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


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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  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.


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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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Proof to Proposition 1

Using Eq. 18, we can define

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

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

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

From Eq. 36 we have

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

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

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

From Eq. 36, we can also derive

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

to which we can substitute Eq. 37 to get

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

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

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

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

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

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

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


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

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

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

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

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

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).

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  • Fertility
  • Economic growth
  • Health expenditures

JEL Classification

  • J13
  • O41