2.1 Data and sample

The data for this study are sourced primarily from the Chinese Household Income Projects (hereafter CHIPs), the Chinese Population Census, and the China City Statistical Yearbook. We utilize data from four CHIPs waves (2002, 2007, 2013, 2018) to gather household-level information. For each wave, the CHIPs datasets include both urban and rural samples, and we utilize data from both to compile comprehensive household-level information. We pool these 4 waves of CHIPs to form the data sample that will be used in reduced-form analysis.

For cleaning household-level data, we limit the age range for husbands and wives to 20–35 years,Footnote ³ focusing on first marriages. The rationale for concentrating on this specific age bracket and first marriages is twofold: Firstly, the majority of Chinese couples enter into their first marriage within the age range of 20–35, as shown in Figure 1. This figure presents the age distribution of first marriages for men and women in China, highlighting that most occur between the ages of 20 and 35. Also, the legal marriage age in China is set at 22 for men and 20 for women. Secondly, examining “young marriages” provides a clearer picture of how the sex ratio in the marriage market influences matching outcomes. This is because marriage decisions for older individuals may be affected by various factors such as children, health conditions, and housing.

Figure 1.

Age at first marriage in China.

Note: We use 1% survey data (individual level) of 2005 China’s population census to plot the age distribution for men and women.

We measure educational correlation, household income inequality, and the sex ratio in the marriage market at the prefecture level, a regional division in China akin to a city. Educational correlation is calculated from the education levels of husbands and wives in the CHIPs survey. The dataset distinguishes nine education categories,Footnote ⁴ and for each prefecture, we use all couples in the 20–35 age cohort to compute the correlation coefficient between spouses’ education levels. This coefficient serves as our measure of assortative marriage.

Household income is calculated as the combined job incomes of both the husband and wife. We then assess income inequality at the prefecture-level using these aggregated household incomes. The sex ratio ( $SR_{jt}$ ) is defined for each prefecture $j$ and wave $t$ . Analyzing data at the prefectural level allows us to capture a more diverse range of sex ratio variations. This heterogeneity is depicted in Figure 2, where we illustrate the sex ratio for the 20 to 35 age cohort at the prefectural level, highlighting regional differences across China. As detailed in Wei et al. (Reference Wei, Zhang and Liu2017), due to policy constraints on internal migration and barriers related to language and culture, marriages in China are predominantly local. This suggests that the sex ratio of male to female residents, as reported in the population census, sufficiently captures the intensity of local marriage market competition. For different waves of CHIPs data, we deduce the corresponding sex ratio from various waves of the population census. Table 1 provides detailed insights into how we determine the sex ratio in the marriage market.

Table 1.

Cohort age selection

Note: The 2000, 2010, and 2015 censuses provide population data on males and females across various age groups (such as 5–9, 10–14, etc.).

Figure 2.

Sex ratio for the 20 to 35 age cohort across prefectures.

Note: The darker the color, the higher the sex ratio. We employ data from the 2010 census to map out the sex ratio for the 20 to 35 age cohort at the prefecture level. The 2010 census provides data on the number of male and female residents in each county, categorized by various age groups (e.g., ages 5–9, 10–14, etc.). We aggregate these figures to the prefecture-level to calculate the sex ratio.

The summary statistics for the key variables are reported in Table 2. The mean educational correlation is 0.659, indicating a generally positive pattern of assortative marriage among couples in China. The Gini coefficient exhibits significant regional variation, with an average value of 0.294 and a standard deviation of 0.076. Its range extends from a minimum of 0.104 to a maximum of 0.605, indicating substantial heterogeneity in income inequality between regions. Similar patterns are observed in the analysis of two other income inequality measures: the logarithm of the income ratio between the 80th and 20th percentiles, and the logarithm of the income ratio between the 90th and 10th percentiles of households, both of which underscore the high regional variability of income inequality. Examining the sex ratio at the prefecture level, we find a standard deviation of 0.082, with the minimum and maximum values being 0.689 and 1.277, respectively, further highlighting its heterogeneity.

Table 2.

Summary statistics

Note: All variables are measured at the prefecture level. The Gini coefficient and the two other income inequality measures are computed by the authors using household income information from four waves of CHIPs data. The sex ratio is derived from the China Population Censuses of 2000, 2010, and 2015. All other control variables are obtained from the China City Statistical Yearbook.

2.2 Sex ratio, assortative marriage, and household income inequality

In this section, we empirically investigate how the sex ratio in the marriage market affects the degree of household assortative marriage and income inequality. Our hypothesis is that the link between the sex ratio in the marriage market and income inequality consists of two parts. First, the gender imbalance in the marriage market influences marital matching outcomes. Du et al. (Reference Du, Wang and Zhang2015) and Yao et al. (Reference Yao, Zhang and Li2023) find that the scarcity of females in China lead to men marry women from a lower social class (i.e., less educated or lower income), suggesting a decrease of assortative mating. Second, the degree of assortative matching in marriage affects household income inequality, as extensively documented in the works of Eika et al. (Reference Eika, Mogstad and Zafar2019); Fernandez et al. (Reference Fernandez, Guner and Knowles2005); Greenwood et al. (Reference Greenwood, Guner, Kocharkov and Santos2014). These studies have found a positive correlation between educational assortative marriage and income inequality.

