Working less for longer: unintended effects of longevity adjustment of retirement age

Abstract

Using a standard macro model with overlapping generations in the manner of Blanchard–Yaari, we show that by broadening the labour supply on the extensive margin through longevity adjustment of the statutory retirement age, labour supply is likely to decrease on the intensive margin. However, this backlash effect depends on the specific design of the pension system. It is significantly higher under a pay-as-you-go scheme with fixed benefits compared to a pay-as-you-go scheme with fixed contributions and a fully funded scheme based on voluntary savings. The findings have implications for responses to the fiscal and macroeconomic challenges raised by an ageing population.

1 Introduction

A policy rule linking the official pension age to life expectancy has been introduced in several countries, including the Nordic countries and the Netherlands (Jensen et al., 2020). In Denmark, for example, the official pension age increases in line with changes in average longevity. While simple and well-motivated, such a rule may have some unintended side effects. For example, as Fig. 1 clearly shows, while both the statutory and the effective retirement age have increased in several OECD countries over the last few decades, there has also been a decrease in average hours worked per employed person.¹ In other words, it seems as though people are working less for a longer period of time, and an important question is where that leaves the net effect on labour supply following an increase in the retirement age.

Fig. 1

Source: OECD.Stat, OECD (2021)

Intensive and extensive margin labour supply in the OECD. Note: Extensive margin labour supply (effective retirement age) is the average effective labour market exit of men and women (Index, 2000 = 100). Extensive margin labour supply (normal retirement age) is the average normal retirement age of men and women. Normal retirement age is defined as the age at which individuals are eligible for retirement benefits from all pension components without penalties. Intensive margin labour supply is proxied by average working hours per employed person (Index, 2000 = 100).

To the best of our knowledge, the literature on such “backlash” effects, with labour supply decreasing on the intensive margin as a possible response to increases on the extensive margin, is still in its infancy.² A few contributions include Börsch-Supan and Ludwig (2010, 2013) , who found backlash effects for public defined benefit (DB) and private, fully funded schemes. Imrohoroğlu and Kitao (2012) simulate the US social security system and analyse the effects of increases in the early retirement age and the normal retirement age, finding a drop in average working hours when assuming fixed prices. Also, Makarski and Tyrowicz (2019) discussed these effects in the context of the minimum eligibility retirement age.

Yet there is a significant body of literature on endogenous retirement decisions. For example, Conde-Ruiz and Galasso (2004) show that early retirement provisions weaken the incentive for human wealth accumulation. Heijdra and Romp (2009) study the effects of demographic shocks in a small open economy setting where individuals make extensive margin retirement decisions, arguing that the pension schemes produce a kink in the lifetime income structure that induces early retirement. French and Jones (2012) find that reforms of public pensions, in the form of reduced generosity of pension schemes and reduced penalties for working past normal retirement age, have a significant effect on labour supply decisions among older workers while affecting younger workers less strongly. Fehr et al. (2012) applied an OLG model to analyse a two-year increase in the normal retirement age in Germany and found that it only results in a one-year delay of effective retirement. While featuring an intensive margin labour supply decision, the authors do not offer more detailed insights on the different channels of backlash effects.

In this study, we seek to add to the literature by developing an analytical framework for demonstrating these effects in a life-cycle model over consumption-leisure choices at the steady-state level.³ Our focus is on exploring how, and to what extent, the backlash effect depends on the design of the pension system. Specifically, two alternative scenarios are investigated: one where no public, mandatory pension scheme is in place, or individuals have voluntary pension savings; and another where there is an unfunded public pay-as-you-go (PAYG) scheme assumed to operate under a period-by-period balanced budget. We distinguish between two PAYG varieties: a scheme with fixed benefits and a scheme with fixed contributions, respectively.

From here the paper is structured as follows. The next section outlines our analytical model, and Sect. 3 shows how the model works under different demographic contingencies. Section 4 features a calibration exercise to quantify some macroeconomic effects of longevity shocks and retirement reforms. Section 5 provides a detailed decomposition of the backlash effect, and Sect. 6 offers an additional robustness analysis. Section 7 concludes and presents some suggestions for future research.

2 The model

2.1 Basic assumptions

We base the analysis on a model with overlapping generations in the style of Yaari (1965) and Blanchard (1985). We consider a small open economy and incorporate endogenous labour supply decisions. The economy is assumed to start in a steady state at time $t=0$. The economy is populated by individuals who face a stochastic risk of death. This hazard rate, or the instantaneous probability of death, is first assumed to be age-independent to preserve tractability. Later, in the numerical section, a more realistic, age-dependent mortality profile is explored. We assume a fixed population size, resulting in the number of individuals “born” into the labour market being equal to the number who died at each moment in time. The hazard rate can be manipulated to simulate longevity, since a lower hazard rate implies that individuals live longer.⁴ In the discussion that follows, we refer to individuals who are not subject to a mandatory pension scheme as partaking in voluntary funded pension savings through actuarial notes.

Workers can choose to spend a portion of their time endowment working, earning a wage at rate $w$. An individual’s productivity is uniform, and the wage is therefore assumed to be constant, both between individuals and throughout their life cycles. However, the labour supply decision of workers is allowed to vary with age. A mandatory retirement age, $g$, is implemented, forcing individuals to leave the labour market at that age. Individuals’ retirement decisions are complex. While financial incentives undoubtedly matter, the empirical evidence on retirement behaviour shows that social factors also affect the decision about when to retire. Evidence suggests that changes in the actual retirement age follow changes in the statutory retirement age, also increases. In this manner, the statutory retirement age serves as a reference point for retirement planning.⁵

To emphasize, our key objective is to shed light on the labour market response on the intensive margin (hours worked per unit of time) following changes on the extensive margin (proportion of lifetime spent in the labour force). To keep this focus, we set aside a number of other important factors, such as a comparative welfare analysis of how retirement reforms perform under alternative pension schemes.

2.2 The pension system

The introduction of a pension system follows the framework set out in Nielsen (1994), thus enabling a comprehensive analysis of how alternative pension systems either mitigate or exacerbate the effects of pension reforms on individual labour supply. This approach provides valuable insights into the effectiveness of various pension strategies in addressing labour market challenges. The pension system is of the pay-as-you-go (PAYG) type with lump-sum transfers.

Since the mortality rate, $\beta$, is age-independent, the fraction of the population that is retired can be expressed as:

$$\begin{array}{c}{\int }_{g}^{\infty }\beta {e}^{-\beta \tau }d\tau ={e}^{-\beta g}\end{array}$$

(1)

Hence the fraction of workers is $(1- {e}^{-\beta g})$. The pension system introduced consists of contributions denoted by $k$ and paid by workers. The collected contributions are transferred as pension benefits, denoted by $b$, to the retired population. These contributions function as lump-sum “taxes” on those who have not reached retirement age, so the pension system is kept balanced period-by-period:

$$\left(1-{e}^{-\beta g}\right)k={e}^{-\beta g}b$$

(2)

This allows us to consider two different PAYG pension schemes: one with fixed contributions, where $b$ is a function of $k$; and one with fixed benefits, where $k$ is a function of $b$. In this PAYG setting, individuals are forced to contribute the same amount to the pension scheme, regardless of how much they work. The specification of the two schemes is similar, the only difference being whether it is the contributions or the benefits that are fixed.

This specification focuses on determining who is paying for rebalancing the pension system. With fixed benefits, it is the workers who must pay higher contributions to maintain the same level of benefits paid to retirees. The case with fixed contributions presents the opposite situation, where it is the retirees who must take smaller benefits in order for contributions to remain unchanged. In the voluntary pension savings case, we simply set $b=k=0$, and saving becomes a purely private affair.

