Top

Published in:

2019 | OriginalPaper | Chapter

6. Pensions

Author : Burkhard Heer

Published in: Public Economics

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this chapter, we first review empirical facts of public pension systems in OECD countries. Subsequently, we introduce a public pension system in the standard two-period overlapping generations (OLG) model of Chap. 2. We consider two different social security systems, pay-as-you-go (PAYG) versus fully funded. While a fully funded pension system does not have any effect on aggregate savings if capital markets are perfect, aggregate savings fall significantly in a PAYG system. Since public pensions are likely to distort household labor supply decisions, we endogenize labor supply below. In addition, we extend the two-period model to a more realistic 70-period model in which the retirement period is smaller than the working period. Next, we derive the optimal amount of pensions in a PAYG system and study how the demographic transition and aging of the population affect the sustainability of social security. We also discuss the findings of the literature on quantitative pension studies in detail. Finally, we introduce the concept of fiscal space and point out its sensitivity with respect to the aging that takes place in many industrialized countries at present.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Income Taxation

next chapter Public Debt

Available only for authorised users

In the United Kingdom, the modern pension system was introduced in 1908 with the help of the Old Age Pensions Act. The United States initially only provided pensions for federal employees under the Civil Service Retirement System in 1920.

In 1916, the retirement age was reduced to 65 years.

Sometimes, the working age is defined over a larger age range, e.g., 15–64, 20–64, or 20–69 years. Accordingly, take care when comparing old-age dependency ratios from different sources.

In this book, we will sometimes distinguish the two old-age dependency ratios OADR2 and OADR3, which refer to the retirees aged 65 (and above) and those aged 70 (and above). While we primarily use OADR2 as our reference value, it may sometimes be important to consider the evolution of OADR3, e.g., when we study pension policies that extend the retirement age to 70 years.

The data are taken from the UN population division and refer to the medium-fertility variant. See Appendix 6.3 for a more detailed description of the data.

Thanks to Hana Ševčíková for providing the data on dependency ratio forecasts in the United States and the EU14 countries.

For the US economy in the year 2050, standard empirical distribution tests such as the Lilliefors, Cramer-von Mises, Anderson-Darling, and Watson tests reject the normality assumption of the OADR2 distribution at the 1% or 5% level of significance. This observation does not hold uniquely for all countries in our data sample (the US and 14 EU-countries).

An excellent non-technical survey of pension designs to meet the demographic challenge is provided by Barr and Diamond (2008).

There are many other public policies that may help to alleviate the pressure on public pension systems during the demographic transition that we do not study in here. Among others, the government could attempt to increase the labor force participation rate, encourage higher immigration (of young individuals), or to use family policies to increase the fertility rate.

However, in recent years, as the baby boomers have begun to retire, many countries have also started to reduce pension benefits, e.g., Italy in 1995 through the Dini reform.

In Germany, the contribution was evenly split between the employer and employee in 2013. Each side pays a contribution rate of 9.45%. In a Walrasian labor market, the legal incidence, i.e., who pays the contribution rate, does not affect the economic incidence, i.e., who actually bears the burden of the tax. If, however, the labor market is subject to a distortion such as a minimum gross wage, the legal incidence does affect the economic incidence.

Börsch-Sopan and Winter (2001) estimate that the pension payments in Germany will increase from 10.0% of GDP in 1995 to 18.4% of GDP in 2030. To finance these higher expenditures resulting from an aging population, the contribution rate will have to increase to 41.1% by 2030.

The authors study a variety of pension reform proposals for the US economy in a general equilibrium model with labor-augmenting technical progress, endogenous bequests, and endogenous labor supply similar to the model we will study in this chapter. They consider different scenarios for the reform of the public pension system. The number reported in the text above corresponds to their experiment 1.

There are other institutional details of the German social insurance system that we will neglect in the following, e.g., there is a social security contribution ceiling, meaning that, in 2013, contributions were only paid on a gross annual income up to an amount of €69.600 (West Germany) or €58.800 (East Germany). In addition, the number of contribution years enters the formula used to compute the pensions. For further details on the German pension system, see Fehr (1999).

In Switzerland, for example, the fully funded pension system is part of a three-part pension system. The other two parts are a mandatory PAYG public pension and an employer-based pension.

From the theory of finance, we know that it is optimal to diversify risk, which implies that pension systems should combine the two different systems as is done, for example, in the three-part public pension system in Switzerland.

