The Impact of Reducing the Pension Generosity on Inequality and Schooling

We investigate the impact of a reduction in the pension replacement rate on the schooling choice and on inequality in an overlapping generations model in which individuals differ by their life expectancy and in their cost of attending schooling. Within our framework we illustrate that many pension systems are ex ante regres‑ sive due to the difference in life expectancy across skill groups. We then derive the level of progressivity that needs to be implemented to restore an equal treatment of the pension system across skill groups.


Introduction
As populations age, government programs that redistribute resources from working age population to the dependent elderly, are increasingly getting under fiscal pressure. Policy adjustments have to be implemented that guarantee the fiscal sustainability of these programs. During the last two decades many OECD countries have passed pension reforms that implicitly reduce the generosity of their pension systems (OECD 2013). However, these reforms are likely to have adverse effects not yet fully investigated. For instance, what will be the effective coverage level for workers that differ by their length of life?
Recent studies for the US (Academies and of Sciences, Engineering, and Medicine 2015;Murtin et al. 2017) and Europe (Mackenbach 2019) show that mortality differences by socio-economic groups have widened over the last decades. Since higher socioeconomic groups survive for more years compared to lower socioeconomic groups, they will receive benefits for a longer time period. As Pestieau and Ponthiere (2016, p. 209) nicely summarize these arguments: "Social insurance systems, like pension systems, which were built to reduce well-being inequalities between the surviving old, tend also to exacerbate well-being inequalities between the surviving old and the prematurely dead." Unless the structure of contributions and benefits is not adjusted to these mortality differentials, welfare programs for the elderly may induce a net transfer from the poor to the rich.
Several papers have recently analyzed the redistributive effects of pension systems when longevity varies across socioeconomic groups both empirically (Academies and of Sciences, Engineering, and Medicine 2015; Haan et al. 2019;Holzmann et al. 2019) and theoretically (Pestieau and Ponthiere 2016;Laun et al. 2019;Sanchez-Romero et al. 2019). The main conclusion from the empirical literature is that pension systems are becoming increasingly regressive due to the increase in the longevity gap across socioeconomic groups. However, in the theoretical models the effects are not straightforward. Mortality differences are to some degree known by the individuals and taken into account when forming life cycle decisions such as education, labor force participation, savings, etc. Moreover individuals differ in further dimensions, such as ability and health. As a result, any social program induces incentives and distortions that differ across different population groups.
For instance, based on the case study of Norway, Laun et al. (2019) find that, when individuals differ by education, health and income, proportionally reducing old-age and disability benefits is preferable to other alternative policies such as raising the early retirement age or increasing social contributions; since raising the retirement age will be avoided by claiming disability benefits. However, the former policy increases inequality across socioeconomic groups. Sanchez-Romero et al. (2019) show that the US Social Security reduces regressivity from longevity differences, but would require group-specific life tables to achieve progressivity. In addition, they find that without separate life tables, despite apparent accounting gains, lower income groups would suffer welfare losses and higher income groups would enjoy welfare gains through indirect effects of pension systems on labor supply. Pestieau and Ponthiere (2016) present an excellent discussion on longevity variations and the social protection system including health, education and pension systems. In their study the authors highlight also the importance of heterogenous attitudes towards the risk of longevity. They also argue when education is endogenously chosen by individuals, and this decision affects their ex ante life expectancy, new challenges may arise with complex implications for the Welfare State. 1 Therefore, it is important to consider behavioral effects when studying implications of alternative reforms of the welfare state.
