Technology

We build upon Lucas (Reference Lucas1988) and Boucekkine et al. (Reference Boucekkine, Martínez and Ruiz-Tamarit2013). The model is set in continuous time, with time running on the interval [ $0,\infty$ ). At any point in time, the production of a homogeneous consumption good, $Y_t$ , is obtained by employing, under constant returns to scale, human capital, $H_{Yt}$ , and physical capital, $K_t$ , as inputs, according to

(1) \begin{eqnarray} Y_t=AK_t^\beta H_{Yt}^{1-\beta }, \qquad \quad \text {with }0\leq \beta \leq 1, \end{eqnarray}

where $A\gt 0$ is the total factor productivity and $\beta$ is a parameter controlling for the factor-input shares in GDP. At any time, the aggregate stock of human capital, $H_t$ , can be used either for producing the final good or for acquiring ‘new’ human capital. The share of $H_t$ that is used for producing the final good is $u_t\in [0,1]$ , whereas that used for attaining new human capital is $1-u_t$ . Since the main goal of this paper is to determine whether (and, eventually, under which conditions) the presence of class-size effects in education can positively contribute to long-run economic growth, we may ignore physical capital. In this respect, compared to Lucas (Reference Lucas1988) and Boucekkine et al. (Reference Boucekkine, Martínez and Ruiz-Tamarit2013), we first set $\beta =0$ in (1) and let the aggregate production function ultimately read as

(2) \begin{equation} Y_t=A H_{Yt}=A(u_tH_t). \end{equation}

Equation (2) now says that the only essential input in production is human capital. Furthermore, we posit that, once produced, total output $Y_t$ is consumed, i.e., $Y_t=C_t$ .Footnote ¹⁰ Through this assumption we can safely neglect any gross investment in physical capital (or any other reproducible resource), as well. In (2), parameter $A\gt 0$ now measures the productivity of human capital, $H_{Yt}$ , in the production of consumables. Let now $N_t$ denote the existing population at time $t$ . In per capita terms, (2) becomes

(3) \begin{equation} y_t= A \left (u_th_t\right )=c_t, \end{equation}

where $y_t\equiv Y_t/N_t$ , $h_t\equiv H_t/N_t$ , and $c_t\equiv C_t/N_t$ . We assume that population grows over time at an exogenous rate $n$ such that

(4) \begin{equation} \frac {\dot {N}_t}{N_t}=n \quad \text {with}\quad n\gtreqless 0\quad \text {and}\quad N(0)\gt 0 . \end{equation}

Per capita human capital accumulation is given by

(5) \begin{equation} \dot {h}_t=\xi (1-u_t)h_t-\delta h_t \quad \text {with}\quad h(0)\gt 0, \end{equation}

where $\xi \gt 0$ is the efficiency of human capital investment and $\delta \in (0,1)$ is the instantaneous depreciation rate of human capital per capita. Unlike Lucas (Reference Lucas1988) and Boucekkine et al. (Reference Boucekkine, Martínez and Ruiz-Tamarit2013), here we account for the novel assumption that the presence of a class–size effect influences the efficiency with which per capita human capital may be augmented over time. Specifically, we assume that $\xi$ increases as the population size $N_t$ (a proxy for class size) decreases and that the faster the population size declines, the more rapidly $\xi$ rises over time, i.e.,

(6) \begin{equation} \xi \equiv \sigma -\varepsilon \left (\frac {\dot {N}_t}{N_t}\right )=\sigma -\varepsilon n \gt 0 \quad \text {with}\quad \sigma \gt 0 \,\text {and}\, \varepsilon \geq 0 . \end{equation}

