Abstract
This paper analyzes how the decisions of individuals to have children and acquire skills affect longterm growth. We investigate a model in which technical progress, human capital, and population arise endogenously. In such an economy, the presence of distortions (such as monopolistic competition, knowledge spillover, and duplication effects) leads the decentralized longrun growth to be either insufficient or excessive. We show that this result depends on the relative contribution of population and human capital in the determination of longterm growth, i.e., on how the distortions affect the tradeoff between the quantity of offsprings and the quality of the family members.
This is a preview of subscription content, access via your institution.
Notes
 1.
The impact of human capital, a form of which being education, on economic growth has been recognized for years in economic theory. See, for instance, the seminal article by Lucas (1988) which underlines the principal importance of human capital formation for growth and development.
 2.
It should be emphasized that the quality–quantity tradeoff on children has been considered as a factor which has contributed to the transition of economies from a stage of stagnation (poverty trap) to perpetual growth. Beside the seminal article by Becker et al. (1990), see also the articles by Galor and Weil (1999) and Doepke (2004) for a thorough analysis of this issue.
 3.
 4.
 5.
Tournemaine (2007) focuses essentially on the effect of population growth on economic development. The author assumes that knowledge is the outcome of a learning by doing process. That is, the costly aspect of R&D is not taken into account. Moreover, in the present paper, we go further in the analysis as we conduct a welfare analysis and examine the policy implication of subsidies to R&D.
 6.
 7.
More precisely, in our model, we formalize a tradeoff between the quantity of offsprings and the quality of the family members.
 8.
An overlapping setup is useful to study the choice of fertility and the decision of education of individuals because the length of a period has an important meaning. However, as pointed out by Barro and SalaiMartin (2004, Chapter 9), for aggregate purposes, a setup in continuous time is more appropriate. As we are interested in the behavior of economywide variables, we then consider a model in continuous time which can be regarded as an approximation of an overlapping framework.
 9.
As in Jones (2003), the number of children enters only indirectly in the utility (Eq. 1) through the number of surviving descendants, L _{ t }. Although this is a simplifying assumption, it does not affect the main insight and results of the paper. The proof of this property is available from the authors upon request. Footnotes 12 and 13 below describe how the results would change in a more general framework. Interested readers can also refer to Tournemaine (2007) who uses a more general utility function in a model where innovations are the outcome of external effects.
 10.
 11.
One could endogenize the mortality rate. Robinson and Srinivasan (1997) suggest that there are connections between the evolution of the gross domestic product and mortality rate. Likewise, it could be argued that the model should include an upper limit to the number of people on earth: Population cannot grow literally beyond all bounds. These issues go beyond the scope of this paper and thus are left for future research.
 12.
It is worth noting that a more general utility function in which the flow of children, n _{ t }, directly affects utility (i.e., a utility of the form \( U_{s}=\int_{s}^{\infty }e^{\rho \left( ts\right) }(\ln c_{t}+\eta \ln n_{t}+\varepsilon \ln L_{t})dt\), with η > 0) would give similar results. The main difference is that the critical condition on parameters which turns the effects of the policies on the tradeoff between the quantity of offsprings and the quality of the family members would be more complicated to determine.
 13.
This result is not general though. Had we assumed a utility function of the form given in Footnote 12, we would obtain that the policy instruments have an influence on n and l _{ h } even in the case ε = 1. The other results, however, would remain unchanged.