In this paper we study whether a reduction in the replacement rate of pension systems may harm differently short versus long lived individuals and whether schooling will increase or decrease. More specifically we propose a model that extends the work by Pestieau and Ponthiere (2016) by introducing heterogeneity in schooling effort, similar to Le Garrec (2015), which endogenously leads to different skill groups in the total population. Higher skill groups are characterized by higher survival and higher levels of income. Following Fehr (2016) we implement a flexible defined benefit pension system in a two-period overlapping generations model that allows to model a flexible level of progressivity of the pension system. Individuals optimally choose the education, consumption and savings. We show that individuals with different educational attainment will face different levels of the implicit tax rate of the pension system. We term the difference between the implicit tax rate of low and high skilled individuals "pension inequality". Moreover, we analytically derive how the difference in the implicit tax rate between the skill groups is a function of the progressivity of the pension system and the ratio of the relative survival to the relative income advantage of high skilled compared to low skilled individuals. The latter ratio has an intuitive economic meaning. The numerator, i.e. the relative survival advantage of high skilled compared to low skilled individuals, indicates to which extent (how many additional years) the high skilled individuals receive pensions relative to the low skilled. The denominator, i.e. the relative income advantage of high skilled to low skilled individuals, reflects the additional contribution that the higher skilled individuals are paying to the system relative to the low skilled individuals. Within this framework we can show that a mortality differential between low and high skilled individuals will induce regressivity for a pension system of the Bismarckian type; i.e., a flat replacement rate. In case of a progressive pension system of the Beveridge type we find that only if the progressivity of the pension system is above the ratio of the relative survival to the relative income advantage, there will be a redistribution from long lived to short lived individuals. Otherwise even a progressive pension system will become regressive in a situation where mortality differs across the low and high skilled individuals. That is, for a given level of income differential, the higher the mortality differential in a society, the more progressive a pension system needs to be in order to avoid that a progressive pension system becomes regressive.
We next study whether and to which extent a pension reform that reduces the replacement rate (i.e. decreases the generosity of the pension system) may influence the pension inequality. As argued above, in a situation where life expectancy differs by education, a change in the replacement rate will change the implicit tax rate differently for low and high skill individuals. In turn, by changing the implicit tax rate differently for high and low skilled individuals, behavioral effects on individual life cycle decisions, such as educational investment, will result. We show that only if the progressivity of the pension system is equal to the ratio of the relative survival to the relative income differential between high and low skilled individuals, will such a pension reform not introduce any type of pension inequality. I.e. given any mortality and income differential in the society, a pension reform that reduces the pension replacement rate has to adjust the progressivity of the pension system to avoid introducing pension inequality. Whether a decrease in the replacement rate will increase or decrease the share of the population that is high skilled will depend on the relative risk aversion of individuals. Overall an increase in education due to a lower level of the replacement rate is prevalent in those pensions systems that have a higher rate of progressivity.
We complement our analytical results by studying the progressivity of the pension systems and the ratio of the relative survival to relative income differential by skill group for a set of 25 selected OECD countries. Based on these two values and our analytical results we show for which of the countries a reduction in the generosity of the pension system may induce pension inequality and a change in the share of the population that acquires skills.