For an individual with $a$ years in the labour market, the discounted stream of future transfers from the pension system is denoted by $p(a)$, and for a worker, it is defined as:

$$\begin{aligned} & p\left( a \right) = \smallint _{a}^{g} - ke^{{\left( {r + \beta } \right)\left( {a - \tau } \right)}} d\tau + \smallint _{g}^{\infty } be^{{\left( {r + \beta } \right)\left( {a - \tau } \right)}} d\tau \\ & = \left\{ {\begin{array}{*{20}c} {\frac{{\left( { - 1 + {\text{e}}^{{\beta g + \left( {r + \beta } \right)\left( {a - g} \right)}} } \right)}}{{r + \beta }}} &\!\!\!k \quad\text{for fixed contributions} \\ {\frac{{\left( { - 1 + {\text{e}}^{{\beta g + \left( {r + \beta } \right)\left( {a - g} \right)}} } \right)}}{{\left( { - 1 + {\text{e}}^{{\beta g}} } \right)\left( {r + \beta } \right)}}} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!b \quad\text{for fixed benefits} \\ {0} & \quad\!\!\!\text{for voluntary savings} \\ \end{array} } \right.\end{aligned}$$

(3)

The first integral represents contributions by workers until retirement, and the second integral represents pension benefits after retirement. In the case of a retiree, the first integral equals zero, and the pension system transfers are found using only the second integral.

For both PAYG cases, note that the value of the pension system at labour market entry (age $0$ in the model) is always negative. This is due to the constant population size assumption, which makes the implicit return on unfunded pension systems zero. Combined with positive returns on investment for the private savings market, the pension contributions are worth less than an equal amount of savings. Our results are thus within the context of a dynamically efficient economy, where long-run growth, as driven by population growth, is less than the interest rate.⁶

2.3 Production and wages

The aggregate production function is given by $Y=\omega L$, where $\omega$ is exogenous and represents the economy's technological level. $L$ represents the aggregate labour supply. Perfect markets are assumed, implying that wages are also exogenous and given by $w=\omega$. This is because the marginal productivity of labour is equal to the producer's cost of employing labour. The labour supply and technology level determine the economy's output.

With wages exogenously given, the potential income over the life cycle of the individual can be determined. The present value of earnings potential, or the potential future wages of an individual aged $a,$ working in all of his or her available time, is defined as (with time allowance normalised to one):

$$h\left(a\right)={\int }_{a}^{g}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau =\frac{\left(1-{e}^{\left(r+\beta \right)\left(a-g\right)}\right)}{\left(r+\beta \right)}w$$

(4)

For retirees, the earnings potential is equal to zero. The actual income of the individual is dependent on the labour/leisure choice of the individual, while the potential income is not. Therefore, the potential earnings separate the exogenous part of the individual’s income from the endogenous part.

2.4 Consumption and labour choices

The individual derives utility from instantaneous consumption, denoted by $c$, and instantaneous leisure, denoted by $1-l$. The instantaneous (CRRA) utility function and the discounted stream of lifetime utility at age $a$ read as follows, respectively:

$$u=\frac{{\left({c}^{\varepsilon }{\left[1-l\right]}^{1-\varepsilon }\right)}^{1-\sigma }-1}{1-\sigma }$$

(5)

$$u\left(a\right)={\int }_{a}^{\infty }\frac{{\left({c\left(\tau \right)}^{\varepsilon }{\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }\right)}^{1-\sigma }-1}{1-\sigma }{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}d\tau$$

(6)

Here, $\rho >0$ is the discount rate, $0\le \varepsilon \le 1$ dictates the relative weight the individual places on consumption and leisure, and $\sigma >0$ is a risk-aversion parameter. The constant coefficient of relative risk aversion is then given by $1-\varepsilon (1-\sigma )$; that is, it is linearly increasing in $\sigma$. Individuals can work until they reach age $g$. At that point, they are forced to retire, $l\left(\tau \right)$ is set to zero, and the term ${\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }$ reduces to one.

When faced with a positive risk of death, an individual is less inclined to save for the future. We thus introduce actuarial notes issued at a fair rate (Yaari, 1965) that include a component of life insurance. The purchaser of an actuarial note receives a constant stream of payments until death, and the assets of those who die are redistributed to those currently living. These assets therefore function as savings for individuals and provide the purchaser with an extra component of life insurance. The rate of return on the actuarial note is then ${r}^{A}=r+\beta$. Due to our adoption of the small open economy case, the rate $r$ is exogenous and equals the world interest rate.

Individuals can freely lend and borrow at the world rate of interest. Parameter values distinguish whether a nation is a debtor nation (with impatient inhabitants), a creditor nation (with relatively patient inhabitants), or the razor edge case of a perfectly balanced economy. In our setting, there is no intrinsic characteristic that distinguishes between foreign and domestic assets. Therefore, it can be assumed that each individual's total wealth is held in an unspecified mix of the two.

Workers are faced with the budget constraint:

$${\int }_{a}^{\infty }c\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau =i\left(a\right)+{\int }_{\text{a}}^{g}wl\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau +p\left(a\right)$$

(7)

where $i\left(a\right)$ denotes the total assets, or stock of actuarial notes, held by the individual at age $a$. For retirees, the integral on the right-hand side reduces to zero. Solving the utility maximisation yields a relationship between labour supply and consumption for a worker⁷:

$$wl\left(a\right)=w-\frac{1-\varepsilon }{\varepsilon }c\left(a\right)$$

(8)

Differentiating the first-order conditions and acknowledging that an individual is forced to retire at age $g$ yields the Euler equations:

$$\frac{\dot{c}\left(a\right)}{c\left(a\right)}=\frac{1}{\sigma \varepsilon -\varepsilon +1}\cdot (r-\rho )$$

(9)

$$\left\{ {\begin{array}{*{20}l} {\frac{{ - \dot{l}\left( a \right)}}{{\left[ {1 - l\left( a \right)} \right]}} = \frac{1}{{\left( { - \varepsilon \sigma + \sigma + \varepsilon } \right)}}\left( {r - \rho } \right){\mkern 1mu} for{\mkern 1mu} a} \hfill & { < g} \hfill \\ { = 0} \hfill & {for{\mkern 1mu} a \ge {\mkern 1mu} g} \hfill \\ \end{array} } \right.$$

(10)

This implies that consumption throughout the life cycle follows the path $c\left(a+\tau \right)=c\left(a\right){e}^{\frac{1}{\sigma \varepsilon -\varepsilon +1}(r-\rho )\tau }$. A similar relationship can be found for the time path of leisure (valid only for workers).

If we apply the first-order condition for consumption with the consumption/labour supply relationship (Eq. 8 ) on the budget constraint and then integrate, we arrive at a relationship for the consumption at each age for the individual:

$$c\left(a\right)=\mu \left(a\right)\varepsilon \frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\left[i\left(a\right)+p\left(a\right)+h\left(a\right)\right]$$

(11)

Here we define $\mu (a)\equiv \frac{1}{1-\left(1-\varepsilon \right)\left({e}^{\frac{\left(a-g\right)}{\sigma }\left(\left(\rho +\beta \right)+\left(r+\beta \right)\left(\sigma -1\right)\right)}\right)}>1$ for the workers, which is a scaling factor for consumption and leisure choices that account for the forced retirement age. For retirees, the term becomes $\mu \equiv \frac{1}{\varepsilon }$. Essentially, the forced retirement age does not allow individuals to apply the first-best solution to the consumption/leisure problem. They now have fewer “degrees of freedom” to acquire income through labour. The individual still prefers a smooth path for consumption and labour supply, as is dictated by the Euler Eqs. (9) and (10). Therefore, with the forced retirement age captured in $\mu (a)$, the individual shifts labour supply at the intensive margin downwards, choosing higher consumption and leisure throughout the working period.

For the workers ($0\le a<g$), the following relationship for leisure is:

$$1-l\left(a\right)=\mu \left(a\right)\frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\frac{\left(1-\varepsilon \right)}{w}\left[i\left(a\right)+p\left(a\right)+h\left(a\right)\right]$$

(12)

Consumption is thus related to the time until retirement (captured in $\mu$), risk of death, risk aversion, the interest rate, and the discount rate. For a worker, the $\mu$ expression always starts at a value above $1$. As retirement approaches, this expression converges to $\frac{1}{\varepsilon }$ and remains there throughout retirement, leading to the standard Blanchard–Yaari style relationship for consumption.