In equation (6), $\xi$ is designed to capture, in a reduced form fashion, the existence of two contrasting forces at work in the accumulation of human capital. On one hand, there is a positive effect resulting from the effort in education, represented by $\sigma$ .Footnote ¹¹ On the other hand, we have a potentially perverse effect due to population dynamics. In fact, for any $\varepsilon \gt 0$ , there exists a negative correlation between the rate of population growth and the efficiency with which an individual can enhance her own stock of embodied knowledge (human capital). This means that the larger (smaller) the value of $n$ , the smaller (larger) the efficiency of investment in human capital per capita. Therefore, in our model, the extent of the class–size effect can be directly measured by the value of $\varepsilon$ .Footnote ¹² When $n=0$ , our model simplifies to the one used in Lucas (Reference Lucas1988) and Boucekkine et al. (Reference Boucekkine, Martínez and Ruiz-Tamarit2013), with $\xi =\sigma \gt 0$ . The same outcome occurs, evidently, also when $\varepsilon =0$ .

Combine (4), (5), and (6) to get

(7) \begin{equation} \dot {h}_t=(\sigma -\varepsilon n)(1-u_t)h_t-\delta h_t \quad \text {with}\quad n\gtreqless 0. \end{equation}

The next section explores the condition(s) for the presence of class-size effects in education to contribute positively to long-run economic growth.

Estimates

Assume for simplicity $\delta =0$ . Then, using (11), equation (7) can be written as:

(14) \begin{equation} \frac {\dot {h}}{h_t}\equiv \gamma =\sigma (1-u_t)-\varepsilon \left [n_t\times (1-u_t)\right ]. \end{equation}

The empirical counterpart of (14) is

(15) \begin{equation} \gamma _{it}=\zeta _{0}+\zeta _1(1-u_{it})+\zeta _2\left [n_{it}\times (1-u_{it})\right ]+\zeta _3 n_{it}+\epsilon _{it}, \end{equation}

where $\gamma _{it}$ is the annual growth rate of the Human Capital Index and $\epsilon _{it}$ is an iid error term.Footnote ²³ Confronting (14) with (15), we see immediately that $\hat {\zeta _1} = \hat {\sigma }$ and $\hat {\zeta _2}=-\hat {\varepsilon }$ . Notice also that in (15) the interaction between population growth ( $n_t$ ) and the investment in education ( $1-u_t$ ) is complete.

Table 1 presents the OLS fixed effects (FE), OLS random effects (RE), and feasible GLS (FGLS) estimates, respectively, for (15). As predicted by the theory, for the interaction $n_t\times (1-u_t)$ , all three models show a negative association with the dependent variable $\gamma$ . The FE and RE models show similar coefficient estimates of -0.19 and -0.22, respectively, both significant at a 1% level. The FGLS model suggests a slightly smaller coefficient of -0.09, also significant at a 1% level. As predicted by the theory, the variable $1-u_t$ is positively associated with $\gamma$ in all models, with the FE model showing the largest coefficient estimate (0.79), and the FGLS showing the smallest (0.24), but again with a remarkably low standard error. Lastly, the variable $n_t$ , which has been added to account for the complete interaction, also has a positive relationship with $\gamma$ , with an approximate coefficient value of 0.1 across all models. Finally, the $R^2$ values indicate that the FGLS model has a considerably better fit ( $R^2$ = 0.74) to the data compared to the FE ( $R^2$ = 0.17) and RE ( $R^2$ = 0.05) models. Time and individual controls are included in all models.

Table 1.

OLS,GLS estimation

Note: $^{*}$ p $\lt$ 0.1; $^{**}$ p $\lt$ 0.05; $^{***}$ p $\lt$ 0.01

$\sigma _{idios}$ : idiosyncratic variance; $\sigma _{id}$ : individual variance; standard errors in brackets.

FGLS model offers the most precise and best-fitting model for the data, given its low standard errors and high $R^2$ value. Notice also that in the OLS models, the inference is influenced by non-normal residuals. This is evident from the Shapiro-Wilk test, which rejects the assumption of normality with a test statistic of 0.988 and a highly significant p-value of 0.000. The FGLS model also accounts for potential correlations among covariates. This flexibility makes it more suitable for datasets where the assumptions of homoscedasticity in the residuals and independence of covariates are violated.

