IDEAS home Printed from https://ideas.repec.org/a/eee/thpobi/v155y2024icp51-66.html
   My bibliography  Save this article

The optimal momentum of population growth and decline

Author

Listed:
  • Feichtinger, Gustav
  • Wrzaczek, Stefan

Abstract

About 50 years ago, Keyfitz (1971) asked how much further a growing human population would increase if its fertility rate were immediately to be reduced to replacement level and remain there forever. The reason for demographic momentum is an age–structure inertia due to relatively many potential parents because of past high fertility. Although nobody expects such a miraculous reduction in reproductive behavior, a gradual decline in fertility in rapidly growing populations seems inevitable. As any delay in fertility decline to a stationary level leads to an increase in the momentum, it makes sense to think about the timing and the quantum of the reduction in reproduction. More specifically, we consider an intertemporal trade-off between costly pro- and anti-natalistic measures and the demographic momentum at the end of the planning period. This paper uses the McKendrick–von Foerster partial differential equation of age–structured population dynamics to study a sketched problem in a distributed parameter control framework. Among the results obtained by applying an appropriate extension of Pontryagin’s Maximum Principle are the following: (i) monotony of adaptation efforts to net reproduction rate and convex decrease/concave increase (if initial net reproduction rate exceeds 1/is below 1); and (ii) oscillating efforts and reproduction rate if, additionally, the size of the total population does not deviate from a fixed level.