We aggregate consumption by applying the consumption path based on the Euler equation. At steady state, aggregate consumption, $C$, and aggregate labour supply, $L$, become⁸:

$$C=\beta {\int }_{0}^{\infty }c\left(a\right){e}^{-\beta a}da=\beta c\left(0\right){\int }_{0}^{\infty }{e}^{\left(\frac{1}{\sigma \varepsilon -\varepsilon +1}\left(r-\rho \right)-\beta \right)a}da=\frac{\beta c\left(0\right)}{\beta +\frac{\rho -r}{\sigma \varepsilon -\varepsilon +1}}$$

(13)

$$L=\beta {\int }_{0}^{g}l\left(a\right){e}^{-\beta a}da=\frac{\beta \left(1-l\left(0\right)\right)\left(1-{e}^{\frac{1}{\left(-\varepsilon \sigma +\sigma +\varepsilon \right)}\left(r-\rho \right)g-\beta g}\right)}{\frac{\left(r-\rho \right)}{\left(-\varepsilon \sigma +\sigma +\varepsilon \right)}-\beta }+\left(1-{e}^{-\beta g}\right)$$

(14)

3 Effects of population ageing and pension reform

3.1 Demographic ageing

The effects of ageing in the form of a decrease in mortality and fertility rates are now analysed for an individual just entering the labour market at the beginning of their economic life ($a=0)$.⁹ This analysis is performed in a steady state only and for the razor-edge case, when $r=\rho$, although this assumption is relaxed in the numerical analysis to follow. Under this assumption, individuals have perfectly flat consumption (and labour supply) smoothing over their life cycle. If we relax this assumption and allow $r>\rho ,$ individuals plan their life cycle consumption and labour supply paths with constant growth when they enter the labour market.

An individual entering the labour market chooses their consumption and leisure according to:

$$c\left(0\right)=\mu \left(0\right)\varepsilon \frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\left[p\left(0\right)+h\left(0\right)\right]$$

(15)

And

$$1-l\left(0\right)=\mu \left(0\right)\frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\frac{\left(1-\varepsilon \right)}{w}\left[p\left(0\right)+h\left(0\right)\right]$$

(16)

Any effects on the mandatory retirement age adjustment factor $\mu (0)$, the pension value $p\left(0\right)$, and earnings potential $h(0)$, stemming from longevity shifts in $\beta$ yield a level impact on consumption and leisure choices. These types of shocks work only through income channels and not through substitution channels between consumption and leisure, as the longevity factor does not enter the Euler equations (Eqs. (9) and (10)).

Starting with the longevity shock related to labour supply on the intensive margin for new incoming generations with greater longevity, the hazard rate $\beta$ appears directly in Eqs. (15) and (16). A decreased hazard rate makes it more likely that an individual survives into a (now longer) retirement period. With longer life expectancy, an individual must now fund consumption for a longer lifetime. Overall, a fall in consumption and leisure during the working period is observed compared to a generation of individuals with lesser longevity.

Two other observations are worth mentioning: First, a decrease in the hazard rate, β, leads to an increase in the present value of earnings potential, $h\left(0\right).$ This makes intuitive sense, since a lower $\beta$ implies that an individual is more likely to survive throughout their working life. This makes the individual feel wealthier, resulting in a desire for higher consumption and leisure. Second, the value of the PAYG pension schemes, described by $p\left(0\right)$, decreases as longevity increases (lower $\beta$). This is clear by noting that the value of an unfunded pension system always starts negatively for an individual just entering the labour market, turning positive when the individual reaches age $a=rg/\left(\beta +r\right)$. This implies that the age at which the value of the unfunded pension scheme turns positive for the individual is now higher, which makes the value of the pension system even more negative at labour market entry, decreasing consumption and leisure.

The fourth component is the scaling factor, $\mu (0)$, which is found to increase with greater longevity (see Appendix C). Overall, a longevity shock, reflected by a reduced hazard rate, reduces leisure choices (see Appendix D):

$$\frac{\partial 1-l\left(0\right)}{\partial \beta }>0$$

This holds for all pension arrangements.

3.2 Increase in the retirement age

Since a decrease in the hazard rate exerts pressure on the pension system, a natural pension reform option would be to increase the retirement age to “rebalance” society’s old-age dependency ratio to its previous state. However, it can have an unintended effect on individuals’ work intensity.

Under both PAYG schemes, the impact of an increase in retirement age on the labour supply at the intensive margin is not clear, and only a numerical analysis can reveal the direction of impact (see Appendix D for details). However, when unfunded pension systems are absent and only voluntary savings are the norm, it is certain that backlash effects will occur, where increasing the retirement age (the extensive margin) will increase leisure (lower labour supply at the intensive margin):

$$\frac{\partial 1-l\left(0\right)}{\partial g}>0\text{ for voluntary savings case}$$

A retirement reform scheme aimed at increasing the labour supply on the extensive margin can have the unintended effect of lowering the labour supply on the intensive margin at steady state, and it will certainly do so if no unfunded pension system is in place. Retirement age increases need to be more than proportional relative to increases in longevity if the intensive margin labour supply is to return to its previous level. In these cases, individuals who are faced with working for more years choose to work less each year.

In addition to this, our analysis indicates that voluntary savings will lead to an increase in aggregate consumption as the retirement age rises, as is demonstrated in Eqs. (13) and (15). The backlash effect does not dominate at the aggregate level, as is shown in Eqs. (14) and (16); it merely diminishes the efficacy of retirement age reform. Under both PAYG schemes, raising the retirement age has ambiguous effects on aggregate consumption and leisure. This is examined further in the numerical exercise.

4 Numerical analysis

We have established the theoretical framework of our model. Now we conduct a numerical exercise to examine the impact of aging and pension reforms across various schemes, including voluntary savings and PAYG with fixed benefits or contributions. This analysis quantifies the backlash effects of increasing labour supply at the extensive margin.

4.1 Calibration of retirement reform

We have Denmark in mind as a “case study”, focusing on expected changes in longevity and corresponding changes in the retirement age.¹⁰ Over the last three decades, longevity at birth has increased by almost one year every four years, with similar developments projected for the future (Statistics Denmark, 2024). This paved the way for a major reform in 2006, implying a scheme with a one-to-one link between changes in longevity and retirement age (see Fig. 2).

Fig. 2

Source: Danish Rational Economic Agents Model, DREAM

Official retirement age in Denmark under current rules and expectations.

The aim of the Danish reform is to target an expected pension period of 14.5 years for the average Dane. The scheme is semi-automatic in the sense that a change must be approved by parliament every fifth year. As forward guidance, changes are preannounced with a lead time of 15 years. To avoid large adjustments, the change each year can neither be below 6 months nor exceed 12 months.

Regarding the number of working years, g, the average labour market entry age in Denmark was just above 22 in 2019 (European Commission, 2021), and the retirement age was 67 (OECD, 2023). Hence, g = 67–22 = 45. Given current expectations and agreed-upon policies, the retirement age in Denmark will increase by approximately 10 years over a period of five decades, or by one year every five years. While this may be regarded as overly ambitious in international context, our baseline longevity shock and subsequent adjustment of the retirement age has been calibrated as a five-year increase in life expectancy and an equal increase in the retirement age.

4.2 Calibration of the model

Starting with the relative weight of consumption in utility, we have chosen a value of $\varepsilon =0.25$. This is a compromise between the works of Kydland and Prescott (1982) and Imrohoroğlu and Kitao (2012), who assume a value of $\varepsilon =\frac{1}{3}$ motivated by the assumption that households' allocation of time to non-market activities is about twice as large as the allocation to market activities; and the work of Eichenbaum et al. (1988), who estimate the parameter to range from 0.12 to 0.18.¹¹

The world interest rate, r, is set at 1.8%, following Benhabib et al. (2014), and the subjective discount rate is set $0.015$. Hence, $r>\rho$, implying growing consumption and leisure choices over time.¹²

As to age-dependent mortality, we adopt a realistic mortality structure proposed by Boucekkine et al. (2002). There is now a finite maximum age, $A=\frac{\text{ln}{\mu }_{0}}{{\mu }_{1}}$, and the unconditional probability for an individual to reach age $\tau \in [0,A]$ is given by $m\left(\tau \right)=\frac{{\mu }_{0}-{e}^{{\mu }_{1}\cdot \tau }}{{\mu }_{0}-1}$. The shape of the mortality profile is determined by the parameters ${\mu }_{0}$ and ${\mu }_{1}$. Both parameters are calibrated in accordance with empirical survival functions for Denmark such that life expectancy is preserved and the sum of squared residuals minimized. The estimated parameters are ${\mu }_{0}=401$ and ${\mu }_{1}=0.0817$.

