3.2.1. The impact of household human capital levels on monetary investment in children

Proof. Taking the partial derivative of Equation (8) with respect to parental human capital levels, we have

(11) ${{\partial {Q_t}} \over {\partial {H_t}}}\;\; = \;\;Q_t^{2 - \varepsilon }{{{\alpha _1}{W_t}v\left[ {\left( {1 - \varepsilon } \right){\alpha _3}H_t^\varepsilon + {\alpha _4}\bar H_t^\varepsilon } \right]} \over {\left( {1 - \varepsilon } \right){{({\alpha _1}{W_t}{H_t}v)}^2}}} \gt 0$

That is, for all households, a higher level of household human capital is associated with greater monetary investment in children’s education. The intuition is that high–human-capital households can provide better living conditions for their children and typically choose higher-quality educational arrangements, which raises the associated costs of child investment.

3.2.2. The impact of household human capital levels on time investment

Proof. Taking the partial derivative of Equation (9) with respect to parental human capital levels, we have

(12) ${{\partial {e_t}} \over {\partial {H_t}}} = e_t^{2 - \varepsilon }{{{\alpha _3}\varepsilon } \over {\left( {\varepsilon - 1} \right){\alpha _2}v}}H_t^{\varepsilon - 1}$

• If $\varepsilon \lt 0$ , ${{\partial {e_t}} \over {\partial {H_t}}} \gt 0$ , indicating that as household human capital increases, time investment in children’s education also rises.

• If $0 \lt \varepsilon \le 1$ , ${{\partial {e_t}} \over {\partial {H_t}}} \le 0$ , implying that as household human capital increases, time investment in children’s education declines.

The underlying mechanism is as follows: when $\varepsilon \lt 0$ , the elasticity of substitution among input factors satisfies $0 \lt {\tilde E_s} \lt 1$ , indicating a complementary relationship between educational input factors. In this case, the accumulation of children’s human capital requires both parental human capital and time investment, making them jointly necessary inputs. In contrast, when $0 \lt \varepsilon \le 1$ , the elasticity of substitution satisfies ${\tilde E_s} \ge1$ , indicating a substitutive relationship. Under this condition, parental human capital and monetary investment can serve as a substitute for direct time investment in children’s education.

3.2.3. The impact of household human capital levels on the number of children

Proof. Taking the partial derivative of Equation (10) with respect to parental human capital levels, we have:

(13) ${{\partial {N_t}} \over {\partial {H_t}}} = {\rho \over {v\left( {1 + \rho + \beta } \right)}}{{\varepsilon \left( {{\alpha _3}H_t^\varepsilon + {\alpha _4}\bar {H}_t^\varepsilon } \right)} \over {\left( {\varepsilon - 1} \right){{\left( {{\alpha _1}Q_t^\varepsilon + {\alpha _2}e_t^\varepsilon + {\alpha _3}H_t^\varepsilon + {\alpha _4}\bar H_t^\varepsilon } \right)}^2}}}A$

where $$A = {{{\alpha _4}{\mkern 1mu} {{{\bar{\,H_t}} }^{{\kern 1pt} \varepsilon }}} \over {{W_t}{\mkern 1mu} v{\mkern 1mu} H_t^2}}{\mkern 1mu} \,{Q_t} - {{{\alpha _3}H_t^{\varepsilon - 1}} \over v}{\mkern 1mu} {e_t}$$ . Furthermore, when $A = 0$ , $${H_t} ={\hat H_t}$$ $$ = {\left( {{{\left( {{{{\alpha _1}} \over {{\alpha _2}}}} \right)}^{{1 \over {\varepsilon - 1}}}}{{{\alpha _3}} \over {{\alpha _4}}}W_t^{{\varepsilon \over {\varepsilon - 1}}}{{\bar H}_t}^{ - \varepsilon }} \right)^{{{1 - \varepsilon } \over {{\varepsilon ^2}}}}}$$ .

• If $\varepsilon \lt 0$ , then:

  1. – When ${H_t} \lt {\widehat H_t}$ , $A \lt 0,$ and ${{\partial {N_t}} \over {\partial {H_t}}} \lt 0$ , indicating that an increase in household human capital reduces the number of children.