All payments are made at the end of the period.

As one exception, the first pillar of the Swiss pension system is also financed by sources other than wage income, e.g., by a levy on the income of the self-employed.

This argument is taken from Chapter 3.2 in Blanchard and Fischer (1989).

Recall that $(1+r)s_t = c^2_{t+1}-(1+n)d$.

This effect is absent in the derivation of (6.15) where we only consider an unexpected change of d _t and d _t+1 on the savings of the generation born in period t. Since the capital stock k _t is predetermined, the wage w _t does not change for the present (working) generation, but only for future generations.

The intuition for this result given in Sect. 3.2.5 was that increasing the capital stock results in lower interest rates r and, with s _r > 0, in lower savings s, such that capital cannot grow without bound.

See Appendix 4.2 for the derivation of the Frisch labor supply elasticity in the case of a Cobb-Douglas utility function.

See also Sect. 4.4.5 for a discussion of the empirical evidence on ν ₁.

You should attempt to compute these values. In the computation, we set ν ₀ = 257.15 as implied by the calibration in the case of elastic labor supply (see the following section).

As an alternative measure of the welfare change, some authors use output equivalent change. In this case, Δ is computed as the percentage of output by which the consumption levels in the old steady state need to be raised to obtain the same utility as in the new steady state.

We will consider the effects of a gradual policy change in Sect. 6.4.3.

To derive the elasticity, simply take the logarithm on both sides of (6.27c) implying

$$\displaystyle \begin{aligned} \ln l_t = \nu_1 \left[ \ln \left((1-\tau) w_t\right) + \ln \lambda_t -\ln \nu_0\right] \end{aligned}$$

and notice that

$$\displaystyle \begin{aligned} \frac{\partial \ln l_t}{\partial \ln \left((1-\tau) w_t \right)} = \frac{\partial l_t}{\partial \left( (1-\tau) w_t\right)} \frac{(1-\tau) w_t}{l_t} = \nu_1.\end{aligned}$$

More specifically, we need to solve (6.21a) for w ₀ and w ₁ in periods 0 and 1, (6.21b) for r ₁ in period 1, and (6.28) and (6.30) in period 0.

See also Appendix 4.1.

The algorithm used in our program is also very sensitive to the number of transition periods. In particular, for a larger number of transition periods, the solution does not converge.

Of course, we are quite idealistic in assuming that the individual is able to determine the parameters ρ _pen and pen _min. Usually, the pension is a complicated function of contributions and the number of contribution years. In Germany, for example, years of university eduction increase your pension despite that students do not contribute to the PAYG system.

In steady state, output and capital both grow at the rate γ. The US growth rate of real GDP per capita amounted to 2.00% during the period 1960–2011.

The results are computed with the help of the GAUSS program Ch6_social_security4.g, which solves the non-linear system of equations (6.43) with the Gauss-Newton Algorithm. In addition, we recalibrate ν ₀ such that, for τ = 0 and γ = 80%, the steady-state labor supply is equal to l = 0.30.

The data for the US economy are taken from Hansen (1993). The efficiency profile for Germany is computed with the help of the average hourly wages of s-year-olds during the period 1990–1997 following the method of Hansen (1993). Average productivity is normalized to one. We further interpolated the productivity-age profile with a polynomial function of order 3. We used data from the Cross National Data Files for West Germany during the period 1990–1997, which are extracted from the German Socio Economic Panel, GSOEP.

The consideration of the highest income households would not significantly affect our welfare results on optimal social security. The welfare effect of social security is a second-order consideration for these households since the pension income from social security is a relatively small share of total savings for the top income earners.

In a recent study, however, Caliendo, Guo, and Hosseini (2014) demonstrate that this result is sensitive to the assumption of whether (1) bequest income is fixed or endogenous and (2) bequest income is redistributed anonymously or through a direct linkage between deceased parents and surviving children.

In addition, Heer (2018) models income uncertainty from unemployment and specifies a more general utility function with Epstein-Zin preferences that include the Cobb-Douglas utility function (6.45) as a special case.

Be careful when you compare our equilibrium conditions to those in the literature. In some cases, the indexation of the survival probabilities is different and ϕ _s denotes the probability to survive up to age s conditional on surviving up to age s − 1 as in İmrohoroğlu, İmrohoroğlu, and Joines (1995) or Huggett (1996), while our notation follows Conesa and Krueger (1999).

