A theory of financial inclusion and income inequality

We develop a theory linking financial inclusion, defined as access to formal loans and financial assets, to income inequality. Initial inequality of households is modeled by a random variable determining initial endowments. These initial endowments can be used to invest instantaneously in human capital and financial assets. Human capital translates into income based on a strictly concave production function, suggesting optimal levels of investment. Financial assets earn yields which do not depend on the amount invested by individuals. Theoretical predictions are tested using the China Household Finance Survey (CHFS) for 2011 and 2013. Initial conditions modeled by a random variable are replaced by an actual distribution of income or assets to derive theoretical predictions regarding the proportion of the population that might benefit from financial inclusion. Financial inclusion does mitigate under-investment in education – but formal loans do not contribute. Income inequality worsens if households rely on formal or informal loans, whereas access to bank accounts improves households' prospects in the future income distribution. However, households below the 40th percentile of household income do benefit from informal loans.


Introduction
Economic theory suggests that financial exclusion can lead to persistent inequality. In the presence of credit market imperfections, individuals' initial wealth determines their ability to invest in human or physical capital, which prevents social mobility of the poor and perpetuates inequality (Banerjee and Newman 1993;Galor and Zeira 1993;Aghion and Bolton 1997;Ghatak and Jiang 2002;Galor and Moav 2004;Mehrotra and Yetman 2014). Seminal theoretical papers by Aghion and Bolton (1997), Banerjee and Newman (1993) and Galor and Zeira (1993) model heterogeneity of individuals using a distribution of initial wealth, which is similar to our approach. Their theories model a binary choice -to be or not to be an entrepreneur (Banerjee and Newman 1993;Aghion and Bolton 1997) or a skilled worker (Galor and Zeira 1993), which requires investment. These models suggest that financial inclusion can reduce income inequality by increasing opportunities for education and entrepreneurship among the poor. However, if one relaxes the assumption of binary choices and introduces a continuous variable of investment in education, theoretical predictions differ. A 'middle class' emerges for whom investing in education by taking loans is not a value maximizing strategy. Even in the absence of credit rationing not all individuals might benefit from financial inclusion. Our theory addresses this research gap by modeling how individuals maximize their future discounted excess income using loans (financial inclusion) to invest in education. As individuals are heterogeneous, i.e. have different initial endowments, these individual investment decisions do influence income inequality. We establish two Theorems, demonstrating that not all individuals can benefit from financial inclusion. Hence, depending on parameter values financial inclusion can increase or decrease inequality.
While most cross-country studies have found a negative relationship between financial inclusion and income inequality (Honohan 2008;Mookerjee and Kalipioni 2010;Kim 2016;Neaime and Gaysset 2018;Turégano and Herrero 2018), it is unclear whether these findings hold for Asia, in general, and China, in particular (Park and Mercado 2018). Furthermore, these studies have not investigated the mechanisms or processes underlying the relationship between financial inclusion and inequality, nor have they examined whether the effects of financial inclusion vary across different financial services. Most importantly, many prior studies use country (Kim 2016) or state level data (Jayaratne and Strahan 1996;Burgess and Pande 2005;Beck, Levine, and Levkov 2007), which cannot reveal heterogeneity among individuals or households. This paper aims to contribute to this literature by investigating the relationship between financial inclusion and income inequality both theoretically and empirically using disaggregated data. Following Allen et al. (2012), financial inclusion is defined as the use of formal financial services. Our theoretical predictions are tested using data from a nationally representative sample of over 8,000 Chinese households collected by the China Household Finance Survey (CHFS) in 2011 and 2013. Our findings suggest that financial inclusion does mitigate under-investment in education -but formal loans do not contribute. Income inequality worsens if households rely on formal or informal loans, whereas access to bank accounts improve households' position in the future income distribution. Yet, households in the bottom 40% of the household income distribution do benefit from informal loans. This empirical finding is expected as a consequence of Theorem 3.2, which derives an upper bound in terms of initial endowments. Hence, households that are 'too rich' are not expected to benefit from access to loans to finance education.