The fraction of working individuals (${N}_{c}$) and retirees (${N}_{b}$) in the economy can be obtained by integrating function $m\left(\tau \right):$

$${N}_{c}={\int }_{0}^{g}m\left(\tau \right)d\tau =\frac{\left({e}^{{\mu }_{1}g}-{\mu }_{0}\cdot {\mu }_{1}\cdot g-1\right)}{\left(1-{\mu }_{0}\right){\mu }_{1}}$$

$${N}_{b}= {\int }_{g}^{A}m(\tau )d\tau =\frac{{\mu }_{0}-{\mu }_{0}\text{ln}{\mu }_{0}-{e}^{{\mu }_{1}g}+{\mu }_{0}\cdot {\mu }_{1}\cdot g}{\left(1-{\mu }_{0}\right){\mu }_{1}}$$

For the PAYG pension benefit and contribution rates, benefits were matched to the actual average mandatory pension replacement rate in Denmark in 2022, which amounted to 73.1% of average earnings (OECD, 2023). In our model, since the wage rate does not have a material effect on our results, we set it to one. The endogenous labour choice and the benefits rate were estimated by iteration until they were at an equilibrium where the model replacement rate corresponded to the observed replacement rate, yielding b ≈ 0.184. Given the proportion of workers and pensioners, the contribution rate was obtained by balancing out the PAYG pension budget,¹³ which yields k ≈ 0.073.

The baseline calibration is summarized in Table 1.

Table 1 Baseline calibration

Given these parameter values, the relevant aggregate values for the baseline case can be calculated. Leisure on the individual level is then given by the following expression:

$$1-l\left(a\right)=\frac{1}{\varepsilon {\xi }_{1}\left(a\right)+(1-\varepsilon ){\xi }_{2}\left(a\right)}\cdot \frac{1-\varepsilon }{w}\cdot \left[i\left(a\right)+h\left(a\right)+p\left(a\right)\right]$$

(17)

Further details along with definitions of ${\xi }_{1}\left(a\right)$ and ${\xi }_{2}\left(a\right)$ can be found in Appendix C.

At this stage, no distinction is made between the two PAYG schemes, as they are identical in the baseline. However, to evaluate the effects of a shock to the economy, in the form of either increased longevity or decreased fertility, each pension system was considered separately.

4.3 Voluntary pension savings

We now induce an increase in life expectancy by five years. This is simulated by re-adjusting the mortality parameters in the following manner: ${\mu }_{0}=407$ and ${\mu }_{1}=0.0758$. The effects of the longevity shock over the life cycle of an individual are shown below in Fig. 3. The baseline labour supply is the dashed line, and the path of labour supply after the shock is the solid line. Individuals who face greater longevity substitute current consumption and leisure for future consumption. This is a combined result of several factors: individuals have a higher chance of surviving and expect to be able to work for longer because they are more likely to survive until retirement, and individuals discount the future less heavily because they are more likely to be alive to enjoy their savings.

Fig. 3

Life cycle path of labour supply in the voluntary pension savings case: baseline and longevity shock

Leisure decreases with increased longevity as individuals finance a longer retirement. In this case, there is an increase in the individual labour supply intensive margin of approximately $2.5\%$, and the aggregate labour supply drops by $4.1\%$. The gap between the individual and aggregate measures is caused by the higher proportion of individuals in retirement, where labour supply is 0.

Next we assume that the government attempts to re-balance the demographic structure such that the old-age dependency ratio approaches its original value prior to the longevity shock. In this case, the one-to-one Danish policy requires a retirement age increase of five years, making $g$ equal to 50, implying that individuals retire at the physical age of 72. This has the effect of lowering labour supply on the intensive margin (the backlash effect) and increasing the rate of consumption, as there is less need to save for retirement. The effects are summarised in Fig. 4.

Fig. 4

Life cycle path of labour supply in the voluntary pension savings case: longevity shock and retirement reform

This causes the aggregate labour to increase by 4.5%, as expected from an increase in the size of the labour force. Nonetheless, there is a significant decrease of 2.3% in the labour supply at the intensive margin due to the existence of the backlash effect. This result is seen in Fig. 4, where the solid line shifts towards the dashed line when the reform is implemented.

The semiautomatic retirement age increases in Denmark typically take the form of a one-year increase every four to five years, as is seen in Fig. 2. A one-year increase in the retirement age would cause a backlash effect corresponding to just over a full workday each year. While this effect seems small, when summed up over the work life, it corresponds to nearly three months. Thus the total effect of increasing the labour supply on the extensive margin by one year is only about $9\frac{1}{4}$ months of increased labour supply for the average worker.

4.4 PAYG with fixed benefits

Under a PAYG scheme with fixed benefits, the contribution rate is a function of the demographic structure. When the size of the retired population changes relative to the working population, the lump-sum contributions of the currently working population automatically change. An individual just entering the labour market will certainly experience this change in contributions but will not necessarily experience pension benefits since they are not guaranteed to survive until retirement.

As before, we start by looking at the case of a higher old-age dependency ratio triggered by a five-year increase in life expectancy. This causes lump-sum pension contributions to increase from 7.3% to 9.2% of the wage rate, as a higher proportion of the population is retired. The demographic shock causes the intensive margin of labour supply to increase by 4.1% (2.5% in the voluntary saving case) and causes the aggregate labour supply to drop by only 2.1%, as compared with 4.1% in the voluntary savings case. This is illustrated in Fig. 5.

Fig. 5

Life cycle path of labour supply in a fixed benefits scheme: baseline and longevity shock

In addition to the effects traced in Sect. 4.3, the retirement reform now has the effect of reducing pension contributions, since the number of retirees receiving pension benefits has shrunk and the number of workers has increased. As a result, working individuals have more disposable income, leading to an increase in both consumption and leisure. This amplifies the backlash effect: Intensive margin labour supply decreases by 4.0%, as compared with 2.3% in the voluntary saving case (see Fig. 6).

Fig. 6

Life cycle path of labour supply in a fixed benefits scheme: longevity shock and retirement reform

For the typical one-year increase in retirement age as planned in Denmark (see Fig. 2), the backlash effect is approximately two working days each year. This decrease in the intensive margin results in a total of almost five months less work over the working life. An increase in the retirement age of one year would therefore result in only $7\frac{1}{4}$ months of extra work for the average worker.

4.5 PAYG with fixed contributions

Finally, the experiment was performed within a PAYG setting with fixed contributions. As longevity increases, the proportional size of the retired population increases; as a result, pension benefits drop from 18.4% of the wage rate in the baseline to 14.6%. The individual is more likely to reach retirement, but when they do, pension benefits are markedly reduced, causing an increase in voluntary savings. In order to finance a longer retirement, individuals increase their labour supply at the intensive margin by approximately the same amount as in the voluntary savings case. This is demonstrated by Fig. 7.

Fig. 7

Life cycle path of labour supply in a fixed contributions scheme: baseline and longevity shock

A five-year increase in the retirement age causes a backlash effect of 1.9% (Fig. 8). This effect is far smaller than in the case with fixed benefits, and even smaller than in the voluntary savings case. The reason is that the increase in retirement age now decreases the value of the pension system to the individual, which softens the backlash effect.¹⁴ Again, this demonstrates how important the existence and type of PAYG pension schemes are for the behaviour of the intensive margin of labour supply. A typical single-year increase in retirement age would, in this case, cause a decrease in the intensive margin labour supply of approximately one workday each year. Over the life cycle, this accounts for roughly two-and-a-half months less work. The retirement age increase of one year therefore achieves an increase in the labour supply of the average worker by approximately 9½ months.

Fig. 8

Life cycle path of labour supply in a fixed contributions scheme: longevity shock and retirement reform

5 Decomposition of the backlash effect

We have seen that when measured against a benchmark of voluntary pension savings, the backlash effect is stronger in the PAYG scheme with fixed benefits and weaker in the PAYG scheme with fixed contributions. Now we decompose the backlash effect and compare between different pension schemes.