The Model
We consider a small open economy populated by overlapping generations. Working age generations contribute to a pay-as-you-go pension system that pays for the pension benefits of old-age generations. The population is assumed to be stationary (i.e., constant population) and is comprised of a continuum of heterogenous individuals in each generation. 2

Individuals
Life is divided into two periods: young and old. In the first period, individuals survive with probability one, they choose their consumption level c and whether to become skilled workers ( e s ) or to stay unskilled ( e u ). Assume the effort of attending school has utility cost ∈ ℝ and differs across individuals (Oreopoulos 2007;Restuccia and Vandenbroucke 2013;Le Garrec 2015;Sanchez-Romero et al. 2016). If > 0 individuals incur a cost by making the effort to continue schooling. A negative value of simply implies that individuals like going to school. 3 Individuals survive with probability (e i ) to the second period, which depends on the skill level, and choose their consumption level d. Throughout, we impose the following assumption Assumption 1 The survival probability increases with the skill level, (e s ) > (e u ).
The effort of attending school also captures the ignorance of future outcomes derived from decisions made during the schooling period, which is observed during adolescence (Oreopoulos 2007). Thus, although some individuals are aware that becoming a skilled worker increases their likelihood of survival to old-age, other individuals might not be aware of the positive effects of education, which can be modeled with a large and positive value of .
The preferences of an individual of type are described by the following utility function: where {e i =e s } is an indicator function that takes the value of one if individuals decide to become skilled workers ( e s ) and takes the value of zero otherwise ( e u ), ∈ (0, 1] is the subjective discount factor, and u(⋅) is the period-utility function (with u > 0 , u ′ > 0 , and u ′′ < 0).
During the first period, individuals consume part of their disposable income and save for retirement by purchasing annuities where s denotes private savings, is the social contribution rate paid to the PAYG pension system, and y(e i ) is the gross labor income earned by a worker with skill level e i , with i ∈ {s, u} . To simplify the exposition of the model, we impose the following assumption: Assumption 2 The income difference between skilled and non-skilled workers is such that the consumption of skilled workers is always greater than the consumption of non-skilled workers.
Assumption 2 implies that our model provides the same result as a model with time costs of schooling, see Le Garrec (2015).
In the second period, individuals consume their wealth, which is equal to the sum of the annuities purchased in the first period and the pension benefits claimed where R ≤ 1 is the market discount factor, is the maximum pension replacement rate-see Fig. 1, and y(e u ) + (1 − )y(e i ) is the pension base used to calculate the pension benefit. 4 Parameter ∈ [0, 1] reflects the extent to which the pension system is more "Beveridgean" (i.e. = 1 ) or "Bismarckian" (i.e. = 0). 5 In order to introduce the pension replacement rate -relative to the gross labor income-of an individual with skill level e i , we express the pension benefits claimed as y(e i )f (e i , ) . Thus, old-age consumption can be rewritten as (2) c + s = (1 − )y(e i ), is the relative income advantage of a skilled worker. The term (e s ) reflects the degree of progressivity of the replacement rate formula. Hence, for = 0 , Eq. (5) shows that the replacement rate is flat at a value of , whereas the replacement rate faced by an individual declines with income as tends to one.
Combining (2) and (4), and rearranging terms, we obtain that the lifetime budget constraint of an individual with skill level e i is The left-hand side of (6) is the present value of lifetime consumption. The righthand side of (6) is the initial wealth of the individual, which includes the gross labor income earned and the social security wealth at the entrance into the labor market, − E (e i )y(e i ) ; where E (e i ) is the effective social security tax/subsidy rate: 6 The effective social security tax/subsidy rate can take positive or negative values. In particular, under an actuarially fair pension system the effective social security tax rate is zero ( E = 0 ), whereas in non-actuarially fair pension systems social contributions paid can generate either implicit taxes ( E > 0 ) or implicit subsidies ( E < 0). Optimal consumption and saving For the life-cycle model given by (1) and (6) individuals with skill level e i optimally choose in the first period to consume ) −1 is the individual's marginal propensity to consume with respect to human wealth. See proof in "Appendix 1". By plugging (8) into (2), and using (7), we denote the optimal saving rate of individuals with skill level e i The first term on the right-hand side of (9) measures the marginal propensity to save, while the negative term in (9) measures the reduction in private savings (i.e., crowding-out effect) caused by the pension system. Thus, as shown in (9), an increase (resp. reductions) in the social security rate, , and/or an increase in the pension replacement rate, f (e i , ) , yield a higher (resp. lower) crowding-out effect on savings. 7 Optimal schooling Individuals choose whether to become skilled workers ( e s ) or remain unskilled ( e u ). This decision depends on the schooling effort , which differs across individuals. The optimal schooling decision satisfies Equation (10) implies that an individual with utility cost of schooling lower (resp. higher) than ̄ will optimally choose to become a skilled (resp. unskilled) worker. The parameter ̄ denotes the threshold utility cost of schooling for which an individual is indifferent between staying unskilled and becoming a skilled worker; i.e, V(e u ;̄) = V(e s ;̄) . Equating the expected utility between a skilled worker and an unskilled worker gives Equation (11) measures the difference between the utility of consumption of a skilled worker, who also has higher life expectancy, and the utility of consumption of an unskilled worker. From (11) it is straightforward to show that the threshold utility cost of schooling increases the higher is the income of skilled workers; i.e.
y(e s ) y(e s ) > 0 . Hence, ceteris paribus the income of unskilled workers, more individuals choose to continue schooling when the income of skilled workers rises.
Differentiating (11) with respect to an increase in the life expectancy of skilled workers we can identify two opposite effects on ̄. 8 On the one hand, skilled workers enjoy higher utility due to the higher probability of surviving to old-age. On the other hand, skilled workers lose utility because they have to reduce consumption (i.e. "years-to-consume effect") in order to finance the additional years lived. To guarantee that the impact of a longer life span on schooling is always positive, also known as the Ben-Porath mechanism (see Ben-Porath 1967), we impose Assumption 3.
Assumption 3 is a sufficient, although not necessary, condition that guarantees that a marginal increase in the longevity of skilled individuals leads to a marginal increase in the threshold utility cost of schooling.