  2. – When ${H_t} \gt {\widehat H_t}$ , $A \gt 0,$ and ${{\partial {N_t}} \over {\partial {H_t}}} \gt 0$ , indicating that an increase in household human capital raises the number of children.

  3. – This implies a U-shaped relationship between household human capital and the number of children.

• If $0 \le \varepsilon \le 1$ , then:

  1. – When ${H_t} \lt {\widehat H_t}$ , $A \lt 0,$ and ${{\partial {N_t}} \over {\partial {H_t}}} \gt 0$ , implying that an increase in household human capital raises the number of children.

  2. – When ${H_t} \gt {\widehat H_t}$ , $A \gt 0,$ and ${{\partial {N_t}} \over {\partial {H_t}}} \lt 0$ , implying that an increase in household human capital reduces the number of children.

  3. – This results in an inverted U-shaped relationship between household human capital and the number of children.

To understand the mechanism behind the above findings, we further analyze the problem through the lens of the quantity–quality trade-off. Differentiating fertility with respect to children’s human capital yields ${{\partial {N_t}} \over {\partial {H_{t + 1}}}} = \left( {{{\partial {N_t}} \over {\partial {H_t}}}} \right)/\left( {{{\partial {H_{t + 1}}} \over {\partial {H_t}}}} \right)$ . Since ${{\partial {H_{t + 1}}} \over {\partial {H_t}}} \gt 0$ , the sign of ${{\partial {N_t}} \over {\partial {H_{t + 1}}}}$ is the same as that of ${{\partial {N_t}} \over {\partial {H_t}}}$ .

When $\varepsilon \lt 0$ , the inputs required to raise children’s human capital are complements. If ${H_t} \lt {\widehat H_t}$ , then ${{\partial {N_t}} \over {\partial {H_{t + 1}}}} \lt 0$ . A complementary educational technology requires multiple inputs to be scaled up jointly. As a result, low–human-capital households tend to choose childbearing rather than intensive child investment, whereas middle–human-capital households reduce fertility in order to invest more in children. If ${H_t} \gt {\widehat H_t}$ , then ${{\partial {N_t}} \over {\partial {H_{t + 1}}}} \gt 0$ . In this region, complementarity enables high–human-capital households to produce higher-quality children, which in turn encourages them to have more children.

When $0 \le \varepsilon \le 1$ , the inputs required to raise children’s human capital are substitutes. If ${H_t} \lt {\widehat H_t}$ , then ${{\partial {N_t}} \over {\partial {H_{t + 1}}}} \gt 0$ . Under a substitutable technology, households can increase their human capital and monetary inputs to substitute for parental time devoted to educating children; consequently, fertility increases as household human capital rises. If ${H_t} \gt {\widehat H_t}$ , then ${{\partial {N_t}} \over {\partial {H_{t + 1}}}} \lt 0$ . In this region, substitutability allows higher–human-capital households to rely heavily on monetary inputs to substitute for time inputs; the resulting high cost of child investment leads fertility to decline as household human capital rises.

4.1.1. Dependent variables

Consistent with the theoretical model and the hypotheses, we focus on fertility and child-education investment decisions across households with different levels of human capital. For household fertility decisions, the dependent variable is the number of children in the household. The CFPS 2020 records a unique identifier for each child in the household; by matching household records with child records, we compute the total number of children for each household.

For household education decisions, the dependent variable is children’s human capital. Following Dai and Li (Reference Dai and Li2022) and Wu and Zhang (Reference Wu and Zhang2024), we use children’s Chinese-language and mathematics performance to proxy comprehension ability and logical reasoning ability, respectively. In the CFPS child module, academic performance is reported on a four-point scale (“poor” = 1, “average” = 2, “good” = 3, and “excellent” = 4). Based on these measures, we construct a composite index of children’s human capital by applying the entropy-weight method to Chinese and mathematics scores.

In addition, to further examine how household human capital affects monetary and time investments in children’s education, we measure educational time investment using the CFPS child-module item on “weekly hours spent helping with homework,” and measure monetary investment using the natural logarithm of “total household educational expenditures over the past 12 months + 1.”