Using Eqs. (3), (4) and (17), when $a=0$ and $w=1$, and letting $\xi \left(a\right)\equiv \frac{1}{\varepsilon {\xi }_{1}\left(a\right)+(1-\varepsilon ){\xi }_{2}\left(a\right)}$ yields the following:

$$\begin{array}{*{20}c} {1 - l\left( 0 \right) = \frac{{\left( {1 - \varepsilon } \right)}}{w}\xi \left( 0 \right)\left[ {\underbrace {{\mathop \smallint \limits_{0}^{g} - km\left( \tau \right)e^{ - \tau r} d\tau }}_{\begin{subarray}{l} {\text{Discounted}}\;{\text{stream}}\;{\text{of}} \\ {\text{ pension }}\;{\text{contributions}} \end{subarray} } + \underbrace {{\mathop \smallint \limits_{g}^{A} bm\left( \tau \right)e^{ - \tau r} d\tau }}_{\begin{subarray}{l} {\text{Discounted }}\;{\text{stream}}\;{\text{of}} \\ {\text{pension}}\;{\text{benefits}} \end{subarray} } + \underbrace {{\mathop \smallint \limits_{0}^{g} m\left( \tau \right)e^{ - \tau r} d\tau }}_{\begin{subarray}{l} {\text{Discounted }}\;{\text{stream}}\;{\text{of}} \\ {\text{potential }}\;{\text{income}} \end{subarray} }} \right]} \\ \end{array} { }$$

(18)

Equation (18) captures how the pension scheme and earnings potential determine the intensive margin of labour supply. Equation (10) portrays how, along with the interest rate $r$, time preference $\rho$, risk aversion parameter $\sigma ,$ and weight of consumption in utility $\varepsilon$, labour supply at birth l(0) determines the leisure path over the life cycle. Since $r$, $\rho$, $\sigma ,$ and $\varepsilon$ are assumed to be unaffected by retirement age, the backlash effect is captured in l(0).

Differentiation shows how retirement age increases affect the leisure decisions of individuals through the earnings potential channel, scaled by factors ξ(0) and $\frac{\left(1-\varepsilon \right)}{w}$:

$$\frac{\partial }{\partial g}\left[ {\frac{{\left( {1 - \varepsilon } \right)}}{w}\xi \left( 0 \right)\mathop \smallint \limits_{0}^{g} wm\left( \tau \right)e^{ - \tau r} d\tau } \right] > 0$$

(19)

As the retirement age increases, the individual has the option of working for longer and feels richer. Since both leisure and consumption are normal goods, the individual chooses to consume more of both. This, in turn, translates directly to less intensive margin labour supply, as is captured by Eq. (19).

By applying the equation that ensures a balanced PAYG system to Eq. (18), the effects of pension contributions and benefits on intensive margin labour supply in both PAYG schemes can be found. Furthermore, by taking the partial derivative with respect to retirement age and applying the scaling factors $\xi \left(0\right)$ and $\frac{\left(1-\varepsilon \right)}{w}$, the contribution of each channel to the backlash effect can be found.

For the PAYG scheme with fixed benefits, we obtain the following:

$$\begin{array}{*{20}c} {{\text{Contributions}}\;{\text{channel:}}\;\;\frac{\partial }{\partial g}\underbrace {{\frac{{\left( {1 - \varepsilon } \right)}}{w}\xi (0)\left( { - \frac{{\mu_{0} - \mu_{0} \ln \mu_{0} - e^{{\mu_{1} g}} + \mu_{0} {\kern 1pt} {\kern 1pt} \cdot{\kern 1pt} \mu_{1} {\kern 1pt} \cdot{\kern 1pt} {\kern 1pt} g}}{{e^{{\mu_{1} g}} - \mu_{0} {\kern 1pt} \cdot{\kern 1pt} \mu_{1} {\kern 1pt} \cdot{\kern 1pt} {\kern 1pt} g - 1}}b\mathop \smallint \limits_{0}^{g} m(\tau )e^{ - \tau r} d\tau } \right)}}_{{{\text{Scaled}}\;{\text{effect }}\;{\text{of }}\;{\text{contrubtions }}\;{\text{on }}\;{\text{leisure}}}}} \\ \end{array}$$

(19a)

$${\text{Benefits}}\;{\text{channel:}}\;\;\frac{\partial }{\partial g}\underbrace {{\frac{{\left( {1 - \varepsilon } \right)}}{w}\xi (0)\left( {b\mathop \smallint \limits_{g}^{A} m\left( \tau \right)e^{ - \tau r} d\tau } \right)}}_{{{\text{Scaled}}\;{\text{effect}}\;{\text{of }}\;{\text{benefits}}\;{\text{on }}\;{\text{leisure}}}}$$

(19b)

For the PAYG scheme with fixed contributions, we obtain:

$$\begin{array}{*{20}c} {{\text{Contributions}}\;{\text{ channel}}:{ }\frac{\partial }{\partial g}\underbrace {{\frac{{\left( {1 - \varepsilon } \right)}}{w}\xi (0)\left( { - k\mathop \smallint \limits_{0}^{g} m\left( \tau \right)e^{ - \tau r} d\tau } \right)}}_{{{\text{Scaled}}\;{\text{ effect }}\;{\text{of }}\;{\text{contributions }}\;{\text{on }}\;{\text{leisure}}}}} \\ \end{array}$$

(19c)

$$\begin{array}{*{20}c} {{\text{Benefits}}\;{\text{channel:}}\;{ }\frac{\partial }{\partial g}\underbrace {{\frac{{\left( {1 - \varepsilon } \right)}}{w}\xi (0)\left( {\frac{{e^{{\mu_{1} g}} - \mu_{0} {\kern 1pt} \cdot{\kern 1pt} \mu_{1} {\kern 1pt} \cdot{\kern 1pt} g - 1}}{{\mu_{0} - \mu_{0} \ln \mu_{0} - e^{{\mu_{1} g}} + \mu_{0} {\kern 1pt} \cdot{\kern 1pt} {\kern 1pt} \mu_{1} {\kern 1pt} \cdot{\kern 1pt} g}}k\mathop \smallint \limits_{g}^{A} m\left( \tau \right)e^{ - \tau r} d\tau } \right)}}_{{{\text{Scaled }}\;{\text{effect}}\;{\text{of }}\;{\text{benefits }}\;{\text{on }}\;{\text{leisure}}}}} \\ \end{array}$$

(19d)

Figure 9 applies the baseline calibration values to the solution of these partial derivatives and Eq. (18) to decompose the backlash effect into the effect of earnings potential and the effects of each component of the pension scheme.

Fig. 9

Decomposition of the backlash effect in the voluntary savings, fixed contributions and fixed benefits cases. Note: size of each effect has been indexed to the backlash effect in the voluntary saving case which is set to 1. Higher bars imply greater increases in leisure when the retirement age is increased. The light grey bars imply a positive contribution to the backlash effect, the black bars imply a negative contribution, and the dark-grey bars are the total backlash effect in each case

In the voluntary savings case in Fig. 9, the backlash effect is determined by both the earnings potential effect and the forced retirement age scaling factor. The intuition behind this effect is traced above (see Eq. 19), and the earnings potential effect is equal in all three schemes.

In addition to the earnings potential effect, the pension scheme with fixed contributions causes a reduction in the backlash effect through two channels. First, as retirement age increases, the individual is forced to contribute to the pension scheme for a longer period without a reduction in the contribution rate. This causes the individual to feel poorer, compensating for this by increasing labour supply on the intensive margin, which leads to a reduction in the backlash effect, captured by the first black striped bar in Fig. 9. Second, as the individual faces a higher retirement age, there is a smaller chance of survival into retirement and collection of pension benefits, which makes the individual feel even poorer, compensating for this by increasing labour supply on the intensive margin even further, captured by the second black bar in Fig. 9. In the case with fixed contributions, the earnings potential effect is partially offset by two relatively smaller pension scheme effects, both of which cause a reduction in the backlash effect compared to the voluntary savings case.