Our contribution to the literature is threefold. First, recent theoretical work on the impact of financial inclusion on the distribution of income has been limited, and has focused on entrepreneurship as the causal mechanism linking access to financial services and inequality (Besley, Burchardi, and Ghatak 2018;Dabla-Norris et al. 2020). Our theoretical model investigates the impact of financial inclusion on income inequality through its effects on education and human capital accumulation. By permitting a continuous control variable, i.e. investment in education, our theory reveals within country disparities, i.e. not all individuals are expected to benefit from financial inclusion. Second, with few exceptions (e.g. Zhang and Posso 2019), most empirical studies on this issue are cross-country studies that use supply-side data on financial inclusion collected from financial regulators. We investigate the relationship between financial inclusion and inequality in China, using demand-side data on financial inclusion collected from households. The choice of China as the setting for the empirical analysis is motivated by the fact that China has the world's largest unbanked population (Demirguc-Kunt, Klapper, and Singer 2017). Moreover, China's inequality levels used to be close to Nordic countries but they are now approaching US levels (Piketty, Yang, and Zucman 2019). Its increase in inequality is unprecedented (Naughton 2018), 1 and unequal access to education has been one of the driving factors (Jain-Chandra et al. 2018). Finally, while a recent review of the literature on financial inclusion suggests that not all financial products are equally effective in reducing inequality (Demirguc-Kunt, Klapper, and Singer 2017, 18-19), existing studies have failed to provide theoretical and empirical evidence showing the potentially differential impact of account ownership (and the savings and payment services they provide) and credit on household income inequality. This paper aims to fill this gap. The rest of the paper is organized as follows. Section 2 reviews the theoretical and empirical literature on the relationship between financial inclusion and inequality. Section 3 derives our theoretical model. In Sections 4 and 5, we discuss our empirical strategy and data. Section 6 presents our results, and Section 7 concludes.

Prior research
Economic theory provides conflicting predictions about the relationship between financial inclusion and inequality. The theoretical models of Galor and Zeira (1993) and Banerjee and Newman (1993) suggest a negative linear relationship between access to finance and income inequality. In contrast, the model of Greenwood and Jovanovic (1990) predicts a non-linear, inverted U-shaped relationship between access to financial services and inequality, which depends on the level of economic development. Work by Claessens and Perotti (2007) contends that causality may run in the opposite direction, from an unequal distribution of income to an unequal distribution of access to finance.
As noted by Aslan et al. (2017, 6), while existing theories on the relationship between finance and inequality refer explicitly to a link between financial access (or inclusion) and income inequality, most of the empirical literature has initially examined the link between financial development and income distribution. 2 Cross-country evidence on the distributional effects of financial development is mixed. Evidence from Beck, Levine, and Levkov (2007) indicates that financial development reduces income inequality. This is consistent with more recent findings by Zhang and Naceur (2019). However, research by Kim andLin (2011) andLaw, Tan, andAzman-Saini (2014) suggests that the relationship between financial development and inequality is non-linear and depends on a country's level of financial development or institutional quality. In contrast to these studies, research by Dabla-Norris et al. (2015) and De Haan and Sturm (2017) shows that financial development increases inequality. Evidence from China is equally mixed, with some studies finding a negative relationship between finance and inequality (Jalil and Feridun 2011); others an inverted-U relationship (Zhang and Chen 2015); and yet others a positive relationship (Koh, Lee, and Bomhoff 2020).
A more recent literature has investigated the relationship between financial inclusion and income inequality. This literature has thus far been dominated by cross-country studies. Most of them point to a negative relationship between financial inclusion and inequality (Honohan 2008;Mookerjee and Kalipioni 2010;Kim 2016;Neaime and Gaysset 2018;Turégano and Herrero 2018) but their findings differ across regions and countries (Park and Mercado 2018;Dabla-Norris et al. 2020). For instance, Mookerjee and Kalipioni (2010), as well as Neaime and Gaysset (2018), find that increasing access to financial services, through bank branches results in a less unequal income distribution. Similarly, findings by Aslan et al. (2017) show that increasing the intensity of use of financial services 3 leads to a reduction in income inequality. However, a more recent study by Park and Mercado (2018) finds that increasing the accessibility, availability and usage of financial services 4 tends to reduce income inequality, except in developing Asia (including China and India).
