2.1 Heterogeneous households

There is a unit continuum of households indexed by $i\in \lbrack 0,1]$ . Within household $i$ , the utility of an individual who works at time $t$ is given byFootnote ⁴

(1) \begin{equation} U^{t}(i)=u\left [ n_{t}(i),h_{t+1}(i),c_{t+1}(i)\right ] =\eta \ln n_{t}(i)+\gamma \ln h_{t+1}(i)+\ln c_{t+1}(i)\text {,} \end{equation}

where $c_{t+1}(i)$ is the individual’s consumption at time $t+1$ , $n_{t}(i)$ denotes the number of children the individual has at time $t$ , $\eta \gt 0$ is the fertility preference parameter, $h_{t+1}(i)$ denotes the level of human capital that the individual passes onto each child, and $\gamma$ is the quality preference parameter. We assume that all individuals within the same household $i$ have the same level of human capital at time 0. Then, they will also have the same level of human capital for all $t$ as an endogenous outcome.

The individual allocates $e_{t}(i)$ units of time to her children’s education. The accumulation equation of human capital is given byFootnote ⁵

(2) \begin{equation} h_{t+1}(i)=\phi (i)e_{t}(i)+(1-\delta )h_{t}(i)\text {,} \end{equation}

where the parameter $\delta \in (0,1)$ is the depreciation rate of human capital that a generation passes onto the next.Footnote ^{6 ,} Footnote ⁷ As for the ability parameter $\phi (i)\gt 0$ of household $i$ ,Footnote ⁸ it is heterogeneous across households and follows a general distribution with the following mean:Footnote ⁹

\begin{equation*} \overline {\phi }\equiv \int _{0}^{1}\phi (i)di\text {.} \end{equation*}

The heterogeneity of households is captured by their differences in $\phi (i)$ , which in turn give rise to an endogenous distribution of human capital. We focus on heterogeneity in $\phi (i)$ because it allows for a stationary distribution of the population share of different households in the long run, whereas heterogeneity in other parameters, such as $\eta$ or $\gamma$ , imply that households with the largest $\eta$ or smallest $\gamma$ would dominate the population in the long run.

An individual in household $i$ allocates $1-e_{t}(i)-\sigma n_{t}(i)$ units of time to work and earns $w_{t}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)$ as real wage income, where the parameter $\sigma \in (0,1)$ determines the time cost $\sigma n_{t}(i)$ of fertility.Footnote ¹⁰ For simplicity, we assume that there are economies of scale in the time spent in educating children within a family, and the cost of having more children is reflected in the time cost of childrearing.Footnote ¹¹

The individual devotes her entire wage income to saving at time $t$ and consumes the return at time $t+1$ :Footnote ¹²

(3) \begin{equation} c_{t+1}(i)=(1+r_{t+1})w_{t}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)\text {,} \end{equation}

where $r_{t+1}$ is the real interest rate. Substituting (2) and (3) into (1), the individual maximizes

\begin{align*} \underset {e_{t}(i),\text { }n_{t}(i)}{\max }\text { }U^{t}(i)&=\eta \ln n_{t}(i)+\gamma \ln \left [ \phi (i)e_{t}(i)+(1-\delta )h_{t}(i)\right ] \\ \nonumber &\quad+ \ln \left \{ (1+r_{t+1})w_{t}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)\right \} \text {,} \end{align*}

taking $\{r_{t+1},w_{t},h_{t}(i)\}$ as given. The utility-maximizing level of fertility $n_{t}(i)$ is

(4) \begin{equation} n_{t}(i)=\frac {\eta }{\sigma (1+\eta +\gamma )}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] \text {,} \end{equation}

which is decreasing in $\phi (i)$ but increasing in $h_{t}(i)$ . In other words, households with a lower ability to accumulate human capital and a higher level of human capital choose to have more children. In (4), fertility $n_{t}(i)$ is decreasing in $\phi (i)/h_{t}(i)$ . As we will show, households with higher $\phi (i)$ have higher $h_{t}(i)$ and also higher $\phi (i)/h_{t}(i)$ before the level of human capital reaches the steady state, at which point all households share the same $\phi (i)/h_{t}(i)$ . Therefore, households with higher ability $\phi (i)$ generally have higher human capital $h_{t}(i)$ and lower fertility $n_{t}(i)$ , generating a negative relationship between these two variables. To understand this negative relationship, we also derive the utility-maximizing level of education $e_{t}(i)$ asFootnote ¹³