The Proportion of Skilled Workers
In this economy there is a continuum of individuals who are heterogeneous by their utility cost of schooling. Let g( ) be the probability density function of the utility cost of schooling within each generation. The corresponding cumulative distribution function of is G( ) = ∫ −∞ g(x)dx . We define the proportion of individuals that choose to become skilled workers by q. Thus, from (11) we have Figure 2 shows a stylized density distribution of the utility cost of schooling in which the gray area represents the value of q = G(̄) . Individuals with >̄ choose to stay unskilled (white area under the curve), while those with <̄ become skilled workers (gray area under the curve). Thus, we can visually observe that only individuals with a close to ̄ are susceptible to a change in ̄. 9 As a consequence, from (13) and Fig. 2 we have that an increase in the threshold value ̄ yields a higher proportion of skilled workers because it becomes optimal for some unskilled to continue schoolling, i.e. G � (̄) > 0.
(13) q = G ̄ . 9 A similar model setting, in which only a set of individuals are affected by a policy change, has been used for analyzing the implication of compulsory schooling on wealth, health and happiness (Oreopoulos 2007). 8 Assuming for simplicity no pension benefits, the partial derivative of (11) with respect to (e s ) gives The first term inside the parenthesis is the additional utility gained by living longer, while the last term inside the parenthesis is the utility cost of living longer. Thus, Assumption 3 guarantees that (12) is positive. Note that the introduction of pension benefits in (12) implies that individuals gain an additional utility from the higher probability of receiving the old-age pension benefits.

Inequality and Pension Systems
Pension systems are designed either to treat all contributors equally (Bismarckian) or to distribute from rich workers to poor workers (Beveridgean). However, this distinction is not so clear when life expectancy differs across skill groups. From Eq. (7) it follows that skilled and unskilled workers do not face the same effective social security tax/subsidy rate when the life expectancy differs across skill groups. Indeed, even progressive pension systems might induce a regressive distribution of income from low-income workers to high-income workers (Pestieau and Ponthiere 2016; Sanchez-Romero and Prskawetz 2017; Ayuso et al. 2017). From Eqs. (6) and (7) it is clear that a pension system that generates the same effective social security tax/subsidy rate for all contributor types does not cause any redistribution of income across skill groups. Hence, the pension system maintains the relative wealth position of all contributors. Instead, if the effective social security tax rate of unskilled workers is higher, or lower, than that of skilled workers, the pension system will change the wealth position between unskilled and skilled workers. From now on we refer to this inequality as "pension inequality". Note that pension inequality is defined for any positive or negative difference between the effective taxes of unskilled and skilled workers.
A simple approach for analyzing whether a pension system induces pension inequality is to calculate the absolute difference between the effective social security tax rate faced by unskilled workers and that of skilled workers, which we denote by | | . From (5) and (7) we have where (e s ) ∈ [0, 1] is the relative survival advantage of a skilled worker with respect to an unskilled worker Assuming the same life expectancy across skill groups (i.e., (e u ) = (e s ) ), Eq. (14) implies that a pension system with a flat replacement rate ( = 0 ) does not redistribute resources across skill groups, while a progressive replacement rate ( > 0 ) redistributes income from skilled workers to unskilled workers. 10 In contrast, once that the life expectancy differs across skill groups, we show in Proposition 1 that there will be a redistribution from unskilled to skilled workers in pension systems with a flat replacement rate, whereas the result is ambiguous in case of a progressive pension system. When (e s ) > (e u ) holds, Proposition 1 shows that a pension system with a flat replacement rate becomes ex ante regressive, transferring income from short-lived and unskilled workers to long-lived and skilled workers. In contrast, by allowing a progressive pension system ( > 0 ), the government is capable of reducing the difference in the effective social security tax rate paid by the two skill groups. Moreover, we obtain from (14) that skilled workers face the same effective social security tax as unskilled workers when the degree of progressivity ( ) is equal to the ratio of the relative mortality advantage of skilled workers and the relative income advantage of skilled workers, which we denote by p = (e s ) (e s ) . 11 Thus, any other degree of progressivity ( ≠ p ) benefits one skill group at the expense of the other. In particular, for a positive gap in life expectancy between skill groups, Fig. 3 shows that a (15) (e s ) = (e s ) − (e u ) (e s ) .
pension system whose degree of progressivity is lower than p (i.e. < p ) redistributives from short-lived and unskilled workers to long-lived and skilled workers. In contrast, a pension system with a degree of progressivity greater than p (i.e. > p ) redistributives from long-lived and skilled workers to short-lived and unskilled workers.