Suggested Citation

  • Feichtinger, Gustav & Wrzaczek, Stefan, 2024. "The optimal momentum of population growth and decline," Theoretical Population Biology, Elsevier, vol. 155(C), pages 51-66.
    • Handle: RePEc:eee:thpobi:v:155:y:2024:i:c:p:51-66
    DOI: 10.1016/j.tpb.2023.12.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0040580923000813
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.tpb.2023.12.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Janna Bergsvik & Agnes Fauske & Rannveig Kaldager Hart, 2021. "Can Policies Stall the Fertility Fall? A Systematic Review of the (Quasi‐) Experimental Literature," Population and Development Review, The Population Council, Inc., vol. 47(4), pages 913-964, December.
    2. Stefan Wrzaczek & Ekaterina Shevkoplyas & Sergey Kostyunin, 2014. "A Differential Game Of Pollution Control With Overlapping Generations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-14.
    3. W. Brian Arthur & Geoffrey McNicoll, 1977. "Optimal Time Paths with Age-Dependence: A Theory of Population Policy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 44(1), pages 111-123.
    4. Kuhn, Michael & Wrzaczek, Stefan & Oeppen, Jim, 2010. "Recognizing progeny in the value of life," Economics Letters, Elsevier, vol. 107(1), pages 17-21, April.
    5. Herbert Dawid & Joshua R. Goldstein & Vladimir Veliov & Gustav Feichtinger, 2009. "Keeping a learned society young," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 20(22), pages 541-558.
    6. Joshua Goldstein, 2002. "Population momentum for gradual demographic transitions: an alternative approach," Demography, Springer;Population Association of America (PAA), vol. 39(1), pages 65-73, February.
    7. Nathan Keyfitz, 1971. "On the momentum of population growth," Demography, Springer;Population Association of America (PAA), vol. 8(1), pages 71-80, February.
    8. Frankovic, Ivan & Kuhn, Michael & Wrzaczek, Stefan, 2020. "Medical innovation and its diffusion: Implications for economic performance and welfare," Journal of Macroeconomics, Elsevier, vol. 66(C).
    9. Gustav Feichtinger & Alexia Prskawetz & Andrea Seidl & Christa Simon & Stefan Wrzaczek, 2017. "A bifurcation analysis of gender equality and fertility," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 1221-1243, November.
    10. Stefan Wrzaczek & Michael Kuhn & Ivan Frankovic, 2020. "Using Age Structure for a Multi-stage Optimal Control Model with Random Switching Time," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 1065-1082, March.
    11. Ivan Frankovic & Michael Kuhn & Stefan Wrzaczek, 2020. "On the Anatomy of Medical Progress Within an Overlapping Generations Economy," De Economist, Springer, vol. 168(2), pages 215-257, June.
    12. Nan Li & Shripad Tuljapurkar, 2000. "The solution of time-dependent population models," Mathematical Population Studies, Taylor & Francis Journals, vol. 7(4), pages 311-329.
    13. Kuhn, Michael & Wrzaczek, Stefan & Prskawetz, Alexia & Feichtinger, Gustav, 2011. "Externalities in a life cycle model with endogenous survival," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 627-641.
    14. Wrzaczek, Stefan & Kuhn, Michael & Prskawetz, Alexia & Feichtinger, Gustav, 2010. "The reproductive value in distributed optimal control models," Theoretical Population Biology, Elsevier, vol. 77(3), pages 164-170.
    15. Gustav Feichtinger & Alexia Prskawetz & Stefan Wrzaczek & Michael Kuhn, 2011. "The reproductive value as part of the shadow price of population," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 24(28), pages 709-718.
    16. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2006. "Anticipation effects of technological progress on capital accumulation: a vintage capital approach," Journal of Economic Theory, Elsevier, vol. 126(1), pages 143-164, January.
    17. Anne Gauthier, 2007. "The impact of family policies on fertility in industrialized countries: a review of the literature," Population Research and Policy Review, Springer;Southern Demographic Association (SDA), vol. 26(3), pages 323-346, June.
    18. Arthur J. Robson & Hillard S. Kaplan, 2007. "Why do We Die? Economics, Biology and Aging," American Economic Review, American Economic Association, vol. 97(2), pages 492-495, May.
    19. Chu, C.Y. Cyrus & Chien, Hung-Ken & Lee, Ronald D., 2008. "Explaining the optimality of U-shaped age-specific mortality," Theoretical Population Biology, Elsevier, vol. 73(2), pages 171-180.
    20. Prskawetz, Alexia & Veliov, Vladimir M., 2007. "Age-specific dynamic labor demand and human capital investment," Journal of Economic Dynamics and Control, Elsevier, vol. 31(12), pages 3741-3777, December.
    21. Arthur J. Robson & Hillard S. Kaplan, 2003. "The Evolution of Human Life Expectancy and Intelligence in Hunter-Gatherer Economies," American Economic Review, American Economic Association, vol. 93(1), pages 150-169, March.
    22. Stefan Wrzaczek, 2021. "An OLG Differential Game of Pollution Control with the Risk of a Catastrophic Climate Change," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 23(04), pages 1-26, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Gustav Feichtinger & Stefan Wrzaczek, 2024. "The optimal transition to a stationary population for concentrated vitality rates," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 50(6), pages 171-184.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stefan Wrzaczek & Michael Kuhn & Ivan Frankovic, 2020. "Using Age Structure for a Multi-stage Optimal Control Model with Random Switching Time," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 1065-1082, March.
    2. Kuhn, Michael & Frankovic, Ivan & Wrzaczek, Stefan, 2017. "Medical Progress, Demand for Health Care, and Economic Performance," VfS Annual Conference 2017 (Vienna): Alternative Structures for Money and Banking 168249, Verein für Socialpolitik / German Economic Association.
    3. Robert Schoen, 2005. "Intrinsically dynamic population models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 12(3), pages 51-76.
    4. Frankovic, Ivan & Kuhn, Michael & Wrzaczek, Stefan, 2016. "Medical care within an OLG economy with realistic demography," ECON WPS - Working Papers in Economic Theory and Policy 02/2016, TU Wien, Institute of Statistics and Mathematical Methods in Economics, Economics Research Unit.
    5. Thomas Espenshade & Analia Olgiati & Simon Levin, 2011. "On Nonstable and Stable Population Momentum," Demography, Springer;Population Association of America (PAA), vol. 48(4), pages 1581-1599, November.
    6. Wrzaczek, Stefan & Kuhn, Michael & Prskawetz, Alexia & Feichtinger, Gustav, 2010. "The reproductive value in distributed optimal control models," Theoretical Population Biology, Elsevier, vol. 77(3), pages 164-170.
    7. Stelter, Robert, 2014. "Over-aging: Are present human populations too old?," Thuenen-Series of Applied Economic Theory 137, University of Rostock, Institute of Economics.
    8. Cyrus Chu, C.Y. & Chien, Hung-Ken & Lee, Ronald D., 2010. "The evolutionary theory of time preferences and intergenerational transfers," Journal of Economic Behavior & Organization, Elsevier, vol. 76(3), pages 451-464, December.
    9. Kredler, Matthias, 2014. "Vintage human capital and learning curves," Journal of Economic Dynamics and Control, Elsevier, vol. 40(C), pages 154-178.
    10. Kuhn, Michael & Wrzaczek, Stefan & Prskawetz, Alexia & Feichtinger, Gustav, 2011. "Externalities in a life cycle model with endogenous survival," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 627-641.
    11. Klaus Prettner & Alexia Prskawetz, 2010. "Demographic change in models of endogenous economic growth. A survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(4), pages 593-608, December.
    12. David N. Koons & Randall Holmes & James B. Grand, 2006. "Population inertia and its sensitivity to changes in vital rates or initial conditions," MPIDR Working Papers WP-2006-040, Max Planck Institute for Demographic Research, Rostock, Germany.
    13. Junji Kageyama & Kazuma Sato, 2021. "Explaining the U-shaped life satisfaction: dissatisfaction as a driver of behavior," Journal of Bioeconomics, Springer, vol. 23(2), pages 179-202, July.
    14. B. Skritek & V. M. Veliov, 2015. "On the Infinite-Horizon Optimal Control of Age-Structured Systems," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 243-271, October.
    15. David E. Bloom & Michael Kuhn & Klaus Prettner, 2022. "Modern Infectious Diseases: Macroeconomic Impacts and Policy Responses," Journal of Economic Literature, American Economic Association, vol. 60(1), pages 85-131, March.
    16. Kuhn, Michael & Wrzaczek, Stefan & Prskawetz, Alexia & Feichtinger, Gustav, 2015. "Optimal choice of health and retirement in a life-cycle model," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 186-212.
    17. Junji Kageyama, 2011. "The intertemporal allocation of consumption, time preference, and life-history strategies," Journal of Bioeconomics, Springer, vol. 13(2), pages 79-95, July.
    18. Grafeneder-Weissteiner, Theresa & Prettner, Klaus, 2010. "Agglomeration processes in aging societies," Department of Economics Working Paper Series 131, WU Vienna University of Economics and Business.
    19. Michal Engelman & Christopher L. Seplaki & Ravi Varadhan, 2017. "A Quiescent Phase in Human Mortality? Exploring the Ages of Least Vulnerability," Demography, Springer;Population Association of America (PAA), vol. 54(3), pages 1097-1118, June.
    20. Feichtinger, Gustav & Lambertini, Luca & Leitmann, George & Wrzaczek, Stefan, 2022. "Managing the tragedy of commons and polluting emissions: A unified view," European Journal of Operational Research, Elsevier, vol. 303(1), pages 487-499.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:thpobi:v:155:y:2024:i:c:p:51-66. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.sciencedirect.com/journal/theoretical-population-biology .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.