(5) \begin{equation} e_{t}(i)=\frac {1}{1+\eta +\gamma }\left [ \gamma -(1+\eta )(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] \text {,} \end{equation}

which is increasing in $\phi (i)$ but decreasing in $h_{t}(i)$ . In summary, for a given $h_{t}(i)$ , households with a larger $\phi (i)$ choose a higher level of education $e_{t}(i)$ but a smaller number $n_{t}(i)$ of children, reflecting the quality-quantity tradeoff. Given the same initial human capital $h_{0}(i)=h_{0}$ , differences in education ability $\phi (i)$ give rise to differences in education level $e_{t}(i)$ .Footnote ¹⁴

Substituting (5) into (2) yields the autonomous and stable dynamics of human capital as

(6) \begin{equation} h_{t+1}(i)=\frac {\gamma }{1+\eta +\gamma }\left [ \phi (i)+(1-\delta )h_{t}(i)\right ] \text {,} \end{equation}

where $h_{t+1}(i)$ is increasing in $\phi (i)$ and $h_{t}(i)$ . The total amount of human capital in the economy at time $t$ is

\begin{equation*} H_{t}=\int _{0}^{1}h_{t}(i)L_{t}(i)di\text {,} \end{equation*}

where $L_{t}(i)$ is the working-age population size of household $i$ . The law of motion for $L_{t}(i)$ is

(7) \begin{equation} L_{t+1}(i)=n_{t}(i)L_{t}(i)=\frac {\eta }{\sigma (1+\eta +\gamma )}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] L_{t}(i)\text {,} \end{equation}

and the size of the aggregate labor force in the economy at time $t$ is

\begin{equation*} L_{t}=\int _{0}^{1}L_{t}(i)di\text {.} \end{equation*}

Let’s define $s_{t}(i)\equiv L_{t}(i)/L_{t}$ as the working-age-population (i.e., labor) share of household $i$ .

Lemma 1. The labor share $s_{t}(i)$ of household $i$ at time $t\geq 1$ is given by

\begin{equation*} s_{t}(i)=\frac {\prod \nolimits _{\tau =0}^{t-1}n_{\tau }(i)L_{0}(i)}{ \int _{0}^{1}\prod \nolimits _{\tau =0}^{t-1}n_{\tau }(i)L_{0}(i)di}\text {,} \end{equation*}

where the fertility decision $n_{t}(i)$ of household $i$ at time $t\geq 1$ is given by

\begin{equation*} n_{t}(i)=\frac {\eta }{\sigma (1+\eta +\gamma )}\left \{ \sum _{\tau =0}^{t-1}\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{\tau }+\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{t}\left [ 1+(1-\delta )\frac {h_{0}(i)}{\phi (i)}\right ] \right \} \text {,} \end{equation*}

which is a decreasing function of $\phi (i)/h_{0}(i)$ .

Notice that changes to $n_{\tau }(i)$ in any one period will affect $s_{t}(i)$ in all future generations. The reason is general and does not depend on the specific assumptions of this model: a temporary growth effect has a permanent level effect. Therefore, if the fertility rate of an ability group drops temporarily, this group would ceteris paribus forever have a lower population share than it would otherwise have had. As we will later see, if the high-ability household experiences a temporary reproduction loss, the economy will have a lower share of high-ability people forever. We will also show that this loss will permanently lower human capital, innovation and growth.

2.2 Final good

Perfectly competitive firms use the following production function to produce final good $Y_{t}$ , which is chosen as the numeraire:

(8) \begin{equation} Y_{t}=H_{Y,t}^{1-\alpha }\int _{0}^{N_{t}}X_{t}^{\alpha }(j)dj\text {,} \end{equation}

where the parameter $\alpha \in (0,1)$ determines production labor intensity $1-\alpha$ , and $H_{Y,t}$ denotes human-capital-embodied production labor. $X_{t}(j)$ denotes a continuum of differentiated intermediate goods indexed by $j\in \lbrack 0,N_{t}]$ . Firms maximize profit, and the conditional demand functions for $H_{Y,t}$ and $X_{t}(j)$ are given by

