete-time, the agent changes state at the end of the period (i.e. year) in which the underlying continuous-time change takes place. In particular, when changing state the agent stays at least one period in the new state before (potentially) leaving it.

For given agent, $A_i$, we start the employment process 6 years prior to the agent's labour market entry. Denote the starting time $t_{min}$ corresponding to age $x_{min}$. The process is started in what would be the stationary distribution if reactivation intensities were constant, rather than decaying, i.e. (A40) $P(E_i(t_{min})=1)=1-P(E_i(t_{min})=0)=\frac{\alpha }{\alpha +\beta }.$(A40) Given its initial state, the E-process is constructed by simulating sojurn times of alternate type until the agent retires. The E-processes are assumed independent between the agents, and also independent of the Υ-process. In particular, it is implicitly assumed that unemployment periods have no effect on the obtainable income upon reentry to the labour market.

The intensities are calibrated to Danish employment data, see Box 5. Most workers will experience at least a short spell of unemployment during their career, and some will become permanently unemployed at some stage. Figure A6 illustrates the dynamics over time when combined with the Υ-process (but without inflation). The aggregate effects on lifetime income for individuals can be seen in Figure A8. We note, in particular, the bimodal income distribution when (long-term) unemployment risk is included.

Figure A8. Distribution of aggregate income (basis year 2020) for the working population. The mass concentrated around the left mode of the income distribution that includes unemployment risk is influenced significantly by the long-term risk of unemployment. Specifically, when the 5-year re-employment probability exceeds $90\mathrm{\%}$ – see Figure A7 – the left mode rapidly diminishes, and the two distributions with and without unemployment risk practically overlap.