In the literature, expected lifetime utility is either stated in the form of (6.44) (e.g., in İmrohoroğlu, İmrohoroğlu, and Joines 1995 or Huggett 1996) or the product of the cumulative survival probabilities, $\prod _{j=1}^s \phi _{t+j-2,j-1}$, is dropped from this expression (e.g., in Conesa and Krueger 1999). In the latter case, expectations are also formed with respect to (stochastic) survival and instantaneous utility of being dead is set equal to zero. We adhere to the former notation so that expectations $\mathbb {E}_t\{.\}$ are only formed with respect to stochastic idiosyncratic productivity. This notation will be useful in a model of Chap. 7.5 where we analyze the demographic transition. In this model, survival of the individuals is stochastic, while individual productivity is deterministic. As a consequence, the derivation of the Euler equation that contains the survival probability of the individual as an additional factor will become more evident.

In Chap. 5, we denoted the labor income tax rate by τ ^L. Notice that the tax on labor income in this model is the sum of the wage income tax and the social security contribution rate, τ ^L = τ ^w + τ ^p.

In contrast to Sect. 6.3.4, we do not assume pensions to be related to the individual’s lifetime social security contributions. Our simplifying assumption is supported by the results of Fehr, Kallweit, and Kindermann (2013) and Heer (2018), who find in their studies with earnings-dependent pensions that pensions should optimally be provided lump-sum rather than earnings-dependent.

The mean of the workers’ efficiency $ \eta \epsilon \bar y_s$ is normalized to one.

Related research that uses such a value for β includes İmrohoroğlu, İmrohoroğlu, and Joines (1995) and Huggett (1996). With this value of β, the effective time discount factor of the newborn for utility at age s, $\beta ^{s-1} \left (\prod _{j=1}^{s} \phi _{j-1}\right )$, displays an increasing weight to instantaneous utility until real lifetime age 63, before it declines again and even falls below one after the real lifetime age 82 (for the survival probabilities for the year 2015).

The concept of the value function is introduced in Appendix 6.2.

An alternative would be to either let government expenditures be a production input or let the government provide a public consumption good. See the applications in Chaps. 4 and 5.

In Problem 6.6, you are asked to test whether consumption habits help to improve the modeling of consumption-age behavior in a standard Auerbach-Kotlikoff model that implies a downward jump in consumption at the age of retirement.

One of the first studies to highlight the role of the OLG model in accounting for observed wealth heterogeneity was Huggett (1996).

See De Nardi (2015) for a survey of modeling wealth heterogeneity in quantitative general equilibrium models.

İmrohoroğlu and Kitao (2009) also study the effect of the Frisch labor supply elasticity on aggregate labor and the labor-age profile. They distinguish between two different scenarios for the pension reform, consisting of the downsizing of the system by 50% or the total elimination of social security. İmrohoroğlu and Kitao show that the effect of pension reforms on aggregate labor is rather insensitive to the Frisch elasticity, while the profile of hours over the life-cycle is highly sensitive. They also find substantial welfare gains from the reduction in pensions even in the case of a low labor supply elasticity. According to their Table 6.2, the long-run welfare gain of half-privatization amounts to 4.3% of total consumption for a low Frisch elasticity equal to η _lw = 0.5. In contrast to our approach, however, they do not model permanent productivity differences between the workers, and thus, income heterogeneity is smaller in their model than in ours.

We neglect one factor that might increase the welfare effects of social security, however. Fuster, İmrohoroğlu, and İmrohoroğlu (2003) find that in the case of two-sided altruism towards ancestors and descendants, the welfare effects of social security are enhanced. Altonij, Hayashi, and Kotlikoff (1997), however, present empirical evidence that rejects the implications of altruism for intergenerational risk-sharing behavior.

More precisely, we assume that the per capita government expenditures grow at the exogenous rate of technological growth.

For example, Kitao (2014) also considers a linear adjustment over a period of 50 years.

Recall that aggregate labor L _t in period t is expressed relative to total population N _t.

For example, young workers in the years 2050–2060 supply the highest number of working hours during the entire transition period, but their wealth peaks only at the end of their working life in later years.

Recall that we assumed that $\tilde G$ would remain at its 2015 level. In addition, government revenue from accidental bequests declines due to higher survival probabilities. The latter effect, however, is rather modest.