(9) \begin{equation} w_{t}=(1-\alpha )\frac {Y_{t}}{H_{Y,t}}\text {,} \end{equation}

(10) \begin{equation} p_{t}(j)=\alpha \left [ \frac {H_{Y,t}}{X_{t}(j)}\right ] ^{1-\alpha }\text {.} \end{equation}

2.3 Intermediate goods

Each intermediate good $j$ is produced by a monopolistic firm, which uses a one-to-one linear production function that transforms $X_{t}(j)$ units of final good into $X_{t}(j)$ units of intermediate good $j\in \lbrack 0,N_{t}]$ . The profit function is

(11) \begin{equation} \pi _{t}(j)=p_{t}(j)X_{t}(j)-X_{t}(j)\text {,} \end{equation}

where the marginal cost of production is constant and equal to one (recall that final good is the numeraire). The monopolist maximizes (11) subject to (10) to derive the monopolistic price as

(12) \begin{equation} p_{t}(j)=\min \left \{ \mu ,\frac {1}{\alpha }\right \} =\mu \gt 1\text {,} \end{equation}

where $\mu \leq 1/\alpha$ is a patent policy parameter as in Li (Reference Li2001) and Goh and Olivier (Reference Goh and Olivier2002). One can show that $X_{t}(j)=X_{t}$ for all $j\in \lbrack 0,N_{t}]$ by substituting (12) into (10). Then, we substitute (10) and (12) into (11) to derive the equilibrium amount of monopolistic profit as

(13) \begin{equation} \pi _{t}=\left ( \mu -1\right ) X_{t}=(\mu -1)\left ( \frac {\alpha }{\mu }\right ) ^{1/(1-\alpha )}H_{Y,t}\text {.} \end{equation}

2.4 R&D

We denote $v_{t}$ as the value of a newly invented intermediate good at the end of time $t$ . The value of $v_{t}$ is given by the present value of future profits from time $t+1$ onwards:

(14) \begin{equation} v_{t}=\sum _{s=t+1}^{\infty }\left [ \pi _{s}/\prod \limits _{\tau =t+1}^{s}(1+r_{\tau })\right ] \text {.} \end{equation}

Competitive R&D entrepreneurs invent new products by employing $H_{R,t}$ units of human-capital-embodied labor. We specify the following innovation process:Footnote ¹⁵

(15) \begin{equation} \Delta N_{t}=\frac {\theta N_{t}H_{R,t}}{L_{t}}\text {,} \end{equation}

where $\Delta N_{t}\equiv N_{t+1}-N_{t}$ . The parameter $\theta \gt 0$ determines R&D productivity $\theta N_{t}/L_{t}$ , where $N_{t}$ captures intertemporal knowledge spillovers as in Romer (Reference Romer1990) and $1/L_{t}$ captures a dilution effect that removes the scale effect.Footnote ¹⁶ If the following free-entry condition holds:

(16) \begin{equation} \Delta N_{t}v_{t}=w_{t}H_{R,t}\Leftrightarrow \frac {\theta N_{t}v_{t}}{L_{t}}=w_{t}\text {,} \end{equation}

then R&D $H_{R,t}$ would be positive at time $t$ . If $\theta N_{t}v_{t}/L_{t}\lt w_{t}$ , then R&D does not take place at time $t$ (i.e., $H_{R,t}=0$ ). Lemma2 provides the condition for $H_{R,t}\gt 0$ , which requires R&D productivity $\theta$ to be sufficiently high in order for innovation to take place.

Lemma 2. R&D $H_{R,t}$ is positive at time $t$ if and only if the following inequality holds:

(17) \begin{equation} \int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)s_{t}(i)di\gt \frac {1}{\theta }\text {.} \end{equation}

2.5 Aggregation

Imposing symmetry on (8) yields $Y_{t}=H_{Y,t}^{1-\alpha }N_{t}X_{t}^{\alpha }$ . Then, we substitute (10) and (12) into this equation to derive the aggregate production function as

(18) \begin{equation} Y_{t}=\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}N_{t}H_{Y,t}\text {.} \end{equation}

Using $N_{t}X_{t}=(\alpha /\mu )Y_{t}$ , we obtain the following resource constraint on final good:

(19) \begin{equation} C_{t}=Y_{t}-N_{t}X_{t}=\left ( 1-\frac {\alpha }{\mu }\right ) Y_{t}\text {,} \end{equation}

where $C_{t}$ denotes aggregate consumption. Finally, the resource constraint on human-capital-embodied labor is

(20) \begin{equation} \int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)L_{t}(i)di=H_{Y,t}+H_{R,t}\text {.} \end{equation}

2.6 Equilibrium

The equilibrium is a sequence of allocations $\{X_{t}(j),Y_{t},e_{t}(i),n_{t}(i),c_{t}(i),C_{t},h_{t}(i),H_{t},H_{Y,t},H_{R,t},L_{t}\}$ and prices $\{p_{t}(j),w_{t},r_{t},v_{t}\}$ that satisfy the following conditions:

• • individuals choose $\{e_{t}(i),n_{t}(i),c_{t}(i)\}$ to maximize utility taking $\{r_{t+1},w_{t},h_{t}(i)\}$ as given;

• • competitive firms produce $Y_{t}$ to maximize profit taking $\{p_{t}(j),w_{t}\}$ as given;

• • a monopolistic firm produces $X_{t}(j)$ and chooses $p_{t}(j)$ to maximize profit;

• • competitive entrepreneurs perform R&D to maximize profit taking $\{w_{t},v_{t}\}$ as given;

• • the market-clearing condition for the final good holds such that $Y_{t}=N_{t}X_{t}+C_{t}$ ;

• • the resource constraint on human-capital-embodied labor holds such that $H_{Y,t}+H_{R,t}=\int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)L_{t}(i)di$ ;

• • total saving equals asset value such that $w_{t}\int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)L_{t}(i)di=N_{t+1}v_{t}$ .

3.1 Stage 1: Human capital accumulation only

The initial level of human capital for each individual in household $i$ is $h_{0}(i)$ . Suppose the following inequality holds at time 0:

(21) \begin{equation} \int _{0}^{1}\left [ 1-e_{0}(i)-\sigma n_{0}(i)\right ] h_{0}(i)s_{0}(i)di=\frac {1}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{0}(i)}{\phi (i)}\right ] h_{0}(i)s_{0}(i)di\lt \frac {1}{\theta }\text {,} \end{equation}

which uses (4) and (5). In (21), both the initial labor share $s_{0}(i)\equiv L_{0}(i)/L_{0}$ and initial human capital $h_{0}(i)$ are exogenously given. Then, Lemma2 implies that $H_{R,0}=0$ and

(22) \begin{equation} H_{Y,0}=\frac {1}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{0}(i)}{\phi (i)}\right ] h_{0}(i)L_{0}(i)di\text {.} \end{equation}

In this stage of development, the economy features only human capital accumulation. Human capital $h_{t}(i)$ accumulates according to the autonomous and stable dynamics in (6), and $s_{t}(i)$ evolves according to Lemma1. However, so long as the following inequality holds at time $t$ :

(23) \begin{equation} \int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)s_{t}(i)di=\frac {1}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] h_{t}(i)s_{t}(i)di\lt \frac {1}{\theta }\text {,} \end{equation}

we continue to have $H_{R,t}=0$ and

(24) \begin{equation} H_{Y,t}=\frac {1}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] h_{t}(i)L_{t}(i)di\text {.} \end{equation}

Substituting (24) into (18) yields the level of output per worker as

(25) \begin{equation} y_{t}\equiv \frac {Y_{t}}{L_{t}}=\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}N_{0}\frac {H_{Y,t}}{L_{t}}=\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}\frac {N_{0}}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] h_{t}(i)s_{t}(i)di\text {,} \end{equation}

where $N_{0}$ remains at the initial level and output increases as human capital accumulates.

