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Rising longevity, fertility dynamics, and R&D-based growth


This study constructs an overlapping-generations model with endogenous fertility, mortality, and R&D activities. We demonstrate that the model explains the observed fertility dynamics of developed countries. When the level of per capita wage income is either low or high, an increase in such income raises the fertility rate. When the level of per capita wage income is in the middle, an increase in such income decreases the fertility rate. The model also predicts the observed relationship between population growth and innovative activity. At first, both the rates of population growth and technological progress increase; that is, there is a positive relationship. Thereafter, the rate of population growth decreases but the rate of technological progress increases, showing a negative relationship.

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  1. HDI measures human development and is composed of per capita gross domestic product (GDP), life expectancy, and school enrolment.

  2. Hirazawa and Yakita (2017) also hint at a similar relationship between TFR and per capita GDP.

  3. Cutler et al. (2006) also emphasize the importance of public health. For example, Chakraborty (2004) incorporates this aspect by assuming that the survival probability in old age depends on public health expenditure. However, for simplicity, we do not consider this aspect.

  4. Bloom et al. (2003) find that an increase in life expectancy leads to higher savings rates. Gehringer and Prettner (2017) find a positive effect of rising longevity on total factor productivity.

  5. See Galor (2005, 2011) for extensive literature reviews on unified growth theory.

  6. To our knowledge, only Strulik et al. (2013) investigate R&D activities in a unified growth framework.

  7. Hirazawa and Yakita (2017) show that fertility rebound does not occur in their model if elderly labor supply is prohibited (see Section 4.1 in Hirazawa and Yakita 2017).

  8. Prettner (2013) integrates the OLG model of Blanchard (1985) with the R&D-based growth models of Romer (1990) and Jones (1995) and investigates the effects of demographic change on long-run growth. However, he does not take into account the transitional dynamics.

  9. The Ben-Porath mechanism works as follows: rising life expectancy prolongs active working lives, which has a positive impact on investment in human capital.

  10. Following Galor (2005) and Strulik et al. (2013), we assume that nt is the number of surviving children. As in Cigno (1998), Strulik (2004), and Galor and Mountford (2008), child mortality is an important factor of fertility and economic development. However, because our focus is adult mortality, we do not consider child mortality.

  11. Cigno and Werding (2007) explain the reason parents obtain their utility from the number of their children based on a constitutional theory of the family. The family constitution, for example, describes unwritten rules that state each of their children pay or give some amount of goods or filial attention to his or her parents. Thus, parents obtain their utility from the number of their children. Moreover, Cigno et al. (2017) show that parents do not invest in each of their children under certain conditions even if parents derive utility from their children’s wellbeing.

  12. In this study, we do not assume that the productivity of labor input to R&D is marginally decreasing (the “stepping-on-toes externality” as in Jones 1995) because it would not change the results qualitatively.

  13. In this study, At+ 1 < At does not hold because there is no product obsolescence.

  14. If we do not impose Assumption 1, Nt+ 1 < Nt holds for all \(A_{t}>\hat {A}_{2}\). In this case, the economy definitely falls into the no R&D region. Thereafter, At and nt become constant and nt falls to below 1. Since Nt+ 1 < Nt, the economy cannot get out of the trap.

  15. Because unified growth models as in Galor (2005, 2011) require that the initial population size must be sufficiently small and be historically given, the sufficient condition of a large initial population size may seem to be contradiction. However, as mentioned in the Introduction, this study is interested in the economy from the post-Malthusian regime, or in other words, does not focus on the Malthusian epoch. Galor (2005) states that technological progress showed a marked acceleration in the post-Malthusian regime. Thus, to ensure this, a large initial population size is needed.

  16. This result can be consistent with the argument of Boserup (1981) which explores the historical positive relationship between population size and technological levels.

