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Endogenous education and the reversal in the relationship between fertility and economic growth

Abstract

To reconcile the predictions of research and development (R&D)-based growth theory regarding the impact of population growth on productivity growth with the available empirical evidence, we propose a tractable, continuous-time, multisector, R&D-based growth model with endogenous education and endogenous fertility. As long as the human capital dilution effect is sufficiently weak, faster population growth may lead to faster aggregate human capital accumulation, to faster technological progress, and, thus, to a higher growth rate of productivity. By contrast, when the human capital dilution effect becomes sufficiently strong, faster population growth slows down aggregate human capital accumulation, dampens the rate of technical change, and, thus, reduces productivity growth. Therefore, the model can account for the possibly negative correlation between population growth and productivity growth in R&D-based growth models depending on the strength of the human capital dilution effect.

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Notes

  1. This term was introduced first by Strulik et al. (2013) to describe that particular phenomenon according to which over the very long run, population growth is first positively and then negatively correlated with R&D-based productivity growth. According to them, in the course of time the increase of individual human capital overcompensates the associated decline of population growth in a way that leads to an ultimate rise of the aggregate human capital stock. Given that human capital is the driving force of R&D activity, this causes at some point higher R&D output and, hence, higher R&D-based economic growth. Unlike Strulik et al. (2013), in our paper the emphasis is on the intensity of the individual human capital dilution effect as a possible alternative source of the population-productivity reversal along a BGP.

  2. Since final output is produced competitively under constant returns to scale with respect to its rival inputs, at equilibrium HY and xi are rewarded according to their own marginal products. Hence, Ζ is the share of GDP accruing to intermediates, and (1 − Ζ) is the share of Y going to human capital.

  3. The variables wY,t, wI,t, and wn,t denote the wage paid to a generic unit of human capital employed in the final output sector, the intermediate sector, and the research sector, respectively.

  4. So, unlike Bucci (2013), we rule out here the possibility that the returns to specialization might be negative. A negative R means that an increase in n would lead to some sort of inefficiency in the economy as, following a rise of the number of intermediate good varieties, total GDP would ceteris paribus decline in this case.

  5. In these papers, if all intermediates are hired in the same amount the returns to specialization are either increasing or at most constant.

  6. For a detailed discussion of the “fishing-out” and “standing-on-shoulders” effects, see Jones (1995, 2005).

  7. Likewise, if μ > 1, increasing the number of researchers would imply an increase (more than proportional) in the total number of innovations produced per unit of time (Eq. 8).

  8. From Hn = (χHΦ/nη)1/μ, it is immediate to observe that ∂Hn/∂H = (1/μ)(Hn/H)Φ. For given H and n, and for given parameters μ, χ, and η, if Φ is positive the larger the size of this parameter the bigger ∂Hn/∂H.

  9. When Φ = 0, Eq. (8) becomes: , with χ > 0, μ > 0, and η < 1. This specification coincides with that employed by Jones (2005, Eq. 16). This means that μ > Φ is the working assumption in this class of models. We will use this assumption below (see Proposition 2 and Remark 1).

  10. As it is well known, the difference between a Millian-type and a Benthamite-type intertemporal utility function is that in the former case the optimizing agent maximizes average utility of the dynasty, whereas in the latter case s/he maximizes aggregate utility of the dynasty. The Benthamite formulation would lead to serious complications in the presence of endogenous fertility decisions given that the endogenous fertility rate appears in the exponent of the discount factor. However, the Millian formulation with a positive utility of the number of children captures the same effects as the Benthamite formulation such that the involved tradeoffs to the household are the same. Our choice of using a logarithmic specification for the instantaneous utility function serves the scope of making the problem even more tractable analytically.

  11. It is possible to show that introducing additional time costs of fertility in Eq. (13) leaves the key results of the model unchanged (see Appendix C for details).

  12. One can easily verify that, under the assumption of decreasing returns to human capital per capita in the production of (new) human capital, in the very long run human capital would no longer be a growth engine since economic growth would be wholly driven by population growth (i.e., fertility).

  13. The presence of gL on the right hand side of Eq. (12) reflects also a sort of dilution-effect that, following agents’ choice of having more children, may ultimately hit per capita asset investment in the form of an additional (consumption-)cost. More precisely, it reveals the cost (in terms of foregone consumption) of bringing the amount of per capita assets of the newcomers up to the average level of the existing population. Therefore, this formulation implies that, as long as Ω > 0, population growth tends to reduce the speed of asset accumulation (by the average individual in the population).

  14. When ξ = 1, \( \overset{\bullet }{h}/h=\overset{\bullet }{H}/H-{g}_L=\sigma \left(1-u\right)-{g}_L \): “…population growth operates like depreciation of human capital per capita (Strulik 2005, Eq. 2, p. 135). Unlike Strulik (2005), we set the rate of physical obsolescence of human capital equal to zero. This is done just for the sake of simplicity.

  15. See Zyngier (2014). According to Bressoux et al. (2009, p. 560): “…The effect of class size is shown to be significant and negative: a smaller class size improves student achievement. The impact is evaluated as being between 2.5% and 3% of a standard deviation of the scores… It is worth noting that the effect of class size seems more beneficial to low-achieving students within classes. The effect is particularly large for classes in priority education areas… This finding shows the complexity of the education production function and proves that it is essential to study how resources impact different students differently…”.

  16. With these parameter values, we also have: r − γH + (1 − R)γn = 0.031 > 0.

  17. We are particularly grateful to a referee for suggesting us to reflect more thoughtfully on this issue.

  18. See Galor and Weil (2000), and Galor (2005, 2011) for detailed descriptions of the Post-Malthusian and Modern Growth regimes, and for an exhaustive analysis of the driving forces of economic growth, population growth, and human capital accumulation in each of the two different regimes.

  19. “The number of scientists and engineers engaged in R&D in the United States has grown dramatically over time, from under 500,000 in 1965 to nearly 1 million in 1989. Other advanced countries have experienced even larger increases in R&D employment…” (Segerstrom 1998, p. 1290).

  20. While (as already said) there seemingly exists no available point estimate of the parameter \( \overline{\alpha} \) (and, more generally, of the degree of returns to specialization, R), there are instead many studies that document very well the decline of the labor share of income, especially in the richest economies (see Fig. 2, p. 71, and Fig. 3, p. 73, in Karabarbounis and Neiman 2014; ILO-OECD 2015). In this regard, the OECD (2012) claims that over the period from 1990 to 2009 the share of labor compensation in national income declined in 26 out of 30 rich countries for which data were available, and calculated that the median (adjusted) labor share of national income across these economies fell from 66.1% to 61.7%. More recent OECD calculations find that the average adjusted labor share in G20 countries went down by about 0.3 percentage points per year between 1980 and the late 2000s. Similar downward trends of the labor share have also been documented by other international institutions (IMF 2007; European Commission 2007; BIS 2006; ILO 2012).

