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R&D, human capital, fertility, and growth


This paper analyzes how the decisions of individuals to have children and acquire skills affect long-term 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 long-run 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 long-term growth, i.e., on how the distortions affect the trade-off between the quantity of offsprings and the quality of the family members.

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  1. 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. 2.

    It should be emphasized that the quality–quantity trade-off 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. 3.

    See, for instance, Kelley (1988) and Ehrlich and Lui (1997) who survey the empirical and theoretical literature dealing with the connections between population and economic growth.

  4. 4.

    See also Young (1998), Peretto (1998), Dinopoulos and Thompson (1998), Howitt (1999), and Chantrel et al. (2010) among others who develop various R&D-based models which share this property.

  5. 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. 6.

    See also Bucci and La Torre (2009) who develop a similar framework as Strulik (2005).

  7. 7.

    More precisely, in our model, we formalize a trade-off between the quantity of offsprings and the quality of the family members.

  8. 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 Sala-i-Martin (2004, Chapter 9), for aggregate purposes, a setup in continuous time is more appropriate. As we are interested in the behavior of economy-wide variables, we then consider a model in continuous time which can be regarded as an approximation of an overlapping framework.

  9. 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. 10.

    It is not crucial to introduce altruism as a causal effect of population growth. As in Palivos (1995), we could assume that individuals simply enjoy to be parents so that the utility function (Eq. 1) depends on the flow of births, n t , rather than on the stock of children, L t .

  11. 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. 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 \( U_{s}=\int_{s}^{\infty }e^{-\rho \left( t-s\right) }(\ln c_{t}+\eta \ln n_{t}+\varepsilon \ln L_{t})dt\), 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 trade-off between the quantity of offsprings and the quality of the family members would be more complicated to determine.

  13. 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.


  1. Aghion P, Howitt P (1992) A model of growth through creative destruction. Econometrica 60(2):323–351

    Article  Google Scholar 

  2. Arnold L (1998) Growth, welfare, and trade in an integrated model of human-capital accumulation and research. J Macroecon 20(1):84–105

    Article  Google Scholar 

  3. Barro RJ (1991) Economic growth in a cross section of countries. Q J Econ 106(2):407–443

    Article  Google Scholar 

  4. Barro RJ, Becker GS (1988) A reformulation of the economic theory of fertility. Q J Econ 53(1):1–25

    Google Scholar 

  5. Barro RJ, Becker GS (1989) Fertility choice in a model of economic growth. Econometrica 57(2):481–501

    Article  Google Scholar 

  6. Barro RJ, Sala-i-Martin X (2004) Economic growth, 2nd edn. MIT, Cambridge

    Google Scholar 

  7. Becker GS (1991) A treatise on the family. Harvard University Press, Cambridge

    Google Scholar 

  8. Becker GS, Murphy KM, Tamura R (1990) Human capital, fertility, and economic growth. J Polit Econ 98(5):S12–S37

    Article  Google Scholar 

  9. Blackburn K, Hung VTY, Pozzolo AF (2000) Research, development and human capital accumulation. J Macroecon 22(2):189–206

    Article  Google Scholar 

  10. Boonprakaikawe J, Tournemaine F (2006) Production and consumption of education in a R&D-based growth model. Scott J Polit Econ 53(5):565–585

    Article  Google Scholar 

  11. Boserup E (1965) The conditions of agricultural growth: the economics of agrarian change under population pressure. Allen, London

    Google Scholar 

  12. Bucci A, La Torre D (2009) Population and economic growth with human and physical capital investments. Int Rev Econ 56(1):17–27

    Article  Google Scholar 

  13. Cass D (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 32(3):233–240

    Article  Google Scholar 

  14. 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)

  15. Connolly M, Peretto P (2003) Industry and the family: two engines of growth. J Econ Growth 8(1):115–148

    Article  Google Scholar 

  16. 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

    Article  Google Scholar 

  17. Dalgaard CJ, Kreiner CT (2001) Is declining productivity inevitable? J Econ Growth 6(3):187–203

    Article  Google Scholar 

  18. Dinopoulos E, Thompson P (1998) Schumpeterian growth without scale effects. J Econ Growth 3(4):313–335

    Article  Google Scholar 

  19. Doepke M (2004) Accounting for fertility decline during the transition to growth. J Econ Growth 9(3):347–383

    Article  Google Scholar 

  20. Easterly W, Rebelo S (1993) Fiscal policy and economic growth: an empirical investigation. J Monet Econ 32(3):417–458