References
Aghion P, Howitt P (1992) A model of growth through creative destruction. Econometrica 60(2):323–351
Arnold L (1998) Growth, welfare, and trade in an integrated model of humancapital accumulation and research. J Macroecon 20(1):84–105
Barro RJ (1991) Economic growth in a cross section of countries. Q J Econ 106(2):407–443
Barro RJ, Becker GS (1988) A reformulation of the economic theory of fertility. Q J Econ 53(1):1–25
Barro RJ, Becker GS (1989) Fertility choice in a model of economic growth. Econometrica 57(2):481–501
Barro RJ, SalaiMartin X (2004) Economic growth, 2nd edn. MIT, Cambridge
Becker GS (1991) A treatise on the family. Harvard University Press, Cambridge
Becker GS, Murphy KM, Tamura R (1990) Human capital, fertility, and economic growth. J Polit Econ 98(5):S12–S37
Blackburn K, Hung VTY, Pozzolo AF (2000) Research, development and human capital accumulation. J Macroecon 22(2):189–206
Boonprakaikawe J, Tournemaine F (2006) Production and consumption of education in a R&Dbased growth model. Scott J Polit Econ 53(5):565–585
Boserup E (1965) The conditions of agricultural growth: the economics of agrarian change under population pressure. Allen, London
Bucci A, La Torre D (2009) Population and economic growth with human and physical capital investments. Int Rev Econ 56(1):17–27
Cass D (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 32(3):233–240
Chantrel E, Grimaud A, Tournemaine F (2010) Pricing knowledge and funding research of new technology sectors in a growth model. J Public Econ Theory (in press)
Connolly M, Peretto P (2003) Industry and the family: two engines of growth. J Econ Growth 8(1):115–148
Crenshaw E, Ameen A, Christenson M (1997) Population dynamics and economic development: age specific population growth and economic growth in developing countries, 1965 to 1990. Am Sociol Rev 62(6):974–984
Dalgaard CJ, Kreiner CT (2001) Is declining productivity inevitable? J Econ Growth 6(3):187–203
Dinopoulos E, Thompson P (1998) Schumpeterian growth without scale effects. J Econ Growth 3(4):313–335
Doepke M (2004) Accounting for fertility decline during the transition to growth. J Econ Growth 9(3):347–383
Easterly W, Rebelo S (1993) Fiscal policy and economic growth: an empirical investigation. J Monet Econ 32(3):417–458
Ehrlich I, Lui F (1997) The problem of population and growth: a review of the literature from Malthus to contemporary models of endogenous population and endogenous growth. J Econ Dyn Control 21(1):205–242
Galor O, Weil DN (1999) From Malthusian stagnation to modern growth. Am Econ Rev 89(2):150–154
Grossman GM, Helpman E (1991) Quality ladders in the theory of growth. Rev Econ Stud 58(1):557–586
Howitt P (1999) Steady endogenous growth with population and R&D inputs growing. J Polit Econ 107(4):715–730
Jones CI (1995a) R&Dbased models of economic growth. J Polit Econ 105(4):759–784
Jones CI (1995b) Time series tests of endogenous growth models. Q J Econ 110(2):495–525
Jones CI (2003) Population and ideas: a theory of endogenous growth. In: Aghion P, Frydman R, Stiglitz J, Woodford M (eds) Knowledge, information, and expectations in modern macroeconomics, in honor of Edmund S Phelps. Princeton University Press, Princeton, pp 498–521
Jones CI, Williams JC (1998) Measuring the social return to R&D. Q J Econ 113(4):1119–1135
Jones CI, Williams JC (2000) Too much of a good thing? The economics of investment in R&D. J Econ Growth 5(1):65–85
Kelley AC (1988) Economic consequences of population change in the third world. J Econ Lit 26(4):1685–1728
Kocherlakota N, Yi K (1997) Is there endogenous longrun growth: evidence from OECD countries. J Money Credit Bank 29(2):235–262
Koopmans TC (1965) On the concept of optimal economic growth. In: The economic approach to development planning. NorthHolland, Amsterdam, pp 225–300
Kortum S (1997) Research, patenting, and technological change. Econometrica 65(6):1389–1419
Kremer M (1993) Population growth and technological change: one million B.C. to 1990. Q J Econ 108(3):681–716
Lee RD (1988) Induced population growth and induced technological progress: their interaction in the accelerating stage. Math Popul Stud 1(3):265–288