  21. We are indebted to a referee for suggesting us this condensed proof.

  22. If Ω = 0, then it is easy to see that an indeterminate form (of the type: gL = 0/0) would arise.

  23. Clearly, one can use the same argument if the initial change has to do with Ω, instead of υ.

  24. At this stage it is worth observing that, while \( \overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}-\sigma \varphi \) is always lower than one, \( \overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}-\sigma \varphi >0 \) whenever \( 0<\varphi <\frac{1}{\sigma}\left(\frac{R\varUpsilon}{1+ R\varUpsilon}\right) \), that is, when φ is positive but sufficiently small. This is the assumption that we implicitly use in the comparative statics results reported above (also in the light of the fact that, as far as we know, reliable point estimates of φ do not exist).

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Acknowledgments

We are indebted to three anonymous referees and to the editors of this journal (in particular Prof. Alessandro Cigno and Prof. Madeline Zavodny) for providing constructive comments and valuable suggestions.

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APPENDICES

APPENDICES

APPENDIX A: DERIVATION OF EQS. (18)–(29), AND ANALYSIS OF THE SPECIAL CASE WHERE Ω = 0

The Hamiltonian function (Jt) related to the inter-temporal optimization problem (11)–(12)–(13) in the text reads as:

Jt = [log(ct) + υ log(gLt)]eρt + λat[rtat + uthtwt − (1 + ΩgLt)ct] + λht[σ(1 − ut) − ξgLt]ht where λat and λht are the co-state variables associated with the two state variables, at and ht, respectively. The (necessary) first order conditions are:

$$ \frac{\partial {J}_t}{\partial {c}_t}=0\kern0.5em \iff \kern0.5em \frac{e^{-\rho t}}{c_t}={\lambda}_{at}\left(1+\Omega {g}_{Lt}\right) $$
(A1)
$$ \frac{\partial {J}_t}{\partial {\mathrm{g}}_{Lt}}=0\kern0.5em \iff \kern0.5em \frac{\upsilon {e}^{-\rho t}}{{\mathrm{g}}_{Lt}}-\Omega {\lambda}_{at}{c}_t-\xi {\lambda}_{ht}{h}_t=0 $$
(A2)
$$ \frac{\partial {J}_t}{\partial {u}_t}=0\kern0.5em \iff \kern0.5em {\lambda}_{at}=\sigma \frac{\lambda_{ht}}{w_t} $$
(A3)
$$ \frac{\partial {J}_t}{\partial {a}_t}=-{\overset{\bullet }{\lambda}}_{at}\kern1em \iff \kern1em {\lambda}_{at}{r}_t=-{\overset{\bullet }{\lambda}}_{at} $$
(A4)
$$ \frac{\partial {J}_t}{\partial {h}_t}=-{\overset{\bullet }{\lambda}}_{ht}\kern1em \iff \kern1em {\lambda}_{at}{u}_t{w}_t+{\lambda}_{ht}\left[\sigma \left(1-{u}_t\right)-\xi {g}_L\right]=-{\overset{\bullet }{\lambda}}_{ht}, $$
(A5)

along with the two transversality conditions:

$$ \underset{t\to +\infty }{\lim }{\lambda}_{at}{a}_t=0,\kern5.25em \underset{t\to +\infty }{\lim }{\lambda}_{ht}{h}_t=0, $$

and the initial conditions:

$$ a(0)>0,\kern5.25em h(0)>0. $$

From (A1) it follows that:

$$ {e}^{-\rho t}={\lambda}_{at}\left(1+\Omega {g}_{Lt}\right){c}_t. $$
(A6)

Plugging this expression into (A2) yields:

$$ {c}_t{\lambda}_{at}\left[\frac{\upsilon }{{\mathrm{g}}_{Lt}}-\Omega \left(1-\upsilon \right)\right]=\xi {\lambda}_{ht}{h}_t. $$
(A7)

Employing (A3) into (A7) delivers:

$$ \left[\frac{\upsilon }{{\mathrm{g}}_{Lt}}-\Omega \left(1-\upsilon \right)\right]=\frac{\xi }{\sigma}\frac{h_t{w}_t}{c_t}. $$
(A8)

Using (A3) in (A5), instead, gives:

$$ \frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}=\left(\xi {g}_{Lt}-\sigma \right). $$
(A9)

Along a BGP all variables depending on time (including Lt) grow at constant, possibly positive, exponential rates. Therefore, in a BGP equilibrium (where gLt = gL, ∀ t ≥ 0) Eq. (A9) suggests that \( {\overset{\bullet }{\lambda}}_{ht}/{\lambda}_{ht} \) is also constant. Eqs. (A3) and (A4) imply, respectively:

$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=\frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}-\frac{{\overset{\bullet }{w}}_t}{w_t}, $$
(A10)
$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=-{r}_t $$
(A11)

Combination of (A9), (A10) and (A11) leads to:

$$ {r}_t=\sigma +\frac{{\overset{\bullet }{w}}_t}{w_t}-\xi {g}_L. $$
(A12)

Since human capital is perfectly mobile across sectors, at equilibrium it will be rewarded according to the same wage, i.e.: wYt = wIt = wnt ≡ wt. Moreover, along the BGP this common wage will grow at a constant exponential rate, implying that \( \frac{\overset{\bullet \kern0.66em }{w_{Yt}}}{w_{Yt}}=\frac{\overset{\bullet \kern0.66em }{w_{It}}}{w_{It}}=\frac{\overset{\bullet \kern0.66em }{w_{nt}}}{w_{nt}}\equiv \frac{\overset{\bullet \kern0.66em }{w_t}}{w_t} \) is constant. Accordingly, Eq. (A12) implies that in the BGP equilibrium the real rate of return on asset holdings, r , will be constant, as well.

From (A1) and (A4) together:

$$ \frac{\overset{\bullet }{c_t}}{c_t}=r-\rho, \kern1em {c}_t\equiv \frac{C_t}{L_t}. $$
(A13)

By combining (A7), (A13), (A4), (A9) and (13) in the main text, after some simple algebra it is finally possible to obtain:

$$ {u}_t=\frac{\rho }{\sigma }=u,\kern0.5em \forall t\ge 0,\kern0.5em \iff \kern1.00em 1-u=\left(\frac{\sigma -\rho }{\sigma}\right). $$
(A14)

Eq. (A14) suggests that along a BGP the allocation of human capital between productive and non-productive activities is also constant.

Making use of Eqs. (6) and (10) in the main text, we find that along a BGP:

$$ {V}_{nt}=Z\left(1-Z\right){\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z\frac{n_t^R}{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]},\kern1.25em R\equiv 1+\overline{\alpha}-Z>0;\kern0.5em \frac{{\overset{\bullet }{n}}_t}{n_t}\equiv {\gamma}_n;\kern0.5em \frac{{\overset{\bullet }{H}}_t}{H_t}\equiv {\gamma}_H $$
(A15)

Notice that, for any 0 < Z < 1, HYt > 0, HIt > 0, and nt > 0, Vnt is always positive as long as:

$$ r>{\gamma}_H-\left(1-R\right){\gamma}_n. $$
(A15.1)

Given Vnt, from Eq. (9.1) in the main text:

$$ {w}_{nt}=\frac{Z}{\chi}\left(1-Z\right){s}_n^{\mu -1}{H}_t^{\mu -1-\varPhi }{n}_t^{\eta }{\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z\frac{n_t^R}{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}, $$
(A16)

where sn ≡ Hnt/Ht is (by definition) constant in a BGP-equilibrium.

