## Abstract

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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## Notes

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

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

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

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

Following Galor (2005) and Strulik et al. (2013), we assume that

*n*_{t}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.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.

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.

In this study,

*A*_{t+ 1}<*A*_{t}does not hold because there is no product obsolescence.If we do not impose Assumption 1,

*N*_{t+ 1}<*N*_{t}holds for all \(A_{t}>\hat {A}_{2}\). In this case, the economy definitely falls into the no R&D region. Thereafter,*A*_{t}and*n*_{t}become constant and*n*_{t}falls to below 1. Since*N*_{t+ 1}<*N*_{t}, the economy cannot get out of the trap.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.

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

Gross domestic income (GDI) is calculated by

$$\text{GDI} = (1-\rho n_{t}) w_{t} N_{t} + \pi_{t} A_{t}. $$From Eqs. 9b, 13, 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

*w*_{t}*L*_{Y,t}= (1 −*α*)*Y*_{t}, 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.

## References

Alvarez-Pelaez MJ, Groth C (2005) Too little or too much R&D?. Eur Econ Rev 49:437–456

Blanchard OJ (1985) Debt, deficits and finite horizons. J Polit Econ 93:223–247

Bloom DE, Canning D, Graham B (2003) Longevity and life-cycle savings. Scand J Econ 105:319–338

Boserup E (1981) Population and technology. Blackwell, Oxford

Chakraborty S (2004) Endogenous lifetime and economic growth. J Econ Theory 116:119–137

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

Cigno A, Komura M, Luporini A (2017) Self-enforcing family rules, marriage and the (non)neutrality of public intervention. J Popul Econ 30:805–834

Cigno A, Werding M (2007) Children and pensions. MIT Press, Cambridge

Cutler DM, Deaton AS, Lleras-Muney A (2006) The determinants of mortality. J Econ Perspect 20:97–120

Day C (2016) Fertility and economic growth: the role of workforce skill composition and child care prices. Oxf Econ Pap 68:546–565

de la Croix D, Licandro O (2013) The child is father of the man: implications for the demographic transition. Econ J 123:236–261

Diamond P (1965) National debt in a neoclassical growth model. Am Econ Rev 55:1126–1150

Galor O (2005) From stagnation to growth: unified growth theory. In: Aghion P, Durlauf S (eds) Handbook of economic growth. North-Holland, Amsterdam, pp 171–293

Galor O (2011) Unified growth theory. Princeton University Press, Princeton

Galor O, Mountford A (2008) Trading population for productivity: theory and evidence. Rev Econ Stud 75:1143–1179

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

Gehringer A, Prettner K (2017) Longevity and technological change. Macroecon Dyn. forthcoming. https://doi.org/10.1017/S1365100517000293

Grossman GM, Helpman E (1991) Innovation and growth in the global economy. MIT Press, Cambridge

Hashimoto K, Tabata K (2016) Demographic change, human capital accumulation and R&D-based growth. Can J Econ 49:707–737

Hirazawa M, Yakita A (2017) Labor supply of the elderly, fertility and economic development. J Macroecon 51:75–96

Jones CI (1995) R&d-based models of economic growth. J Polit Econ 103:759–784

Luci-Greulich A, Thévenon O (2013) The impact of family policies on fertility trends in developed countries. Eur J Popul 29:387–416

Luci-Greulich A, Thévenon O (2014) Does economic advancement ‘cause’ a re-increase in fertility? an empirical analysis for OECD countries (1960–2007). Eur J Popul 30:187–221

Maruyama A, Yamamoto K (2010) Variety expansion and fertility rates. J Popul Econ 23:57–71

Myrskylä M, Kohler HP, Billari F (2009) Advances in development reverse fertility declines. Nature 460:741–743

Prettner K (2013) Population aging and endogenous economic growth. J Popul Econ 26:811–834

Prettner K, Trimborn T (2017) Demographic change and R&D-based economic growth. Economica 84:667–681

Romer PM (1990) Endogenous technological change. J Polit Econ 98:71–102

Strulik H (2004) Economic growth and stagnation with endogenous health and fertility. J Popul Econ 17:433–453

Strulik H, Prettner K, Prskawetz A (2013) The past and future of knowledge-based growth. J Econ Growth 18:411–437

Yaari M (1965) Uncertain lifetime, life insurance, and the theory of the consumer. Rev Econ Stud 32:137–150

## Acknowledgments

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.