In the following, we apply a FMM to the equation (14) to cluster countries according to the unobserved heterogeneity. In general, mixture models allow to test the behavior of human capital growth dynamics, under the assumption that unobserved heterogeneity affects parameters estimation. In particular, we employ a FMM, relaxing the hypothesis of iid residuals and allowing for correlated random terms.Footnote ²⁴ The random component captures the impact of unobserved country-specific variables, limiting the effects of the omitted variable bias. This approach allows dealing with the unobserved heterogeneity due to the non-monotonic, non-uniform relationship between regressors and response, as predicted by our theory. Moreover, through this estimation procedure, we are able to perform a cluster analysis: we sort countries into groups based on the homogeneity of the conditional distribution of their long-run growth rates with respect to the estimated unobservable factors.

According to the generalized linear model framework (McCullagh and Nelder, Reference McCullagh and Nelder1989), equation (14) can be written as:

(16) \begin{equation} \gamma _{it}=\zeta _{0i}+\zeta _{1i}(1-u_{it})+\zeta _{2i}\left [{n}_{it}\times (1-u_{it})\right ]+\zeta _{3i} n_{it}+\epsilon _{it}, \end{equation}

where parameters $\zeta _i=\zeta _{0i},\zeta _{1i},\zeta _{2i},\zeta _{3i}$ are now country specific. Specifically, they capture the country-specific unobserved factors that affect human capital growth, through population change, the share of human capital dedicated to fostering new human capital, and its interaction with population growth.

We assume that parameters in $\zeta _{i}$ can be empirically described by random variables, with unspecified probability function, and cluster-specific variances $\sigma _i$ . In this way, equation (16) takes explicitly into account the between countries random terms correlation.

The nonparametric maximum likelihood estimator (NPMLE) of the distribution is discrete,Footnote ²⁵ with a finite number of locations and masses. This implies that the country-specific latent variables are modeled as measures of the difference between country i-th’s covariates and their sample mean. We assume that $\gamma _{it}$ is a conditionally independent realization of the potential human capital growth, given the set of random factors, which varies across countries and accounts for both individual variation and dependence among country-specific rates of growth.

Table 2 presents the results, while Table 3 provides the corresponding country classification. From the previous specification, we obtain a classification of countries into 5 clusters.Footnote ²⁶ The first noteworthy result is the positive impact of the coefficient $1-u_t$ across the different clusters. Consistently to the theory, investment in human capital is positively associated with the HC index’s growth rate, ranging from 0.30 (Cluster 5) to a maximum of 1.91 (Cluster 3). This variance suggests that, despite the consistent positive relationship between investment in human capital and its growth rate, this relationship is heterogeneous across clusters, underscoring the influence of unobserved factors.

Table 2.

Finite mixture model

Note: $^{***}p\lt 0.001$ ; $^{**}p\lt 0.01$ ; $^{*}p\lt 0.05$ . Standard errors in brackets.

Table 3.

Country classification, model in Table 2

A distinct pattern emerges when examining the interaction term capturing the class-size effects ( $\varepsilon$ ), revealing three key trends. Only Cluster 5 exhibits a positive effect. All the remaining clusters share a comparably negative coefficient, ranging from -0.26 to -0.42, with the exception of Cluster 3 which displays a significantly stronger negative relationship, with a coefficient of -0.76. The negative sign aligns with the theoretical model, where a negative relationship exists between population growth and the efficiency of human capital (see equation (6)). However, the positive sign found for one group contradicts the class-size effects, indicating that for some countries, larger population growth is associated with more efficient human capital. Below we will try to better understand this effect by studying the composition of the different groups.

When considering the impact of the coefficient linked to the population growth variable, a careful interpretation is required. Unlike the previous two coefficients, this one is included in the model to account for empirical interactions but does not have a direct implication derived from the theoretical framework.Footnote ²⁷ Finally, although frequently not statistically significant, most clusters exhibit a negative intercept. This indicates a tendency toward a lower baseline growth rate of the HC index.