We can now use Eqs. (5), (2) and (4.1) in the main text, obtaining:

$$ {w}_{It}={Z}^2{\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^{Z-1}{n}_t^R. $$
(A17)

From Eq. (15) in the main text, by equalizing (A16) and (A17) in this appendix one gets:

$$ {s}_I\equiv \frac{H_{It}}{H_t}=\frac{Z\chi}{\left(1-Z\right)}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}}\frac{n_t^{1-\eta }}{H_t^{\mu -\varPhi }}. $$
(A18)

Combining Eqs. (1) and (4.1) in the text yields:

$$ {w}_{Yt}\equiv \frac{\partial {Y}_t}{\partial {H}_{Yt}}=\left(1-Z\right){\left(\frac{H_{Yt}}{n_t}\right)}^{-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z{n}_t^R. $$
(A19)

From (16) in the main text and (A18) above, equalization of (A17) and (A19) in this appendix delivers:

$$ {s}_Y\equiv \frac{H_{Yt}}{H_t}=\left(\frac{1-Z}{Z^2}\right){s}_I=\frac{\chi }{Z}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}}\frac{n_t^{1-\eta }}{H_t^{\mu -\varPhi }}. $$
(A20)

Along a BGP all variables depending on time grow at constant (possibly positive) exponential rates and the sectoral shares of human capital employment are also constant. Therefore, from Eq. (8) in the main text it follows that:

$$ \frac{{\overset{\bullet }{n}}_t}{n_t}\equiv {\gamma}_n=\left(\frac{\mu -\varPhi }{1-\eta}\right){\gamma}_H. $$
(A21)

If μ − Φ = 1 − η, we have a very special case of the model in which human capital and technology grow at the same rate γn = γH ≡ γ in the long-run BGP equilibrium. Here we allow for the most general possible case in which μ ≠ Φ ≠ Φ + 1 − η.

Using Eqs. (A16), (A17), (A19), and (A21), we observe that along a BGP wages grow at a common and constant rate:

$$ \frac{{\overset{\bullet }{w}}_{nt}}{w_{nt}}=\frac{{\overset{\bullet }{w}}_{It}}{w_{It}}=\frac{{\overset{\bullet }{w}}_{Yt}}{w_{Yt}}\equiv \frac{{\overset{\bullet }{w}}_t}{w_t}=R{\gamma}_n. $$
(A22)

From (17) in the text and (A15) in this appendix we conclude that along a BGP:

$$ \frac{{\overset{\bullet }{a}}_t}{a_t}\equiv {\gamma}_a={\gamma}_H+R{\gamma}_n-{g}_L,\kern1em {a}_t\equiv {A}_t/{L}_t,\kern1em {g}_L\equiv {\overset{\bullet }{L}}_t/{L}_t. $$
(A23)

Merging (12) in the main text and (A11) in this appendix yields:

$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=-{\gamma}_a+u\frac{h_t{w}_t}{a_t}-\left(1+\varOmega {g}_L\right)\frac{c_t}{a_t}. $$
(A24)

Similarly, from the combination of (13) in the body-text and (A9) in this appendix we get:

$$ \frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}=-{\gamma}_h-\sigma u,\kern1em {\gamma}_h\equiv {\overset{\bullet }{h}}_t/{h}_t,\kern1em {h}_t\equiv {H}_t/{L}_t. $$
(A25)

Note that (A25) and (A9), taken together, confirm that in this economy:

\( {\overset{\bullet }{h}}_t=\left[\sigma \left(1-u\right)-\xi {g}_L\right]{h}_t \), see Eq. (13) in the text.

Combination of Eqs. (A9), (A10), (A22), (A23), (A24), and Eq. (13) in the body-text, finally leads to:

$$ \frac{c_t}{a_t}=\frac{u}{\left(1+\varOmega {g}_L\right)}\left[\sigma +\frac{h_t{w}_t}{a_t}\right], $$
(A26)

where γh = γH − gL has been used.

By employing, again, Eq. (13) in the main text, Eq. (A22), Eq. (A23) and the fact that γh = γH − gL, from (A26) one immediately concludes that:

$$ \frac{c_t}{a_t}=\frac{u}{\left(1+\Omega {g}_L\right)}\left[\sigma +\underset{\equiv \Gamma (0)>0}{\underbrace{\frac{h(0)w(0)}{a(0)}}}\right] $$
(A27)

where h(0) > 0, a(0) > 0 and w(0) > 0 are the given initial values (i.e., at t = 0) of ht, at and wt, respectively.

Eqs. (A27) implies that along a BGP (where u and gL are constant):

$$ {\gamma}_c={\gamma}_a. $$
(A28)

Eq. (A26) can be recast as:

$$ {h}_t{w}_t=-\sigma {a}_t+\left(\frac{1+\varOmega {g}_L}{u}\right){c}_t. $$
(A29)

Similarly, Eq. (A8) can be rewritten as:

$$ {h}_t{w}_t=\frac{\sigma }{\xi}\left[\frac{\upsilon }{g_L}-\varOmega \left(1-\upsilon \right)\right]{c}_t. $$
(A8.1)

After equating (A29) and (A8.1), we are able to compute the endogenous value of ct/at:

$$ \frac{c_t}{a_t}=\frac{\sigma }{\left(\frac{1+\varOmega {g}_L}{u}\right)-\frac{\sigma }{\xi}\left[\frac{\upsilon }{g_L}-\varOmega \left(1-\upsilon \right)\right]}. $$
(A30)

Eq. (A30) confirms that, along a BGP, ct and at grow at a common rate, see (A27) and (A28) above. This happens because the ratio of these two variables is constant in the BGP equilibrium. Finally, we equate (A30) and (A27) and get:

$$ \varGamma (0)\xi \varOmega {g}_L^2+\left\{\varGamma (0)\left[\xi +\rho \varOmega \left(1-\upsilon \right)\right]+\rho \sigma \varOmega \left(1-\upsilon \right)\right\}{g}_L-\rho \upsilon \left[\sigma +\varGamma (0)\right]=0. $$
(A31)

Eq. (A31) gives the endogenous value of gL along a BGP-equilibrium. If we simplify the analysis by normalizing all the relevant initial conditions to one, in such a way that h(0) = w(0) = a(0) = 1 > 0, and therefore \( \varGamma (0)\equiv \frac{h(0)w(0)}{a(0)}=1 \), then Eq. (A31) easily becomes:

$$ \xi \varOmega {g}_L^2+\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]{g}_L-\rho \upsilon \left(1+\sigma \right)=0. $$
(A31.1)

