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The concavity of the value function of the extended Barro-Becker model

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  • Qi, Ling
  • Kanaya, Sadao

Abstract

In this paper, the model of endogenous fertility proposed by Becker and Barro (1988), and extended by Benhabib and Nishimura (1989) is considered. However, in their model, the uniqueness of the optimal path may fail to hold. Furthermore, the value function may not be concave, because of the variable discount factor with respect to the choice made about fertility. In this paper, we show that under a set of mild conditions, based on the assumption that the cost of raising a child is non-constant, there exists a unique optimal path and the value function is concave and continuously differentiable. We also show the existence of the unique steady state and a monotonically optimal path, and confirm that the steady state is saddle point stable.

Suggested Citation

  • Qi, Ling & Kanaya, Sadao, 2010. "The concavity of the value function of the extended Barro-Becker model," Journal of Economic Dynamics and Control, Elsevier, vol. 34(3), pages 314-329, March.
    • Handle: RePEc:eee:dyncon:v:34:y:2010:i:3:p:314-329
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    References listed on IDEAS

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    1. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    2. Gary S. Becker & Casey B. Mulligan, 1997. "The Endogenous Determination of Time Preference," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 112(3), pages 729-758.
    3. Barro, Robert J & Becker, Gary S, 1989. "Fertility Choice in a Model of Economic Growth," Econometrica, Econometric Society, vol. 57(2), pages 481-501, March.
    4. Gary S. Becker & Robert J. Barro, 1988. "A Reformulation of the Economic Theory of Fertility," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 103(1), pages 1-25.
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    Cited by:

    1. Pérez-Nievas, Mikel & Conde-Ruiz, José I. & Giménez, Eduardo L., 2019. "Efficiency and endogenous fertility," Theoretical Economics, Econometric Society, vol. 14(2), May.
    2. Córdoba, Juan Carlos & Liu, Xiying & Ripoll, Marla, 2016. "Fertility, social mobility and long run inequality," Journal of Monetary Economics, Elsevier, vol. 77(C), pages 103-124.
    3. Inyong Shin, 2016. "Change and prediction of income and fertility rates across countries," Cogent Economics & Finance, Taylor & Francis Journals, vol. 4(1), pages 1119367-111, December.
    4. Liu, Xiying, 2015. "Optimal population and policy implications," ISU General Staff Papers 201501010800005546, Iowa State University, Department of Economics.
    5. Córdoba, Juan Carlos & Liu, Xiying & Ripoll, Marla, 2016. "Fertility, social mobility and long run inequality," Journal of Monetary Economics, Elsevier, vol. 77(C), pages 103-124.
    6. Shin, Inyong, 2013. "The Effect of Compressed Demographic Transition and Demographic Gift on Economic Growth," MPRA Paper 45003, University Library of Munich, Germany.

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