3.2 Does innovation occur?

Equation (6) shows that human capital $h_{t}(i)$ converges to a steady state given by

(26) \begin{equation} h^{\ast }(i)=\frac {\gamma \phi (i)}{1+\eta +\gamma \delta }\text {,} \end{equation}

which is increasing in household $i$ ’s ability $\phi (i)$ . Substituting (26) into (4) and (5) yields the steady-state levels of education and fertility given by

(27) \begin{equation} e^{\ast }(i)=e^{\ast }=\frac {\gamma \delta }{1+\eta +\gamma \delta }\text {,} \end{equation}

(28) \begin{equation} n^{\ast }(i)=n^{\ast }=\frac {\eta }{\sigma (1+\eta +\gamma \delta )}\text {,} \end{equation}

where $n^{\ast }$ is the same across all households because they are independent of $\phi (i)$ . In other words, the negative effect of $\phi (i)$ and the positive effect of $h^{\ast }(i)$ on $n^{\ast }(i)$ cancel each other. As a result, the distribution of the population share of different households is stationary in the long run.

In the long run, we may have positive or negative population growth. If we assume $\eta \gt (1+\gamma \delta )\sigma /(1-\sigma )$ , then the long-run population growth rate would be positive (i.e., $n^{\ast }\gt 1$ ). If we assume $\eta \lt (1+\gamma \delta )\sigma /(1-\sigma )$ instead, then the long-run population growth rate would be negative (i.e., $n^{\ast }\lt 1$ ). Even with negative population growth in the long run, the economy may still experience economic growth (i.e., long-run growth in $y_{t}$ ) driven by innovation.Footnote ¹⁸

Does innovation occur? It depends on R&D productivity $\theta$ . Lemma2 implies that if the following inequality holds:

(29) \begin{equation} \left ( 1-e^{\ast }-\sigma n^{\ast }\right ) \int _{0}^{1}h^{\ast }(i)s^{\ast }(i)di=\frac {\gamma }{(1+\eta +\gamma \delta )^{2}}\int _{0}^{1}\phi (i)s^{\ast }(i)di\gt \frac {1}{\theta }\text {,} \end{equation}

then human capital accumulation eventually triggers the activation of innovation, under which the R&D condition in (16) holds and R&D $H_{R,t}$ becomes positive. Therefore, the endogenous activation of innovation requires a sufficiently large R&D productivity parameter $\theta$ , such that (29) holds before human capital converges to a steady state. If innovation does not occur, then the economy features only human capital accumulation and converges to the following steady-state level of output per worker:

\begin{equation*} y^{\ast }=\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}\frac {\gamma N_{0}}{(1+\eta +\gamma \delta )^{2}}\int _{0}^{1}\phi (i)s^{\ast }(i)di\text {,} \end{equation*}

which uses (26) in (25).

3.3 Stage 2: Innovation and human capital accumulation

We now consider the case in which the activation of innovation has occurred and derive the equilibrium growth rate in the presence of innovation. Substituting (18) into (9) yields the equilibrium wage rate as

(30) \begin{equation} w_{t}=(1-\alpha )\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}N_{t}\text {.} \end{equation}

Then, substituting (30) into (16) yields the equilibrium invention value as

(31) \begin{equation} \frac {v_{t}}{L_{t}}=\frac {1-\alpha }{\theta }\left ( \frac {\alpha }{\mu }\right ) ^{\alpha /(1-\alpha )}\text {.} \end{equation}

The structure of overlapping generations implies that the value of assets at the end of time $t$ must equal the amount of saving at time $t$ given by wage income at time $t$ :

(32) \begin{equation} N_{t+1}v_{t}=w_{t}\int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)L_{t}(i)di=w_{t}(H_{Y,t}+H_{R,t})\text {,} \end{equation}

where the second equality uses (20). Substituting (30) and (31) into (32) yields

(33) \begin{equation} N_{t+1}=\frac {\theta N_{t}}{L_{t}}(H_{Y,t}+H_{R,t})\text {.} \end{equation}

Combining (15) and (33) yields the equilibrium level of $H_{Y,t}$ as

(34) \begin{equation} \frac {H_{Y,t}}{L_{t}}=\frac {1}{\theta } \end{equation}

for all $t$ . Substituting (4), (5) and (34) into (20) yields the equilibrium level of $H_{R,t}$ as

(35) \begin{align} \frac {H_{R,t}}{L_{t}}&=\int _{0}^{1}\left [ 1-e_{t}(i)-\sigma n_{t}(i)\right ] h_{t}(i)s_{t}(i)di-\frac {H_{Y,t}}{L_{t}}\\ \nonumber \quad &=\frac {1}{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] h_{t}(i)s_{t}(i)di-\frac {1}{\theta }\text {.} \end{align}