The solution to this equation is:

$$ {g}_L=\frac{-\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]\pm \sqrt{{\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\xi \varOmega \rho \upsilon \left(1+\sigma \right)}}{2\xi \varOmega} $$
(A31.2)

One root is positive and the other one is negative. The positive root is:

$$ {g}_L=\frac{-\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]+\sqrt{{\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\xi \varOmega \rho \upsilon \left(1+\sigma \right)}}{2\xi \varOmega}. $$

Using Eqs. (1) and (4.1) in the body-text and the definitions of \( {y}_t\equiv \frac{Y_t}{L_t} \), \( R\equiv 1+\overline{\alpha}-Z>0 \), and \( {\overset{\bullet }{L}}_t/{L}_t\equiv {g}_L \), we obtain the growth rate of real per capita output along a BGP:

$$ {\gamma}_y\equiv \frac{{\overset{\bullet }{y}}_t}{y_t}=\frac{{\overset{\bullet }{Y}}_t}{Y_t}-{g}_L={\gamma}_H+R{\gamma}_n-{g}_L={\gamma}_a={\gamma}_c, $$
(A32)

see Eqs. (A23) and (A28).

Eq. (A32) is important because it says that along a BGP, per capita income (y), per capita asset holdings (a) and per capita consumption (c) all grow at the same rate.

We are now able to compute the BGP-equilibrium values of sn, sI and \( {s}_{y.} \) Eq. (14) in the main text suggests that:

$$ u={s}_Y+{s}_I+{s}_n. $$
(A33)

Using the fact that (see A20) \( {s}_Y=\left(\frac{1-Z}{Z^2}\right){s}_I \) in the expression above we obtain:

$$ {s}_I=\left(\frac{Z^2}{1-Z+{Z}^2}\right)\left(u-{s}_n\right). $$
(A34)

Hence:

$$ {s}_Y=\left(\frac{1-Z}{1-Z+{Z}^2}\right)\left(u-{s}_n\right). $$
(A35)

According to (A20), however, it is also true that:

$$ {s}_Y\equiv \frac{H_{Yt}}{H_t}=\frac{\chi }{Z}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}}\frac{n_t^{1-\eta }}{H_t^{\mu -\varPhi }}. $$

Equating the last expression to (A35) yields:

$$ \frac{H_t^{\mu -\varPhi }}{n_t^{1-\eta }}=\frac{\chi \left(1-Z+{Z}^2\right)}{Z\left(1-Z\right)}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}\left(u-{s}_n\right)}. $$
(A36)

From Eq. (8) in the body-text we get:

$$ \frac{H_t^{\mu -\varPhi }}{n_t^{1-\eta }}=\frac{\chi }{s_n^{\mu }}{\gamma}_n. $$
(A37)

Equalization of (A36) and (A37) finally leads to:

$$ {s}_n=\frac{Z\left(1-Z\right){\gamma}_n}{\left(1-Z+{Z}^2\right)\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]+Z\left(1-Z\right){\gamma}_n}\cdot u. $$
(A38)

Given sn above, it is now possible to compute the BGP ratio \( \frac{H_t^{\mu -\varPhi }}{n_t^{1-\eta }} \) (by using either Eq. A37 or Eq. A36), along with sI and sY (see Eqs. A34 and A35).

Note that, when γn > 0, Z ∈ (0; 1), and Eq. (A15.1) is satisfied, then the following inequality does hold:

$$ 0<\varXi \equiv \frac{Z\left(1-Z\right){\gamma}_n}{\left(1-Z+{Z}^2\right)\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]+Z\left(1-Z\right){\gamma}_n}<1. $$

With u ∈ (0; 1), this implies that:

$$ 0<{s}_n\equiv \varXi u<1,\kern1.50em \mathrm{and}\kern1.50em 0<\left(u-{s}_N\right)=u-\varXi u=u\left(1-\varXi \right)<1. $$

The last result allows sY and sI to be also strictly between zero and one along a BGP. In Eq. (A37) it is evident that, with χ > 0, γn > 0, and sn ∈ (0; 1), the ratio \( {H}_t^{\mu -\varPhi }/{n}_t^{1-\eta } \) is always positive.

Finally, using Eqs. (A9), (A10), (A22), (A23), (13.1) in the body-text, and the definition of ht ≡ Ht/Lt, it can be easily shown that along a BGP the two transversality conditions:

$$ \underset{\kern0.65em t\to +\infty }{\lim }{\lambda}_{at}{a}_t=0\kern2.15em \mathrm{and}\kern2.15em \underset{\kern0.65em t\to +\infty }{\lim }{\lambda}_{ht}{h}_t=0, $$

are simultaneously checked when:

$$ \sigma \cdot u>0, $$

for any λa(0) > 0, λh(0) > 0, a(0) > 0, and h(0) > 0.

The condition σ ⋅ u > 0 always holds in our model, as σ ⋅ u = ρ > 0 (see A14).

In the model, as a particular case, Ω can also be equal to zero (see Eq. 12). When Ω = 0 the law of motion of per capita asset holdings becomes:

$$ {\overset{\bullet }{a}}_t=\left({r}_t{a}_t+{u}_t{h}_t{w}_t\right)-{c}_t,\kern1.50em a(0)>0. $$

Economically this means that, following an agent’s choice of having more children, no dilution effect of population growth (in the form of an additional consumption cost) would hit per capita asset investment. In other words, as to the speed at which investment in asset-holdings by an average individual in the population occurs, a change in the population size would play no role in this specific case.

When Ω = 0, from Eq. (18) in the main text, one can easily observe that:

$$ \underset{\varOmega \to 0}{\lim }{g}_L=\frac{0}{0}. $$

Since all the major endogenous variables of the model depend on gL (see Proposition 1 in the text), such variables would inevitably take an indeterminate value when gL does so. As a consequence, studying the behavior of our model when Ω = 0 appears to be extremely relevant from an economic as well as an algebraic point of view. In this regard, we apply de l’Hôpital’s rule to Eq. (18) in the text. More formally, call by B(Ω) and E(Ω) the numerator and the denominator of Eq. (18) in the body-text, respectively. Then, since \( \underset{\varOmega \to 0}{\lim }{g}_L\equiv \frac{B\left(\varOmega \right)}{E\left(\varOmega \right)}=\frac{0}{0} \), by the de l’Hôpital’s rule:Footnote 21

$$ \underset{\varOmega \to 0}{\lim }{g}_L=\underset{\varOmega \to 0}{\lim}\frac{B\hbox{'}\left(\varOmega \right)}{E\hbox{'}\left(\varOmega \right)}=\frac{\rho \upsilon \left(1+\sigma \right)}{\xi }. $$

This means that an equilibrium growth rate of the population (gL) does exist, is finite, and is strictly positive when Ω = 0.

