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Endogenous fertility in a growth model with public and private health expenditures

Abstract

We build an overlapping-generations model that incorporates endogenous fertility choices, in addition to public and private expenditures on health. Following the seminal analysis of Bhattacharya and Qiao (J Econ Dyn Control 31:2519–2535, 2007) we assume that the effect of public health investment is complementary to private health expenditures. We find that this effect reinforces the positive impact of the capital stock on aggregate saving. Furthermore, we show that this complementarity can provide an additional explanation behind a salient feature of demographic transition; that is, the fertility decline along the process of economic growth.

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Notes

  1. For empirical support on the negative relation between improved health status and fertility rates, see Finlay (2007).

  2. This particular extension is not critical for the subsequent results. These remain similar even if individuals care only about old-age consumption.

  3. Of course, the intuition differs as it will transpire in Section 5 where we provide the formal analysis.

  4. For a similar assumption on health expenses being incurred during old adulthood, see Gutiérrez (2008).

  5. We assume that \(\delta\bar{\varepsilon} < 1\) in order to ensure the concavity of h t + 1 with respect to x t + 1. A functional form that satisfies all these conditions is \(Z(p_{t+1}) = 1 + \frac{\varphi p_{t+1}}{1 + p_{t+1}}\) with \(\varphi = \bar{\varepsilon} - 1\).

  6. The idea that public health spending contributes to some type of capital formation is intuitive once we think of spending on hospitals, medical equipment, support for medical research and training etc.

  7. We can think of many examples that justify this assumption. The presence of qualified professionals—in the national health system—that offer support and advice on various difficulties that may emerge while people are trying to quit smoking (e.g., cravings etc.) may provide an incentive for smokers to seek and buy treatments that support Nicotine Replacement Therapy (patches, gums etc.). Clinical depression can be combated more effectively if sufferers combine antidepressant medication with appropriate counselling by qualified psychiatrists—counselling that is sometimes offered by professionals employed in the national health system. See Bhattacharya and Qiao (2007) for further examples in support of this conjecture.

  8. Depending on different parameter values, n t can become less than one in the steady state. However, this possibility can be ruled out by an appropriate parameter restriction—particularly, if the parameter q is sufficiently low. Alternatively, rather than normalising the total endowed time to unity, we could have endowed young individuals with a larger time endowment—so large, to guarantee that fertility is always above one. Qualitatively, none of our results would be affected.

References

  • Becker GS, Murphy KM, Tamura R (1990) Human capital, fertility, and economic growth. J Polit Econ 98(5):12–37

    Article  Google Scholar 

  • Bhattacharya J, Qiao X (2007) Public and private expenditures on health in a growth model. J Econ Dyn Control 31(8):2519–2535

    Article  Google Scholar 

  • Blackburn K, Cipriani (2002) A model of longevity, fertility and growth. J Econ Dyn Control 26(2):187–204

    Article  Google Scholar 

  • Cigno A (1998) Fertility decisions when infant survival is endogenous. J Popul Econ 11(1):21–28

    Article  Google Scholar 

  • de la Croix D, Doepke M (2003) Inequality and growth: why differential fertility matters? Am Econ Rev 93(4):1091–1113

    Article  Google Scholar 

  • Ehrlich I, Lui F (1997) The problem of population and growth: a review of the literature from Malthus to contemporary models of endogenous population and endogenous growth. J Econ Dyn Control 21(1):205–242

    Article  Google Scholar 

  • Finlay J (2007) The role of health in economic development. PGDA working paper no. 21, Harvard University

  • Frankel M (1962) The production function in allocation and growth: a synthesis. Am Econ Rev 52(5):996–1022

    Google Scholar 

  • Galor O (2012) The demographic transition: causes and consequences. Cliometrica 6(1):1–28

    Article  Google Scholar 

  • Galor O, Weil D (1996) The gender gap, fertility, and growth. Am Econ Rev 86(3):374–387

    Google Scholar 

  • Galor O, Weil D (2000) Population, technology, and growth: from the Malthusian regime to the demographic transition and beyond. Am Econ Rev 90(4):806–828

