A mathematical model for social security systems with dynamical systems

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Gilberto González Parra
Abraham J Arenas https://orcid.org/0000-0003-3106-1271


Retirement model, Mathematical modeling, Social Security, Dynamical system


In this paper it is proposed a mathematical approach based on dynamic systems to study the effect of the increase in the Social Security normal retirement age on the worker and on the dynamics of retiree populations. In order to simplify this initial effort, the proposed model does not include some economic variables, such as wage growth, earnings or productivity. Here, we employ numerical simulations of the model to investigate the dynamics of the labor force under different demographic scenarios. Analysis of this type of model with numerical simulations can help government economic planners make optimal strategies to sustain pension systems and forecast future trends of pensioner and worker populations.

 MSC:  97M10, 97M40.


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[1] A. Okamoto, “Simulating progressive expenditure taxation in an agingJapan,” J. Policy Model., vol. 27, no. 3, pp. 309–325, 2005. 34, 43

[2] A. Borsch-Supan, “A model under siege: A case study of the German retirementinsurance system,” Econ. J., vol. 110, no. 461, pp. 24–45, 2000. 34, 35,42

[3] ——, “Incentive effects of social security on labor force participation: evidencein Germany and across Europe,” J. Public Econ., vol. 78, no. 1-2, pp.25–49, 2000. 34, 35, 42

[4] F. El-Hamidi, “Early retirement in the government sector in Egypt: Preferences,determinants, and policy implications,” J. Dev. Areas, vol. 43, no. 2,pp. 79–110, 2010. 35

[5] M. M. Hamid Faruqee, “Population aging in Japan: demographic shock andfiscal sustainability,” JPN. World Econ., vol. 15, no. 2, pp. 185–210, 2003.35, 42

[6] D. Roseveare, W. Leibfritz, D. Fore, and E. Wurzel, “Ageing populations,pensions systems and government budgets: simulations for 20 OECD countries,”OECD Economics Department, Working Paper, vol. 168, 1996. 35

[7] J. F. Jimeno, J. A. Rojas, and S. Puente, “Modelling the impact of agingon social security expenditures,” Econ. Model., vol. 25, no. 2, pp. 201–224,2008. 35, 41, 43, 44

[8] A. Borsch-Supan, A. Heiss, F. Ludwig, and A. Winter, “Pension reform,capital, markets and rate of return,” University of Mannheim, manuscript.35, 41, 44

[9] H. Hock and D. N. Weil, “On the dynamics of the age structure, dependency,and consumption,” J. Popul. Econ., vol. 25, no. 3, pp. 1019–1043, 2012. 35,38

[10] M. Toossi, “Labor force projections to 2020: a more slowly growing workforce,”Monthly Lab. Rev., vol. 135, pp. 43–64, 2012. 36

[11] J. D. Murray, Mathematical Biology I. An Introduction. Springer, Berlin,2002. 37

[12] P. Manfredi and L. Fanti, “Demography in macroeconomic models: whenlabour supply matters for economic cycles,” Metroeconomica, vol. 57, no. 4,pp. 536–563, 2006. 37

[13] G. Feichtinger and G. Sorger, “Self-generated fertility waves in a non-linearcontinuous overlapping generations model,” J. Popul. Econ., vol. 2, no. 4,pp. 267–280, 1989. 37

[14] D. F. Aranda, R. J. Villanueva, A. J. Arenas, and G. C. G. Parra, “Mathematicalmodeling of toxoplasmosis disease in varying size populations,” Comput.Math. Appl., vol. 56, no. 3, pp. 690–696, 2008. 39

[15] SSA, “Social Security. The Official Website of the U.S. Social Security Administration,”Available from: http://www.ssa.gov. 42, 43

[16] OECD, “Ageing and employment policies, Synthesis Report, OECD, Paris(2005),” Available from:http://www.oecdbookshop.org, 2005. 42, 43

[17] INE, “Instituto Nacional de Estadística,” Available from:http://www.ocei.gov.ve/, 2010. 42

[18] D. E. Bloom, D. Canning, R. K. Mansfield, and M. Moore, “Demographicchange, social security systems, and savings,” J. Monetary. Econ., vol. 54,no. 1, pp. 92–114, 2007. 42

[19] A. Ang and A. Maddaloni, Do Demographic changes Affect Risk Premiums?Evidence From International Data. Cambridge,MA: National Bureau ofEconomic Research Working Paper, 2003, vol. 9677. 42

[20] D. E. Bloom, D. Canning, R. K. Mansfield, and M. Moore, “Demographicchange, social security systems, and savings,” J. Monetary. Econ., vol. 54,no. 1, pp. 92–114, 2007. 42

[21] M. Bouzahzah, D. D. la Croix, and F. Docquier, “Policy reforms and growthin computable OLG economies,” J. Econ. Dyn. Control, vol. 26, no. 12, pp.2093–2113, 2002. 42

[22] H. S. Buscher, H. Buslei, K. Goggelmann, H. Koschel, T. F. N. Schmidt,V. Steiner, and P. Winker, “Empirical macro models under test. A comparativesimulation study of the employment effects of a revenue neutral cutin social security contributions,” Econ. Model., vol. 18, no. 3, pp. 455–474,2001. 43

[23] R. Beetsma, L. Bettendorf, and P. Broer, “The budgeting and economicconsequences of ageing in the Netherlands,” Econ. Model., vol. 20, no. 5, pp.987–1013, 2003. 43

[24] L. J. Bettendorf and B. J. Heijdra, “Population ageing and pension reformin a small open economy with non-traded goods,” J. Econ. Dyn. Control,vol. 30, no. 12, pp. 2389–2424, 2006. 43

[25] V. Canudas-Romo, “Three measures of longevity: Time trends and recordvalues,” Demography, vol. 47, pp. 299–312, 2010, referenced in. 46

[26] S. J. Olshansky, D. J. Passaro, R. C. Hershow, J. Layden, B. A. Carnes,J. Brody, L. Hayflick, R. N. Butler, D. B. Allison, and D. S. Ludwig, “Apotential decline in life expectancy in the United States in the 21st century,”N. Engl. J. Med., vol. 352, no. 11, pp. 1138–1145, 2005. 46

[27] J. C. Mico, D. Soler, , and A. Caselles, “Age-structured human populationdynamics,” J. Math. Sociol., vol. 30, no. 1, pp. 1–31, 2006. 47

[28] N. Keyfitz, Applied Mathematical Demography. Springer, 2005. 47

[29] P. Hensen, “The "Bologna process" in European higher education: Impact ofbachelor’s and master’s degrees on german medical education,” Teach. Learn.Med., vol. 22, no. 2, pp. 142–147, 2010. 48