Recall that we consider only the average lifetime utility of the individual generations. The welfare effects might vary considerably across the different productivity types (see also Table 6.7 for the steady-state analysis for the year 2015).

A study that focuses on the political implementability of a transition from the status quo to a reduction in PAYGO pensions in the US is provided by Conesa and Krueger (1999). In accordance with our argument, they find that although the transition to a fully funded pension system would imply substantial welfare gains, a majority of voters would be worse off from this option and thus favor the status quo.

See pages 174–177 in Auerbach and Kotlikoff (1987).

Recall (if your age allows for it) that computer technology in these years was less capable of handling such numerical problems with a high dimension of (individual) state variables.

As noted by İmrohoroğlu, İmrohoroğlu, and Joines (1999), this high value for optimal pensions results from the fact that their model is characterized by dynamic inefficiency in the absence of social security. Higher pensions and, hence, lower savings actually increase total consumption at low replacement rates. In addition, İmrohoroğlu, İmrohoroğlu, and Joines (1999) argue that the US economy is dynamically efficient, as shown by Abel, Mankiw, Summers, and Zeckhauser (1989). In our model above, we only consider dynamically efficient economies in which the population growth rate is below the economic growth rate. İmrohoroğlu, İmrohoroğlu, and Joines (1999) also include land as a (constant) production factor in addition to capital and labor and, as a consequence, their economy is dynamically efficient. They find the optimal unfunded PAYG public pensions in the US to be zero in the stationary state.

Another study with exogenous labor supply that focuses on the distortion of social security contributions affecting the accumulation of capital is Storesletten, Telmer, and Yaron (1999). The main channel emphasized in their model is the financing of pensions with a distortionary income tax that is levied on labor and capital income. Since labor supply is exogenous, the distortion only affects capital accumulation. The authors compare the current system (as of 1996) to alternative scenarios including the abolition of the social security system and a system that is partially PAYG and partially fully funded. They find the alternatives to imply significant welfare gains if general equilibrium effects are taken into account.

İmrohoroğlu, İmrohoroğlu, and Joines (1995) find that the optimal level of social security “appears to be zero when … we incorporate exogenous technological progress in the model”.

Notice that another asset variable in the form of government debt enters our model, and in general equilibrium, the sum of debt (equivalently, government assets) and capital is equal to aggregate savings.

You are asked to compute the Hicksian efficiency gain for the two-period OLG model from Sect. 6.3.2 in Problem 6.4.

To keep the model tractable, Krueger and Ludwig (2007) assume that pensions depend only on the permanent efficiency type, not on the stochastic individual component. In addition, the authors study the transition dynamics under the assumption that contribution rates freeze in 2004. Beyond these assumptions, the model closely resembles that in the previous section.

There are two different tax rates that fulfill the condition of a balanced budget (the two points of interception of the Laffer curve and the line of government expenditures G ₂₀₁₅); naturally, the government chooses the lower tax rate on the upward-sloping side of the Laffer curve.

However, they simplify the model by not considering income uncertainty.

Their model is in some respects more elaborate than ours, in particular with respect to the projection of public medical expenditures; it also includes age-specific fertility rates. In addition, the authors assume that the interest rate on government debt is 1.145% lower than the rate of return on capital throughout the transition.

We will discuss the role of debt in the next chapter.

In fact, we could drop the expectational operator $\mathbb {E}_t$ in the Bellman equation for the retired and replace it by the survival probability ϕ _s,t because they do not face income uncertainty (in contrast to workers).

A detailed description can be found in Chapter 11.6.1 in Heer and Maußner (2009).

This value for the final year is found by trial and error. We choose 2250 because the transition of the endogenous values is complete by then. In Fig. 6.15, we drop the presentation of the final periods to better illustrate the transition.

Our households are naive in the sense that they ignore their future behavior in the optimization decision in period 1; in other words, in period 1, they do not realize that they will behave the same way in period 2 (applying quasi-hyperbolic discounting to the discounted utility of the remaining lifetime).

A seminal paper that introduces you to quasi-hyperbolic discounting and commitment technologies is Laibson (1997).

In addition, İmrohoroğlu, İmrohoroğlu, and Joines (2003) find that social security is not effective in correcting for under-saving that results from time-inconsistent preferences.

The following description is taken from Chapter 9.1 in Heer and Maußner (2009).