We can now substitute (35) into (15) to derive the equilibrium growth rate of $N_{t}$ as

(36) \begin{equation} g_{t}\equiv \frac {\Delta N_{t}}{N_{t}}=\frac {\theta H_{R,t}}{L_{t}}=\frac {\theta }{1+\eta +\gamma }\int _{0}^{1}\left [ 1+(1-\delta )\frac {h_{t}(i)}{\phi (i)}\right ] h_{t}(i)s_{t}(i)di-1\text {,} \end{equation}

which is also the equilibrium growth rate of output per worker $y_{t}=\left ( \alpha /\mu \right ) ^{\alpha /(1-\alpha )}N_{t}/\theta$ . Finally, the steady-state equilibrium growth rate of $N_{t}$ and $y_{t}$ is

(37) \begin{equation} g^{\ast }=\frac {\theta \gamma }{(1+\eta +\gamma \delta )^{2}}\int _{0}^{1}\phi (i)s^{\ast }(i)di-1\text {.} \end{equation}

In the steady state, $s^{\ast }(i)$ is also the population share of household $i$ and still depends on the initial distribution of $h_{0}(i)$ and the exogenous distribution of $\phi (i)$ as shown in Lemma1.

4.1 A parametric example

In this section, we provide a simple parametric example to illustrate our results more clearly and prepare for the empirical analysis in Section 5. We consider two types of households. Specifically, $\phi (i)=\overline {\phi }+\varsigma$ for $i\in \lbrack 0,0.5]$ and $\phi (j)=\overline {\phi }-\varsigma$ for $j\in \lbrack 0.5,1]$ . As before, the households own the same initial amount of human capital (i.e., $h_{0}(i)=h_{0}$ for $i\in \lbrack 0,1]$ ). Their initial population shares are also the same (i.e., $s_{0}(i)=1$ for $i\in \lbrack 0,1]$ ); in this case, the mean of $\phi (i)$ is simply $\overline {\phi }$ and the coefficient of variation in $\phi (i)$ is $\varsigma /\overline {\phi }$ . Therefore, for a given $\overline {\phi }$ , an increase in $\varsigma$ raises the coefficient of variation in $\phi (i)$ and also makes (40) more likely to hold by raising $\int _{0}^{1}[1/\phi (i)]di=1/(\overline {\phi }-\varsigma ^{2}/\overline {\phi })\gt 1/\overline {\phi }$ .

From (26), their steady-state levels of human capital are different and given by $h^{\ast }(i)=\gamma (\overline {\phi }+\varsigma )/(1+\eta +\gamma \delta )$ for $i\in \lbrack 0,0.5]$ and $h^{\ast }(j)=\gamma (\overline {\phi }-\varsigma )/(1+\eta +\gamma \delta )$ for $j\in \lbrack 0.5,1]$ . From (42), the steady-state growth rate $g^{\ast }$ is given by

(43) \begin{align} g^{\ast }&=\frac {\theta \gamma }{(1+\eta +\gamma \delta )^{2}}\left [ (\overline {\phi }+\varsigma )s_{H}^{\ast }+(\overline {\phi }-\varsigma )s_{L}^{\ast }\right ] -1\\ \nonumber \quad &=\frac {\theta \gamma }{(1+\eta +\gamma \delta )^{2}}\left \{ \overline {\phi }+\varsigma \left [ s_{H}^{\ast }(\underset {-}{\varsigma })-s_{L}^{\ast }(\underset {+}{\varsigma })\right ] \right \} -1\text {,} \end{align}

where $s_{L}^{\ast }\equiv \int _{0.5}^{1}s^{\ast }(j)dj=s^{\ast }(j)/2$ is the steady-state population share of household $j\in \lbrack 0.5,1]$ with low ability $\phi (j)=\overline {\phi }-\varsigma$ whereas $s_{H}^{\ast }\equiv \int _{0}^{0.5}s^{\ast }(i)di=s^{\ast }(i)/2$ is the steady-state population share of household $i\in \lbrack 0,0.5]$ with high ability $\phi (i)=\overline {\phi }+\varsigma$ . We note that $s_{H}^{\ast }+s_{L}^{\ast }=1$ . Then, from Lemma1, we have