    Article  Google Scholar 

  • Gutiérrez M (2008) Dynamic inefficiency in an overlapping generation economy with pollution and health costs. J Public Econ Theory 10(4):563–594

    Article  Google Scholar 

  • Kalemli-Ozcan S (2003) A stochastic model of mortality, fertility, and human capital investment. J Dev Econ 70(1):103–118

    Article  Google Scholar 

  • Kirk D (1996) Demographic transition theory. Popul Stud 50(3):361–387

    Article  Google Scholar 

  • Manuelli RE, Seshadri A (2009) Explaining international fertility differences. Q J Econ 124(2):771–807

    Article  Google Scholar 

  • Pitt MM, Rosenzweig MR, Nazmul Hassan M (1990) Productivity, health, and inequality in the intrahousehold distribution of food in low-income countries. Am Econ Rev 80(5):1139–1156

    Google Scholar 

  • Romer P (1986) Increasing returns and long-run growth. J Polit Econ 94(5):1002–1037

    Article  Google Scholar 

  • Soares RR (2005) Mortality reductions, educational attainment, and fertility choice. Am Econ Rev 95(3):580–601

    Article  Google Scholar 

  • Strulik H (2008) Geography, health, and the pace of demo-economic development. J Dev Econ 86(1):61–75

    Article  Google Scholar 

  • Tamura R (1996) From decay to growth: a demographic transition to economic growth. J Econ Dyn Control 20(6–7):1237–1261

    Article  Google Scholar 

  • World Health Organisation (2010) World Health Statistics

  • Zhang J, Zhang J (2005) The effect of life expectancy on fertility, saving, schooling and economic growth: theory and evidence. Scand J Econ 107(1):45–66

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to two anonymous referees and an Editor of this journal for numerous comments and suggestions that have significantly improved the content and substance of our paper. Any remaining errors or omissions are solely our responsibility. Intan Zanariah Zakaria acknowledges financial support from the Malaysian Ministry of Higher Education and the International Isamic University of Malaysia. The usual disclaimer applies.

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Correspondence to Dimitrios Varvarigos.

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Appendix

Appendix

Proof to Proposition 1

Using Eq. 18, we can define

$$ J\left( {qn_t ,k_t } \right)=qn_t -\frac{\gamma }{1+\beta \left[ {1+\delta Z\left( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn_t } \right)^{1-\alpha }} \right)} \right]+\gamma }. $$
(36)

We can also use Eqs. 6, 10 and 15 to write

$$ \tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn_t } \right)^{1-\alpha }=p_{t+1} . $$
(37)

From Eq. 36 we have

$$ J_{k_t}{\kern1pt} ({\cdot ,\cdot}) {\kern-1pt}={\kern-1pt}\frac{\gamma \beta \delta {Z}' {\kern-1pt}\left( {\tau {\kern-1pt}\left( {1{\kern-1pt}-{\kern-1pt}\alpha } \right){\kern-1pt} Ak_t^\alpha {\kern-1pt}\left( {1-qn_t } \right)^{1-\alpha }} \right){\kern-1pt}\tau{\kern-1pt}\left( {1{\kern-1pt}-{\kern-1pt}\alpha } \right)A\alpha k_t^{\alpha -1} \left( {1{\kern-1pt}-{\kern-1pt}qn_t } \right)^{1-\alpha }}{\left\{ {1{\kern-1pt}+{\kern-1pt}\beta \left[ {1{\kern-1pt}+{\kern-1pt}\delta Z\left( {\tau \left( {1{\kern-1pt}-{\kern-1pt}\alpha } \right)Ak_t^\alpha \left( {1{\kern-1pt}-{\kern-1pt}qn_t } \right)^{1-\alpha }} \right)} \right]{\kern-1pt}+{\kern-1pt}\gamma } \right\}^2}>0. $$
(38)