Abel, A. B., Mankiw, N. G., Summers, L. H., & Zeckhauser, R. J. (1989). Assessing dynamic efficiency: Theory and evidence. Review of Economic Studies, 56, 1–20.CrossRef

Alkema, L., Raftery, A. E., Gerland, P., Clark, S. J., Pelletier, F., Buettner, T., & Heilig, G. K. (2011). Probalistic projections of the total fertility rate for all countries. Demography, 48, 815–839.CrossRef

Altonij, J. G., Hayashi, F., & Kotlikoff, L. (1997). Parental altruism and intervivos transfers: Theory and evidence. Journal of Political Economy, 105, 1121–1166.CrossRef

Attanasio, O., Kitao, S., & Violante, G. L. (2007). Global demographic trends and social security reform. Journal of Monetary Economics, 54, 144–198.CrossRef

Auerbach, A. J., & Kotlikoff, L. (1987). Dynamic fiscal policy. New York: Cambridge University Press.

Barr, N., & Diamond, P. (2008). Reforming pensions—Principles and policy choices. Oxford: Oxford University Press.CrossRef

Blanchard, O. J., & Fischer, S. (1989). Lectures on macroeconomics. Cambridge: MIT Press.

Börsch-Sopan, A. H., & Winter, J. (2001). Population aging, savings behavior and capital markets (NBER working paper No. 8561).

Braun, R. A., & Joines, D. H. (2015). The implications of a graying Japan for government policy. Journal of Economic Dynamics and Control, 57, 1–23.CrossRef

Budría Rodriguez, S., Díaz-Giménez, J., & Quadrini, V. (2002). Updated facts on the U.S. distributions of earnings, income, and wealth. Federal Reserve Bank of Minneapolis Quarterly Review, 26, 2–35.

Caliendo, F. N., Guo, N. L., & Hosseini, R. (2014). Social security is NOT a substitute for annuity markets. Review of Economic Dynamics, 17, 739–755.CrossRef

Conesa, J. C., & Krueger, D. (1999). Social security reform with heterogeneous agents. Review of Economic Dynamics, 2, 757–795.CrossRef

De Nardi, M. (2015). Quantitative models of wealth inequality: A survey (NBER Working Paper, 21106).CrossRef

De Nardi, M., & Yang, F. (2016). Wealth inequality, family background, and estate taxation. Journal of Monetary Economics, 77, 130–145.CrossRef

De Nardi, M., Imrohoroğlu, S., & Sargent, T. J. (1999). Projected US demographics and social security. Review of Economic Dynamics, 2, 575–615.CrossRef

Fehr, H. (1999). Welfare effects of dynamic tax reforms. Tübingen: Mohr Siebeck.

Fehr, H., Kallweit, M., & Kindermann, F. (2013). Should pensions be progressive? European Economic Review, 63, 94–116.CrossRef

Fernández-Villaverde, J., & Krueger, D. (2007). Consumption over the life cycle: Facts from consumer expenditure survey data. Review of Economics and Statistics, 89(3), 552–565.CrossRef

Fuster, L., İmrohoroğlu, A., & İmrohoroğlu, S. (2003). A welfare analysis of social security in a dynastic framework. International Economic Review, 44, 1247–1274.CrossRef

Gerland, P., Raftery, A. E., Sevčíková, H., Li, N., Gu, D., Spoorenberg, T., Alkema, L., Fosdick, B. K., Chunn, J., Lalic, N., Bay, G., Buettner, T., Heilig, G. K., & Wilmoth, J. (2014). World population stabilization unlikely this century. Science, 346, 234–237.CrossRef

Guvenen, F., Karahan, F., Ozkan, S., & Sang, J. (2015). What do data on millions of U.S. workers reveal about life-cycle earnings risk? (NBER Working Paper Series, 20913).

Hansen, G. (1993). The cyclical and secular behavior of the labor input: Comparing efficiency units and hours worked. Journal of Applied Econometrics, 8, 71–80.CrossRef

Hansen, G., & İmrohoroğlu, S. (2008). Consumption over the life cycle: The role of annuities. Review of Economic Dynamics, 11(3), 566–583.CrossRef

Heer, B. (2018). Optimal pensions in aging economies. B.E. Journal of Macroeconomics, 18, 1–19.