This finding is in contrast to those of China-specific studies (Zhang, Zhang, and He 2018;Huang and Zhang 2019;Zhang and Posso 2019). While the empirical evidence on the potential inequality-reducing effects of financial inclusion in China is limited, it generally suggests that financial inclusion can contribute to reducing inequality, by facilitating entrepreneurship among the rural poor. Focusing on rural China, Zhang and Posso (2019) find that poor households benefit more from financial inclusion than rich ones. Using household data from the 2011 China Household Finance Survey, they construct a multidimensional index of financial inclusion, which includes measures of account ownership, savings, credit, and insurance, and investigate its impact on household income. Their findings show that financial inclusion has a positive effect on income, and that this effect is larger for households at the lower quantiles of the income distribution, indicating that it reduces income inequality within rural areas. Financial inclusion has also been found to reduce rural-urban income inequality 5 (Huang and Zhang 2019). Findings by Zhang, Zhang, and He (2018) indicate that digital financial inclusion is positively associated with household income, and this positive association is only significant for rural households. Similarly,  find that Fintech-driven financial inclusion has a positive effect on household income, and this positive effect is larger for rural than for urban households. These studies suggest that one of the mechanisms underlying the inequality-reducing effect of (digital) financial inclusion is entrepreneurship by low-income rural households (Zhang, Zhang, and He 2018). Our paper contributes to this literature by investigating theoretically and empirically: (i) whether financial inclusion, understood as the use of formal financial services, can reduce income inequality by facilitating investment in human capital; and (ii) whether different types of financial services are equally effective in reducing inequality.

Theoretical considerations
The theory aims to understand the process of financial inclusion, i.e. the demand for formal financial services. These services include debt denoted D and financial assets (e.g. savings) A. Debt incurs costs in the form of interest payments with a rate r D , while financial assets yield r A . Banks, which are not modeled explicitly, act as intermediaries taking deposits A and providing loans D. To cover their transaction costs, there needs to be a positive net interest margin ν = r D − r A > 0 even in the absence of default risk. This interest margin could be regarded as a measure of the quality of financial services.
Demand for finance arises due to the need of households to invest in human capital (K), i.e. education. This investment yields income (Y), i.e. there is a trade-off between consumption now and consumption later, governed by inter-temporal preferences. These preferences are captured in the discount rate r. The link between income and human capital is modeled using a strictly concave production function f (K), hence, f (K) > 0 and f (K) < 0. This implies that there exists an optimal capital stock K * where marginal benefits equal marginal costs. Individuals maximize their excess consumption denoted C, i.e. basic needs such as food are assumed to be covered. They can choose to invest an amount I into human capital K or consume any excess income.
The most important feature of financial inclusion is the fact that households exhibit heterogeneity with respect to access to finance. The model, hence, captures this heterogeneity in the initial condition. Individuals enter the stage with an initial endowment W 0 , which is a realization of the random process W with probability density ρ and support on R + . At time t = 0 (the start of the model) individual i's endowment is drawn and public knowledge. The latter assumption can be relaxed, leading to information asymmetry (adverse selection). Moreover, households are risk-neutral, which simplifies the model without loss of generality in the absence of uncertainty after t = 0. We assume that individuals do not receive an initial income Y(0) = 0, which suggests the following budget constraint at t = 0.