In the case with fixed benefits, the pension scheme causes two opposing effects. First, as the retirement age increases, the contribution rate decreases, as there are more workers paying for the benefits of fewer retirees. This causes the individual to feel richer, leading to increased leisure, captured by the tall light grey striped bar in Fig. 9. Second, just as in the case with fixed contributions, this entails that the individual is less likely to survive until retirement to enjoy pension benefits, making the individual feel poorer and causing an increase in labour supply, captured by the black bar in Fig. 9. The contribution effect dominates the benefit effect, and we end up with the most pronounced backlash effect.

Finally, the scaling factor is addressed.$\xi \left(a\right)\equiv \frac{1}{{\xi }_{1}\left(a\right)+\frac{1-\varepsilon }{\varepsilon }{\xi }_{2}\left(a\right)}$ accounts for the effect of the forced retirement age on consumption and leisure for working individuals. $\xi \left(a\right)$ is a multiplicative term present in all channels of the backlash effect, as is traced in Eqs. (17), (18) and (19). In Sect. 4, it was shown that the scaling factor for age-independent mortality decreases as retirement age increases. The same result holds true for age-dependent mortality setting, whereby we have that $\frac{\partial \xi \left(a\right)}{\partial g}<0$. The negative pressure exerted on labour supply at the intensive margin becomes milder when retirement age increases, as forced retirement negatively influences a smaller portion of the individual’s life now and makes decision-making less suboptimal. This means that the forced retirement channel in $\xi$ attenuates every other channel’s impact on labour supply decisions.

6 Sensitivity analysis and further discussion

In this section, we elaborate on which factors determine the magnitude of the backlash effect, in addition to discussing model characteristics.

6.1 Key determinants of the magnitude of the backlash effect

The numerical experiments performed, and the relevant parameter changes, are summarised in Table 2, while all parameters other than those specified are kept equal to the baseline calibration.

Table 2 Alternative parameters in robustness checks

This new set of parameters is applied to Eqs. (18) and (19), replicating the decomposition above. The results are then indexed to the backlash effect in the voluntary saving case, to give a better sense of the relative order of magnitude of each channel. In the voluntary saving case, a retirement age increase was followed by an intensive margin labour supply reduction of 2.3%. Each unit on the vertical axis of Fig. 10 is associated with a backlash effect of this magnitude.

Fig. 10

Relative size and decomposition of backlash effect under alternative parameters. Note: size of each effect has been indexed to the backlash effect in the voluntary saving case which is set to 1. Higher bars imply greater increases in leisure when the retirement age is increased (higher backlash effect). The channels of each effect are calculated by applying the baseline calibration (along with appropriate parameter changes from Table 2) to Eq. (18) along with the partial derivatives of Eq. (19)

Increasing the interest rate magnifies the (relatively) patient character of the individuals. The higher interest rate decreases the present value of labour income, pension contributions, and pension benefits, which attenuates the backlash effects. Interestingly, the backlash effect in the voluntary saving and fixed contribution cases is virtually nil. In the case with fixed benefits, the effect of lower contributions is relatively strong, as the lower contribution payments are discounted less heavily than the future benefits. The takeaway is that if the difference between interest rates and subjective discount rates is large enough, backlash effects tend to be mitigated or disappear altogether.

Given a higher interest rate, the loss in wealth by working less, and hence, saving less, is greater than for the baseline interest rate. As such, individuals are more reluctant when considering a reduction in labour supply at birth. Individuals subject to a higher interest rate experience a smaller increase in lifetime earnings, which reduces the income effect that leads to increased leisure at birth.

Increasing the discounting rate has the effect that individuals now choose a decreasing path of consumption and leisure over their life cycle. They choose to consume more while younger and less while older (see Eqs. 9 and 10). The higher discounting factor has the opposite effect on the backlash effect compared to the interest rate. The economy is now composed of impatient individuals, making it a debtor nation, and the backlash effect increases in all three pension schemes under consideration. The greatest impact is on earnings potential, which nearly doubles. The effects related to the pension scheme do not change as much. The discounting rate influences the individual’s leisure decision through the $\xi (a)$ parameter, which is multiplied by every channel of the backlash effect. This effect is not caused by the present value of income, benefits, or contributions, but by the sensitivity of labour/leisure choices to retirement age.

Stronger unfunded pensions, manifesting in a higher replacement ratio, do not change the earnings potential from the baseline case. When a scheme with fixed benefits is in place, a later retirement age increases the value of the pension system to the individual. When the replacement ratio is increased by ten percentage points, this effect is amplified, so the backlash effects are now stronger compared to the baseline case. If a scheme with fixed contributions is in place, a later retirement age reduces the value of the pension transfers. This means that with a higher pension replacement ratio, the individual feels even poorer compared to the baseline scenario when the retirement age is increased.

Increasing individuals’ level of risk aversion by increasing the CRRA coefficient is the same as decreasing the elasticity of intertemporal substitution, strengthening individuals’ consumption smoothing motive. The baseline values for the interest and discounting rate indicate that $r>\rho$. This implies an increasing path of consumption and leisure over the life cycle of individuals at a rate that is slowed down by the increased risk aversion. However, the present values of income and the pension scheme are unchanged. The higher level of risk aversion increases the scaling factor associated with the impact of the mandatory retirement age on labour/leisure choices, leading to an increase in the backlash effect for every pension scheme.

6.2 Further discussion of model characteristics

The model features a realistic mortality profile with a finite maximum age, as proposed by Boucekinne. Nonetheless, given the assumption of constant population size, the fertility rate is endogenous to the choice of the model’s mortality rate. This factor affects the order of magnitude of the backlash effect, owing mainly to its evident impact on the dependency ratio, but it does not undermine our key findings: that the potential for the backlash effect exists and the setup of the pension scheme matters.

The model features a flat wage profile, implying that individuals’ productivity is constant over their life cycle. By applying age-dependent wages, the labour supply over the life cycle would feature a hump shape where individuals work more during the ages where they are relatively more productive (see Eq. (8)). This, in turn, would have an interesting impact on the backlash effect. As labour supply is increased on the extensive margin, those extra years available to the individual are characterised by low productivity, diminishing the backlash effect generally but making it more pronounced at specific ages.

Diminished labour productivity at higher ages, or simply low labour market attachment at higher ages, introduces a particularly important policy dimension to the retirement age reform. The importance of coordinating all of the institutional features between early retirement options, health issues leading to the inability to work, and the option of using unemployment insurance as a way to retire earlier are important to understanding the path for successful reform. Increasing the extensive margin of labour supply for everyone, irrespective of health status, requires that people with health problems work further. This undermines the capacity of retirement age increases to ensure pension system sustainability effectively while exacerbating problems among those in poor health. In general, unemployment is an alternative to working longer. If people adjust the length of their working career by leaving the labour market early, our model is likely to overestimate the size of the backlash effect, although the underlying backlash effect would still be present.

In the context of our model, there is voluntary unemployment, in the sense that individuals choose to decrease their labour force participation on the intensive margin. This is similar to the interpretation of Kliponen et al. (2006), where the employment rate is modelled through individuals’ intensive margin labour supply decisions. This is also comparable to the intensive margin decisions in Börsch-Supan et al. (2006), where there is no involuntary unemployment, but individuals transition slowly into retirement as they decrease their intensive margin labour force participation. This, in effect, is what the backlash captures: as the retirement age increases, the individuals choose to work less (on the intensive margin).

Our treatment of employment is also comparable to de la Croix et al. (2013), where individuals are forced to retire no later than an exogenously set maximum age, with the possibility of retiring one or two five-year periods before then, mirroring the voluntary unemployment present in our model. The individuals are then subject to a similar shock in our model: retirement age increases, which causes short-run unemployment to rise due to incomplete labour markets, while long-run unemployment returns to the natural rate.

Generally, involuntary employment requires modelling mechanisms, such as market failures or frictions. Baksa and Munkacsi (2016) apply hiring costs to analyse the effect of demographic change and retirement reforms on unemployment. Both increased longevity (modelled through a lower mortality rate) and increased retirement age (modelled by a decrease in retirement probability) are found to have no effect on unemployment. Our model is a steady-state analysis without market frictions and does not try to capture the transition mechanism between steady states. We therefore opt for markets without frictions or failures, which implies that there is no involuntary unemployment.