Lucas RE (1988) On the mechanics of economic development. J Monet Econ 22(1):3–42
Mankiw NG, Romer D, Weil DN (1992) A contribution to the empirics of economic growth. Q J Econ 107(2):407–437
Mendoza EG, MilesiFerretti GM, Asea P (1997) On the ineffectiveness of tax policy in altering longrun growth: Harbergers superneutrality conjecture. J Public Econ 66(1):99–126
Palivos T (1995) Endogenous fertility, multiple growth paths, and economic convergence. J Econ Dyn Control 19(8):1489–1510
Peretto P (1998) Technological change and population growth. J Econ Growth 3(4):283–311
Peretto P (2003) Fiscal policy and longrun growth in R&Dbased models with endogenous market structure. J Econ Growth 8(3):325–347
Redding S (1996) The lowskill, lowquality trap: strategic complementarities between human capital and R&D. Econ J 106(435):458–470
Robinson JA, Srinivasan TN (1997) Longterm consequences of population growth: technological change, natural resources, and the environment. In: Rozenzweig MR, Stark O (eds) Handbook of population and family economics. NorthHolland, Amsterdam, pp 1175–1298
Romer P (1990) Endogenous technological change. J Polit Econ 98(5):S71–S102
Schou P (2002) Pollution externalities in a model of endogenous fertility and growth. Int Tax Public Financ 9(6):709–725
Segerstrom P (1998) Endogenous growth without scale effects. Am Econ Rev 88(5):1290–1310
Simon JL (1981) The ultimate resource. Princeton University Press, Princeton
Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 70(1):65–94
Strulik H (2005) The role of human capital and population growth in R&Dbased models of economic growth. Rev Int Econ 13(1):129–145
Tournemaine F (2007) Can population promote income percapita growth? a balanced perspective. Econ Bull 15(8):1–7
Tournemaine F (2008) Social aspirations and choice of fertility: why can status motive reduce percapita growth? J Popul Econ 21(1):49–66
Tournemaine F, Tsoukis C (2010) Status, fertility, growth and the great transition. Singap Econ Rev 55(3):553–574
Young A (1998) Growth without scale effects. J Polit Econ 106(1):41–63
Acknowledgements
We would like to thank Juntip Boonprakaikawe and two anonymous referees for helpful comments and suggestions. The first author is grateful to the University of Nottingham (UK) for its support during the redaction of this paper.
Author information
Affiliations
Corresponding author
Additional information
Responsible editor: Alessandro Cigno
Appendix
Appendix
Problem of the social planner
In this part, we present the problem of the social planner. We derive the firstorder conditions and derive some preliminary results that will be useful to compute the growth rate of variables in steady state and the allocation of labor across sectors.
First of all, we must note that intermediate goods are treated symmetrically in the final sector. Then, the social planner allocates the same amount of each to the output sector. One has x _{ t }(i) = x _{ t }. The production function is \(Y_{t}=A_{t}\left( H_{Y_{t}}\right) ^{1\alpha }\left( x_{t}\right) ^{\alpha }\). The resource constraint is Y _{ t } = L _{ t } c _{ t } + A _{ t } x _{ t }. After substitutions, the current value Hamiltonian of the problem is:
where λ _{ t }, μ _{ t }, ν _{ t }, ξ _{ t }, and ς _{ t } are costate variables.
The firstorder conditions are:
along with the transversality conditions:
and
We focus on the balanced growth path, i.e., on the path such that the growth rates and the shares of labor allocated to the different sectors of the economy are constant (\(l_{Y_{t}}=l_{Y}\), \(l_{A_{t}}=l_{A}\), \(l_{n_{t}}=l_{n}\), \( l_{h_{t}}=l_{h}\), n _{ t } = n for all t). From Eq. 26, one has:
Using the resource constraint (Eq. 8), one gets:
and
Moreover, using the preceding property with Eq. 22, one has g _{ ς } = 0 at steady state. Thus, one computes − g _{ μ } = g _{ A }, − g _{ ν } = g _{ h }, and − g _{ ξ } = g _{ L }. Now, we can compute the level of growth of variables and the allocation of labor across sectors. These tasks are carried out next.