Equation (1) states that initial endowment (e.g. wealth or initial human capital) can be invested into human capital K(0 + ) and the financial asset A(0 + ). If the initial endowment is insufficient, debt D(0 + ) can be used to reach desired levels of human capital. Note that ν = r D − r A > 0 implies that debt is not used to acquire the financial asset. Individuals maximize their future discounted excess consumption C, where human capital (K), financial assets (A) and debt (D) are state variables. Individuals have an infinite time horizon. We ignore taxes. Note that investment refers to I =K and repayments of loans S =Ḋ. In most applications, we consider the case where r D > r > r A , i.e. after the initial adjustment at t = 0 governed by (1) individuals do not increase their financial assets. Continuous discounting is applied. Capital does not depreciate.
Note that using the initial endowment as described in the budget constraint at t = 0 in (1) suggests an impulse control problem because the state variables debt, assets and human capital can change instantaneously as suggested by (1). After the initial impulse, continuous processes captured in the two differential equations in (2) apply. This setup refers to an impulse control problem. In addition, control constraints apply as ; hence, the state variables K and D enter the control constraint. To solve this problem analytically, we apply Theorem 1 in Kling (2020). A detailed mathematical discussion of this type of impulse control problem can be found in Kling (2020). The proof of Theorem 1 in Kling (2020) shows that the optimal time for an impulse is at t = 0 as suggested by (1). Hence, relaxing the assumption that the initial endowment W 0 is invested instantaneously (i.e. at n points in time denoted t j with j = 1, 2, . . . , n) does not affect optimal time paths of states (e.g. human capital). Furthermore, maximizing the value of the future discounted excess consumption C can be written as follows.
Theorem 3.1: The optimization problem (2) can be rewritten as follows, where κ(D(0 + )) is the time into steady state and K * is the steady state capital stock.
Theorem 3.1 refers to the case W 0 < K * ; hence, the individual does not have sufficient endowment to 'jump' into the preferred state of human capital K * . Therefore, the individual can take a loan D(0 + ) at t = 0, which constitutes an impulse. As r D > r, this impulse is used to move human capital from zero (the assumed initial value) to K * poor , which is determined f K * poor = r D . Hence, K * poor < K * and it constitutes a transient steady-state of human capital for poor individuals. The initial optimal debt level is then D(0 + ) = K * poor − W 0 . Hence, being 'poor' in our theory means that the initial endowment W 0 is not sufficient to instantaneously reach the optimal level of human capital K * . The initial endowment W 0 is exogenously drawn from the probability density ρ, i.e. 'rolling the dice at birth'. However, the optimal level of human capital K * depends on the interplay between marginal benefits of human capital and the cost of debt as derived in Theorem 3.2. Consequently, poverty is relative and partly endogenous in our theory.
As r D > r, a poor individual has an incentive to reduce the debt level and uses all resources to do so until debt is repaid. The time when debt is repaid is labeled κ D . The repayment time can be found from the following inhomogeneous first order differential equation using the condition D(κ D ) = 0. The integrating factor is exp(− r D dt) = exp(−r D t), and C is an arbitrary constant.
The constant C follows from the initial condition D(0 Obviously, the cash flow from human capital in the transient steady state exceeds interest payments on the initial loan. Therefore, κ D > 0 as required based on (5). Once all debt is repaid at time t = κ D , a poor individual has an incentive to invest using income from human capital until K * , the steady-state human capital stock, is reached. The time it takes to move to the final capital stock starting at time t = κ D follows from the following first-order differential equation.
Equation (6) can be solved analytically depending on the production function f (K). After the steady-state is reached at t = κ, any income goes into excess consumption assuming r > r A , i.e. no financial asset will be accumulated. The future discounted value of excess consumption refers to the perpetuity cash flow discounted back from the time of the steady-state t = κ to t = 0 as shown in Theorem 3.1.  Figure 1 plots the initial distribution of endowment, where ρ(W 0 ) refers to the probability density. The poor with initial endowment below optimal levels of human capital K * = 7.5 can take loans D(0) to reach the transient steady-state K poor = 6. The rich with W 0 > K * = 7.5 can invest in financial assets A(0). Individuals with initial endowment from K poor = 6 to K * = 7.5 cannot benefit from financial inclusion.