Heer, B., & Irmen, A. (2014). Population, pensions and endogenous economic growth. Journal of Economic Dynamics and Control, 46, 50–72.CrossRef

Heer, B., & Maußner, A. (2009). Dynamic general equilibrium modeling: Computational methods and applications (2nd ed.). Heidelberg: Springer.CrossRef

Heer, B., Polito, V., & Wickens, M. R. (2017). Population aging, social security and fiscal limits (CEPR Discussion Paper Series, DP11978).

Hubbard, R. G., & Judd, K. L. (1987). Social security and individual welfare: Precautionary saving, borrowing constraints, and the payroll tax. American Economic Review, 77, 630–646.

Huggett, M. (1996). Wealth distribution in life-cycle economies. Journal of Monetary Economics, 38, 373–396.CrossRef

Hurd, M. D. (1989). Mortality risk and bequests. Econometrica, 57, 799–813.CrossRef

İmrohoroğlu, S., & Kitao, S. (2009). Labor supply elasticity and social security reform. Journal of Public Economics, 93, 867–878.CrossRef

İmrohoroğlu, A., İmrohoroğlu, S., & Joines, D. H. (1995). A life cycle analysis of social security. Economic Theory, 6, 83–114.CrossRef

İmrohoroğlu, A., İmrohoroğlu, S., & Joines, D. H. (1999). Social security in an overlapping generations economy with land. Review of Economic Dynamics, 2, 638–665.CrossRef

İmrohoroğlu, A., İmrohoroğlu, S., & Joines, D. H. (2003). Time-inconsistent preferences and social security. Quarterly Journal of Economics, 118, 745–784.CrossRef

Judd, K. L. (1998). Numerical methods in economics. Cambridge: MIT Press.

Kitao, S. (2014). Sustainable social security: Four options. Review of Economic Dynamics, 17, 756–779.CrossRef

Krueger, D., & Ludwig, A. (2007). On the consequences of demographic change for rates of returns to capital, and the distribution of wealth and welfare. Journal of Monetary Economics, 54, 49–87.CrossRef

Krueger, D., & Perri, F. (2006). Does income inequality lead to consumption inequality? Evidence and theory. Review of Economic Studies, 73, 163–193.CrossRef

Laibson, D. (1997). Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, 112, 443–477.CrossRef

Mendoza, E. G., Razin, A., & Tesar, L. L. (1994). Effective tax rates in macroeconomics: Cross country estimates of tax rates on factor incomes and consumption. Journal of Monetary Economics, 34, 297–323.CrossRef

Nishiyama, S., & Smetters, K. (2007). Does social security privitization produce efficiency gains? Quarterly Journal of Economics, 122, 1677–1719.CrossRef

OECD. (2015). Pensions at a Glance 2015: OECD and G20 indicators. Paris: OECD Publishing.

OECD. (2017). Pensions at a Glance 2017: OECD and G20 indicators. Paris: OECD Publishing.

Quadrini, V. (2000). Entrepreneurship, saving, and social mobility. Review of Economic Dynamics, 3, 1–40.CrossRef

Raftery, A. E., Li, N., Sevčíková, H., Gerland, P., & Heilig, G. K. (2012). Bayesian probabilitic population projections for all countries. Proceedings of the National Academy of Sciences, 109, 13915–13921.CrossRef

Raftery, A. E., Chunn, J., Gerland, P., & Sevčíková, H. (2013). Bayesian probabilitic projections of life expectancy for all countries. Demography, 50, 777–801.CrossRef

Stokey, N., Lucas, R. E., & Prescott, E. C. (1989). Recursive methods in economic dynamics. Cambridge: Harvard University Press.

Storesletten, K., Telmer, C. I., & Yaron, A. (1999). The risk-sharing implications of alternative social security arrangements. Carnegie-Rochester Series on Public Policy, 50, 213–259.CrossRef

Storesletten, K., Telmer, C. I., & Yaron, A. (2004). Consumption and risk sharing over the business cycle. Journal of Monetary Economics, 51, 609–633.CrossRef

UN (2015). World population prospects: The 2015 revision, Methodology of the United Nations population estimates and projections (ESA/P/WP. 242).

Title: Pensions
Author: Burkhard Heer
Publisher: Springer International Publishing
Book: Public Economics
Print ISBN: 978-3-030-00987-8

Electronic ISBN: 978-3-030-00989-2

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-00989-2_6