    Article  Google Scholar 

  21. 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

    Article  Google Scholar 

  22. Galor O, Weil DN (1999) From Malthusian stagnation to modern growth. Am Econ Rev 89(2):150–154

    Article  Google Scholar 

  23. Grossman GM, Helpman E (1991) Quality ladders in the theory of growth. Rev Econ Stud 58(1):557–586

    Google Scholar 

  24. Howitt P (1999) Steady endogenous growth with population and R&D inputs growing. J Polit Econ 107(4):715–730

    Article  Google Scholar 

  25. Jones CI (1995a) R&D-based models of economic growth. J Polit Econ 105(4):759–784

    Article  Google Scholar 

  26. Jones CI (1995b) Time series tests of endogenous growth models. Q J Econ 110(2):495–525

    Article  Google Scholar 

  27. 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

    Google Scholar 

  28. Jones CI, Williams JC (1998) Measuring the social return to R&D. Q J Econ 113(4):1119–1135

    Article  Google Scholar 

  29. 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

    Article  Google Scholar 

  30. Kelley AC (1988) Economic consequences of population change in the third world. J Econ Lit 26(4):1685–1728

    Google Scholar 

  31. Kocherlakota N, Yi K (1997) Is there endogenous long-run growth: evidence from OECD countries. J Money Credit Bank 29(2):235–262

    Article  Google Scholar 

  32. Koopmans TC (1965) On the concept of optimal economic growth. In: The economic approach to development planning. North-Holland, Amsterdam, pp 225–300

    Google Scholar 

  33. Kortum S (1997) Research, patenting, and technological change. Econometrica 65(6):1389–1419

    Article  Google Scholar 

  34. Kremer M (1993) Population growth and technological change: one million B.C. to 1990. Q J Econ 108(3):681–716

    Article  Google Scholar 

  35. Lee RD (1988) Induced population growth and induced technological progress: their interaction in the accelerating stage. Math Popul Stud 1(3):265–288

    Article  Google Scholar 

  36. Lucas RE (1988) On the mechanics of economic development. J Monet Econ 22(1):3–42

    Article  Google Scholar 

  37. Mankiw NG, Romer D, Weil DN (1992) A contribution to the empirics of economic growth. Q J Econ 107(2):407–437

    Article  Google Scholar 

  38. Mendoza EG, Milesi-Ferretti GM, Asea P (1997) On the ineffectiveness of tax policy in altering long-run growth: Harbergers superneutrality conjecture. J Public Econ 66(1):99–126

    Article  Google Scholar 

  39. Palivos T (1995) Endogenous fertility, multiple growth paths, and economic convergence. J Econ Dyn Control 19(8):1489–1510

    Article  Google Scholar 

  40. Peretto P (1998) Technological change and population growth. J Econ Growth 3(4):283–311

    Article  Google Scholar 

  41. Peretto P (2003) Fiscal policy and long-run growth in R&D-based models with endogenous market structure. J Econ Growth 8(3):325–347

    Article  Google Scholar 

  42. Redding S (1996) The low-skill, low-quality trap: strategic complementarities between human capital and R&D. Econ J 106(435):458–470

    Article  Google Scholar 

  43. Robinson JA, Srinivasan TN (1997) Long-term consequences of population growth: technological change, natural resources, and the environment. In: Rozenzweig MR, Stark O (eds) Handbook of population and family economics. North-Holland, Amsterdam, pp 1175–1298

    Chapter  Google Scholar 

  44. Romer P (1990) Endogenous technological change. J Polit Econ 98(5):S71–S102

    Article  Google Scholar 

  45. Schou P (2002) Pollution externalities in a model of endogenous fertility and growth. Int Tax Public Financ 9(6):709–725

    Article  Google Scholar 

  46. Segerstrom P (1998) Endogenous growth without scale effects. Am Econ Rev 88(5):1290–1310

    Google Scholar 

  47. Simon JL (1981) The ultimate resource. Princeton University Press, Princeton

    Google Scholar 

  48. Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 70(1):65–94

    Article  Google Scholar 

  49. Strulik H (2005) The role of human capital and population growth in R&D-based models of economic growth. Rev Int Econ 13(1):129–145