A similar analysis can be conducted for different types of individuals depending on the initial endowment W 0 . Thus far, we explored CASE 3 in detail. Considering the initial condition and interest rates, the following cases emerge.
i.e. individuals in the 'middle' do not have an incentive to take debt to invest in human capital. They will use income to grow to the optimal level of human capital K * rich assuming r > r A . • CASE 3: W 0 < K * poor , i.e. this individual lacks endowment to invest optimally. Optimal investment is determined by f K * poor = max(r, r D ). Demand for finance refers to D(0) = K * poor − W(0). If r D > r the transient steady state lasts until t = κ D when all debt is repaid. From t = κ D to t = κ the individual funds further investment into human capital using income until K * is reached.
As the net interest margin ν = r D − r A is positive, it follows that K * rich ≥ K * poor . All households have the same function f (K) transforming capital stock into income. Thus, heterogeneity stems from initial endowments.
In the absence of uncertainty after t = 0, information asymmetry and permitting an open economy (investments can exceed savings), households 'jump' at t = 0 into their steady state capital stock. Note that (2) does not permit an increase in debt after t = 0, which is not optimal due to the fact that f (K) is strictly concave. This also suggests that debt must be sustainable. These assumptions can be relaxed. For instance, if debt and financial assets have to be balanced then r D and r A could be derived endogenously. These extensions go beyond the scope of this paper. Figure 1 illustrates the expected patterns for rich and poor individuals. Poor individuals have a demand for debt to reach their optimal capital stock K * poor = 6, and rich households 'jump' into K * rich = 7.5 instantaneously. Any excess endowment creates financial assets if r A > r or is consumed. Interestingly, Figure 1 demonstrates that there is a third group for whom debt is not attractive as r D is too high. This 'middle class' will grow using their income to reach K * rich = 7.5 over time. Figure 2 illustrates optimal paths for rich individuals in red and poor individuals in green. Asset accumulation only occurs if r A > r, otherwise any excess is consumed.  After repaying debt at time τ 1 , they use their incomes to invest in human capital to reach the optimal capital stock K * at time τ 2 . Any additional income will be invested in the financial asset A * (τ 2 ). The rich (in green) 'jump' into the optimal capital stock K * . Any additional endowment at t = 0 is invested in the financial asset A * (0), which cumulates over time with income.
Theorem 3.2 establishes the impact of financial inclusion on individuals in different parts of the initial endowment distribution. It is apparent that the distributional effects are ambiguous. Hence, the claim that financial inclusion even in ideal settings reduces inequality cannot be easily established. To obtain closed-form solutions, we choose the production function f (K) = K α with α ∈ (0, 1), which implies that f (K) > 0 and f (K) < 0, i.e. the production function is strictly concave.

Theorem 3.2:
Under ideal settings (no frictions, no uncertainty after t = 0 + , no capital controls, r D > r > r A ) and given f ( poor , then CASE 3 applies, implying the following optimal time path of human capital. First, at t = 0 + poor individuals jump to Then the time until debt is repaid κ D follows from (5).
Then from (6), we obtain the first-order differential equation, where C is an arbitrary constant.
The constant C gives.
Finally, the time into the stead-state κ follows.
Then from Theorem 3.1.
A rich individual (CASE 1), i.e. W 0 > K * can reach K * instantaneously and save excess initial endowment, which generates interest income in perpetuity.
Finally, CASE 2, i.e. K * poor ≤ W 0 ≤ K * , suggests that individuals in the middle category do not take debt and invest their initial endowment in human capital, i.e. K(0 Then the time into the steady-state follows. Figure 3. Proportion of the poor who can benefit from financial inclusion (Theorem 3.2).
Theorem 3.2 states that the poor who benefit from financial inclusion have initial wealth where r D is cost of capital and α is the production parameter. Figure 3 plots the proportion of the poor that fulfills this condition. Cost of debt, r D , is in the range 0.73-12% in line with descriptive statistics for China.