4.1.2. Core explanatory variables

The key explanatory variable in this study is household human capital. Existing research (Epo et al., Reference Epo, Baye, Mwabu, Manda, Ajakaiye and Kipruto2025; Su & Guo, Reference Su and Guo2022) typically measures parental human capital along three dimensions: education, health, and work-related characteristics. Following the approach of Su and Guo (Reference Su and Guo2022), we construct an individual-level human capital index based on education, health, and work experience using the entropy-weight method, and then aggregate to the household level by taking the average of the parents’ human capital indices, yielding our measure of household human capital.

Specifically, for the education dimension, we use the highest completed years of schooling reported in the CFPS 2020 individual module, coded as: illiterate = 0, primary school = 6, junior high school = 9, high school = 12, junior college = 15, bachelor’s degree = 16, master’s degree = 19, and PhD = 21. For the health dimension, we use self-rated health, coded as: very healthy = 1, healthy = 2, relatively healthy = 3, fair = 4, and unhealthy = 5. For the work-experience dimension, following Su and Guo (Reference Su and Guo2022), we proxy actual years of work experience as (age-years of schooling-6).

4.1.3. Control variables

Following Ning et al. (Reference Ning, Tang, Huang, Tan, Lin and Sun2022) and Singletary et al. (Reference Singletary, Justice, Baker, Lin, Purtell and Schmeer2022), we include a set of control variables capturing child characteristics, household characteristics, and regional characteristics to more accurately identify household fertility and education decisions.

For household characteristics, we control for parents’ education, parents’ health, parents’ employment status, and residential location. Parents’ education is measured as the maximum years of schooling between the two parents. Parents’ health is constructed from self-rated health: it equals 1 if at least one parent reports being healthy, and 0 otherwise. Parents’ employment status is based on the CFPS 2020 classification of occupational status; we take the household-level average of parents’ employment indicators to capture overall parental employment. Residential location is controlled for using an urban–rural dummy, equal to 1 for urban households and 0 for rural households.

For child characteristics, in households with multiple children, child age is measured as the average age of all children. For regional characteristics, we include province fixed effects to absorb systematic cross-province differences in economic development, public service provision, and fertility- and education-related policies.

Detailed variable definitions are reported in Table 1.

Table 1.

Descriptive statistics of variables

Table 1 Long description

The table presents descriptive statistics of variables related to children and households. It includes three main categories: dependent variables, core explanatory variables, and control variables. The table has 12 rows and 6 columns. Column headers are Variable name, Obs., Mean, Std. Dev., Min, and Max. Row labels are grouped under the three main categories. Row 1: Number of Children, 4891, 2.10, 0.95, 0, 8. Row 2: Child human capital index, 3685, 2.83, 0.90, 1, 4. Row 3: Monetary investment, 4744, 6.28, 3.16, 0, 11. Row 4: Time investment, 4744, 3.86, 5.50, 0, 26. Row 5: Household human capital index, 4891, 0.39, 0.10, 0.07, 0.68. Row 6: Household health status, 4891, 0.89, 0.31, 0, 1. Row 7: Household employment status, 4891, 3.42, 2.49, 1, 9. Row 8: Parents' maximum years of schooling, 4891, 9.56, 4.34, 0, 21. Row 9: Urban residence, 4891, 0.48, 0.50, 0, 1. Row 10: Children's Age, 4891, 7.64, 4.28, 0, 16.

4.2.1. Regression model for household fertility

To analyze fertility patterns across households with different levels of human capital, we estimate the following baseline model:

(14) $n\_chil{d_i} = {\beta _0} + {\beta _1}capita{l_i} + {\beta _2}capital_i^2 + {\beta _3}{X_i} + {\mu _i}$

The dependent variable $n\_chil{d_i}$ denotes the number of children in household $i$ . Since the number of children is a nonnegative count outcome, we estimate Equation (14) using a Poisson model. Motivated by the theoretical predictions, household human capital and fertility may exhibit a nonlinear (U-shaped or inverted-U-shaped) relationship; we therefore include both household human capital $capita{l_i}$ and its squared term $capital_i^2$ . The vector ${X_i}$ includes a set of control variables capturing child characteristics, household characteristics, and the regression also controls for province fixed effects. ${\mu _i}$ is an error term.