The Impact of Reducing the Pension Replacement Rate
In the next decades it is expected that many pension schemes will introduce reforms that reduce the generosity of their systems in order to improve its long-run sustainability. Finland, Germany, Japan, and Spain, for instance, have already introduced automatic adjustment mechanisms, which reduce the replacement rate, to guarantee its sustainability (OECD 2017b). In this section, we study the impact of this policy on our measure of pension inequality and also on the incentives for becoming a skilled worker.

Impact on Pension Inequality
Given that the replacement rate affects pension inequality in a multiplicative way, Eq. (14) implies that a reduction in the replacement rate, , leads to a less regressive pension system if < p (lower pension inequality), while this policy diminishes the progressivity of the pension system if > p (higher pension inequality). 12 If a pension system aims at avoiding any pension inequality, while reducing the generosity of the pension system, the progressivity of the pension system should satisfy that = p. To see the relevance of this policy we compare the degree of progressivity of the pension system ( ) to the ratio of the relative mortality to the relative income advantage of skilled workers (p) for a selection of OECD countries. 13 We derive the value of p by combining information on relative earnings of men aged 55-65 by educational attainment from OECD (2017a, 2019a) with male life expectancy at age 65 by educational attainment from Murtin et al. (2017). For the case of the Netherlands, we use data on life expectancy at age 65 by educational attainment from CBS for the period 2015-2018. The degree of progressivity of each pension system ( ) is calculated using the gross pension replacement rate from mandatory pension schemes (public and private) by percentage of individual earnings from OECD (2017b, 2019b). Therefore, we restrict our analysis to the unfunded component of the pension system in each country. The information is provided for low, median, and high income earners. High income earners are individuals with a wage above 1.5 times the median wage, whereas low income earners are individuals with a wage less than 0.5 times the median wage. 14 Figure 4 shows that despite the fact that many pension schemes include some degree of progressivity in the replacement rate formula (i.e., > 0 ), the existing longevity gap by socioeconomic status (Murtin et al. 2017) leads many pension systems to be ex ante regressive (see light grey dots). As a consequence, the fall in the replacement rate will yield a reduction in pension inequality-as measured in (14)-in the ex ante regressive pension systems ( < p ), while it will increase pension inequality in the ex ante progressive pension systems ( > p ) (see dark gray dots). From Fig. 4 we can also observe that the minimum value of p is close to 8% (Mexico), the maximum is 83% (Hungary), and the most frequent value ranges between 20 and 46% for the selection of OECD countries, with an average value close to 36%. 15

Impact on Education
Pension systems may also affect the optimal schooling decision of individuals through changes in the effective social security tax/subsidy rate. This is because the effective social security rate has an impact on the expected income earned by workers, on the marginal benefit of education, and ultimately on the educational distribution of the population.
To study the impact of reducing the generosity of the pension system on education, we differentiate the proportion of skilled workers, q, with respect to a fall in . From (11), (13), and (28), we have where = u � (c * (e u ))y(e u ) u � (c * (e s ))y(e s ) is the ratio of the marginal utility of work between unskilled and skilled workers. 16 See "Appendix 3" for a detailed derivation. Equation (16) measures the marginal (utility) gain/loss of reducing the replacement rate to those individuals who are at the margin between staying unskilled or becoming skilled. Note that the right-hand side of (16) is multiplied by g(̄) . Hence, for those individuals whose effort of attending schooling is close to ̄ , the fall in the replacement rate leads to a change in the difference between the effective social contribution rate paid by both skill groups (i.e., − ) as well as to an income effect and a substitution effect caused by the increase in the disposable income during the working period. This is represented by the second term inside the squared brackets; i.e., ( − 1) − E (e u ) . On the one hand, individuals use the increase in disposable income to avoid the effort of attending school (income effect). On the other hand, since the fall in reduces the effective tax rate and hence raises the disposable income, this policy makes it more attractive to become a skilled worker (substitution effect). We can distinguish three cases depending on whether the income effect is lower, equal to, or greater than the substitution effect.
For expositional simplicity, we first study the case in which the income effect is equal to the substitution effect ( = 1 ). According to Eq. (16), in this case we just need to differentiate between the case where the progressivity of the pension system is below and alternatively above p. We know that for < p a decrease in the replacement rate makes the pension system less regressive and hence less individuals will invest in education-since the unskilled are now better off-, implying a decrease in the share of skilled workers. On the other hand if > p a decrease in the replacement rate makes the pension system less progressive, which implies that more individuals will have an incentive to become skilled-since the skilled are now better off-, thereby increasing the share of skilled workers. It is also important to note that the extent to which a decrease in the replacement rate changes the share of skilled workers depends on the absolute difference between and p.
Next we relax the assumption of the income effect and substitution effect to be equal.
In case the income effect dominates (see Fig. 5a) and > p , the benefit that skilled workers experience from the reduction in the replacement rate should be large enough to compensate for the effort of attending school. As a consequence, when the income effect dominates, a reduction in the replacement rate will increase the share of skilled people only if is much larger than p (i.e., ≫ p ); otherwise