Then from Theorem 3.1.
To illustrate Theorem 3.2, we write a Python code that calculates the proportion of the poor that benefits from financial inclusion. The calculation assumes that individual's endowment is drawn from a normal distribution. As outlined in Section 5, our sample refers to the China Household Finance Survey (CHFS), which is used to derive parameter values to compute Figure 3. For instance, cost of debt has a lower bound of 0.73%, which refers to the bottom 10% of our sample, and an upper bound of 12.00%, which is the top 10% of our sample. Figure 3 plots the proportion of the poor that benefits from financial inclusion as a function of cost of debt r D and the production parameter α. Access to finance must be cheap to ensure that a large section of society benefits from financial inclusion.

Empirical strategy
To test theoretical predictions, information on individuals' income (I), financial assets (A), debt (D), and human capital (K) is required. To understand the impact of financial inclusion on income inequality, it is ideal to explore a country such as China, which has undergone a rapid expansion of financial services, economic and social transition. Hence, we use data from the China Household Finance Survey (CHFS), which provides information on individuals and households. Control variables such as age, gender and location are available.
First, the empirical approach tries to estimate the link between human capital and income using various measures of education. Theoretically, in line with endogenous growth theory one would expect that investment in education K increases income I. The theory assumes an unknown functional form I = f (K). It is plausible to assume (and also leads to a steady-state) that f (K) in concave, i.e. f (K) > 0 and f (K) < 0. One could follow White (1980), which models non-linear relationships -but only works for marginal changes in variables as it basically refers to a Taylor approximation. We follow another approach using a Cobb-Douglas type function of various assets that generate a broad measure of income. Second, we require marginal costs of financing based on formal and informal loans, i.e. using the cost of debt r D , optimal education K * follows from f (K * ) = r. Third, comparing marginal costs and benefits of investment determines under-investment in education. This might be due to financial constraints, i.e. lack of access to formal bank loans, which can be tested empirically. Fourth, we try to explain under-investment in education to understand underlying drivers including controls (e.g. gender, age), regional-effects and measures of financial inclusion. Finally, we try to establish whether financial inclusion contributes to more or less income inequality comparing households' position in the income distribution over time.

Data and variables
The paper uses data from the China Household Finance Survey (CHFS), a nationally representative data set collected by Southwestern University of Finance and Economics. The data set includes information about household demographics, (non-financial and financial) assets, liabilities (including education loans); income and expenditures; social security and commercial insurance. The CHFS conducted in 2011 and 2013 covers 127,230 individuals in 29,733 households in the year 2011 or 2013. For some variables such as gender (FEMALE) or location (RURAL) data for all 127,230 cases are available; however, due to missing data some variables are only observed for a subset of cases as shown in the second column of Table 2. For instance, income (INC) is only available for 80,778 individuals.
Variables derived from the CHFS refer to income measures, education, assets, financial inclusion, motives for taking loans, individual controls and region-specific variables. To link individuals to households, we use a unique identifier (HHID). The YEAR of the survey is indicated. We use a broad measure of income (INC) that includes income from employment, second jobs, bonuses, income from farming and businesses. The dummy FARM INC indicates whether individuals receive income from farming. The dummy IND INC flags if individuals rely on income from businesses, which mostly refers to small businesses and self-employment.
Education variables refer to dummies that take value one if the person has no formal education (NO EDU), primary education (PRIM EDU), secondary education (SEC EDU), and whether they attended college or university (HIGH EDU). In addition, the dummy FOREIGN EDU takes value one if the person studied abroad. We quantify assets as fixed assets (FIX A), which includes business assets, property (HOUSE VAL) and the value of cars (CAR VAL). The purchase price of properties is recorded (HOUSE PAY ) but does not deviate substantially from current values on average. We indicate whether assets include business assets as this signals ownership of usually small companies (BUS A). We use a broad definition of financial assets (FIN A) including checking accounts, deposits, stocks, bonds, futures, warrants and wealth management products. Cash holding of households (CASH), and lending from households (LEND) are recorded.