The following Lemma introduces explicit constraints on the (relation among the feasible) values of the model’s parameters such that the resultant endogenous variables are economically meaningful when Ω = 0.

LEMMA

Assume Ω = 0. Then,

  • γH, γh, γn, γw, γy = γa = γc, and r are all positive;

  • 0 < (1 − u) < 1;

  • r > γH − (1 − R)γn, (see A15.1)

if the following assumptions on the parameters’ values hold true:

  1. (i)

    0 < ξ < 1;

  2. (ii)

    \( \varUpsilon >\frac{1}{1-\xi }; \)

  3. (iii)

    (σ − ρ) > 0;

  4. (iv)

    \( 0<\upsilon <\frac{\left(\sigma -\rho \right)}{\rho \left(1+\sigma \right)}<\frac{\sigma +\left(\sigma -\rho \right) R\varUpsilon}{\rho \left(1+\sigma \right)\left(1+ R\varUpsilon \right)}. \)

Proof: Immediate from Eqs. (19)–(25) in the main text when Ω = 0 and \( {g}_L=\frac{\rho \upsilon \left(1+\sigma \right)}{\xi } \). ■

From Proposition 2 in the text, ξ ∈ (0; 1) seems the most interesting situation to be analyzed. This is the reason why we still focus on this case here (assumption i above). Also notice that supposing \( \varUpsilon >\frac{1}{1-\xi }>0 \) (assumption ii) is equivalent to postulating \( \mu -\varPhi >\frac{1-\eta }{1-\xi }>0 \), which again implies μ > Φ (exactly as in Proposition 2 in the text).

The next Proposition analyzes the interaction between population growth and economic growth in the BGP equilibrium of this simpler model (Ω = 0).

PROPOSITION

Assume Ω = 0 and\( \varUpsilon >\frac{1}{1-\xi }>0 \)(which still implies μ > Φ). The sign of the relation between population growth and economic growth in the BGP equilibrium crucially continues to depend on the magnitude of ξ. Again, we observe that:

  • $$ \frac{\partial {\gamma}_y}{\partial {g}_L}>0\kern1em \mathrm{if}\kern1em 0<\xi <\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<1; $$
  • $$ \frac{\partial {\gamma}_y}{\partial {g}_L}=0\kern1em \mathrm{if}\kern1em 0<\xi =\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<1; $$
  • $$ \frac{\partial {\gamma}_y}{\partial {g}_L}<0\kern1em \mathrm{if}\kern1em 0<\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<\xi <1. $$

Proof: Again, suppose that there is a change in υ such that gL ultimately varies while at the same time R, ϒ, σ, ρ, and ξ do not. Then:

$$ \frac{\partial {\gamma}_y}{\partial {g}_L}=\frac{\frac{\partial {\gamma}_y}{\partial \upsilon }}{\frac{\partial {g}_L}{\partial \upsilon }}=\frac{-\left[\xi \left(1+ R\varUpsilon \right)- R\varUpsilon \right]\frac{\partial {g}_L}{\partial \upsilon }}{\frac{\partial {g}_L}{\partial \upsilon }}=-\left[\xi \left(1+ R\varUpsilon \right)- R\varUpsilon \right]. $$
$$ {\displaystyle \begin{array}{cc}\mathrm{Therefore}:& \frac{\partial {\gamma}_y}{\partial {g}_L}>0\kern1em \mathrm{if}\kern1em 0<\xi <\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<1,\\ {}& \frac{\partial {\gamma}_y}{\partial {g}_L}=0\kern1em \mathrm{if}\kern1em 0<\xi =\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<1,\\ {}& \kern1.2em \frac{\partial {\gamma}_y}{\partial {g}_L}<0\kern1em \mathrm{if}\kern1em 0<\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}<\xi <1.\kern1em \blacksquare \end{array}} $$

This proposition (obtained when Ω = 0) confirms the results of the most general case of the model (Ω > 0, Proposition 2 in the text): when ξ > 0 population growth operates like a form of depreciation in the per capita human capital investment function. In addition, if ξ is sufficiently small (0 < ξ < 1 – see Proposition 2 in the text and the proposition above), a threshold level of this parameter, \( \overline{\xi}\equiv \left[ R\varUpsilon /\left(1+ R\varUpsilon \right)\right]\in \left(0;1\right) \), does exist such that if ξ is below (respectively, above or equal to) this threshold, then economic growth depends positively (respectively, negatively, or does not depend at all) on population growth.

APPENDIX B: gLas a function of Ω, υ, ρ, σ, and ξ

The graph of:

$$ {g}_L=\frac{-\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]+\sqrt{{\left[\xi +\rho \varOmega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\xi \varOmega \rho \upsilon \left(1+\sigma \right)}}{2\xi \varOmega}, $$

as a function of Ω, υ, ρ, σ, and ξ, is reported below. In what follows, we use:

$$ {\displaystyle \begin{array}{l}\rho =0.038\\ {}\sigma =0.0505\\ {}\xi =0.75\\ {}\varOmega =0.5\\ {}\upsilon =0.2\end{array}} $$

Under this parameter-constellation, realistic values for the main endogenous variables of the model can be obtained (see Table 1 in the text).

It is apparent from the graphs above that in a BGP-equilibrium the endogenous birth rate, gL, increases with the preference of parents for children (υ), but decreases with the consumption-cost of each child (Ω), as stated in the main text.

The intuition for these results goes as follows. If Ω increases, this raises the costs of each child in terms of foregone consumption such that individuals would choose to reduce fertility (Figures 1 and 2); if υ increases, parents derive more utility per child and therefore fertility rises (Figures 3 and 4); if ρ increases, parents are more impatient, educate themselves less and choose to have more children because this raises utility (Figure 5); if σ is higher, education for the children is easier and this mitigates the human capital dilution effect of population growth such that parents choose to have more children (Figure 6); finally, if ξ increases, the dilution effect of population growth on human capital accumulation becomes stronger such that parents substitute quality for quantity in their decision of having children and therefore reduce fertility (Figure 7).