Fig. 5
Impact of a reduction in the replacement rate on the proportion of skilled workers by degree of progressivity of the pension system ( ). Notes: For < 1 the substitution effect dominates over the income effect, while for > 1 the income effect dominates over the substitution effect they would opt to stay unskilled. In contrast, in case that the substitution effect dominates (see Fig. 5b) and < p , unskilled workers might find that the additional income they gain from the reduction in the replacement rate is not large enough to compensate for the increase in disposable income they would obtain if they would become skilled. As a consequence, when the substitution effect dominates, a reduction in the replacement rate will increase the share of unskilled people only if is much smaller than p (i.e., ≪ p ); otherwise they would opt to become skilled. To better understand the impact of a fall in the replacement rate on the distribution of skilled workers for the selection of OECD countries, we assume that the marginal utility of consumption follows a power utility function u � (x) = x − , where is the relative risk aversion coefficient. We choose two alternative values for the relative risk aversion ∈ {0.5, 1.5} , which are within the lower and upper bounds for estimated by Chetty (2006). 17 A relative risk aversion of 0.5 implies a value of that ranges between 0.7 and 0.8 across the countries analyzed, with an average value of 0.75. Hence, the substitution effect dominates over the income effect. A relative risk aversion of 1.5 implies a value of that ranges between 1.1 and 1.3 across the countries analyzed, with an average value of 1.2, and therefore the income effect dominates over the substitution effect. Moreover, we assume each period lasts forty years, the annual subjective discount rate is 1%, and the annual market discount rate is 1.5%, which is the result of calculating the difference between an interest rate of 3% and a productivity growth rate of 1.5%. 18 Figure 6 shows that if the relative risk aversion is 0.5 ( < 1 ) a fall in the replacement rate will lead to an increase in the number of skilled workers in most countries (see black triangles), except in Hungary, Latvia, and Sweden. If the relative risk aversion is 1.5 ( > 1 ) a fall in the replacement rate will only lead to an increase in the proportion of skilled workers in countries with a sufficiently high degree of progressivity (Australia, Canada, Denmark, Israel, New Zealand, and United Kingdom). This is because the decline in pension inequality is not large enough to compensate for the effort of attending school. If we assume instead a relative risk aversion of 1 (log utility), a fall in the replacement rate will yield an increase the proportion of skilled workers in countries that are ex ante progressive ( > p ) and a decline in the proportion of skilled workers in countries that are ex ante regressive ( < p). 19

The Combined Effect
In the last subsections, we have discussed whether a reduction in the replacement rate of the pension system will increase or decrease pension inequality and schooling. We have analytically shown that these effects will depend on the degree of progressivity of the pension system. Combining the results from (14) and (16), Proposition 2 summarizes the impact of a fall in the replacement rate of the pension system on schooling and on pension inequality jointly.

Proposition 2 Given
(1)-(6), and Assumptions 1 and 3, a fall in the replacement rate leads to one of the following four alternatives

