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

  7. More precisely, in our model, we formalize a trade-off 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 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. 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. 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. 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 \( U_{s}=\int_{s}^{\infty }e^{-\rho \left( t-s\right) }(\ln c_{t}+\eta \ln n_{t}+\varepsilon \ln L_{t})dt\), 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. 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.


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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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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