Introducing endogenous factor prices could also deepen the analysis. Decreasing returns on labour would imply a diminished backlash effect and have an impact on substitution effects, as the relative price of consumption and leisure would change. As the labour supply on the extensive margin is increased, the total stock of the labour supply would increase, causing a drop in wages. This, in turn, would cause the interest rate to increase, as the marginal productivity of capital has risen. This would affect the saving rate of the individual, as they would now receive higher returns on their savings. An effect is therefore observed that changes the outcome of the individual’s life cycle planning, affecting the size of the backlash effect, but it does not eliminate it or change its direction.

7 Concluding remarks

Longevity adjustment of the retirement age has become a key component of pension reform in several countries. In the absence of such a rule, the financial viability of the pension system, and fiscal policy more generally, could be cast in doubt.

In this paper, a model of leisure-consumption choices was proposed to demonstrate that policies designed to increase the labour supply on the extensive margin may well have the unintended effect of decreasing the labour supply on the intensive margin; i.e., more years of work could lead to less work each year. Our study has demonstrated how important the design of the pension system is when increases in retirement age are implemented. Specifically, whether the PAYG pension system has fixed contributions or fixed benefits can be crucial to the decision-making process of individuals as they choose the intensity of labour supply throughout their lives. The existence and type of pension scheme can either weaken or strengthen the unintended effects of increased extensive margin labour supply on intensive margin labour supply.

In view of the underlying demographic trends, characterised not least by increased longevity at age 60, our findings could potentially have important implications for fiscal policy in general and the pension system in particular. The key takeaway from this study is that pension reforms aimed at either indexing retirement age increases in a one-to-one manner with respect to increases in life expectancy or increasing the retirement age to preserve the same old-age dependency ratio may lead policymakers to overestimate the impact of such measures on aggregate labour supply. Understanding that a scheme with fixed contributions experiences a less pronounced backlash effect is valuable, although admittedly, the magnitude of the difference may not be sufficient to motivate a pension scheme transition.

Nonetheless, if policymakers account for the backlash effect and the ways to mitigate it through pension scheme design, it could benefit economies as they navigate the narrow road of fiscal sustainability. Indeed, as populations age, individuals might want to work for longer while younger professionals could simultaneously benefit from fewer working hours to alleviate pressure from, say, childrearing. In fact, fewer working hours per week could boost fertility, improving fiscal sustainability in the long run. Fewer working hours per week could improve overall wellbeing through a reduction in strain-related illnesses, which are associated with some professions, while allowing for time with family and benefiting children who need their parents’ attention. This view harmonizes well with the backlash effect, as the effect is driven by utility maximization.

Given the results presented in this article, a further look at these backlash results is warranted, where other features should be considered. The question of the actual size of the backlash effect seems to be an empirical one. Future theoretical work should consider features such as endogenous retirement decisions, endogenous interest rate setting, alternative wage profiles, and especially, proportional taxation for its relevance to fiscal sustainability. It is acknowledged that the addition of these features would provide a more accurate measurement of the backlash effects. The question of economic transition to the new steady state is also a relevant question left for future work.

Appendices

Appendix A

See Fig. 11.

Fig. 11

Source: OECD, Macrobond

Intensive and extensive margin labour supply in the OECD, by country. Note: Intensive margin labour supply (dashed line) is proxied by average working hours per employed person (Index, 2000 = 100). Extensive margin labour supply (solid line) is proxied by the average effective labour market exit of men and women (Index, 2000 = 100).

Appendix B: Utility maximization problem

In this appendix, the solution of the utility maximisation problem is presented in detail. Stating the utility maximisation problem formally:

$$\begin{array}{c}\text{max}u\left(a\right)\\ \left\{c\left(\tau \right),l\left(\tau \right)\right\}\end{array}={\int }_{a}^{\infty }\frac{{\left({c\left(\tau \right)}^{\varepsilon }{\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }\right)}^{1-\sigma }-1}{1-\sigma }*{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}d\tau$$

$$\text{s}.\text{t}. {\int }_{a}^{\infty }c\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau =i\left(a\right)+{\int }_{a}^{g}wl(\tau ){e}^{\left(r+\beta \right)(a-\tau )}d\tau +p(a)$$

Solving with the Lagrangian:

$$\mathcal{L}={\int }_{a}^{\infty }\frac{{\left({c(\tau )}^{\varepsilon }{\left[1-l(\tau )\right]}^{1-\varepsilon }\right)}^{1-\sigma }-1}{1-\sigma }*{e}^{\left(\rho +\beta \right)(a-\tau )}d\tau +\lambda \left(a\right)\left[ i\left(a\right)+p\left(a\right)+{\int }_{a}^{g}wl\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau -{\int }_{a}^{\infty }c\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau \right]$$

$$\frac{\partial \mathcal{L}}{\partial c(\tau )}=\frac{\left(1-\sigma \right)\cdot {\left({c\left(\tau \right)}^{\varepsilon }{\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }\right)}^{-\sigma }\cdot {\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }\cdot \varepsilon \cdot c{\left(\tau \right)}^{\varepsilon -1}}{1-\sigma }\cdot {e}^{(\rho +\beta )(a-\tau )}-\lambda \left(a\right){e}^{\left(r+\beta \right)(a-\tau )}=0$$

(B1)

$$\frac{\partial \mathcal{L}}{\partial l(\tau )}=\frac{\left(1-\sigma \right)\cdot {\left({c\left(\tau \right)}^{\varepsilon }{\left[1-l\left(\tau \right)\right]}^{1-\varepsilon }\right)}^{-\sigma }\cdot {c{\left(\tau \right)}^{\varepsilon }\cdot \left(1-\varepsilon \right)\left[1-l\left(\tau \right)\right]}^{-\varepsilon }\cdot \left(-1\right)}{1-\sigma }{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}+\lambda \left(a\right)w{e}^{\left(r+\beta \right)\left(a-\tau \right)}=0$$

(B2)

By isolating $\lambda \left(a\right)$ from (A1) and (A2):

$$wl\left(\tau \right)=w-\frac{1-\varepsilon }{\varepsilon }c(\tau )$$

By isolating $\lambda (a)$ from (A1), setting $a=\tau$ and integrating on both sides, the following is obtained:

$${\int }_{a}^{\infty }{c\left(a\right)\left[{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}\cdot {\left({e}^{\left(r+\beta \right)\left(a-\tau \right)}\right)}^{\sigma -1}\right]}^{\frac{1}{\sigma }}d\tau ={\int }_{a}^{\infty }c\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau$$

$$=i\left(a\right)+p\left(a\right)+{\int }_{a}^{\text{max}\{a,g\}}wl\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau$$

$$=i\left(a\right)+p\left(a\right)+{\int }_{a}^{\text{max}\{a,g\}}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau -\frac{1-\varepsilon }{\varepsilon }{\int }_{a}^{\text{max}\{a,g\}}c\left(\tau \right){e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau$$

Using the same trick again and solving the integrals:

$$i\left(a\right)+p\left(a\right)+{\int }_{a}^{\text{max}\{a,g\}}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau$$

$$i\left(a\right)+p\left(a\right)+{\int }_{a}^{\text{max}\{a,g\}}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau =c\left(a\right){\int }_{a}^{\infty }{\left[{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}\cdot {\left({e}^{\left(r+\beta \right)\left(a-\tau \right)}\right)}^{\sigma -1}\right]}^{\frac{1}{\sigma }}d\tau +\frac{1-\varepsilon }{\varepsilon }c\left(a\right){\int }_{a}^{\text{max}\{a,g\}}{\left[{e}^{\left(\rho +\beta \right)\left(a-\tau \right)}\cdot {\left({e}^{\left(r+\beta \right)\left(a-\tau \right)}\right)}^{\sigma -1}\right]}^{\frac{1}{\sigma }}d\tau$$

$$=c\left(a\right)\cdot \frac{\sigma }{\rho +r\sigma -r+\beta \sigma }+\frac{1-\varepsilon }{\varepsilon }c\left(a\right)\frac{\sigma }{\rho +r\sigma -r+\beta \sigma }\left[1-{e}^{\frac{\left(a-\text{max}\{a,g\}\right)}{\sigma }\left(\left(\rho +\beta \right)+\left(r+\beta \right)\left(\sigma -1\right)\right)}\right]$$

$$=\frac{c\left(a\right)\cdot \sigma }{\rho +r\sigma -r+\beta \sigma }\left\{\frac{1-\left(1-\varepsilon \right)\left({e}^{\frac{\left(a-\text{max}\{a,g\}\right)}{\sigma }\left(\left(\rho +\beta \right)+\left(r+\beta \right)\left(\sigma -1\right)\right)}\right)}{\varepsilon }\right\}$$