4.2.2. Regression model for household education investment

To further examine how household human capital affects the mode of educational investment, we specify separate regressions for monetary and time investments in children’s education:

(15) $mone{y_i} = {\beta _0} + {\beta _1}capita{l_i} + {\beta _2}ag{e_i} + {\beta _3}age_i^2 + {\beta _4}{X_i} + {\mu _i}$

(16) $tim{e_i} = {\beta _0} + {\beta _1}capita{l_i} + {\beta _2}ag{e_i} + {\beta _3}age_i^2 + {\beta _4}{X_i} + {\mu _i}$

The dependent variables $mone{y_i}$ and $tim{e_i}$ denote household $i$ ’s monetary and time investments in children’s education, respectively. Descriptive statistics show that approximately 16.7% of households report zero monetary educational spending and about 50.39% report zero educational time input. Because educational investment is an active household choice, these zeros are not merely due to censoring or missing values; rather, they carry meaningful economic content. We therefore estimate Equations (15) and (16) using Poisson pseudo-maximum likelihood (PPML), which is well suited to outcomes with a large mass at zero and is robust to heteroskedasticity.

Moreover, during primary and lower-secondary schooling, households typically devote substantial time to homework assistance and incur higher educational expenditures. Educational time and monetary inputs may therefore vary with children’s age in an inverted-U-shaped manner. Accordingly, we include children’s age $ag{e_i}$ and its squared term $age_i^2$ in Equations (15) and (16). The vector ${X_i}$ includes additional household- and province-level controls, and ${\mu _i}$ is an error term.

4.2.3. Regression model for children’s CES human-capital production function

To quantify substitutability versus complementarity among educational inputs, we further specify a CES-type child human-capital production function:

(17) ${\rm{ln}}\left( {child\_capita{l_i}} \right) = {1 \over \varepsilon }{\rm{ln}}\left( {{\alpha _1}money_i^\varepsilon + {\alpha _2}time_i^\varepsilon + {\alpha _3}capital_i^\varepsilon + {\alpha _4}avg\_capita{l^\varepsilon }} \right) + {\mu _i}$

The dependent variable ${\rm{ln}}\left( {child\_capita{l_i}} \right)$ is the log human capital level of child $i$ . The variables $mone{y_i}$ and $tim{e_i}$ denote the household’s monetary and time investments in education, $capita{l_i}$ is household human capital, and $avg\_capital$ is the economy-wide average level of human capital capturing the external human-capital environment. The parameter $\varepsilon $ governs the degree of substitutability (or complementarity) among inputs in the CES technology, and ${\mu _i}$ is an error term.

Following the estimation strategies for CES functions in Koesler and Schymura (Reference Koesler and Schymura2015) and Attanasio et al. (Reference Attanasio, Meghir and Nix2020), we estimate Equation (17) using nonlinear least squares (NLS) to identify the substitutability or complementarity among educational inputs.

5.1.1. Regression results for the number of children

We first examine the effect of household human capital on fertility using the baseline specification in Equation (14). The corresponding estimates are reported in Table 2, columns (1) and (2). Column (1) presents results without any control variables, while column (2) adds a full set of controls, including child characteristics, household characteristics, and province fixed effects.

Table 2.

Estimation results of baseline regression

Table 2 Long description

A table with 10 rows and 8 columns showing estimation results of baseline regression. The columns are labeled as (1) Poisson Number of children, (2) Poisson Number of children, (3) PPML Monetary investment, and (4) PPML Time investment. The rows are labeled as Household human capital, Human capital squared, Child age, Child age squared, Household health status, Household employment status, Maximum years of schooling, Urban residence, Province fixed effects, and Constant. Each cell contains numerical values with standard errors in parentheses. Notable trends include significant negative values for Household human capital in columns (1) and (2), and significant positive values for Child age in columns (2), (3), and (4).

Notes: Robust standard errors are reported in parentheses. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.

As shown in Table 2, in column (1), the estimated coefficient on household human capital is -2.287 and the coefficient on its squared term is 2.701; both are statistically significant at the 1% level. After adding the control variables in column (2), the signs and statistical significance of household human capital and its squared term remain stable. These results indicate a significant U-shaped relationship between household human capital and fertility: as household human capital increases, the number of children first decreases and then increases.