Lower pension inequality and less skilled workers (Case A)
Higher pension inequality and less skilled workers Lower pension inequality and more skilled workers (Case C) Higher pension inequality and more skilled workers (Case D)  Fig. 4 is complemented with the share of total labor income earned by skilled workers. This additional variable is calculated combining information on the share of men aged 55-64 by educational attainment with the relative earnings of men aged 55-64 by educational attainment from OECD (2017a, 2019a). Calculations done assuming each period lasts forty years, a power marginal utility function u � (x) = x − , where is the relative risk aversion coefficient, a constant annual real interest rate of 3%, a productivity growth rate of 1.5%, and a subjective discount rate of 1% In Proposition 2, columns indicate the impact of lowering the replacement rate on pension inequality, while rows indicate the impact of lowering the replacement rate on education. Figure 7 illustrates the four alternative cases stated in Proposition 2, as they depend on the degree of progressivity of the pension system and the parameter of the relative risk aversion. Each panel is divided in three shaded areas (light gray, gray, and dark gray), which are the results of combining (14) and (16). If a pension system lies within the light gray area, a fall in the replacement rate leads not only to a reduction in pension inequality but also to a reduction in the proportion of skilled workers. If a pension system lies within the dark gray area, a fall in the replacement rate leads to an increase the proportion of skilled workers and to an increase in pension inequality. However, if a pension system lies within the gray area, the impact of a fall in the replacement rate on inequality and education depends on the whether the substitution effect dominates over the income effect. In particular, for < 1 , a lower replacement rate not only reduces pension inequality between education groups, but it also increases the proportion of skilled workers. In contrast, for > 1 , a fall in the replacement rate leads to a reduction in the number of skilled workers and an increase in pension inequality.
Combining the numerical results shown in Figs. 4 and 6, and using Proposition 2, we obtain for a relative risk aversion of 0.5 (see Fig. 8a) that a fall in replacement rate: (1) will increase both the proportion of skilled workers and pension inequality (see green triangles) in ex ante progressive pension systems; (2) will increase the proportion of skilled workers and reduce pension inequality (see blue diamonds) in countries with ∈ (0, p) and − > (1 − ) − E (e u ) ; and (3) will lead to less skilled workers and  However, when the income effect dominates over the substitution effect (i.e., relative risk aversion of 1.5), a fall in the replacement rate (1) will lead in countries with a sufficiently high degree of pension progressivity (i.e., > p ) and to an increase in the proportion of skilled workers and in pension inequality (see green triangles); (2) will reduce the proportion of skilled workers and raise pension inequality in ex ante progressive countries and

Conclusion
We set up a small-open economy with overlapping generations in which heterogeneous individuals optimally choose their consumption path and their educational attainment. We assume a positive correlation between the length of schooling and the survival probability at old age. To study the impact of a reduction in the generosity of the pension system, we introduce a pay-as-you-go pension system that allows for any combination between a fully Beveridgean pension system and a fully Bismarckian pension system. Within our framework, we show that a pension system with a flat replacement rate redistributes resources from unskilled workers with short lives to skilled workers with long lives. By reducing the generosity of the pension system with a flat replacement rate, our model shows that the difference between the effective social security tax rate of both skill groups will diminish, but also the proportion of skilled workers. However, if the pension system is sufficiently progressive, a reduction in the pension replacement rate may increase the proportion of skilled workers and reduce wealth inequality.
where i > 0 is the corresponding Lagrange multiplier. The optimal schooling decision is given by The first-order conditions (FOCs) are: Combining the FOCs we obtain the standard Euler condition Assuming that the marginal utility of consumption is a standard power function u � (c) = c − we have Substituting into the budget constraint, we have Equation (22) is the initial consumption of an individual with skill level e i . The first term on the right-hand side is the marginal propensity to consume out of gross labor income of an individual of type e i , while the second term is the gross labor income of an individual of type e i .

Appendix 2: The Pension System
Consider a stable and mature defined-benefit PAYG pension system with a balanced budget. Given the population and economic characteristics, the budget constraint of the pension system is where the left-hand side of Eq. (23) stands for the total social contributions paid by unskilled and skilled workers, respectively, and the right-hand side stands for the total benefits claimed by the surviving retirees of both skill groups. Dividing both sides of Eq. (23) by the total labor income, the social contribution rate, , is given by where is the share of total labor income earned by skilled workers The first term on the right-hand side of Eq. (24) represents the contribution rate necessary to pay for the pension benefits of unskilled workers, while the second term on the right-hand side accounts for the contribution rate to pay for the pension benefits (20) u � (d * (e * i )) = u � (c * (e * i ))R∕ .

Substituting the FOCs gives
Differentiating the budget constraint (6)  where E (e i ) is given by (28).
. The qualitative results presented in this paper are calculated assuming that unskilled workers are exclusively comprised of individuals with "below upper secondary education", whereas skilled workers have "upper secondary or higher education". In this alternative simulation we assume that 50% of individuals with "upper secondary and post-secondary non-tertiary education" are assigned to the unskilled group, whereas the other 50% are assigned to the skilled group.