Table 3 reports the regression results, where both region and year fixed effects are controlled. To study the relationship between gender imbalances in the marriage market and the degree of assortative matching between the spouses, column (1) reports the regression results where education correlation between the spouses is used as a dependent variable. In columns (2)–(4), we study the relationship between the sex ratio and household income inequality. Household income inequality measures are represented by three metrics: the Gini coefficient (Gini), the logarithm of the income ratio between the 90th and 10th percentiles (Wage9010), and the logarithm of the income ratio between the 80th and 20th percentiles (Wage8020). In column (1), we find that the regression coefficient for the sex ratio is $-0.297$ , indicating that a higher sex ratio in the marriage market is associated with a lower degree of assortative matching. This finding is consistent with the existing literature. Turning to income inequality, the results in columns (2)–(4) also show negative and significant effects across all three measures, suggesting that an increase in the sex ratio reduces household income inequality. However, these OLS estimates are descriptive correlations rather than causal effects; to address potential endogeneity concerns, we turn to an instrumental-variable approach in the next section.

Table 3.

Sex ratio, degree of assortative matching, and household income inequality

Note: Robust standard errors in parentheses. $^{***}$ $p\lt 0.01$ , $^{**}$ $p\lt 0.05$ , $^{*}$ $p\lt 0.1$ . We have consolidated data from four waves of CHIPs, encompassing both rural and urban samples, to gather household income information. In our analysis, we focus on couples where both the husband and wife are between 20 and 35 years old and are in their first marriage. The sex ratio in the marriage market is derived from population census data.

2.3 Instrumental variable approach

To resolve the potential endogeneity concern, we use a leave-one-out spatial instrument: the average sex ratio across other cities within the same province (excluding the target city). This variable serves as a proxy for shared regional demographic pressures, policy enforcement intensity (e.g., local implementation of the one-child policy), and cultural norms that jointly influence sex ratio imbalances across cities within a province.

This instrument satisfies the two key requirements for a valid instrumental variable: (i) Relevance: cities within the same province typically share similar demographic policies, migration patterns, birth planning norms, and economic conditions. As a result, the sex ratio in other cities is highly predictive of the target city’s own sex ratio. Our first-stage regression confirms this strong relationship, with F-statistics well above conventional thresholds. This ensures that the instrument is informative for explaining variation in the endogenous regressor (local sex ratio). (ii) Exogeneity: the leave-one-out construction ensures that the instrument does not mechanically contain information about the target city’s own outcome (i.e., educational correlation, household income inequality). Furthermore, by controlling for prefecture fixed effects and year fixed effects, we purge the influence of time-invariant local characteristics and national trends. Under these conditions, any direct effect of other cities’ sex ratios on local income inequality is highly unlikely, leaving the local sex ratio as the only plausible channel through which the instrument affects the outcome.

This approach aligns with the empirical logic of spatial and contextual instruments employed in previous studies, such as Chetty and Hendren (Reference Chetty and Hendren2018); Kleven et al. (Reference Kleven, Landais and Søgaard2019), where leave-one-out neighborhood averages are used to isolate local causal effects without contamination from the target unit’s own data.

The 2SLS estimates are reported in Table 4. Across all specifications, the coefficients on the sex ratio remain negative, consistent with the baseline regressions.Footnote ⁵ The effects on household income inequality are negative and statistically significant across all three measures. Moreover, the magnitudes are larger in absolute terms compared with the OLS estimates, implying that failing to account for endogeneity biases the baseline estimates toward zero. For instance, the IV coefficient for the 90/10 income ratio is −5.627, substantially larger than its OLS counterpart. This suggests that unobserved factors—such as selective migration or local shocks correlated with both sex ratios and income distribution—may have attenuated the baseline results.

Table 4.

The effect of sex ratio on assortative marriage and household income inequality: 2SLS estimates

Note: All regressions control for prefecture and year fixed effects. Robust standard errors clustered at the prefecture level are in parentheses. Panel B reports first-stage results and instrument diagnostics. $^{***}$ $p\lt 0.01$ , $^{**}$ $p\lt 0.05$ , $^{*}$ $p\lt 0.1$ . The number of observations is smaller in the IV regressions because the construction of the instrument (prefecture–year average sex ratio) and the absorption of fixed effects drop prefecture–year cells with missing instruments or singleton groups. The reduction in sample size is therefore mechanical and does not reflect selective exclusion of data.

Taken together, these findings reinforce the causal interpretation of our main result: demographic imbalances in the marriage market systematically reduce household income inequality.

3.1 Setup

The economy is populated by individuals of two genders, men and women. Both men and women can be either skilled, “ $s$ ”, or unskilled, “ $u$ ”. There are four types of agents: skilled men, skilled women, unskilled men, and unskilled women. Following Fernandez et al. (Reference Fernandez, Guner and Knowles2005), we employ a two-stage matching model. In the first stage, matches are formed randomly across genders, while in the second stage, unmatched individuals are paired within their education group (i.e., skilled with skilled and unskilled with unskilled).

Before entering the marriage market, each individual decides what education level to acquire: they can become skilled or unskilled. Once they have finished their education, all agents enter the marriage market, where they go through one or two rounds of matching. They observe each other’s skill type, which perfectly predicts the future income of an individual. In addition, a matched couple draws a match-specific quality (“love”) shock, and jointly decide whether to form a married couple or to remain single.