Fig. 1
figure 1

The graph of gL(vertical axis) as a function of Ω (horizontal axis), Ω ∈ [0.000000000001; 1000]

Fig. 2
figure 2

The graph of gL(vertical axis) as a function of Ω (horizontal axis), Ω ∈ [0.000000000001; 10]

Fig. 3
figure 3

The graph of gL(vertical axis) as a function of υ (horizontal axis), υ ∈ [0.000000000001; 1000]

Fig. 4
figure 4

The graph of gL(vertical axis) as a function of υ (horizontal axis), υ ∈ [0.000000000001; 1]

Fig. 5
figure 5

The graph of gL(vertical axis) as a function of ρ (horizontal axis), ρ ∈ [0.000000000001; 1]

Fig. 6
figure 6

The graph o f gL(vertical axis) as a function of σ (horizontal axis), σ ∈ [0.000000000001; 1000]

Fig. 7
figure 7

The graph of gL(vertical axis) as a function of ξ (horizontal axis), ξ ∈ [0.000000000001; 1000]

APPENDIX C: INTRODUCING CHILD–COSTS IN TERMS OF PARENTAL TIME

The objective of this appendix is to see what happens when one introduces additional time-costs of fertility. At this aim, assume that the original law of motion of per capita human capital (Eq. 13 in the text) becomes:

$$ {\overset{\bullet }{h}}_t=\left[\sigma \left(1-\varphi {g}_{Lt}-{u}_t\right)-\xi {g}_{Lt}\right]{h}_t,\kern2em \sigma >0,\kern2em \xi >0,\kern2em \varphi >0 $$
(13a)

where the term φgLt identifies another source of costs (in terms of time subtracted to the activity of skill acquisition, 1 − ut) related to the choice of having more children. As in the original version of the paper, here we continue to assume that:

  • Individuals have a Millian-type intertemporal utility function of the form:

$$ U\equiv \underset{0}{\overset{\infty }{\int }}\left[\log \left({c}_t\right)+\upsilon \log \left({g}_{Lt}\right)\right]{e}^{-\rho t} dt,\kern1em \rho >0,\kern1em \upsilon >0. $$
  • The flow budget constraint is:

$$ {\overset{\bullet }{a}}_t={r}_t{a}_t+{u}_t{h}_t{w}_t-\left(1+\varOmega {g}_{Lt}\right){c}_t,\kern2em \varOmega \ge 0. $$

Finally, we continue to use the same definition of Balanced Growth Path (BGP) equilibrium provided in the main text (Section 3).

The Hamiltonian function (Jt) of the dynamic, intertemporal problem now reads as:

$$ {J}_t=\left[\log \left({c}_t\right)+\upsilon \log \left({g}_{Lt}\right)\right]{e}^{-\rho t}+{\lambda}_{at}\left[{r}_t{a}_t+{u}_t{h}_t{w}_t-\left(1+\varOmega {g}_{Lt}\right){c}_t\right]+{\lambda}_{ht}\left[\sigma \left(1-\varphi {g}_{Lt}-{u}_t\right)-\xi {g}_{Lt}\right]{h}_t, $$

where λat and λht continue to be the co-state variables associated with the two state variables (at and ht, respectively). The necessary first order conditions are:

$$ \frac{\partial {J}_t}{\partial {c}_t}=0\kern1em \iff \kern1em \frac{e^{-\rho t}}{c_t}={\lambda}_{at}\left(1+\varOmega {g}_{Lt}\right) $$
(C1)
$$ \frac{\partial {J}_t}{\partial {g}_{Lt}}=0\kern1em \iff \kern1em \frac{\upsilon {e}^{-\rho t}}{g_{Lt}}-\varOmega {\lambda}_{at}{c}_t-{\lambda}_{ht}\left(\sigma \varphi +\xi \right){h}_t=0 $$
(C2)
$$ \frac{\partial {J}_t}{\partial {u}_t}=0\kern1em \iff \kern1em {\lambda}_{at}=\sigma \frac{\lambda_{ht}}{w_t} $$
(C3)
$$ \frac{\partial {J}_t}{\partial {a}_t}=-{\overset{\bullet }{\lambda}}_{at}\kern1em \iff \kern1em {\lambda}_{at}{r}_t=-{\overset{\bullet }{\lambda}}_{at} $$
(C4)
$$ \frac{\partial {J}_t}{\partial {h}_t}=-{\overset{\bullet }{\lambda}}_{ht}\kern1em \iff \kern1em {\lambda}_{at}{u}_t{w}_t+{\lambda}_{ht}\left[\sigma \left(1-\varphi {g}_{Lt}-{u}_t\right)-\xi {g}_{Lt}\right]=-{\overset{\bullet }{\lambda}}_{ht}, $$
(C5)

along with the two transversality conditions:

$$ \underset{t\to +\infty }{\lim }{\lambda}_{at}{a}_t=0,\kern5.25em \underset{t\to +\infty }{\lim }{\lambda}_{ht}{h}_t=0, $$

and the initial conditions:

$$ a(0)>0,\kern5.25em h(0)>0. $$

From (C1) it follows that:

$$ {e}^{-\rho t}={\lambda}_{at}\left(1+\Omega {\mathrm{g}}_{Lt}\right){c}_t. $$
(C6)

Plugging this expression into (C2) yields:

$$ {c}_t{\lambda}_{at}\left[\frac{\upsilon }{{\mathrm{g}}_{Lt}}-\Omega \left(1-\upsilon \right)\right]={\lambda}_{ht}\left(\sigma \varphi +\xi \right){h}_t. $$
(C7)

Employing (C3) into (C7) delivers:

$$ \left[\frac{\upsilon }{{\mathrm{g}}_{Lt}}-\Omega \left(1-\upsilon \right)\right]=\frac{\left(\sigma \varphi +\xi \right)}{\sigma}\frac{h_t{w}_t}{c_t}. $$
(C8)

Using (C3) in (C5), instead, gives:

$$ \frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}=\left(\sigma \varphi +\xi \right){\mathrm{g}}_{Lt}-\sigma $$
(C9)

Along a BGP all variables depending on time (including Lt) grow at constant (possibly positive) exponential rates. Therefore, in a BGP equilibrium (where gLt = gL, . ∀ t ≥ 0) Eq. (C9) suggests that \( {\overset{\bullet }{\lambda}}_{ht}/{\lambda}_{ht} \) is also constant. Eqs. (C3) and (C4) imply, respectively:

$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=\frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}-\frac{{\overset{\bullet }{w}}_t}{w_t}, $$
(C10)
$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=-{r}_t. $$
(C11)

Combining together (C9), (C10), and (C11) leads immediately to:

$$ {r}_t=\sigma +\frac{{\overset{\bullet }{w}}_t}{w_t}-\left(\sigma \varphi +\xi \right){\mathrm{g}}_L $$
(C12)

Since human capital is perfectly mobile across sectors, at equilibrium it will be rewarded according to the same wage, i.e.: wYt = wIt = wnt ≡ wt. Moreover, along the BGP this common wage will grow at a constant exponential rate, implying that \( \frac{{\overset{\bullet }{w}}_{Yt}}{w_{Yt}}=\frac{{\overset{\bullet }{w}}_{It}}{w_{It}}=\frac{{\overset{\bullet }{w}}_{nt}}{w_{nt}}\equiv \frac{{\overset{\bullet }{w}}_t}{w_t} \) is constant. Accordingly, Eq. (C12) implies that in a BGP equilibrium the real rate of return on asset holdings, r , will be constant, as well.