(44) \begin{equation} \frac {s_{L}^{\ast }}{s_{H}^{\ast }}=\frac {\prod \nolimits _{t=0}^{\infty }n_{t}(j)}{\prod \nolimits _{t=0}^{\infty }n_{t}(i)}\gt 1\text {,} \end{equation}

where

\begin{eqnarray*} n_{t}(j) &=&\frac {\eta }{\sigma (1+\eta +\gamma )}\left \{ \sum _{\tau =0}^{t-1}\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{\tau }+\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{t}\left [ 1+(1-\delta )\frac {h_{0}}{\overline {\phi }-\varsigma }\right ] \right \} \text {,} \\ n_{t}(i) &=&\frac {\eta }{\sigma (1+\eta +\gamma )}\left \{ \sum _{\tau =0}^{t-1}\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{\tau }+\left [ \frac {\gamma (1-\delta )}{1+\eta +\gamma }\right ] ^{t}\left [ 1+(1-\delta )\frac {h_{0}}{\overline {\phi }+\varsigma }\right ] \right \} \text {.} \end{eqnarray*}

Therefore, $s_{L}^{\ast }/s_{H}^{\ast }$ is increasing in $\varsigma$ , which together with $s_{H}^{\ast }+s_{L}^{\ast }=1$ implies that $s_{L}^{\ast }$ is increasing in $\varsigma$ and $s_{H}^{\ast }$ is decreasing in $\varsigma$ as stated in (43).

In summary, an increase in $\varsigma$ leads to an immediate increase in the coefficient of variation in $\phi (i)$ given by $\varsigma /\overline {\phi }$ and a subsequent decrease in the steady-state growth rate $g^{\ast }$ given by (43) by reducing the average level of human capital and the level of innovation in the long run due to the temporary evolutionary disadvantage of the high-ability households. In the next section, we will test this theoretical prediction using cross-country data.

4.1.1 Education policy

We now consider a simple policy experiment. Suppose the government designs a set of policies (which may be public investment in education or an institutional reform in the education system) that improve the schooling system for all households. If these policies are nondiscriminatory, then we can treat them as a proportional shock $\lambda _{e}\gt 1$ that scales up the education abilities $\phi (i)$ of all households. High-ability households’ ability will become $\lambda _{e}\left ( \overline {\phi }+\varsigma \right )$ , whereas low-ability households’ ability will become $\lambda _{e}\left ( \overline {\phi }-\varsigma \right )$ . Since $\overline {\phi }-\varsigma \gt 0$ , the effects on fertility $n_{t}(j)$ and $n_{t}(i)$ are all negative. This result means that education facilities and support will reduce population growth by increasing the family’s potential for education. For example, after decades of education policies, China’s fertility rate has dropped despite the 2016 abandonment of the single-child policy. Our model allows arguing that China’s recent population decline is not easily revertible because the country’s fertility transition to quality children is a byproduct of its inclusive and meritocratic education tradition. Will it hamper economic growth? According to our model, it will not. The reader can easily prove that

\begin{equation*} \frac {1+(1-\delta )\frac {h_{0}}{\lambda _{e}\left ( \overline {\phi }-\varsigma \right ) }}{1+(1-\delta )\frac {h_{0}}{\lambda _{e}\left ( \overline {\phi }+\varsigma \right ) }} \end{equation*}

decreases in $\lambda _{e}$ , which implies - by (44) - that $s_{L}^{\ast }/s_{H}^{\ast }$ decreases as well, thereby leading to an increase in $g^{\ast }$ due to the evolutionary process giving rise to a larger population share of high-ability households. Therefore, we can state that:

Corollary 2. A policy that proportionally raises all education abilities will lead to a decrease in fertility and an increase in long-term economic growth.

5.1 Empirical results

Table 1 presents the results from our baseline cross-country analysis. Column 1 starts with a bivariate regression, showing that educational ability heterogeneity is negatively and significantly associated with GDP per capita growth rates. This relationship is visually depicted in Figure 2 of Appendix C, where we plot average annual economic growth rates against educational ability heterogeneity across countries. The scatter plot clearly illustrates a negative relationship, observed across various continents, as indicated by distinct color-coded data points.

Table 1.