Alternatively, we can substitute Eq. 37 in Eq. 38 to write

$$ J_{k_t } \left( {\cdot ,\cdot } \right)=\frac{\gamma \beta \delta \alpha {Z}'\left( {p_{t+1} } \right)p_{t+1} }{\left\{ {1+\beta \left[ {1+\delta Z\left( {p_{t+1} } \right)} \right]+\gamma } \right\}^2k_t }>0. $$
(39)

From Eq. 36, we can also derive

$$ J_{qn_t } \left( {\cdot ,\cdot } \right)=1-\frac{\gamma \beta \delta {Z}'\left( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn_t } \right)^{1-\alpha }} \right)\tau \left( {1-\alpha } \right)^2Ak_t^\alpha \left( {1-qn_t } \right)^{-\alpha }}{\left\{ {1+\beta \left[ {1+\delta Z\left( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn_t } \right)^{1-\alpha }} \right)} \right]+\gamma } \right\}^2}, $$
(40)

to which we can substitute Eq. 37 to get

$$ J_{qn_t } \left( {\cdot ,\cdot } \right)=1-\frac{\gamma \beta \delta \left( {1-\alpha } \right){Z}'\left( {p_{t+1} } \right)p_{t+1} }{\left\{ {1+\beta \left[ {1+\delta Z\left( {p_{t+1} } \right)} \right]+\gamma } \right\}^2\left( {1-qn_t } \right)}. $$
(41)

Substituting Eqs. 18 and 37 in Eq. 41, we can write the latter as

$$ J_{qn_t } \left( {\cdot ,\cdot } \right)=1-\frac{\gamma }{1+\beta \left[ {1+\delta Z\left( {p_{t+1} } \right)} \right]+\gamma }\frac{\left( {1-\alpha } \right)\beta \delta {Z}'\left( {p_{t+1} } \right)p_{t+1} }{1+\beta \left[ {1+\delta Z\left( {p_{t+1} } \right)} \right]}>0, $$
(42)

which is positive because \({Z}^{\prime}\left( {p_{t+1} } \right)p_{t+1} <Z\left( {p_{t+1} } \right)\) holds. Now, we can combine the results in Eqs. 42 and 38, and apply the implicit function theorem to Eq. 36. This yields

$$ \frac{dqn_t }{dk_t }=-\frac{J_{k_t } \left( {\cdot ,\cdot } \right)}{J_{qn_t } \left( {\cdot ,\cdot } \right)}<0, $$
(43)

Therefore, given that q is a fixed parameter, we can conclude that \(n_t =n\left( {k_t } \right)\) such that \({n}'\left( {k_t } \right)<0\). Finally, we can use the previous analysis to write Eq. 17 as

$$ \begin{array}{rll} s_t &=& \frac{\beta \big[ {1+\delta Z\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}{1+\beta \big[ {1+\delta Z\big ( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}\left( {1-\tau } \right)\left( {1-\alpha } \right) \\ && \times Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }, \end{array} $$

from which it is straightforward to establish that ds t /dk t  > 0. □

Proof to Proposition 2

From Eqs. 18 and 20, we can establish that \(n\left( 0 \right)=\frac{\gamma /q}{1+\beta \left( {1+\delta } \right)+\gamma }\) and \(n\left( \infty \right)=\frac{\gamma /q}{1+\beta \left( {1+\delta \bar{{\varepsilon }}} \right)+\gamma }\). Combining these with Eq. 22, we see that \(\psi \left( 0 \right)=0\) and \(\psi \left( \infty \right)=\infty \). Furthermore, it is