  17. Gross domestic income (GDI) is calculated by

    $$\text{GDI} = (1-\rho n_{t}) w_{t} N_{t} + \pi_{t} A_{t}. $$

    From Eqs. 9b13, and 30, we obtain

    $$\text{GDI} = w_{t} \left( L_{Y,t} + A_{t} x_{t} + L_{A,t} \right) + \frac{1-\alpha}{\alpha} w_{t} A_{t} x_{t} = \frac{1}{1-\alpha} w_{t} L_{Y,t} + w_{t} L_{A,t}. $$

    Because Eq. 6a implies wtLY,t = (1 − α)Yt, GDI is as given below:

    $$\text{GDI} = Y_{t} + w_{t} L_{A,t}. $$

    Thus, we confirm that the value of GDI is equivalent to that of GDP.


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We would like to express our sincere gratitude to Prof. Alessandro Cigno and two anonymous referees for their constructive comments and suggestions. We would like to thank Real Arai, Masaru Inaba, Koji Kitaura, Kazutoshi Miyazawa, Akihisa Shibata, Kouki Sugawara, Kizuku Takao, Takashi Unayama, Akira Yakita, and the seminar participants at the Kansai University, Kyoto University, and Nagoya Gakuin University for their useful comments. Any errors are our responsibility.


This study was funded by a grant from the Japan Society for the Promotion of Science (grant number 16J09472).

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Correspondence to Kunihiko Konishi.

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Responsible editor: Alessandro Cigno


Appendix A: Proof of Proposition 1

To investigate the sign of \(\frac {\partial n_{t}}{\partial w_{t}}\), we rearrange (20) as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial n_{t}}{\partial w_{t}} = \gamma \frac{{\Phi} (w_{t})}{\left[1 + \beta \lambda(w_{t}) + \gamma \right]^{2} (\rho w_{t} + \delta)^{2}}, \end{array} $$

where Φ(wt) ≡ δ[1 + βλ(wt) + γ] − βλ(wt)wt(ρwt + δ). The sign of \(\frac {\partial n_{t}}{\partial w_{t}}\) is determined by that of Φ(wt), that is, \({\Phi }(w_{t}) \gtreqless 0\) implies \(\frac {\partial n_{t}}{\partial w_{t}} \gtreqless 0\). Differentiating Φ(wt) with respect to wt and using Eqs. 19a and 19b yield

$$\begin{array}{@{}rcl@{}} {\Phi}^{\prime} (w_{t}) &=& - \beta w_{t} \left[ 2\rho \lambda^{\prime}(w_{t}) + \lambda^{\prime\prime}(w_{t}) (\rho w_{t} + \delta) \right],\\ &=& - \frac{\beta \nu \psi \chi w_{t} e^{-\psi w_{t}}}{\left( 1 + \chi e^{-\psi w_{t}}\right)^{3}} \left\{ 2\rho - \psi (\rho w_{t} + \delta) + \chi \left[2\rho + \psi(\rho w_{t} + \delta) \right] e^{-\psi w_{t}} \right\}, \\ &\equiv& - \frac{\beta \nu \psi \chi w_{t} e^{-\psi w_{t}}}{\left( 1 + \chi e^{-\psi w_{t}}\right)^{3}} {\Omega} (w_{t}). \end{array} $$

Here, Ω(wt) satisfies

$$\begin{array}{@{}rcl@{}} {\Omega} (0) &=& 2\rho (1 + \chi ) + \psi \delta (\chi - 1), \\ {\Omega}^{\prime} (w_{t}) &=& - \psi \rho - \psi \chi \left[ \rho + \psi(\rho w_{t} + \delta) \right] e^{-\psi w_{t}} < 0. \end{array} $$

If Ω(0) < 0, we have Φ(wt) > 0 for all wt. On the other hand, if Ω(0) > 0, we have Φ(wt) ⪌ 0 for any \(w_{t} \lesseqgtr \tilde {w}\), where \(\tilde {w}\) is defined as \({\Phi }^{\prime }(\tilde {w}) = 0\). Here, the condition of Ω(0) > 0 is as follows:

$$\begin{array}{@{}rcl@{}} \chi > \frac{\psi \delta - 2\rho}{\psi \delta + 2\rho}. \end{array} $$

These results imply that the relationship between Φ(wt) and wt has a U-shape and \(\tilde {w}\) minimizes the value of Φ(wt).