Growth rates of variables
Using the R&D technology (Eq. 7), one has g _{ A } = γ(g _{ h } + g _{ L })/(1 − ϕ), where g _{ h } = ψl _{ h } (see Eq. 4) and g _{ L } = n − m (see Eq. 3). Using the production of output (Eq. 6) with the fact that g _{ c } + n − m = g _{ x } + g _{ A } = g _{ Y }, one gets g _{ c } = g _{ A } + g _{ h }.

Allocation of labor
Combining Eqs. 22, 23, and 27, one gets:
$$ \frac{\gamma g_{A}l_{Y}}{l_{A}}+(\phi 1)g_{A}=\rho . $$(31)Combining Eqs. 23, 24, and 28 yields:
$$ \frac{g_{h}l_{Y}}{l_{h}}+\frac{g_{h}l_{A}}{l_{h}}=\rho , $$which simplifies to give:
$$ l_{Y}+l_{A}=\frac{\rho }{\psi }. $$(32)Combining Eqs. 23–25, 29, and 30 leads to:
$$ \frac{(\varepsilon 1)l_{Y}}{\lambda _{t}(1\alpha )Y_{t}}+l_{Y}+l_{A_{t}}= \frac{\rho \left( n/b\right) ^{1/\theta 1}}{\theta b}. $$Using Eqs. 21 and 30, the preceding result yields:
$$ \varepsilon l_{Y}+l_{A_{t}}=\frac{\rho \left( n/b\right) ^{1/\theta 1}}{\theta b}. $$(33)Equations 31–33, the production function of children (Eq. 2), and the labor constraint (Eq. 5) are the equations given in the Proposition 1. They allow to determine the exact values of l _{ Y }, l _{ A }, l _{ n }, l _{ h }, and n.
Steadystate equilibrium
In this section, we characterize the steadystate equilibrium. The growth rate of variables have the same form as those given in “Problem of the social planner” above. The main difference between the two solutions relies on the allocation of labor across sectors. Combining Eqs. 15 and 18, one gets:
Manipulation of Eqs. 16, 19, and 34 leads to:
Using the value of the growth rates computed above, one gets \( (1(n/b)^{1/\theta }l_{h})=\rho /\psi \). Using this condition with Eq. 5, one gets:
Combining Eqs. 17 and 20, one gets:
Using this result with Eq. 16 differentiated with respect to time, Eq. 34 and the budget constraint, one gets:
Using Eq. 8 with Eq. 10 yields:
Using Eq. 9, one obtains:
Then, plugging the two preceding results in Eq. 36 leads to:
Using Eqs. 12 and 13, one gets:
Using Eq. 9 and simplifying, one gets:
Differentiating Eq. 13 with respect to time, one obtains g _{ V } = g _{ Y } − g _{ A }. Using r _{ t } = π _{ xt }/V _{ t } + g _{ V } = g _{ c } + (n − m) + ρ (see Eq. 14), one gets:
Equations 35, 37, and 38 depend on l _{ Y }, l _{ A }, l _{ h }, and n. Therefore, using the individual constraint of labor (Eq. 5) with the technology of production of children (Eq. 2), one can determine the values of l _{ Y }, l _{ A }, l _{ h }, l _{ n } , and n.
Rights and permissions
About this article
Cite this article
Tournemaine, F., Luangaram, P. R&D, human capital, fertility, and growth. J Popul Econ 25, 923–953 (2012). https://doi.org/10.1007/s0014801003464
Received:
Accepted:
Published:
Issue Date:
Keywords
 R&D
 Human capital
 Fertility
JEL Classification
 O31
 O41