Men and women in a married couple share a common joint utility function. Each agent is assumed to derive utility from consumption and from the quality of his (her) match. The indirect utility function for a couple with match quality $q$ and household income $I$ is:

(1) \begin{equation} V(I,q)=U(I)+q=U[e(w_{m}^{z}+w_{f}^{z})]+q \end{equation}

where $U$ is a continuous and strictly increasing function of $I$ ; $w_{m}^{z}$ denotes the wage of males with skill type $z$ ( $z$ is either skilled “ $s$ ” or unskilled “ $u$ ”) and $w_{f}^{z}$ denotes the wage of females with skill type $z$ . $e$ is used to measure the consumption equivalence scale, which is a measure of the cost of living for a household given family size and demographic composition. We follow the literature and set $e=0.7$ in this study. Household income is:

(2) \begin{equation} I=\left \{\begin{array}{ll} w^{s}_{m}+w^{s}_{f}, \\ w^{s}_{m}+w^{u}_{f}, \\ w^{u}_{m}+w^{s}_{f}, \\ w^{u}_{m}+w^{u}_{f}, \\ \end{array}\right . \end{equation}

The total number of men is normalized to 1 ( $N^m=1$ ). As a result of the “male abundance,” the number of women is smaller than the number of men ( $N^f\lt 1$ ). In the model, $\frac {N^m}{N^f}=\frac {1}{N^f}$ denotes the sex ratio in the marriage market. $\psi _m$ is the share of skilled men among all men, $\psi _f$ is the share of skilled women among all women.

3.2 Matching

There are two rounds of matching in the marriage market. In the first round, agents meet randomly and draw a random match-specific quality $q$ . We assume that $q$ is non-negative and follows a continuous distribution, taking the possible values in the $[0,q_{\max }]$ interval.Footnote ⁶ The cumulative distribution function (CDF) of $q$ is $Q$ , with the expected value equal to $\mu$ . The match can be accepted by both agents (resulting in a marriage), or rejected by at least one of the agents. In the latter case, both agents go to the second round of matching.

In the second round, all remaining unmatched women are guaranteed a partner within their own education group (skilled unmarried women are matched with skilled unmarried men, while unskilled unmarried women are matched with unskilled unmarried men). A new random match quality is drawn from the same distribution $Q$ . Since the number of men exceeds the number of women ( $N^m \gt N^f$ ), all women will eventually be matched, but some men may remain single after the second stage. This feature ensures the consistency of the model and reflects the demographic imbalance that motivates our analysis.

Let $\bar {q}_{g}^{zz}$ be the cutoff match quality for an agent with gender $g$ and in a match type $zz$ , where $g \in \{m,f\}$ , and $zz \in \{ss,su,us,uu\}$ . Below this cutoff, the agent will not accept the match. Thus, the probability of agents getting married is $1-Q\big[\max (\bar {q}_{m}^{zz},\bar {q}_{f}^{zz})\big]$ .

In the first round, the total number of meetings is $N^f$ . Since $N^m\gt N^f$ , some men will not meet a potential partner. For example, the number of meetings between skilled male and unskilled female is: $N^{f}\psi _{m}(1-\psi _{f})$ .

In the second round, some part of the remaining unmarried skilled males match (and get married) with skilled females, and some part of the remaining unmarried unskilled males match (and get married) with unskilled females. Let $\theta _{g}^{z}$ stand for the number of unmarried agents after the first round of the match with gender $g$ and skill type $z$ . For instance, the number of unmarried unskilled males after round 1 is:

(3) \begin{eqnarray} \theta _{m}^{u}&=&\big(1-N^{f}\big)(1-\psi _{m})+N^{f}(1-\psi _{m})\psi _{f}Q\big[\max \big(\bar {q}^{us}_{m},\bar {q}^{us}_{f}\big)\big]\nonumber\\&&+N^{f}(1-\psi _{m})\big(1-\psi _{f}\big)Q\big[\max \big(\bar {q}^{uu}_{m},\bar {q}^{uu}_{f}\big)\big] \end{eqnarray}

There is a trade-off between accepting the first round of matching and rejecting it, in which case the agent goes to the second round of matching. In particular, a skilled male who meets an unskilled female in the first round and draws a high match quality will encounter a trade-off between forming a lower-income household with high match quality, or rejecting this match and forming a higher-income household with a skilled female in the second round but with an expected match quality that is $\mu$ .

3.3 Equilibrium

Given $V(I,q)=U(I)+q$ , a skilled male will be indifferent between accepting a first-round match with an unskilled female and rejecting that match and entering into second round if:

\begin{equation*}V\big(I, \bar {q}_{m}^{su}\big)=E[V(I, \mu )]\end{equation*}

where $\bar {q}_{m}^{su}$ denotes the reservation match quality for the skilled male when he meets with an unskilled female. The first character in the superscript “ $s$ ” is used to denote the male’s skill type, whereas the second character “ $u$ ” is used to denote the female’s skill type (we follow the same notations in the rest of the paper).

In the following part, we use this notation to represent the reservation match quality for each type of agent. Thus, there are eight reservation match qualities in total. The equilibrium is determined by eight match quality cutoffs in the first round. Solving for the value of $q$ at the equilibrium yields the reservation quality of $\bar {q}_{g}^{zz}$ .