From (C1) and (C4) together:

$$ \frac{\overset{\bullet }{c_t}}{c_t}=r-\rho, \kern1em {c}_t\equiv \frac{C_t}{L_t}. $$
(C13)

By combining (C7), (C13), (C4), (C9), and (13a) in this appendix, after some simple algebra it is finally possible to obtain:

$$ {u}_t=\frac{\rho }{\sigma }=u,\kern0.5em \forall t\ge 0, $$
(C14)

Eq. (C14) suggests that along a BGP the fraction of time allocated to productive activities is also constant. Making use of Eqs. (6) and (10) in the main text, we find that along a BGP:

$$ {V}_{nt}=Z\left(1-Z\right){\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z\frac{n_t^R}{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]},\kern1em R\equiv 1+\overline{a}-Z>0;\kern1em \frac{{\overset{\bullet }{n}}_t}{n_t}\equiv {\gamma}_n;\kern1em \frac{\overset{\bullet }{H_t}}{H_t}\equiv {\gamma}_{H\cdotp } $$
(C15)

Given Vnt, from Eq. (9.1) in the main text:

$$ {w}_{nt}=\frac{Z}{\chi}\left(1-Z\right){s}_n^{\mu -1}{H}_t^{\mu -1-\Phi}{n}_t^{\eta }{\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z\frac{n_t^R}{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}, $$
(C16)

where sn ≡ Hnt/Ht is (by definition) constant in a BGP-equilibrium.

We can now use Eqs. (5), (2) and (4.1) in the main text, obtaining:

$$ {w}_{It}={Z}^2{\left(\frac{H_{Yt}}{n_t}\right)}^{1-Z}{\left(\frac{H_{It}}{n_t}\right)}^{Z-1}{n}_t^R. $$
(C17)

From Eq. (15) in the main text, by equalizing (C16) and (C17) in this appendix one gets:

$$ {s}_I\equiv \frac{H_{It}}{H_t}=\frac{Z\chi}{\left(1-Z\right)}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}}\frac{n_t^{1-\eta }}{H_t^{\mu -\varPhi }}. $$
(C18)

Combining Eqs. (1) and (4.1) in the body-text yields:

$$ {w}_{Yt}\equiv \frac{\partial {Y}_t}{\partial {H}_{Yt}}=\left(1-Z\right){\left(\frac{H_{Yt}}{n_t}\right)}^{-Z}{\left(\frac{H_{It}}{n_t}\right)}^Z{n}_t^R. $$
(C19)

From (16) in the main text and (C18) above, equalization of (C17) and (C19) in this appendix delivers:

$$ {s}_Y\equiv \frac{H_{Yt}}{H_t}=\left(\frac{1-Z}{Z^2}\right){s}_I=\frac{\chi }{Z}\frac{\left[r+\left(1-R\right){\gamma}_n-{\gamma}_H\right]}{s_n^{\mu -1}}\frac{n_t^{1-\eta }}{H_t^{\mu -\Phi}}. $$
(C20)

Along a BGP all variables depending on time grow at constant (possibly positive) exponential rates, and the sectorial shares of human capital employment are also constant. Therefore, from Eq. (8) in the main text it follows that:

$$ \frac{{\overset{\bullet }{n}}_t}{n_t}\equiv {\gamma}_n=\left(\frac{\mu -\varPhi }{1-\eta}\right){\gamma}_H. $$
(C21)

If μ − Φ = 1 − η, we have a very special case of the model in which human capital and technology grow at the same rate γn = γH ≡ γ in the long-run BGP equilibrium. Here we continue to allow for the most general possible case in which: μ ≠ Φ ≠ Φ + 1 − η.

Using Eqs. (C16), (C17), (C19) and (C21), we observe that along a BGP wages grow at a common and constant rate:

$$ \frac{{\overset{\bullet }{w}}_{nt}}{w_{nt}}=\frac{{\overset{\bullet }{w}}_{It}}{w_{It}}=\frac{{\overset{\bullet }{w}}_{Yt}}{w_{Yt}}\equiv \frac{{\overset{\bullet }{w}}_t}{w_t}=R{\gamma}_n $$
(C22)

From (17) in the main text and (C15) in this appendix we conclude that along a BGP:

$$ \frac{{\overset{\bullet }{a}}_t}{a_t}\equiv {\gamma}_a={\gamma}_H+R{\gamma}_n-{g}_L,\kern1em {a}_t\equiv {A}_t/{L}_t,\kern1em {g}_L\equiv \overset{\bullet }{L_t}/{L}_t. $$
(C23)

Merging (12) in the main text and (C11) in this appendix yields:

$$ \frac{{\overset{\bullet }{\lambda}}_{at}}{\lambda_{at}}=-{\gamma}_a+u\frac{h_t{w}_t}{a_t}-\left(1+\varOmega {g}_L\right)\frac{c_t}{a_t}. $$
(C24)

Similarly, from the combination of (13a) and (C9) in this appendix we get:

$$ \frac{{\overset{\bullet }{\lambda}}_{ht}}{\lambda_{ht}}=-{\gamma}_h-\sigma u,\kern1em {\gamma}_h\equiv \overset{\bullet }{h_t}/{h}_t,\kern1em {h}_t\equiv {H}_t/{L}_t. $$
(C25)

Note that (C25) and (C9), taken together, confirm that in this economy:

\( {\overset{\bullet }{h}}_t=\left[\sigma \left(1-\varphi {g}_L-u\right)-\xi {g}_L\right]{h}_t \), see Eq. (13a) above.

Combination of Eqs. (C9), (C10), (C22), (C23), (C24), and (13a) leads to:

$$ \frac{c_t}{a_t}=\frac{u}{\left(1+\Omega {\mathrm{g}}_L\right)}\left[\sigma +\frac{h_t{w}_t}{a_t}\right], $$
(C26)

where γh = γH − gL has been used.

By employing again Eqs. (C22) and (C23), along with the fact that γh = γH − gL, from (C26) one immediately concludes that:

(C27)

where h(0) > 0, a(0) > 0 and w(0) > 0 are the given initial values (i.e., at t = 0) of ht, at and wt, respectively.

Eqs. (C27) implies that along a BGP (where u and gL are constant):

$$ {\gamma}_c={\gamma}_a. $$
(C28)

Eq. (C26) can be recast as:

$$ {h}_t{w}_t=-\sigma {a}_t+\left(\frac{1+\Omega {g}_L}{u}\right){c}_t. $$
(C29)

Similarly, Eq. (C8) can be rewritten as:

$$ {h}_t{w}_t=\frac{\sigma }{\left(\sigma \varphi +\xi \right)}\left[\frac{\upsilon }{{\mathrm{g}}_L}-\Omega \left(1-\upsilon \right)\right]{c}_t. $$
(C8.1)

After equating (C29) and (C8.1), we are able to compute the endogenous value of ct/at:

$$ \frac{c_t}{a_t}=\frac{\sigma }{\left(\frac{1+\Omega {\mathrm{g}}_L}{u}\right)-\left(\frac{\sigma }{\sigma \varphi +\xi}\right)\left[\frac{\upsilon }{{\mathrm{g}}_L}-\Omega \left(1-\upsilon \right)\right]}. $$
(C30)