To quantify financial inclusion, dummies signal whether individual have checking account and time deposits (ACCOUNT) and whether they take formal LOAN or informal loans LOAN INF. The amount of outstanding formal (DEBT) and informal debt (DEBT INF) is measured. We obtain annual interest rates on formal loans (INTER) and informal loans (INTER INF). To understand the motives of individuals for taking loans, we use a set of dummy variables. These dummies signal whether formal loans are used for financial investments (INVEST), consumption including buying cars (CONSUME), or education (EDU). Informal borrowing used to finance education is flagged (INF EDU) and the amount is measured (INFDEBT EDU). Finally, we use a dummy to indicate whether the main purpose of time deposits is to finance education (DEPOSIT EDU).
The AGE of individuals and their gender (FEMALE) serve as control variables. Location is controlled by the province and a dummy to capture rural environment (RURAL). Table 1 provides variable names and their definitions. Not all variables listed are used in multivariate models; however, they are used to derive additional variables. These variables are introduced in subsequent sections. Table 2 shows descriptive statistics for all variables in the sample including the number of observations N, mean, median, percentiles and the range. As there are two points in time for which we obtained survey data, namely 2011 and 2013, descriptive statistics reported in Table 2 refer to pooled data. Dummy variables such as no education (NO EDU) take the value one if a condition is met or zero otherwise. By construction, the mean of a dummy variable refers to the proportion of the population that has a certain characteristic. For instance, Table 2 states that the mean of the dummy variable for no education is 0.086; hence, 8.6% of the individuals in our sample have no formal education.

Descriptive findings
To obtain initial insights into patterns of income, education and other forms of investment, we provide some descriptive findings. As age is an important factor, individuals are classified into ten birth cohorts of equal size  defined based on age deciles. Means of medians of variables in each birth cohort are calculated to reveal birth cohort related patterns. Figure 4 plots average income and education for ten birth cohorts. Income peaks at median age, revealing an inverted U-shaped relationship between birth cohorts and income. On the right-hand side of Figure 4, categories of education are depicted, where the vertical axis refers to the proportion of individuals in the respective birth cohort that exhibit the respective level of education. For instance, a value of 0.4 means that 40% have the respective characteristic. Education has undergone a profound shift exhibiting a strong birth cohort effect. The older generation at most attended primary school, whereas younger generations undergo vocational or university education.
We define a broad measure of investing in education labeled INVEST EDU, which combines investment using time deposits (savings), formal and informal loans taken to fund education. Investment into education differs between birth cohorts, where the parents' generation and the young invest in education. Hence, there is a bipolar distribution of investment into education as shown in Figure 5. The vertical axes in both panel refer to the proportion of individuals with a respective characteristic, e.g. 0.05 means that 5% of a respective birth cohort use savings to invest in education. Savings are the predominant source of investment into education followed by informal and formal loans. Accordingly, financial inclusion might enhance investment in education by providing access to bank accounts including time deposits and formal bank loans. The following sections explore multivariate models to establish the relationships between income, investment in education and other asset classes as well as the link between education, financial inclusion and inequality suggested by our theoretical considerations.   Note: All models refer to OLS regressions with robust standard errors. Model A3 account for the year of the survey, and specifications A4 and A5 control for provinces and years. * p < 0.05, * * p < 0.01, * * * p < 0.001.

The impact of education on income
Theoretically, we expect that education should increase individuals' income. To quantify the partial impact of education on income, we run several regression models. Table 3 shows regression models that explain log income using measures of education and controls. Model [A3] controls for the year of the survey, and specifications [A4] and [A5] account for year and province level effects using a set of dummies. These three models exhibit higher explanatory power compared to [A1] and [A2] measured by adjusted R-squared and information criteria (AIC and BIC).