By rearranging, the following expression is:

$$c\left(a\right)=\frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\left\{\frac{\varepsilon }{1-\left(1-\varepsilon \right)\left({e}^{\frac{\left(a-\text{max}\{a,g\}\right)}{\sigma }\left(\left(\rho +\beta \right)+\left(r+\beta \right)\left(\sigma -1\right)\right)}\right)}\right\}\left[i\left(a\right)+{\int }_{a}^{\text{max}\{a,g\}}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau +p\left(a\right)\right]$$

For workers, we therefore have:

$$c\left(a\right)=\frac{\left(\rho +r\sigma -r+\beta \sigma \right)}{\sigma }\left\{\frac{\varepsilon }{1-\left(1-\varepsilon \right)\left({e}^{\frac{\left(a-g\right)}{\sigma }\left(\left(\rho +\beta \right)+\left(r+\beta \right)\left(\sigma -1\right)\right)}\right)}\right\}\left[i\left(a\right)+{\int }_{a}^{g}w{e}^{\left(r+\beta \right)\left(a-\tau \right)}d\tau +p\left(a\right)\right]$$

The solution for leisure is similar.

Appendix C: Derivatives of scaling factor to longevity shocks and retirement age increases

As life expectancy increases (lower $\beta$), the worker is more likely to reach retirement, causing $\mu$ to increase:

$$\begin{array}{c}\frac{\partial \mu \left(a\right)}{\partial \beta }= {\left.-\frac{{\text{e}}^{-g\left(\beta +\rho \right)}\left(1-\varepsilon \right)g}{{\left(1+{\text{e}}^{-g\left(\beta +\rho \right)}\left(-1+\varepsilon \right)\right)}^{2}}\right|}_{a=0\wedge \rho =r}<0\#\end{array}$$

(C1)

The effect of a retirement age increase on the value of $\mu$ is determined by the partial derivative:

$$\begin{array}{c}\frac{\partial \mu \left(a\right)}{\partial g}={\left.-\frac{{\text{e}}^{-g\left(\beta +\rho \right)}\left(1-\varepsilon \right)\left(\beta +\rho \right)}{{\left(1+{\text{e}}^{\left(-g\right)\left(\beta +\rho \right)}\left(-1+\varepsilon \right)\right)}^{2}}\right|}_{a=0\wedge \rho =r}<0\#\end{array}$$

(C2)

Given the introduction of age-dependent mortality, the elements of the scaling factor have become more complex:

$${\xi }_{1}\left(a\right)\equiv {\int }_{a}^{A}m\left(\tau \right)\left[{e}^{\frac{\left(a-\tau \right)}{\sigma }\left(\rho +r\left(\sigma -1\right)\right)}\right]d\tau =\frac{\sigma }{{\mu }_{0}-1}\left[{\mu }_{0}\left(-\frac{{e}^{\frac{\left(a-A\right)}{\sigma }\left(\rho +r\left(\sigma -1\right)\right)}-1}{\rho +r\left(\sigma -1\right)}\right)-\left(\frac{{e}^{\frac{a\left(\rho +r\left(\sigma -1\right)\right)-A\left(\rho +r\left(\sigma -1\right)-{\mu }_{1}\sigma \right)}{\sigma }}-{e}^{{\mu }_{1}\cdot a}}{{\mu }_{1}\sigma -\left(\rho +r\left(\sigma -1\right)\right)}\right)\right]$$

$${\xi }_{2}(a)\equiv {\int }_{a}^{\text{max}\{a,g\}}m\left(\tau \right)\left[{e}^{\frac{\left(a-\tau \right)}{\sigma }\left(\rho +r\left(\sigma -1\right)\right)}\right]d\tau =\frac{\sigma }{{\mu }_{0}-1}\left[{\mu }_{0}\left[-\frac{{e}^{\frac{\left(a-\text{max}\{a,g\}\right)}{\sigma }\left(\rho +r\left(\sigma -1\right)\right)}-1}{\rho +r\left(\sigma -1\right)}\right]-\frac{\left({e}^{\frac{(a-\text{max}\{a,g\})\left(\rho +r\left(\sigma -1\right)\right)+\text{max}\{a,g\}\cdot {\mu }_{1}\sigma }{\sigma }}\right)-{e}^{a{\mu }_{1}}}{{\mu }_{1}\sigma -\left(\rho +r\left(\sigma -1\right)\right)}\right]$$

Appendix D: Partial derivatives of leisure choices to longevity shocks and retirement age increases

Here, we show the derivatives of leisure, [$1-l\left(0\right)]$, at labour market entry with respect to the hazard rate $\beta$ for the razor-edge case with $r=\rho$:

$$\left\{\begin{array}{c}\frac{{e}^{g\left(\beta +r\right)}(1-\varepsilon )\varepsilon g}{{({e}^{g\left(\beta +r\right)}+\varepsilon -1)}^{2}}>0 for voluntary savings case\\ \frac{{e}^{\beta g}\left(1-\varepsilon \right)g\left\{{e}^{gr}\left[\varepsilon +k\left(1-\varepsilon \right)\right]-k\left(1-\varepsilon \right)\right\}}{{\left[{e}^{g\left(\beta +r\right)}-\left(1-\varepsilon \right)\right]}^{2}}>0 for DC case\\ \frac{{e}^{\beta g}\left(1-\varepsilon \right)\left[{e}^{gr}{\left({e}^{\beta g}-1\right)}^{2}\varepsilon +b\left({e}^{gr}-1\right)\left({e}^{g\left(2\beta +r\right)}-\left(1-\varepsilon \right)\right)\right]g}{{\left({e}^{\beta g}-1\right)}^{2}{\left[{e}^{g\left(\beta +r\right)}-\left(1-\varepsilon \right)\right]}^{2}}>0 for DB case\end{array}\right.$$

(D1)

Now, the derivative of leisure with respect to retirement age $g$ is still in the razor-edge case.

For the voluntary savings system:

$$\frac{\partial \left[1-l\left(0\right)\right]}{\partial g}=\frac{{e}^{g\left(\beta +r\right)}(1-\varepsilon )\varepsilon (\beta +r)}{{({e}^{g\left(\beta +r\right)}+\varepsilon -1)}^{2}}>0$$

(D2)

For the unfunded fixed-benefit case:

$$\frac{\partial \left[1-l\left(0\right)\right]}{\partial g}=\frac{{e}^{\beta g}\left(1-\varepsilon \right)\left(b\beta \left({e}^{gr}-1\right)\left({e}^{g\left(2\beta +r\right)}+\varepsilon -1\right)-b{e}^{gr}\left({e}^{\beta g}-1\right)\left({e}^{\beta g}+\varepsilon -1\right)r+{e}^{gr}{\left({e}^{\beta g}-1\right)}^{2}\varepsilon \left(\beta +r\right)\right)}{{({e}^{\beta g}-1)}^{2}{({e}^{g\left(\beta +r\right)}+\varepsilon -1)}^{2}}$$

(D3)

As argued in the text, the sign of this expression is ambiguous and only determinable numerically.

For the unfunded fixed-contributions case:

$$\frac{\partial \left[1-l\left(0\right)\right]}{\partial g}=\frac{{e}^{\beta g}(-1+\varepsilon )(\beta ({e}^{gr}(\varepsilon (-1+k)-k)+k-\varepsilon k)+{e}^{gr}(\varepsilon (-1+k)+(-1+{e}^{\beta g})k)r)}{{(-1+{e}^{g(\beta +r)}+\varepsilon )}^{2}}$$

(D4)

As in the fixed benefits case above, the sign of this expression is also ambiguous, and a numerical exercise is required to determine its sign.