    Article  Google Scholar 

  50. Tournemaine F (2007) Can population promote income per-capita growth? a balanced perspective. Econ Bull 15(8):1–7

    Google Scholar 

  51. Tournemaine F (2008) Social aspirations and choice of fertility: why can status motive reduce per-capita growth? J Popul Econ 21(1):49–66

    Article  Google Scholar 

  52. Tournemaine F, Tsoukis C (2010) Status, fertility, growth and the great transition. Singap Econ Rev 55(3):553–574

    Article  Google Scholar 

  53. Young A (1998) Growth without scale effects. J Polit Econ 106(1):41–63

    Article  Google Scholar 

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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.

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Correspondence to Frederic Tournemaine.

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Responsible editor: Alessandro Cigno



Problem of the social planner

In this part, we present the problem of the social planner. We derive the first-order 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:

$$ \begin{array}{rll} \Gamma &=&\ln c_{t}+\varepsilon \ln L_{t}+\lambda _{t}\left[A_{t}\left( l_{Y_{t}}h_{t}L_{t}\right) ^{1-\alpha }\left( x_{t}\right) ^{\alpha }-L_{t}c_{t}-A_{t}x_{t}\right]\\ &&+\,\mu _{t}\delta \left( l_{A_{t}}h_{t}L_{t}\right) ^{\gamma }\left( A_{t}\right) ^{\phi } +\nu _{t}\psi l_{h_{t}}h_{t}+\xi _{t}(n_{t}-m)L_{t}\\ &&+\,\varsigma_{t}\left[1-l_{Y_{t}}-l_{A_{t}}-\left( n_{t}/b\right) ^{1/\theta }-l_{h_{t}}\right], \end{array} $$

where λ t , μ t , ν t , ξ t , and ς t are co-state variables.

The first-order conditions are:

$$ \frac{\partial \Gamma }{\partial c_{t}}=\frac{1}{c_{t}}-\lambda _{t}L_{t}=0, $$
$$ \frac{\partial \Gamma }{\partial l_{Y_{t}}}=\frac{\lambda _{t}(1-\alpha )Y_{t}}{ l_{Y_{t}}}-\varsigma _{t}=0, $$
$$ \frac{\partial \Gamma }{\partial l_{A_{t}}}=\frac{\mu _{t}\gamma \overset{ \bullet }{A_{t}}}{l_{A_{t}}}-\varsigma _{t}=0, $$
$$ \frac{\partial \Gamma }{\partial l_{h_{t}}}=\frac{\nu _{t}\overset{\bullet }{ h_{t}}}{l_{h_{t}}}-\varsigma _{t}=0, $$
$$ \frac{\partial \Gamma }{\partial n_{t}}=\xi _{t}L_{t}-\frac{\varsigma _{t}\left( n_{t}/b\right) ^{1/\theta -1}}{\theta b}=0, $$
$$ \frac{\partial \Gamma }{\partial x_{t}}=\frac{\alpha Y_{t}}{x_{t}}-A_{t}=0, $$
$$ \frac{\partial \Gamma }{\partial A_{t}}=\frac{\lambda _{t}Y_{t}}{A_{t}}+ \frac{\mu _{t}\phi \overset{\bullet }{A_{t}}}{A_{t}}=-\overset{\bullet }{\mu _{t}}+\rho \mu _{t}, $$
$$ \frac{\partial \Gamma }{\partial h_{t}}=\frac{\lambda _{t}(1-\alpha )Y_{t}}{ h_{t}}+\frac{\mu _{t}\gamma \overset{\bullet }{A_{t}}}{h_{t}}+\frac{\nu _{t} \overset{\bullet }{h_{t}}}{h_{t}}=-\overset{\bullet }{\nu _{t}}+\rho \nu _{t}, $$
$$ \frac{\partial \Gamma }{\partial L_{t}}=\frac{\varepsilon }{L_{t}}+\frac{ \lambda _{t}(1-\alpha )Y_{t}}{L_{t}}-\lambda _{t}c_{t}+\frac{\mu _{t}\gamma \overset{\bullet }{A_{t}}}{L_{t}}+\xi _{t}(n_{t}-m)=-\overset{\bullet }{\xi _{t}}+\rho \xi _{t}, $$

along with the transversality conditions:

$$ \underset{t\rightarrow \infty }{\lim}\mu _{t}A_{t}e^{-\rho t}=0, $$
$$ \underset{t\rightarrow \infty }{\lim}\nu _{t}h_{t}e^{-\rho t}=0, $$


$$ \underset{t\rightarrow \infty }{\lim}\xi _{t}L_{t}e^{-\rho t}=0. $$

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:

$$ \alpha Y_{t}=A_{t}x_{t}. $$

Using the resource constraint (Eq. 8), one gets:

$$ L_{t}c_{t}=(1-\alpha )Y_{t}, $$


$$ g_{c}+n-m=g_{x}+g_{A}=g_{Y}. $$

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:

    $$ \frac{\gamma g_{A}l_{Y}}{l_{A}}+(\phi -1)g_{A}=\rho . $$

    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:

    $$ l_{Y}+l_{A}=\frac{\rho }{\psi }. $$

    Combining Eqs. 2325, 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:

    $$ \varepsilon l_{Y}+l_{A_{t}}=\frac{\rho \left( n/b\right) ^{1/\theta -1}}{\theta b}. $$

    Equations 3133, 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.

Steady-state equilibrium

In this section, we characterize the steady-state 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:

$$ g_{c}=r-\left( n-m\right) -\rho . $$

Manipulation of Eqs. 16, 19, and 34 leads to:

$$ \psi \lbrack 1-(n/b)^{1/\theta }]=g_{c}-g_{w}+\rho . $$

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:

$$ l_{Y}+l_{A}=\frac{\rho }{\psi }. $$

Combining Eqs. 17 and 20, one gets:

$$ \varepsilon +\frac{\mu _{t}T_{t}}{L_{t}}+\overset{\bullet }{\left( \xi _{t}L_{t}\right) }=\rho \mu _{t}\left[ d_{t}+\frac{w_{t}h_{t}\left( n/b\right) ^{1/\theta -1}}{b\theta }\right] . $$

Using this result with Eq. 16 differentiated with respect to time, Eq. 34 and the budget constraint, one gets:

$$ \theta b\left( n/b\right) ^{1-1/\theta }\left[ \frac{(\varepsilon -1)c_{t}}{ w_{t}h_{t}}+(1-(n/b)^{1/\theta }-l_{h})\right] +g_{w}+g_{h}-g_{c}=\rho . $$

Using Eq. 8 with Eq. 10 yields:

$$ L_{t}c_{t}=[1-\alpha ^{2}/(1-\tau )]Y_{t}. $$

Using Eq. 9, one obtains:

$$ w_{t}h_{t}L_{t}=\frac{(1-\alpha )Y_{t}}{l_{Y}}. $$

Then, plugging the two preceding results in Eq. 36 leads to:

$$ (\varepsilon -1)l_{Y}\left\{ \frac{1}{(1-\alpha )}-\frac{\alpha ^{2}}{ (1-\alpha )(1-\tau )}\right\} +l_{Y}+l_{A}=\frac{\rho \left( n\right) ^{1/\theta -1}}{\theta b^{1/\theta }}. $$

Using Eqs. 12 and 13, one gets:

$$ \frac{\pi _{xt}}{V_{t}}=\frac{(1-\alpha )\alpha g_{A}Y_{t}}{(1-\tau )(1-\sigma )w_{t}H_{A_{t}}}. $$

Using Eq. 9 and simplifying, one gets:

$$ \frac{\pi _{xt}}{V_{t}}=\frac{\alpha g_{A}l_{Y}}{(1-\tau )(1-\sigma )l_{A}}. $$

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:

$$ \frac{\alpha g_{A}l_{Y}}{(1-\tau )(1-\sigma )l_{A}}-g_{A}=\rho . $$

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.

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Tournemaine, F., Luangaram, P. R&D, human capital, fertility, and growth. J Popul Econ 25, 923–953 (2012).

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  • R&D
  • Human capital
  • Fertility

JEL Classification

  • O31
  • O41