Turning to the control variables, the estimates suggest that, holding other factors constant, households with better health status tend to have more children. In contrast, households with better employment status and higher parental education have fewer children. Moreover, rural households have significantly more children than urban households.

5.1.2. Regression results for monetary and time investments in children’s education

Based on the baseline specifications in Equations (15) and (16), we further examine how household human capital affects households’ monetary and time investments in children’s education. The corresponding estimates are reported in Table 2, columns (3) and (4). Column (3) captures the effect of household human capital on monetary educational investment, while column (4) reports the effect on time investment.

As shown in Table 2, column (3), the estimated coefficient on household human capital is 0.292 and is statistically significant at the 1% level, indicating that households with higher human capital spend more on children’s education. This finding supports Hypothesis 1, suggesting a significant positive association between household human capital and monetary educational investment.

In Table 2, column (4), the coefficient on household human capital is 0.375 and is significant at the 10% level, implying that educational time investment also increases with household human capital. However, patterns in input choices alone are not sufficient to determine whether different educational inputs are substitutes or complements. To identify substitutability versus complementarity, we further estimate the child human capital production function.

5.1.3. Estimation results for the child human capital production function

Based on Equation (17), we estimate the CES-type child human capital production function using NLS. The estimation results are reported in Table 3.

Table 3.

Estimation results for the child human capital production function

Table 3 Long description

A table with seven rows and five columns. The columns are labeled Coefficient, Std. Err., z-statistic, and 95% CI. The rows are labeled Substitution parameter epsilon, Elasticity of substitution E sub s, Contribution of household human capital, Contribution of monetary educational investment, Contribution of educational time investment, and Contribution of average social human capital. Each row contains values for Coefficient, Std. Err., z-statistic, and 95% CI. The table presents statistical data related to the estimation results for the child human capital production function.

Notes: *** and ** denote significance at the 1% and 5% levels, respectively.

Table 3 indicates that specifying the CES function as the technology for children’s human capital formation is empirically reasonable. The estimated value of the substitution parameter $\varepsilon $ is −0.234, which is statistically significantly below zero. The implied elasticity of substitution, $\tilde E$ , is 0.811, significantly less than one, indicating strong complementarity among household human capital, educational time investment, and educational monetary investment. Examining the relative contributions of each input to children’s human capital, educational time investment contributes the most, followed by monetary investment, while the direct contribution of household human capital is comparatively smaller.

Taken together with the earlier regression results for fertility and educational time investment, Table 2 column (4) and Table 3 jointly support Hypothesis 2a: when educational inputs are complementary, households with higher human capital devote more time to children’s education. Moreover, Table 2 column (2) and Table 3 together provide evidence consistent with Hypothesis 3a: under complementarity in the inputs to children’s human capital formation, household human capital and fertility exhibit a U-shaped relationship – fertility first declines and then increases as household human capital rises.

5.3.1. Grouped regressions by household human capital

The baseline and robustness results presented above already provide strong evidence of a robust U-shaped relationship between household human capital and the number of children. Accordingly, we further examine whether the complementarity among educational inputs remains consistent across households with different human-capital endowments. Specifically, we use the estimated minimum point of the U-shaped relationship in Table 2, column (2), denoted by $H{C^{\rm{*}}}$ , as the threshold to divide the sample into low and high-human-capital households, and then estimate the CES production function separately for the two subsamples. The corresponding results are reported in Table 5.

Table 5.

Estimated substitution elasticities of educational inputs

Table 5 Long description

A table with four rows and four columns comparing substitution elasticities for low and high-human-capital households. The columns are labeled Coefficient, Std. Err., z-statistic, and 95% CI. The rows are labeled Low-human-capital households: ε, Low-human-capital households: Ês, High-human-capital households: ε, and High-human-capital households: Ês. Row 1: Coefficient, -0.179; Std. Err., 0.1317; z-statistic, -1.36; 95% CI, [-0.437, 0.079]. Row 2: Coefficient, 0.848***; Std. Err., 0.0948; z-statistic, 8.95; 95% CI, [0.662, 1.034]. Row 3: Coefficient, -0.257***; Std. Err., 0.0737; z-statistic, -3.48; 95% CI, [-0.401, -0.112]. Row 4: Coefficient, 0.796***; Std. Err., 0.0470; z-statistic, 17.04; 95% CI, [0.704, 0.887].

Notes: Robust standard errors are reported in parentheses. *** denotes significance at the 1% levels.

Table 5 shows that the estimated elasticities of substitution are below one in both subsamples, indicating that educational time inputs, monetary inputs, and household human capital are complementary across different household human-capital groups. This input structure substantially intensifies the quantity-quality trade-off faced by low-human-capital households, so that within the low-human-capital range, fertility declines as household human capital increases. By contrast, high-human-capital households are better able to produce higher-quality children; therefore, within the high-human-capital range, fertility tends to increase as household human capital rises further.

5.3.2. Grouped regressions by region (Eastern, Central, and Western China)

Building on the subgroup analysis above, we further split the sample into China’s eastern, central, and western regions to examine whether (i) the U-shaped relationship between household human capital and the number of children and (ii) complementarity among educational inputs are prevalent across regions. The corresponding results are reported in Tables 6 and 7.

Table 6.

Heterogeneous effects of household human capital on fertility

Table 6 Long description

A table comparing the effects of household human capital on the number of children across different regions in China. The table has three columns and four rows. Column headers are Poisson Eastern China, Poisson Central China, and Poisson Western China. Row labels are Household human capital, Household human capital squared, Control variables, Observations, and R squared. Row 1: Household human capital, -2.002***, -1.035*, -2.297***, (0.770), (0.572), (0.843). Row 2: Household human capital squared, 2.737***, 1.818**, 2.710**, (0.980), (0.732), (1.126). Row 3: Control variables, Yes, Yes, Yes. Row 4: Observations, 1,780, 1,803, 1,308. Row 5: R squared, 0.0171, 0.0243, 0.0173.

Notes: See Table 2.

Table 7.

Estimated substitution elasticities of educational inputs

Table 7 Long description

A table with six rows and four columns comparing estimated substitution elasticities of educational inputs across regions in China. The columns are labeled Coefficient, Std. Err., z-statistic, and 95% CI. The rows are labeled Eastern China: ε, Eastern China: Ẽs, Central China: ε, Central China: Ẽs, Western China: ε, and Western China: Ẽs. Row 1: Coefficient, -0.145; Std. Err., 0.0659; z-statistic, -2.20; 95% CI, [-0.274, 0.016]. Row 2: Coefficient, 0.874; Std. Err., 0.0503; z-statistic, 17.36; 95% CI, [0.775, 0.972]. Row 3: Coefficient, -0.510; Std. Err., 0.3156; z-statistic, -1.62; 95% CI, [-1.129, 0.109]. Row 4: Coefficient, 0.662; Std. Err., 0.1384; z-statistic, 4.78; 95% CI, [0.391, 0.933]. Row 5: Coefficient, -0.263; Std. Err., 0.1004; z-statistic, -2.62; 95% CI, [-0.460, -0.067]. Row 6: Coefficient, 0.791; Std. Err., 0.0629; z-statistic, 12.59; 95% CI, [0.668, 0.915].

Notes: Robust standard errors are reported in parentheses. *** and ** denote significance at the 1% and 5% levels, respectively.

Table 6 shows that, in all three regional subsamples, household human capital exhibits a consistent nonlinear pattern – the linear term is negative and the squared term is positive – and most coefficients are statistically significant at conventional levels. This suggests that the U-shaped relationship between household human capital and fertility is broadly consistent across regions. In terms of magnitude, the coefficient on the squared term is relatively larger in the eastern and western regions, indicating a more pronounced degree of nonlinearity in fertility patterns, whereas the curvature of the U-shape is comparatively flatter in the central region. In addition, Table 7 reports that the estimated elasticities of substitution among educational inputs are all significantly below one in the eastern, central, and western regions, implying that educational inputs are complementary across regions.

Taken together, these findings indicate a high degree of regional consistency in the U-shaped relationship between household human capital and fertility, and that the complementary structure of educational inputs is likewise pervasive at the regional level. Because raising children’s educational “quality” requires households to jointly allocate parental time and monetary expenditures, this investment mode amplifies differences in the quantity–quality trade-off across households with different levels of human capital. As a result, household fertility behavior endogenously gives rise to a robust U-shaped fertility structure at the micro level.