We then substitute Eqs. 18 and 19a into Φ(wt) as follows:

$$\begin{array}{@{}rcl@{}} {\Phi}(w_{t}) = \delta(1+\gamma) + \frac{\delta \beta \nu}{1 + \chi e^{-\psi w_{t}}} - \frac{\beta \nu \psi w_{t} (\rho w_{t} + \delta) \chi e^{-\psi w_{t}}}{\left( 1 + \chi e^{-\psi w_{t}}\right)^{2}}. \end{array} $$

Using this, we obtain

$$\begin{array}{@{}rcl@{}} \lim\limits_{w_{t} \to 0} {\Phi}(w_{t}) &=& \delta(1+\gamma) + \frac{\delta \beta \nu}{1 + \chi} > 0, \\ \lim\limits_{w_{t} \to \infty} {\Phi}(w_{t}) &=& \delta(1+\gamma) + \delta \beta \nu > 0. \end{array} $$

In addition, if \({\Phi } (\tilde {w}) < 0\), we obtain the following relationship:

$$\begin{array}{@{}rcl@{}} {\Phi} (w_{t}) &>& 0 \hspace{10pt} when \hspace{10pt} 0<w_{t}<\bar{w}_{1} \hspace{7.5pt} or \hspace{7.5pt} w_{t} >\bar{w}_{2}, \\ {\Phi} (w_{t}) &<& 0 \hspace{10pt} when \hspace{10pt} \bar{w}_{1}<w_{t}<\bar{w}_{2}, \end{array} $$

where \(\bar {w}_{1}\) and \(\bar {w}_{2}\) are defined as \({\Phi }(\bar {w}_{1})= 0\) and \({\Phi }(\bar {w}_{2})= 0\) hold. In contrast, if \({\Phi } (\tilde {w}) > 0\), we have Φ(wt) > 0 for all wt.

Next, we examine the condition of \({\Phi } (\tilde {w}) < 0\). Because \(\tilde {w}\) satisfies \({\Phi }^{\prime } (\tilde {w})= 0\), we obtain

$$\begin{array}{@{}rcl@{}} && 2\rho - \psi (\rho \tilde{w} + \delta) + \chi \left[2\rho + \psi(\rho \tilde{w} + \delta) \right] e^{-\psi \tilde{w}} = 0, \\ &\Leftrightarrow \ & 1 + \chi e^{-\psi \tilde{w}} = \frac{2\psi(\rho \tilde{w} + \delta)}{2\rho + \psi (\rho \tilde{w} + \delta)}. \end{array} $$

An investigation of Eq. 25 gives us the following properties:

$$\begin{array}{@{}rcl@{}} &&\frac{d \tilde{w}}{d \chi} = \frac{[2\rho + \psi(\rho\tilde{w} + \delta)] e^{-\psi \tilde{w}}}{\psi\rho + \psi\chi [\rho + \psi(\rho\tilde{w} + \delta)] e^{-\psi \tilde{w}}} > 0, \end{array} $$
$$\begin{array}{@{}rcl@{}} &&\tilde{w} \to \infty \hspace{15pt} when \hspace{15pt} \chi \to \infty. \end{array} $$

Using Eq. 25, we rearrange the condition of \({\Phi } (\tilde {w}) < 0\) as given below:

$$\begin{array}{@{}rcl@{}} &&{\Phi}(\tilde{w}) < 0, \\ &\Leftrightarrow \ & \delta(1\,+\,\gamma) \,+\, \frac{\delta \beta \nu}{1 \,+\, \chi e^{-\psi \tilde{w}}} - \frac{\beta \nu \psi \tilde{w} (\rho \tilde{w} \,+\, \delta) \chi e^{-\psi \tilde{w}}}{\left( 1 \,+\, \chi e^{-\psi \tilde{w}}\right)^{2}} < 0, \\ &\Leftrightarrow \ & \delta(1\,+\,\gamma) \,+\, \delta \beta \nu \frac{2\rho \,+\, \psi (\rho \tilde{w} \,+\, \delta)}{2\psi(\rho \tilde{w} + \delta)} \\ && \hspace{62pt} - \beta \nu \psi \tilde{w} (\rho \tilde{w} + \delta) \frac{\psi(\rho \tilde{w} + \delta) - 2\rho}{2\rho + \psi (\rho \tilde{w} + \delta)} \!\left[ \frac{2\rho + \psi (\rho \tilde{w} + \delta)}{2\psi(\rho \tilde{w} + \delta)} \right]^{2} \!< 0, \\ &\Leftrightarrow \ & 4\delta(1\,+\,\gamma) \psi(\rho \tilde{w} \,+\, \delta) \,+\, 2\delta \beta \nu [2\rho \,+\, \psi (\rho \tilde{w} + \delta)] < \beta \nu \tilde{w} \!\left\{ [\psi (\rho \tilde{w} + \delta)]^{2} - 4\rho^{2} \right\}, \\ &\Leftrightarrow \ & \psi\rho \!\left[ \!\frac{2(1\,+\,\gamma)}{\beta\nu} \,+\, 1 \!\right] \,+\, \left[\!\frac{2(1\,+\,\gamma)\psi\delta}{\beta\nu} \,+\, (2\rho\,+\,\psi\delta)\right] \!\frac{1}{\tilde{w}} \!<\! \frac{\psi^{2}(\rho\tilde{w} \,+\, \delta)^{2} \,-\, 4\rho^{2}}{2\delta}. \end{array} $$

Let us define the left- and right-hand sides of Eq. 28 as \(\eta _{L}(\tilde {w})\) and \(\eta _{R}(\tilde {w})\). \(\eta _{L}(\tilde {w})\) is decreasing in \(\tilde {w}\) and \(\lim _{\tilde {w}\to 0} \eta _{L}(\tilde {w}) = \infty \) and \(\lim _{\tilde {w}\to \infty } \eta _{L}(\tilde {w}) = \psi \rho \left [ \frac {2(1+\gamma )}{\beta \nu } + 1\right ]\) hold. On the other hand, \(\eta _{R}(\tilde {w})\) is increasing in \(\tilde {w}\) and \(\lim _{\tilde {w}\to 0} \eta _{R}(\tilde {w}) = \frac {\psi ^{2}\delta ^{2} - 4\rho ^{2}}{2\delta }\) and \(\lim _{\tilde {w}\to \infty } \eta _{R}(\tilde {w}) = \infty \) hold. From Eqs. 26 and 27, we find that there is a unique \(\tilde {\chi }\) that satisfies \(\eta _{L}(\tilde {w}) = \eta _{R}(\tilde {w})\). Hence, \(\chi >\tilde {\chi }\) implies \(\eta _{L}(\tilde {w}) < \eta _{R}(\tilde {w})\). In contrast, if \(0<\chi <\tilde {\chi }\), \(\eta _{L}(\tilde {w}) > \eta _{R}(\tilde {w})\) holds (that is, \({\Phi } (\tilde {w}) > 0\) holds).

Appendix B: Properties of Γ(A t)

Using Eqs. 1819a, and 23, we obtain

$$\begin{array}{@{}rcl@{}} {\Gamma}^{\prime} (A_{t}) &=& \frac{A_{t}^{-\phi}}{\beta\nu} \left\{ (1\,-\,\phi)(1+\gamma+\beta\nu) + \left[ (1-\phi) - (1\,-\,\alpha)(1+\gamma)\psi w_{t} \right] \chi e^{-\psi w_{t}} \right\}, \\ &\equiv& \frac{A_{t}^{-\phi}}{\beta\nu} {\Theta}(w_{t}). \end{array} $$

Here, Θ(wt) satisfies

$$\begin{array}{@{}rcl@{}} {\Theta}(0) &=& (1-\phi)(1+\gamma+\beta\nu) + (1-\phi) \chi > 0, \\ {\Theta}^{\prime}(w_{t}) &=& - \left[ (1-\alpha)(1+\gamma) + 1-\phi - (1-\alpha)(1+\gamma) \psi w_{t} \right] \psi \chi e^{-\psi w_{t}}. \end{array} $$

Hence, Θ(wt) ⪌ 0 when wtwΘ, where \(w_{{\Theta }} \equiv \frac {(1-\alpha )(1+\gamma ) + 1-\phi }{(1-\alpha )(1+\gamma ) \psi }\). Using these results, we can define the following two cases: When Θ(wΘ) > 0, we obtain

$$\begin{array}{@{}rcl@{}} {\Theta} (w_{t}) > 0 \hspace{15pt} for \hspace{5pt} all \hspace{5pt} w_{t}. \end{array} $$

On the other hand, when Θ(wΘ) < 0, we obtain

$$\begin{array}{@{}rcl@{}} {\Theta} (w_{t}) &>& 0 \hspace{10pt} when \hspace{10pt} 0<w_{t}<w_{{\Gamma},1}, \hspace{2pt} w_{t} >w_{{\Gamma},2}, \\ {\Theta} (w_{t}) &<& 0 \hspace{10pt} when \hspace{10pt} w_{{\Gamma},1}<w_{t}<w_{{\Gamma},2}, \end{array} $$

where wΓ,1 and wΓ,2 are defined as Θ(wΓ,1) = 0 and Θ(wΓ,2) = 0. Because the sign of Γ(At) is the same as that of Θ(wt) and wt is determined by At, we can state the properties of Γ(At) as in Section 5.1.

We then investigate the condition Θ(wΘ) < 0. Rearranging Θ(wΘ) < 0 yields

$$\begin{array}{@{}rcl@{}} &&{\Theta}(w_{{\Theta}}) < 0, \\ &\Leftrightarrow \ & (1-\phi)(1+\gamma+\beta\nu) - (1-\alpha)(1+\gamma) \chi e^{-\psi w_{{\Theta}}} < 0. \end{array} $$

This implies that Θ(wΘ) < 0 holds when \(\chi >\hat {\chi }\), where \(\hat {\chi }\) is defined as follows:

$$\begin{array}{@{}rcl@{}} \hat{\chi} \equiv \frac{(1-\phi)(1+\gamma+\beta\nu)}{(1-\alpha)(1+\gamma) e^{-\psi w_{{\Theta}}}}. \end{array} $$

In contrast, we obtain Θ(wΘ) > 0 when \(0<\chi <\hat {\chi }\).

Appendix C: Proof of Lemma 1

From Proposition 1, nt has a local maximum (minimum) value at \(w_{t} = \bar {w}_{1}\) (\(w_{t} = \bar {w}_{2}\)). Because \(N_{t + 1} \gtreqless N_{t}\) is equivalent to \(n_{t} \gtreqless 1\), we examine the following three cases. First, if nt > 1 at \(w_{t}=\bar {w}_{2}\), we obtain

$$\begin{array}{@{}rcl@{}} N_{t + 1} \gtreqless N_{t} \hspace{15pt} when \hspace{10pt} w_{t} \gtreqless \hat{w}_{1}, \end{array} $$

where \(\hat {w}_{1}\) is defined as \(n_{t} |_{w_{t}=\hat {w}_{1}} = 1\) and \(\hat {w}_{1}<\bar {w}_{1}<\bar {w}_{2}\). This case corresponds to the panels on the left side of Fig. 4. Second, if nt > 1 at \(w_{t}=\bar {w}_{1}\) and nt < 1 at \(w_{t}=\bar {w}_{2}\), we obtain

$$\begin{array}{@{}rcl@{}} N_{t + 1} &>& N_{t} \hspace{10pt} when \hspace{10pt} \hat{w}_{1}<w_{t}<\hat{w}_{2} \hspace{7.5pt} or \hspace{7.5pt} w_{t} >\hat{w}_{3},\\ N_{t + 1} &<& N_{t} \hspace{10pt} when \hspace{10pt} 0<w_{t}<\hat{w}_{1} \hspace{7.5pt} or \hspace{7.5pt} \hat{w}_{2}<w_{t}<\hat{w}_{3}, \end{array} $$

where \(\hat {w}_{2}\) and \(\hat {w}_{3}\) are defined as \(n_{t} |_{w_{t}=\hat {w}_{j}} = 1\) (j = 2, 3) and \(\hat {w}_{1}<\bar {w}_{1}<\hat {w}_{2}<\bar {w}_{2}<\hat {w}_{3}\). This case corresponds to the panels on the right side of Fig.4. Finally, if nt < 1 at \(w_{t}=\bar {w}_{1}\), we obtain

$$\begin{array}{@{}rcl@{}} N_{t + 1} \gtreqless N_{t} \hspace{15pt} when \hspace{10pt} w_{t} \gtreqless \hat{w}_{1}. \end{array} $$

In this case, \(\bar {w}_{1}<\bar {w}_{2}<\hat {w}_{1}\) holds. Therefore, the population size of young adults Nt decreases during economic development except for sufficiently high levels of wt. Because this is contrary to the observed fact, we rule out this case.

Using Eq. 4b, we obtain the following relationship:

$$\begin{array}{@{}rcl@{}} n_{t} \gtreqless 1 \hspace{8pt} \Leftrightarrow \hspace{8pt} \gamma w_{t} \gtreqless \left[1 + \beta \lambda(w_{t}) + \gamma \right] (\rho w_{t} + \delta). \end{array} $$

Hence, the case where nt > 1 at \(w_{t}=\bar {w}_{2}\) corresponds to \(\gamma \bar {w}_{2} > \left [1 + \beta \lambda (\bar {w}_{2}) + \gamma \right ] (\rho \bar {w}_{2} + \delta )\). On the other hand, the case where nt > 1 at \(w_{t}=\bar {w}_{1}\) and nt < 1 at \(w_{t}=\bar {w}_{2}\) corresponds to \(\gamma \bar {w}_{1} > \left [1 + \beta \lambda (\bar {w}_{1}) + \gamma \right ] (\rho \bar {w}_{1} + \delta )\) and \(\gamma \bar {w}_{2} < \left [1 + \beta \lambda (\bar {w}_{2}) + \gamma \right ] (\rho \bar {w}_{2} + \delta )\).

Appendix D: Calculation of Y t

Substituting Eq. 16 into Eq. 15, we obtain

$$\begin{array}{@{}rcl@{}} Y_{t} = \left( \frac{\alpha^{2}}{1-\alpha} \right)^{\alpha} A_{t}^{1-\alpha} L_{Y,t}, \end{array} $$

We consider the labor market clearing condition to derive LY,t. By using Eqs. 9a and 16, the labor input into production of intermediate goods becomes

$$\begin{array}{@{}rcl@{}} A_{t} x_{t} = \frac{\alpha^{2}}{1-\alpha} L_{Y,t}. \end{array} $$

From Eqs. 13 and 30, we obtain

$$\begin{array}{@{}rcl@{}} L_{Y,t} = \frac{1-\alpha}{1-\alpha+\alpha^{2}} \left[ \left( 1 - \rho n_{t} \right) N_{t} - L_{A,t} \right] \hspace{10pt} if \hspace{10pt} L_{A,t} > 0. \end{array} $$

Furthermore, Eqs. 1011, and 17 yield

$$\begin{array}{@{}rcl@{}} L_{A,t} = \frac{\beta \lambda(w_{t}) N_{t}}{1 + \beta \lambda(w_{t}) + \gamma} - A_{t}^{1-\phi}. \end{array} $$

By using Eqs. 4b162931, and 32, we can calculate the output of final goods Yt.

Appendix E: Derivation of the growth rates along the BGP

E.1 Derivation of \(g^{\ast }_{M}\)

Because Nt+ 1 = nNt and \(\lambda _{t} = \bar {\lambda }\) hold along the BGP, the growth rate of population along the BGP is given as follows:

$$\begin{array}{@{}rcl@{}} 1 + g^{\ast}_{M} = \frac{n^{\ast} N_{t + 1} + N_{t + 1} + \bar{\lambda} N_{t}}{n^{\ast} N_{t} + N_{t} + \bar{\lambda} N_{t-1}} = \frac{(n^{\ast})^{2} + n^{\ast} + \bar{\lambda}}{n^{\ast} + 1 + \frac{\bar{\lambda}}{n^{\ast}}} = n^{\ast}. \end{array} $$

E.2 Derivation of \(g^{\ast }_{A}\)