$$ \begin{array}{rll} {\psi }'(k_t ) &=& \eta \left\langle {\frac{\beta \big[ {1+\delta Z\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}{1+\beta \big[{1+\delta Z\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}} \right. \\ &&{\kern13.5pt} \times \frac{\theta \left( {k_t } \right)}{n\left( {k_t } \right)} +\, \frac{k_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }}{n\left( {k_t } \right)} \\ &&{\kern13.5pt} \times \frac{\beta \delta {Z}'\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)\tau \left( {1-\alpha } \right)A\theta \left( {k_t } \right)}{\big\{1+\beta \big[1+\delta Z\big( {\tau (1-\alpha )Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)\big]\big\}^2} \\ &&{\kern13.5pt} - \left. \frac{\beta \big[1+\delta Z\left( {\tau (1-\alpha )Ak_t^\alpha (1-qn(k_t ))^{1-\alpha }} \right)\big]}{1+\beta \big[1+\delta Z\left( {\tau (1-\alpha )Ak_t^\alpha (1-qn(k_t ))^{1-\alpha }} \right)\big]} \right. \\ &&{\kern13.5pt} \left. \times \frac{k_t^\alpha (1-qn(k_t ))^{1-\alpha }}{[n(k_t )]^2}{n}'(k_t ) \vphantom{{\frac{\beta \big[ {1+\delta Z\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}{1+\beta \big[{1+\delta Z\big( {\tau \left( {1-\alpha } \right)Ak_t^\alpha \left( {1-qn\left( {k_t } \right)} \right)^{1-\alpha }} \big)} \big]}}}\right\rangle , \end{array} $$
(44)

where

$$\theta (k_t )=\alpha k_t^{\alpha -1} (1-qn(k_t ))^{1-\alpha }-k_t^\alpha (1-\alpha )q{n}'(k_t )(1-qn(k_t ))^{-\alpha }. $$
(45)

Clearly, ψ′(k t ) > 0. Now, combine Eqs. 39 and 42 to write Eq. 43 as

$$ \begin{array}{rll} \frac{dqn_t }{dk_t } &{\kern-1pt}={\kern-1pt}& -\frac{\gamma \beta \delta \alpha {Z}'\left( {p_{t+1} } \right)p_{t+1} \{1+\beta [1+\delta Z\left( {p_{t+1} } \right)]\}}{\{1{\kern-1pt}+{\kern-1pt}\beta [1{\kern-1pt}+{\kern-1pt}\delta Z\left( {p_{t+1} } \right)]{\kern-1pt}+{\kern-1pt}\gamma \}\{1{\kern-1pt}+{\kern-1pt}\beta [1{\kern-1pt}+{\kern-1pt}\delta Z\left( {p_{t+1} } \right)]\}{\kern-1pt}-{\kern-1pt}\gamma \beta \delta (1{\kern-1pt}-{\kern-1pt}\alpha ){Z}'\left( {p_{t+1} } \right)p_{t+1} } \\ && \times \frac{1}{k_t \{1+\beta [1+\delta Z\left( {p_{t+1} } \right)]+\gamma \}}. \\ \end{array} $$

Given Z′(0) = ϕ and Z′( ∞ ) = 0, we can combine the expression above together with Eq. 37, \(n(0)=\frac{\gamma /q}{1+\beta (1+\delta )+\gamma }\) and \(n(\infty )=\frac{\gamma /q}{1+\beta (1+\delta \bar{{\varepsilon }})+\gamma }\) to establish that

$$ {n}'\left( {k_t } \right)=\left\{ {{\begin{array}{*{20}c} {-\infty } & {\mbox{for}} & {k_t =0} \\ 0 & {\mbox{for}} & {k_t \to \infty } \\ \end{array} }} \right.. $$
(46)

Combining Eqs. 45 and 46 with Eq. 44, we infer that

$$ {\psi }'\left( {k_t } \right)=\left\{ {{\begin{array}{*{20}c} \infty & {\mbox{for}} & {k_t =0} \\ 0 & {\mbox{for}} & {k_t \to \infty } \\ \end{array} }.} \right. $$
(47)

Thus, we conclude that there must be at least one \(\hat{{k}}\in \left( {0,\infty } \right)\) such that \(\hat{{k}}=\psi (\hat{{k}})\) and \({\psi}^{\prime}(\hat{{k}})<1\), i.e. \(\hat{{k}}\) is a stable steady-state equilibrium. □

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Varvarigos, D., Zakaria, I.Z. Endogenous fertility in a growth model with public and private health expenditures. J Popul Econ 26, 67–85 (2013). https://doi.org/10.1007/s00148-012-0412-1

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Keywords

  • Fertility
  • Economic growth
  • Health expenditures

JEL Classification

  • J13
  • O41