## Funding

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

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## Appendices

### Appendix A: Proof of Proposition 1

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

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

Here, Ω(*w*_{t}) satisfies

If Ω(0) < 0, we have Φ^{′}(*w*_{t}) > 0 for all *w*_{t}. On the other hand, if Ω(0) > 0, we have Φ^{′}(*w*_{t}) ⪌ 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:

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

We then substitute Eqs. 18 and 19a into Φ(*w*_{t}) as follows:

Using this, we obtain

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

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(*w*_{t}) > 0 for all *w*_{t}.

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

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

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

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. 18, 19a, and 23, we obtain

Here, Θ(*w*_{t}) satisfies

Hence, Θ^{′}(*w*_{t}) ⪌ 0 when *w*_{t} ⪌ *w*_{Θ}, 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(*w*_{Θ}) > 0, we obtain

On the other hand, when Θ(*w*_{Θ}) < 0, we obtain

where *w*_{Γ,1} and *w*_{Γ,2} are defined as Θ(*w*_{Γ,1}) = 0 and Θ(*w*_{Γ,2}) = 0. Because the sign of Γ^{′}(*A*_{t}) is the same as that of Θ(*w*_{t}) and *w*_{t} is determined by *A*_{t}, we can state the properties of Γ(*A*_{t}) as in Section 5.1.

We then investigate the condition Θ(*w*_{Θ}) < 0. Rearranging Θ(*w*_{Θ}) < 0 yields

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

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

### Appendix C: Proof of Lemma 1

From Proposition 1, *n*_{t} 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 *n*_{t} > 1 at \(w_{t}=\bar {w}_{2}\), we obtain

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 *n*_{t} > 1 at \(w_{t}=\bar {w}_{1}\) and *n*_{t} < 1 at \(w_{t}=\bar {w}_{2}\), we obtain

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 *n*_{t} < 1 at \(w_{t}=\bar {w}_{1}\), we obtain

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

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

Hence, the case where *n*_{t} > 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 *n*_{t} > 1 at \(w_{t}=\bar {w}_{1}\) and *n*_{t} < 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

We consider the labor market clearing condition to derive *L*_{Y,t}. By using Eqs. 9a and 16, the labor input into production of intermediate goods becomes

From Eqs. 13 and 30, we obtain

Furthermore, Eqs. 10, 11, and 17 yield

By using Eqs. 4b, 16, 29, 31, and 32, we can calculate the output of final goods *Y*_{t}.

### Appendix E: Derivation of the growth rates along the BGP

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

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

### 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:

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 *A*_{t} along the BGP is as follows:

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

From Eq. 29, the growth rate of *Y*_{t} is determined by those of *A*_{t} and *L*_{Y,t}. Equation 31 implies

Because the employment share of final goods production becomes a constant, the growth rate of *L*_{Y,t} along the BGP is equal to *n*^{∗}. Therefore, the growth rate of *Y*_{t} along the BGP is given by

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

From the definition of *Z*_{t}, the growth rate of *Z*_{t} is as follows:

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

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

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Futagami, K., Konishi, K. Rising longevity, fertility dynamics, and R&D-based growth.
*J Popul Econ* **32**, 591–620 (2019). https://doi.org/10.1007/s00148-018-0691-2

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DOI: https://doi.org/10.1007/s00148-018-0691-2

### Keywords

- Fertility
- Mortality
- R&D
- Economic development

### JEL Classification

- J13
- O30
- O40