Model [A1] focuses on educational achievements using dummy variables for primary, secondary, and higher education, where the latter category includes vocational training at colleges and degree level education at universities. Most degrees refer to undergraduate degrees since postgraduate and doctorates are uncommon. No education serves as a reference category. Compared to no education, having secondary education provides the highest increase in expected earnings. Attending college or university has a slightly less pronounced effect. Obtaining a degree from a foreign university provides an additional boost to income. Specification [A2] accounts for the non-linear impact of age suggested by analyzing income and education patterns across birth cohorts (see Figure 4). Again an inverted U-shaped relationship prevails. Females and individuals living in rural settings are disadvantaged in the labor market, i.e. similar education leads to lower levels of income. Controlling for year and province level effects in the remaining three columns of Table 3 does not alter these results. In summary, education increases income and the effect diminishes with higher levels of education. Location and gender matter, resulting in significantly lower earnings potential, which might undermine incentives to invest in education.

Optimal investment in education
To derive optimal investment in education, fixed and financial assets, we estimate the following Cobb-Douglas type function that determines income (INC), where A jt refers to province j and year t effects (captured using dummies) and B i are individual control variables including gender, age and rural settings.
Taking logs of (7) leads to a linear model, which is estimated using OLS and the Huber-White sandwich estimator for robust standard errors. The coefficients sum up to less than one, i.e. α + β + γ < 1, (see Table 4) suggesting diminishing returns to scale. This confirms our theoretical consideration suggesting that the function linking human capital to income is concave. Table 4 demonstrates that investing in education, fixed and financial assets has diminishing returns as the three coefficients add up to less than one. Hence, doubling all three asset classes Note: All models refer to OLS regressions with robust standard errors. All specifications control for provinces and years. * p < 0.05, * * p < 0.01, * * * p < 0.001.
increases income by less than 100%. Yet again, there is evidence of a gender disparity and an urban-rural divide. However, women benefit from investing in education as their returns are the same indicated by an insignificant interaction term (FEMxEDU). In contrast, in rural settings investing in education does not yield returns in terms of higher income, demonstrated by the interaction term RURxEDU.
Using specification [C4] as reference model, marginal returns of investing in education, fixed and financial assets can be calculated from the estimated coefficientsα,β, andγ by taking partial derivatives of income with respect to assets (7).
From (8), it is obvious that ∂INC i /∂INVEST EDU i > 0 and ∂ 2 INC i /∂INVEST EDU 2 i < 0 suggesting an optimal level of investment in education. Similarly, marginal return of investing in fixed and financial assets can be derived. Comparing these marginal returns of investment with marginal costs such as interest rates on formal and informal loans establishes whether individuals would benefit from more investment. Table 5 shows the 25th percentile, the median and the 75th percentile of estimated returns from education and interest rates in different cohorts. Cohorts are either defined by quantiles based on income or fixed assets. The latter is closer to our theoretical concept of endowment as fixed assets includes a wide range of assets such as property. A certain proportion of individuals in each class should benefit from investing in education; however, we observe a tendency that richer individuals, both in terms of income and assets, exhibit higher returns from education. This empirical finding is due to inherent disparities in the labor market as lower income or asset categories have a higher share of females and individuals living in rural settings. Accordingly, a further multivariate analysis is needed to disentangle these interrelationships. Figure 6 illustrates that investing in education yields on average lower returns than the average return from investing in financial assets for both women and men in urban settings. As females exhibit lower income levels, they also experience lower marginal benefits from investing in education or financial assets. Interest rates in the informal sector exceed for some birth cohorts the marginal benefits of investing in education. As returns from education are diminishing with the current level of education, individuals with low levels of educational attainment do benefit more from a marginal increase of their education. Accordingly, we calculated marginal returns of investing education for each individual.
Based on our reference model [C4] in Table 4 that derives the production function of income, we determine the marginal product of investing in education as in (8). Comparing the marginal benefit of education and interest rates on loans, we establish whether individuals under-invest in education (UNDER). Using logistic  regressions with robust standard errors and controlling for provinces, years and birth cohorts, