R&D, human capital, fertility, and growth

Abstract

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.

Appendix

1.1 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:

$$ \frac{\partial \Gamma }{\partial c_{t}}=\frac{1}{c_{t}}-\lambda _{t}L_{t}=0, $$

(21)

$$ \frac{\partial \Gamma }{\partial l_{Y_{t}}}=\frac{\lambda _{t}(1-\alpha )Y_{t}}{ l_{Y_{t}}}-\varsigma _{t}=0, $$

(22)

$$ \frac{\partial \Gamma }{\partial l_{A_{t}}}=\frac{\mu _{t}\gamma \overset{ \bullet }{A_{t}}}{l_{A_{t}}}-\varsigma _{t}=0, $$

(23)

$$ \frac{\partial \Gamma }{\partial l_{h_{t}}}=\frac{\nu _{t}\overset{\bullet }{ h_{t}}}{l_{h_{t}}}-\varsigma _{t}=0, $$

(24)

$$ \frac{\partial \Gamma }{\partial n_{t}}=\xi _{t}L_{t}-\frac{\varsigma _{t}\left( n_{t}/b\right) ^{1/\theta -1}}{\theta b}=0, $$

(25)

$$ \frac{\partial \Gamma }{\partial x_{t}}=\frac{\alpha Y_{t}}{x_{t}}-A_{t}=0, $$

(26)

$$ \frac{\partial \Gamma }{\partial A_{t}}=\frac{\lambda _{t}Y_{t}}{A_{t}}+ \frac{\mu _{t}\phi \overset{\bullet }{A_{t}}}{A_{t}}=-\overset{\bullet }{\mu _{t}}+\rho \mu _{t}, $$

(27)

$$ \frac{\partial \Gamma }{\partial h_{t}}=\frac{\lambda _{t}(1-\alpha )Y_{t}}{ h_{t}}+\frac{\mu _{t}\gamma \overset{\bullet }{A_{t}}}{h_{t}}+\frac{\nu _{t} \overset{\bullet }{h_{t}}}{h_{t}}=-\overset{\bullet }{\nu _{t}}+\rho \nu _{t}, $$

(28)

$$ \frac{\partial \Gamma }{\partial L_{t}}=\frac{\varepsilon }{L_{t}}+\frac{ \lambda _{t}(1-\alpha )Y_{t}}{L_{t}}-\lambda _{t}c_{t}+\frac{\mu _{t}\gamma \overset{\bullet }{A_{t}}}{L_{t}}+\xi _{t}(n_{t}-m)=-\overset{\bullet }{\xi _{t}}+\rho \xi _{t}, $$

(29)

along with the transversality conditions:

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

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

and

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

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:

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

(30)

and

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

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:

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

  (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.

1.2 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:

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

(34)

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:

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

(35)

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:

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

(36)

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:

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

(37)

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:

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

(38)

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.