Heterogeneity of educational ability and economic growth

Notes: ^*** p $\lt$ 0.01, ^** p $\lt$ 0.05, ^* p $\lt$ 0.1. Standard errors in parentheses are clustered by country. The dependent variable is the growth rate of GDP per capita.

To ensure the robustness of our findings, we progressively introduce additional controls. Column 2 adds year fixed effects to account for time-varying factors, while Column 3 incorporates continent fixed effects to control for regional differences. The baseline model presented in Column 4 includes controls for time preferences, geographical characteristics such as proximity to waterways, absolute latitude, mean and standard deviation of elevation, mean and standard deviation of land suitability, and an island dummy.

Despite these controls, concerns about endogeneity remain. The heterogeneity of educational ability and the spatial distribution of economic growth could be jointly influenced by unobserved cultural, institutional, or human factors. For instance, in societies that highly value education, individuals might strive for academic excellence, leading to less variation in test scores and faster economic growth. Ignoring these unobserved factors could bias our results. To address this, we employ two instrumental variables: ancestry adjusted population diversity and migratory distance from East Africa.

Table 2.

Impacts of heterogeneity of educational ability using instrumental variables

Note: p $\lt$ 0.01, ^** p $\lt$ 0.05, ^* p $\lt$ 0.1. Standard errors in parentheses are clustered by country. The dependent variables in columns 1 and 2 are the growth rate of GDP per capita. Column 1 shows the second-stage results using population diversity (ancestry adjusted) as the instrumental variable, with the corresponding first-stage results presented in Column 3. Column 2 presents the second-stage results using prehistoric migratory distance from East Africa as the instrumental variable, with the corresponding first-stage results shown in Column 4. Controls include time preference, distance to the nearest waterway, absolute latitude, mean elevation, mean agricultural land suitability, standard deviation in elevation, standard deviation in agricultural land suitability, and an island dummy.

Ancestry-adjusted population diversity, as discussed by Ashraf and Galor (Reference Ashraf and Galor2013) and Arbatlı et al. (Reference Arbatlı, Ashraf, Galor and Klemp2020), relates to differences in educational ability and is exogenous to contemporary economic conditions. Migratory distance from East Africa that was used in previous studies, such as Ashraf and Galor (Reference Ashraf and Galor2013), Arbatlı et al. (Reference Arbatlı, Ashraf, Galor and Klemp2020), Ashraf et al. (Reference Ashraf, Galor and Klemp2021) and Galor et al. (Reference Galor, Klemp and Wainstock2023), captures the historical migration patterns that shaped genetic diversity, which could influence educational ability heterogeneity.

Table 2 presents the results of the two-stage least squares (2SLS) regressions. In the second stage, Columns 1 and 2 show that the coefficient for educational ability heterogeneity remains significantly negative, confirming its detrimental impact on economic growth. The first-stage results in Columns 3 and 4 validate our instruments, showing a significant positive correlation between educational ability heterogeneity and ancestry adjusted population diversity, and a significant negative correlation with migratory distance from East Africa.

Next, we consider education and innovation as alternative proxies for economic growth. In Columns 1 and 2 of Table 3, the dependent variables are the share of the population with at least some primary education and the logarithm of the average years of education, respectively, capturing the average education level. Columns 3 and 4 use the logarithm of the number of researchers in R&D (per million people) and the logarithm of patent applications as dependent variables, representing innovation rates. The negative coefficients for educational ability heterogeneity across these specifications suggest that it adversely affects both education and innovation, further supporting our theoretical predictions.Footnote ²⁷ We have also conducted several robustness checks to affirm the reliability of our findings.Footnote ²⁸

Table 3.

Impacts of heterogeneity of educational ability on education and innovation

Note: p $\lt$ 0.01, ^** p $\lt$ 0.05, ^* p $\lt$ 0.1. Standard errors in parentheses are clustered by country. In columns 1–2, the dependent variables are the share of the population with primary schooling and the log of average years of education in the first two columns, respectively. In columns 3–4, the dependent variables are the log of the number of researchers in R&D (per million people) and the log of the number of patent applications, respectively. Controls include time preference, distance to the nearest waterway, absolute latitude, mean elevation, mean agricultural land suitability, standard deviation in elevation and agricultural land suitability, and an island dummy.