Overall, these estimates suggest that the relationship between population growth, human capital investment, and the growth rate of the HC index is complex and varies across clusters. The interaction effect is particularly varied, with some clusters showing a positive relationship and others negative. This could reflect different demographic dynamics or varying effectiveness of human capital investment in different sub-populations or contexts represented by the clusters. The direct effect of human capital investment is consistently positive across all clusters, emphasizing its importance in the growth of the HC index. The varied signs and magnitudes of the population growth rate indicate that its impact on the HC index’s growth may depend on other factors captured by the cluster distinctions.

We seek a deeper understanding of our findings by examining the countries grouped into various clusters in our analysis, as detailed in Table 3. Among the 34 OECD countries in our sample, there is an uneven distribution of countries across groups. The largest contingent falls into Cluster 5 (12 countries), followed by Cluster 2 (10 countries) and Cluster 4 (6 countries). The remaining groups (Cluster 1 and Cluster 3) consist of 3 countries each. Initial observations make it challenging to discern any consistent patterns within the classification.Footnote ²⁸ Indeed, we find no substantial differences in terms of investment of human capital $(1-u_t)$ or economic growth across groups during the period under study, and although there are some differences in terms of the HC index, there is not a direct relationship with the interaction coefficient from Table 2.

Figure 1.

Educational proxies and class-size effects.

Remarkably, the results change when we compare them to two proxies for educational quality: the pupil-teacher ratio and government expenditure per student (as a percentage of GDP per capita), both obtained from the World Bank. Figure 1 shows the correlation between the average values of these variables using our econometric sample and the interaction coefficients. Figure 1a demonstrates a clear negative relationship between the pupil–teacher ratio and the class-size effects, indicating that population increases more negatively affect human capital efficiency through the class-size effects in countries with a higher number of students per teacher.Footnote ²⁹ In contrast, Figure 1b shows a strong positive correlation between the interaction coefficients and government expenditure per student, suggesting that the class-size effects is stronger in countries with lower expenditure per student.Footnote ³⁰ These results suggest that higher educational quality, indicated by a lower pupil-teacher ratio or higher expenditure per student, can help reduce or even reverse the class-size effects.

Interestingly, the cluster with higher population growth (Cluster 3) exhibits more pronounced class-size effects. In this cluster we find Luxembourg ( $\partial \gamma /\partial n=-0.147$ ), Mexico ( $\partial \gamma /\partial n=-0.091$ ) and Republic of Korea ( $\partial \gamma /\partial n=-0.104$ ).Footnote ³¹ In 2021, the OECD reports that Korea and Mexico have some of the largest class sizes in both primary (with an average of 22.45 and 22.57 students per teacher, respectively) and secondary education (with an average of 26.49 and 26.17 students per teacher, respectively).Footnote ³² In 2022, instead, Luxembourg had one of the highest proportions of young adults aged 18–24 in education or training, either full-time or part-time, despite having one of the lower average student-to-teacher ratios in both primary and secondary schools.Footnote ³³ However, the relationship between population growth and class size is evidently non-linear. Despite having the lowest average population growth rate, Cluster 4 displays the second-most negative coefficient in the interaction term.

Countries in line with the theoretical model, showing a negative effect of the interaction term are the majority (22 out of 34), with a large heterogeneity among them. Within these groups, we find the Mediterranean and Middle-Eastern countries, most Nordic countries in the sample (except for Norway), along with the Anglo-Saxon countries (Canada, UK, and US) outside Oceania, Chile, and Japan. In examining the group of countries deviating from our theoretical predictions, we find a cluster comprising mostly European nations, Australia, New Zealand, and Norway. It is worth noting that the European block in this cluster excludes Germany, Poland, and Slovakia, making it more difficult to find a common pattern across the clusters.

In summary, our results highlight the presence of class-size effects in some countries, while in others, population growth does not seem to affect human capital efficiency. Our findings suggest that the quality of the education system may partially explain these differences. However, the relationship remains heterogeneous and requires further investigation, as it carries important implications for future economic policy decisions.