In round 1, a skilled male matches with a skilled female will be indifferent if:

\begin{equation*} U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+\bar {q}^{ss}_{m}=\min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\big[U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+\mu \big]+\left [1-\min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\right ]U\big(w_{m}^{s}\big) \end{equation*}

And if

\begin{equation*} U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+q_{\min }\gt \min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\big[U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+\mu \big]+\left [1-\min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\right ]U\big(w_{m}^{s}\big) \end{equation*}

then all the skilled male that are matched with a skilled female will want to get married in round 1.

If

\begin{equation*} U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+q_{\max }\lt \min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\left [U\big[e\big(w_{m}^{s}+w_{f}^{s}\big)\big]+\mu \right ]+\left [1-\min \left(\frac {\theta _{f}^{s}}{\theta _{m}^{s}},1\right)\right ]U(w_{m}^{s}) \end{equation*}

then no skilled male matched with a skilled female will want to get married in round 1. Where $q_{\max }$ , $q_{\min }$ represent the upper bound and lower bound of match quality.

In summary, the equilibrium is characterized by eight indifference conditions. As an illustration, consider the indifference condition for an unskilled male when paired with a skilled female; the remaining conditions are presented in Appendix B.

(4) \begin{equation} U\big[e\big(w_{m}^{u}+w_{f}^{s}\big)\big]+\bar {q}^{us}_{m}=\min \left(\frac {\theta _{f}^{u}}{\theta _{m}^{u}},1\right)\left [U\big[e\big(w_{m}^{u}+w_{f}^{u}\big)\big]+\mu \right ]+\left [1-\min \left(\frac {\theta _{f}^{u}}{\theta _{m}^{u}},1\right)\right ]U(w_{m}^{u}) \end{equation}

3.4 Equilibrium distribution of households

Since we have solved the reservation match quality for different types of match, we can generate the equilibrium distribution of household types. Let $\pi _{zz}$ denotes the number of different types of households in the population, thus we have:

The number of skilled husband with skilled wife in the population:

\begin{equation*}\pi _{ss}=\underbrace {N^{f}\psi _{m}\psi _{f}\left (1-Q \big[ \! \max \big(\bar {q}^{ss}_{m},\bar {q}^{ss}_{f}\big)\big]\right )}_{\text{marriage at the first round}} \ \ \ \ +\underbrace {\min (\theta _{m}^{s},\theta _{f}^{s})}_{\text{marriage at the second round}}\end{equation*}

The number of skilled husband with unskilled wife in the population:

\begin{equation*}\pi _{su}=N^{f}\psi _{m}(1-\psi _{f})\left (1-Q\big[ \! \max \big(\bar {q}^{su}_{m},\bar {q}^{su}_{f}\big)\big]\right )\end{equation*}

For other types of households in the population, we present them in Appendix B.

Note that we have:

\begin{equation*}\pi _{ss}+\pi _{su}+\pi _{us}+\pi _{uu}=N^{f}\end{equation*}

Also have:

\begin{equation*}\pi _{ss}+\pi _{su}+\pi _{us}+\pi _{uu}+\pi _{s}+\pi _{u}=1\end{equation*}

where the total number of marriages equals the number of females in the population.

3.5 Education correlation between husband and wife

Definition: we define the education correlation between husband and wife $\rho$ as:

(5) \begin{equation} \rho =\frac {\frac {\pi _{ss}}{N^{f}}-\frac {\pi _{ss}+\pi _{su}}{N^{f}}\frac {\pi _{ss}+\pi _{us}}{N^{f}}}{\sqrt {\left(\frac {\pi _{ss}+\pi _{su}}{N^{f}}\right)-\left(\frac {\pi _{ss}+\pi _{su}}{N^{f}}\right)^2}\sqrt {\left(\frac {\pi _{ss}+\pi _{us}}{N^{f}}\right)-\left(\frac {\pi _{ss}+\pi _{us}}{N^{f}}\right)^2}} \end{equation}

We show the detailed derivations and descriptions in Appendix B.

4.1 Calibration

The model is calibrated to match the empirical evidence and reproduce quantitative properties. The utility function takes the form of:

\begin{equation*}U(I)=log(I)\end{equation*}

We assume the match quality $q$ follows a continuous uniform distribution. The support of $q$ is $[0,1]$ , where the expected value is $\mu =0.5$ in this setting. A skilled agent is defined as someone who possesses educational qualifications beyond junior middle school, including those from senior middle school, vocational senior secondary school, polytechnic colleges, undergraduate programs, and graduate degrees.Footnote ⁷

The wages for different types of agents are exogenously calibrated. We employ CHIPs 2018 data to project lifetime wages for four types of agents, as it covers the most number of provinces in China compared with other waves of CHIPs data.Footnote ⁸ The life cycle of an individual is divided into 5-year intervals, the age range of men is from 20 to 60, and the age range of women is from 20 to 55, the different selection is due to the mandatory retirement age in China, which is different for men and women (the restricted age is 60 for men and 55 for women). For the projections of lifetime wage of males, the interval is 20–24, 25–29, 30–34, …, 56–60. For females, it’s up to 55. The wage we used is the before-tax net labor wage, as tax may play a role in redistributing incomes. We initially compute the average wage of unskilled women in the age group of 20–24, and do so for other age groups until the group of 51–55, then use the annual discount factor of 0.96 to get the present value of unskilled women’s lifetime wages.Footnote ⁹ We follow the same procedures to compute the lifetime wages of skilled women, unskilled men, and skilled men. The wage of the unskilled female is normalized to 1, and we use the wage premiums to obtain those for skilled females, unskilled males, and skilled males. Where the wage premium of a skilled female is equal to the ratio of the projected lifetime wage for a skilled female over an unskilled female, the same setting applies to other types of agents.

The skilled shares of men ( $\psi ^{m}$ ) and women ( $\psi ^{f}$ ) are also calibrated to match empirical evidence. The study uses the Chinese population census 2015 to measure the fraction of skilled men and skilled women.

We vary the number of women in the model, $N^{f}$ , in the range from 0.81 to 0.98, which implies the sex ratio varies from 1.031 to 1.235. The detailed calibration results can be found in Table 5.

Table 5.

Calibrated parameters

Table 6.

How model fit to data

Note: We compute the mean of the education correlation using the data sample from the empirical analysis.

Figure 3.

Assortative matching and sex ratio.

4.2 Education correlation between couples

To study the impact of gender imbalances on the degree of assortative matching, we vary the sex ratio in our model and investigate what happens to the education correlation (computed using the formula in equation (5)) in equilibrium. Using education correlation between couples as a proxy for the degree of assortative matching in marriages follows the idea of Eika et al. (Reference Eika, Mogstad and Zafar2019); Greenwood et al. (Reference Greenwood, Guner, Kocharkov and Santos2014).

As illustrated in Figure 3, the education correlation between couples is decreasing with the increase in the sex ratio. The intuition here is that the rising number of males in the marriage market makes men less likely to find a wife, thus males are less selective, which results in a decrease in assortative matching.Footnote ¹⁰ This finding is consistent with the results of other papers; Du et al. (Reference Du, Wang and Zhang2015); Yao et al. (Reference Yao, Zhang and Li2023) find that when females are scarce, males tend to marry females of a lower social class, i.e., those who are less educated, have lower personal income and come from families with lower wealth.

To check how the model fits with the data, we compare the moments of education correlation between model output and data sample. The results are reported in Table 6, we find the mean of education correlation in the model output is close to the value in the data sample, which is 0.6256 and 0.6183 respectively.

4.3 Implication for household income inequality

This section reports on how the sex ratio in the marriage market impacts household income inequality. Multiple indices are used to measure income inequality. First, we use the Gini coefficient to reveal the relationship between the sex ratio in the marriage market and household income inequality. By constructing the Lorenz curve (formulating the cumulative household share and cumulative income share), the Gini coefficient can be derived analytically, the derivations are reported in Appendix D.

Figure 4.

Gini coefficient and sex ratio.

Figure 5.

Sex ratio and household income inequality.

Figure 4 illustrates that in the marriage market, the sex ratio is inversely correlated with the Gini coefficient, suggesting that a higher sex ratio correlates with lower household income inequality. The underlying rationale for this pattern relates to the positive association between assortative marriage and household income inequality, a theme explored in several studies (e.g., Eika et al. (Reference Eika, Mogstad and Zafar2019); Fernandez et al. (Reference Fernandez, Guner and Knowles2005); Greenwood et al. (Reference Greenwood, Guner, Kocharkov and Santos2014)). Conversely, our prior analysis reveals that an elevated sex ratio corresponds to reduced assortative matching. This inverse relationship indicates that a higher sex ratio may lead to decreased household income inequality. Intuitively, a decline in assortative marriage suggests a reduction in the proportion of couples with similar education levels (positive assortative mating) and an increase in couples with differing education levels, as shown in Table C2. These shifts in household formation contribute to the reduction in household income inequality, reinforcing our conclusion that the sex ratio is negatively associated with household income disparity.

We also use other measures to study how the sex ratio impacts the inequality of household income. Figure 5 presents four measures of income inequality, which are the coefficient of variation, Theil L index, Theil T index, and logarithmic variance, respectively. These indices provide a more comprehensive understanding of the income distribution and its skewness. For instance, the coefficient of variation emphasizes the standard deviation of income relative to the mean, offering insights into the overall variability of household income. On the other hand, Theil’s indices, particularly the L and T variants, offer a decomposition feature that allows for a detailed analysis of inequality within and between different population subgroups. Similarly, the logarithm of variance addresses scale dependency issues by normalizing the variance, thus providing a measure of dispersion that is not influenced by the income level itself (see Cowell, Reference Cowell2011).

Our analysis, as shown in Figure 5, demonstrates that the sex ratio in the marriage market exhibits a consistent inverse relationship with these indices. Specifically, the coefficient of variation, which gauges the dispersion of income around the mean, shows a downward trend as the sex ratio increases, indicating a contraction in relative income disparities. Furthermore, the Theil indices, both L and T, show a similar downward trend with an increasing sex ratio. The Theil L index, which places greater emphasis on the lower tail of the income distribution, suggests that a higher sex ratio may contribute to raising incomes at the lower end, thereby reducing inequality. The Theil T index, more sensitive to the upper tail, implies that the top earners’ share of total income decreases as the sex ratio rises. This could be interpreted as a decline in the income advantage that typically accrues to households with high educational or occupational homogamy, as the sex ratio increases. Lastly, the logarithmic variance, which mitigates the scale dependency of variance, also shows a reduction with an elevated sex ratio. This indicates that income differences between households are becoming less pronounced, leading to a more even spread of earnings among families.

4.3.1 Quantitative comparison between the model and data

In this section, we present a quantitative comparison between the model and the empirical results. Specifically, we replicate the regressions reported in Table 3 using the simulated data from the calibrated model and contrast the resulting coefficients with those obtained from the empirical sample.

The comparison highlights two key findings (see Table 7). First, the model reproduces the relationship between sex ratios and assortative marriage well. The estimated coefficient of the sex ratio on educational correlation is −0.289, closely aligned with the empirical estimate of −0.297. This correspondence suggests that the model successfully captures the mechanism by which a male-biased marriage market increases the share of random matches, thereby weakening assortative matching.

Second, the model predicts a smaller impact of sex ratios on household income inequality than observed in the data. For the Gini coefficient, the model delivers an estimate of −0.0052, compared with an empirical coefficient of −0.175, While the direction of the effect is consistent, the magnitude is underestimated. This limitation stems from the parsimonious structure of the model, which includes only four discrete household types. Such structure flattens the Lorenz curve and may dampen the sensitivity of inequality measures to changes in household composition. We view this as a promising direction for future research: extending the framework to incorporate richer heterogeneity—such as multiple skill levels or continuous wage distributions—would allow the model to generate a more nuanced income distribution and better match the empirical magnitudes of inequality responses.

Table 7.

Model-implied coefficients vs. data estimates

Note: Coefficients are estimated from regressions of each outcome on the sex ratio. “Model coeff.” is based on simulated data from the calibrated model; “Data coeff.” is based on the empirical sample.

4.4 Inspecting the mechanisms

The negative effect of the sex ratio on assortative matching in our model operates through the interaction of reservation thresholds and the two-stage matching structure. When the number of men increases, competition in the marriage market intensifies. As a result, men become less selective: they lower the reservation match quality that they are willing to accept in the first round. Figure 6a illustrates this pattern across different skill pair types, showing that the reservation cutoff declines as the sex ratio rises.

Figure 6.

Reservation match quality for men and first-stage marriage rate.

This decline in reservation thresholds has direct implications for the share of first-stage marriages. Because the first round involves random matching across education groups, a lower reservation cutoff increases the probability that men accept these random matches. As shown in Figure 6b, the first-stage marriage rate rises monotonically with the sex ratio. More marriages are therefore concluded in the random stage, reducing the role of the second-stage within-group (assortative) matching.

The shift toward a greater share of random matches has important distributional consequences. Since random matches weaken the positive correlation between spouses’ education levels, the overall degree of assortative matching falls. With less assortative sorting by education, income differences across households narrow, leading to lower household income inequality in equilibrium.

4.5 Robustness check: alternative matching function

To examine the robustness of our results, we consider a more general specification of the matching function. Following Den Haan et al. (Reference Den Haan, Ramey and Watson2000), we allow the meeting probabilities on both sides of the market to vary with the sex ratio rather than assuming that each woman is guaranteed to meet a partner in the first stage. The general matching function is defined as:

(6) \begin{equation} M(X,Y) = \frac {XY}{\Big(X^{\tfrac {1}{\omega }}+Y^{\tfrac {1}{\omega }}\Big)^{\omega }} \end{equation}

where $M$ denotes the number of matches between two groups, $X$ and $Y$ represent the number of agents on each side of the market (men and women), and $\omega \gt 0$ is the elasticity of substitution between them. In our context, the formulation becomes:

(7) \begin{equation} M(N^{m},N^{f}) = \frac {N^{m}N^{f}}{\bigg ((N^{m})^{\tfrac {1}{\omega }}+(N^{f})^{\tfrac {1}{\omega }}\bigg)^{\omega }} \end{equation}

Our baseline assumption corresponds to the limiting case as $\omega \to 0$ , in which each woman is guaranteed to meet a man in the first stage. By allowing finite values of $\omega$ , this specification relaxes the baseline assumption and makes women’s meeting probabilities responsive to demographic imbalances.

Quantitative experiments under different values of $\omega$ show that our main results remain robust. Although a higher probability of meeting a partner allows women to be more selective, the overall pattern persists: a higher sex ratio reduces assortative matching and, in turn, lowers household income inequality. This confirms that our key insights are not an artifact of the baseline assumption but hold under a broader class of matching technologies. Detailed derivations and numerical results are reported in Appendix E.

Figure 7.

Decision tree for the male who wants to become skilled.

4.6 Education decision

In previous sections, the skilled labor shares for both males and females were exogenously calibrated. This section, however, explores how agents choose their education levels. Specifically, it aims to conduct a counterfactual experiment examining the choices of males and females in education, considering their expectations of intensified competition in the marriage market among men and reduced competition among women due to changing sex ratios. To investigate this, we model agents making educational investment decisions before entering the marriage market. We denote the period when agents decide on their education as $t$ and the period when they enter the marriage market competition as $t+1$ .

In period $t$ , agents form expectations regarding the utility gains from matches occurring in the following period, $t+1$ . These expectations guide their decision to pursue either a skilled or unskilled path. The process of deciding to become a skilled or unskilled agent involves various scenarios, and we have delineated the expected value function for clarity. Initially, we elucidate the expected payoff for a skilled male using a decision tree diagram, followed by a formal expression of the expected payoff equation. The expected payoff equations for the other three agent types are detailed in Appendix F, along with corresponding decision tree diagrams. These decision trees carefully map out each step’s potential outcomes, allowing for the calculation of the associated expected utility.

Figure 7 presents a decision tree for a man who wants to become skilled. In the first round of matching, he encounters one of three possible outcomes: meeting a skilled female ( $f_{s}$ ), an unskilled female ( $f_{u}$ ), or no one at all (“-”), a scenario that arises when the number of males ( $N^{m}=1$ ) exceeds the number of females ( $N^{f}$ ). If he meets a skilled female ( $f_{s}$ ), he may either agree to marry (“Yes”) or decline the match (“No”), moving to the second round. In the second round, he can either successfully pair with a skilled female and marry or remain single. Similarly, if he meets an unskilled female ( $f_{u}$ ), he faces the choice to either form a household with her (“Yes”) or refuse (“No”) and proceed to the second round. Here, he may marry a skilled female ( $f_{s}$ ) or end up single (“Single”). If he meets no one in the first round (“-”), he directly advances to the second round, where he either finds a skilled female to marry or remains single (“Single”).

The expected utility of being a skilled male given the fraction of $\psi ^{f}_{t+1}$ skilled female is:

(8) \begin{align} \mathrm E_{t}{V_{m}^{s}} & \big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big) = \psi _{t+1}^{f}{N^{f}_{t+1}} \left \{ \int _{\max (\bar {q}_{m}^{ss},\bar {q}_{f}^{ss})}^{q_{\max }} \left[U\big(e\big[w_{m}^{s}+w_{f}^{s}\big]\big)+q_{m}^{ss}\right]dq\right . \nonumber\\[8pt] & \quad \left .\ +Q\big[ \! \max \big(\bar {q}_{m}^{ss},\bar {q}_{f}^{ss}\big)\big] \min\! \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\!\int _{0}^{q_{\max }}\big[U\big(e\big[w_{m}^{s}+w_{f}^{s}\big]\big)+q_{m}^{ss} \big]dq\right .\nonumber\\[8pt] &\quad\left . \ +Q\big[ \! \max \big(\bar {q}_{m}^{ss},\bar {q}_{f}^{ss}\big)\big] \left [1-\min \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\right ]U[w_{m}^{s}] \right \} \nonumber\\[8pt] &\quad+(1-\psi _{t+1}^{f})N^{f}_{t+1}\left \{\int _{\max (\bar {q}_{m}^{su},\bar {q}_{f}^{su})}^{q_{\max }}\left [U\big(e[w_{m}^{s}+w_{f}^{u}]\big)+q_{m}^{su}\right ]dq\right .\nonumber\\[8pt] &\quad\left .\ +Q\big[ \! \max \big(\bar {q}_{m}^{su},\bar {q}_{f}^{su}\big)\big]\min \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\int _{0}^{q_{\max }}\left [U\big(e\big[w_{m}^{s}+w_{f}^{s}\big]\big)+q_{m}^{ss}\right ]dq\right .\nonumber\\[8pt] &\quad\left .\ +Q\big[ \! \max \big(\bar {q}_{m}^{su},\bar {q}_{f}^{su}\big)\big]\left [1-\min \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\right ]U\left [w_{m}^{s}\right ]\right \}\nonumber\\[8pt] &\quad +(1-N^{f}_{t+1})\left \{\min \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\int _{0}^{q_{\max }} \left [U\big(e\big[w_{m}^{s}+w_{f}^{s}\big]\big)+q_{m}^{ss}\right ]dq\right .\nonumber\\[8pt] &\quad\left . \ +\left [1-\min \left(\frac {\theta _{f}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)}{\theta _{m}^{s}\big(\psi _{t+1}^{m},\psi _{t+1}^{f}\big)},1\right)\right ]U\left [w_{m}^{s}\right ] \right \} \end{align}

4.6.1 Education decision and sex ratio

Becoming a skilled agent requires some effort $\phi _{g}\ (g \in \{male, female\})$ , which is $\in [0,+\infty )$ , the effort is i.i.d. for all men and women, with the CDF of $\Phi _{g}$ . An agent $i$ wants to become skilled if:

(9) \begin{equation} \mathrm EV^{s}_{i}-\mathrm EV^{u}_{i}\geq \phi _{i} \end{equation}

Where $\mathrm EV^{s}_{i}$ is the expected payoff of becoming skilled for agent $i$ . The skilled-unskilled payoff difference is generated when a fraction of $\psi$ is skilled:

(10) \begin{align} \bar {\phi }\big(\psi _{t+1}^{m}\big)\equiv \mathrm E_{t}V^{s}_{m}\big(\psi _{t+1}^{m}\big)-\mathrm E_{t}V^{u}_{m}\big(\psi _{t+1}^{m}\big)\nonumber\\[5pt] \bar {\phi }\big(\psi _{t+1}^{f}\big)\equiv \mathrm E_{t}V^{s}_{f}\big(\psi _{t+1}^{f}\big)-\mathrm E_{t}V^{u}_{f}\big(\psi _{t+1}^{f}\big) \end{align}

For any individual $i$ : when $\phi _{i} \leq \bar {\phi }$ (the effort that has to be devoted to becoming a skilled individual is smaller or equal to the reservation effort $\bar {\phi }$ , an agent want to become skilled).