By using Eqs. 32 and 19d, we obtain the employment share of R&D along the BGP as follows:

$$\begin{array}{@{}rcl@{}} \frac{L_{A,t}}{N_{t}} = \frac{\beta \bar{\lambda}}{1 + \beta \bar{\lambda} + \gamma} - \frac{A_{t}^{1-\phi}}{N_{t}}. \end{array} $$

Because the employment share of R&D is constant, the term \(A_{t}^{1-\phi }/N_{t}\) also becomes constant along the BGP. Thus, the growth rate of At along the BGP is as follows:

$$\begin{array}{@{}rcl@{}} \left( 1 + g_{A}^{\ast} \right)^{1-\phi} = \left( \frac{A_{t + 1}}{A_{t}} \right)^{1-\phi} = \frac{N_{t + 1}}{N_{t}} = n^{\ast}. \end{array} $$

E.3 Derivation of \(g^{\ast }_{Y}\)

From Eq. 29, the growth rate of Yt is determined by those of At and LY,t. Equation 31 implies

$$\begin{array}{@{}rcl@{}} \frac{L_{Y,t}}{N_{t}} = \frac{1-\alpha}{1-\alpha+\alpha^{2}} \left( 1 - \rho n^{\ast} - \frac{L_{A,t}}{N_{t}} \right). \end{array} $$

Because the employment share of final goods production becomes a constant, the growth rate of LY,t along the BGP is equal to n. Therefore, the growth rate of Yt along the BGP is given by

$$\begin{array}{@{}rcl@{}} 1 + g_{Y}^{\ast} = \left( 1 + g_{A}^{\ast} \right)^{1-\alpha} n^{\ast} = \left( n^{\ast} \right)^{\frac{1-\alpha}{1-\phi}+ 1}. \end{array} $$

E.4 Derivation of \(g^{\ast }_{Z}\) and \(g^{\ast }_{Z}\)

From the definition of Zt, the growth rate of Zt is as follows:

$$\begin{array}{@{}rcl@{}} 1 + g_{Z,t} &=& \frac{Z_{t + 1}}{Z_{t}} = \frac{Y_{t + 1} + w_{t + 1} L_{A,t + 1}}{Y_{t} + w_{t} L_{A,t}} = \frac{w_{t + 1}}{w_{t}} \frac{\frac{1}{1-\alpha} L_{Y,t + 1} + L_{A,t + 1}}{\frac{1}{1-\alpha} L_{Y,t} + L_{A,t}}, \\ &=& \frac{w_{t + 1}}{w_{t}} \frac{\frac{1}{1-\alpha} \frac{L_{Y,t + 1}}{N_{t + 1}} + \frac{L_{A,t + 1}}{N_{t + 1}}}{\frac{1}{1-\alpha} \frac{L_{Y,t}}{N_{t}} + \frac{L_{A,t}}{N_{t}}} \frac{N_{t + 1}}{N_{t}}. \end{array} $$

Along the BGP, \(\frac {L_{Y,t}}{N_{t}}\) and \(\frac {L_{A,t}}{N_{t}}\) become a constant. These results and (16) imply

$$\begin{array}{@{}rcl@{}} 1 + g_{Z}^{\ast} = \left( 1+g_{A}^{\ast} \right)^{1-\alpha} n^{\ast} = \left( n^{\ast} \right)^{\frac{1-\alpha}{1-\phi}+ 1}. \end{array} $$

Because \(z_{t} = \frac {Z_{t}}{M_{t}}\), the growth rate of zt along the BGP is given by

$$\begin{array}{@{}rcl@{}} 1 + g_{z}^{\ast} = \frac{z_{t + 1}}{z_{t}} = \frac{Z_{t + 1}}{Z_{t}} \frac{M_{t}}{M_{t + 1}} = \frac{1 + g_{Z}^{\ast}}{1 + g_{M}^{\ast}} = \left( n^{\ast} \right)^{\frac{1-\alpha}{1-\phi}}. \end{array} $$

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Futagami, K., Konishi, K. Rising longevity, fertility dynamics, and R&D-based growth. J Popul Econ 32, 591–620 (2019).

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  • Fertility
  • Mortality
  • R&D
  • Economic development

JEL Classification

  • J13
  • O30
  • O40