Eq. (C30) confirms that, along a BGP, ct and at grow at a common rate, see Eqs. (C27) and (C28) above. This happens because the ratio of these two variables is constant in the BGP equilibrium. Finally, we equate (C30) and (C27) and (after some algebra) get:

$$ \varGamma (0)\left(\sigma \varphi +\xi \right)\Omega {g}_L^2+\left\{\varGamma (0)\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\right]+\rho \sigma \Omega \left(1-\upsilon \right)\right\}{\mathrm{g}}_L-\rho \upsilon \left[\sigma +\varGamma (0)\right]=0. $$
(C31)

Eq. (C31) gives the endogenous value of gL (as a function of some of the model’s parameters and initial conditions) along a BGP-equilibrium. If we simplify the analysis by normalizing all the relevant initial conditions to one, in such a way that \( h(0)=w(0)=a(0)=1=\frac{h(0)w(0)}{a(0)}\equiv \varGamma (0)>0 \), then Eq. (C31) easily becomes:

$$ \left(\sigma \varphi +\xi \right)\Omega {\mathrm{g}}_L^2+\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]{\mathrm{g}}_L-\rho \upsilon \left(1+\sigma \right)=0. $$
(C31.1)

The solution to this equation is:

$$ {g}_L=\frac{-\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]\pm \sqrt{{\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\left(\sigma \varphi +\xi \right)\Omega \rho \upsilon \left(1+\sigma \right)}}{2\left(\sigma \varphi +\xi \right)\Omega}. $$
(C31.2)

One root is positive and the other one is negative. The positive root is:

$$ {\displaystyle \begin{array}{c}{g}_L=\frac{-\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]+\sqrt{{\left[\left(\sigma \varphi +\xi \right)+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\left(\sigma \varphi +\xi \right)\Omega \rho \upsilon \left(1+\sigma \right)}}{2\left(\sigma \varphi +\xi \right)\Omega}\\ {}=\frac{-\left[\hat{\xi}+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]+\sqrt{{\left[\hat{\xi}+\rho \Omega \left(1-\upsilon \right)\left(1+\sigma \right)\right]}^2+4\hat{\xi}\Omega \rho \upsilon \left(1+\sigma \right)}}{2\hat{\xi}\Omega},\kern3.25em \hat{\xi}\equiv \left(\sigma \varphi +\xi \right)\kern0.75em \end{array}} $$

Clearly, in the expression above we are considering the specific case where Ω > 0.Footnote 22 This expression coincides exactly with Eq. (18) in the main text whenever the parameter ξ is re-scaled to take explicitly into account also the time-costs of fertility (in the production of per capita human capital), that is, when \( \hat{\xi} \) is defined as: \( \hat{\xi}\equiv \left(\sigma \varphi +\xi \right) \).

Using Eqs. (1) and (4.1) in the main text and the definitions of \( {y}_t\equiv \frac{Y_t}{L_t} \), \( R\equiv 1+\overline{\alpha}-Z>0 \), and \( {\overset{\bullet }{L}}_t/{L}_t\equiv {g}_L, \) we obtain the growth rate of real per capita output along a BGP:

$$ {\gamma}_y\equiv \frac{{\overset{\bullet }{y}}_t}{y_t}=\frac{{\overset{\bullet }{Y}}_t}{Y_t}-{g}_L=\left({\gamma}_H-{g}_L\right)+R{\gamma}_n={\gamma}_h+R{\gamma}_n={\gamma}_a={\gamma}_c, $$
(C32)

see Eqs. (C23) and (C28) above.

Eq. (C32) is important because it says that along a BGP, per capita income (y), per capita asset holdings (a), and per capita consumption (c) all grow at the same rate.

Using (13a), (C14) and (C21) in this appendix, along with the definition of γH = γh + gL, it is possible to re-state (C32) as:

(C33)

where ϒ continues to be defined as (see the main text):

$$ \varUpsilon =\frac{\mu -\varPhi }{1-\eta }. $$

Notice that, while some parameters of the model (namely, ξ, σ, ρ, and now also φ) affect the BGP growth rate of the economy (γy) both directly and indirectly (i.e., through their impact on the birth rate, gL), other parameters (υ and Ω) influence economic growth (γy) only indirectly through their sole effect on gL. Finally, there exists a third set of technological parameters (more precisely: \( \overline{\alpha} \), Ζ, μ, Φ, and η – contributing to define R and ϒ) that show a direct impact on γy, while at the same time having no effect on gL. Using the fact that υ and Ω influence economic growth (γy) only indirectly (i.e., through their effect on gL), suppose that there occurs a change in υFootnote 23 such that gL ultimately varies while at the same time R, ϒ, σ, ρ, ξ, and φ do not change at all. Then, it is possible to observe that:

$$ \frac{\partial {\gamma}_y}{\partial {\mathrm{g}}_L}=\frac{\frac{\partial {\gamma}_y}{\partial \upsilon }}{\frac{\partial {\mathrm{g}}_L}{\partial \upsilon }}=\frac{-\left[\left(\sigma \varphi +\xi \right)\left(1+ R\varUpsilon \right)- R\varUpsilon \right]\frac{\partial {\mathrm{g}}_L}{\partial \upsilon }}{\frac{\partial {\mathrm{g}}_L}{\partial \upsilon }}=-\left[\left(\sigma \varphi +\xi \right)\left(1+ R\varUpsilon \right)- R\varUpsilon \right]. $$

In the model \( R\equiv 1+\overline{\alpha}-Z>0 \). If we continue to assume (as we do in the main text) that ϒ > 0, then we conclude:

  • \( \frac{\partial {\gamma}_y}{\partial {\mathrm{g}}_L}>0\kern0.5em \mathrm{if}\kern0.5em 0<\xi <\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}-\sigma \varphi <1, \)

  • \( \frac{\partial {\gamma}_y}{\partial {\mathrm{g}}_L}=0\kern0.5em \mathrm{if}\kern0.5em 0<\xi =\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}-\sigma \varphi <1, \)

  • \( \frac{\partial {\gamma}_y}{\partial {\mathrm{g}}_L}<0\kern0.5em \mathrm{if}\kern0.5em 0<\overline{\xi}\equiv \frac{R\varUpsilon}{1+ R\varUpsilon}-\sigma \varphi <\xi <1. \)

After noticing that −[(σφ + ξ)(1 + ) − ] < 0 for any ξ ≥ 1, we also have:

  • \( \frac{\partial {\gamma}_y}{\partial {g}_L}<0\kern0.5em \mathrm{if}\kern0.80em \xi \ge 1. \)

These results are qualitatively similar to those presented and discussed in the main text (see Proposition 2, where φ has been taken exactly equal to zero).Footnote 24

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Bucci, A., Prettner, K. Endogenous education and the reversal in the relationship between fertility and economic growth. J Popul Econ 33, 1025–1068 (2020). https://doi.org/10.1007/s00148-019-00762-5

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Keywords

  • Human capital
  • Endogenous fertility
  • R&D-driven productivity growth
  • Non-monotonic growth effects of fertility

JEL classification

  • I25
  • J13
  • O30
  • O41