Without considering the education costs, agents can borrow freely, and contingent on their efforts, the young agent will make the same decision irrespective of their family wealth type (family provides the funding for education). This implies that in equilibrium: a fraction $\Phi (\bar {\phi }_{g})$ of individuals would become skilled, then the equilibrium distribution of education fraction for two genders are:

(11) \begin{align} \psi _{t+1}^{m} & =\Phi \big(\bar {\phi }_{m}\big)\nonumber \\[5pt] \psi _{t+1}^{f} & =\Phi \big(\bar {\phi }_{f}\big) \end{align}

Thus:

(12) \begin{align} \bar {\phi }\big(\Phi \big(\bar {\phi }_{m}\big)\big) & = \mathrm E_{t}V^{s}_{m}\big(\Phi \big(\bar {\phi }_{m}\big)\big)-\mathrm E_{t}V^{u}_{m}\big(\Phi \big(\bar {\phi }_{m}\big)\big)\nonumber\\[6pt] \bar {\phi }\big(\Phi \big(\bar {\phi }_{f}\big)\big) & = \mathrm E_{t}V^{s}_{f}\big(\Phi \big(\bar {\phi }_{f}\big)\big)-\mathrm E_{t}V^{u}_{f}\big(\Phi \big(\bar {\phi }_{f}\big)\big) \end{align}

We firstly assume that $\phi _{g}$ is a uniform distribution for both male and female, which is $\in [0,1]$ . Under this assumption, we have:

(13) \begin{align} \psi _{t+1}^{m} & = \mathrm E_{t}V^{s}_{m}\big(\psi _{t+1}^{m}\big)-\mathrm E_{t}V^{u}_{m}\big(\psi _{t+1}^{m}\big)\nonumber\\[6pt]\psi _{t+1}^{f} & = \mathrm E_{t}V^{s}_{f}\big(\psi _{t+1}^{f}\big)-\mathrm E_{t}V^{u}_{f}\big(\psi _{t+1}^{f}\big) \end{align}

Figure 8.

Education decision and sex ratio.

Note: The results presented in the left-hand figure are derived from model outputs, while those on the right-hand side are based on four waves of population census data (census 2000, 2005, 2010, 2015). We measure the fraction of skilled men and women respectively, by restricting cohorts’ age between 20 to 35.

After integrating both sides of equation (10) and evaluating it on the whole range of $\phi$ (that is $\in [0,1]$ ), we can get equation (13). After solving it, we generate the results plotted in the Figure 8a. In our matching model, agents of each gender make a binary decision regarding their education level, opting to be skilled or unskilled. This choice leads to an equilibrium distribution in the proportion of skilled agents within the population.

As illustrated in Figure 8a, we find that both men and women increasingly become skilled as the sex ratio in the marriage market increases, where the trend of men is above the trend of women, suggesting that men spend more effort in becoming skilled compared with women. Additionally, when we examine the results of the model alongside the data trends shown in Figure 8b, we observe an upward trajectory in the proportion of skilled men and women over time. This pattern suggests that our model offers insight into the increasing levels of human capital in China, indicating that the increase in male abundance induces higher human capital.

The reasons behind the trend of both men and women becoming more skilled as the sex ratio increases are as follows. For men, firstly, being skilled typically means higher earnings, which can boost their appeal in the increasingly competitive marriage market. Second, when examining the data in Table C1, we see that the probabilities of an unskilled man ( $\pi _{u}$ ) and a skilled man ( $\pi _{s}$ ) of remaining single rise with an increasing sex ratio. However, the likelihood of being single is higher for unskilled men ( $\pi _{u}\lt \pi _{s}$ ), suggesting that it’s riskier for them to find a partner. Additionally, should an unskilled man marry, he is more likely to marry an unskilled woman, as indicated by the higher values of ( $\pi _{uu}$ ) in comparison to other household types, leading to a lower utility gain in forming a household. These factors motivate men to become skilled.

For women, firstly, choosing to remain unskilled means they are more likely to marry an unskilled husband (as $\pi _{uu}\lt \pi _{su}$ ), which generally offers less benefit. Secondly, if a woman becomes skilled, she is more likely to marry a skilled husband than an unskilled one (as $\pi _{ss}\gt \pi _{us}$ ). Third, as the sex ratio in the marriage market increases, even though an unskilled woman’s chances of marrying a skilled husband increase, the likelihood of forming a household with skilled men is still lower than marrying an unskilled husband (as $\pi _{uu}\gt \pi _{su}$ ). Therefore, women are more inclined to become skilled.

This finding offers a potential explanation for the observed increase in education levels in China. Analysis of census data suggests that a growing surplus of men is associated with higher educational attainment for both men and women. In addition, we enrich the literature that has studied the interconnection between premarital investment in education and marriage market competition. Lafortune (Reference Lafortune2013) finds that males increased their human capital investment when faced with an increase in the sex ratio. Bhaskar and Hopkins (Reference Bhaskar and Hopkins2016) indicate that if there is an unbalanced sex ratio, the abundant sex invests more, while the scarcer one invests less. Additionally, Bhaskar et al. (Reference Bhaskar, Li and Yi2023) illustrate that parents of boys invest less in education in regions with a more male-biased sex ratio but they invest more in housing. Our findings enrich these studies, showing that in response to an increasing surplus of males, both men and women enhance their educational investments. Notably, men’s investments in education are more pronounced compared to those of women.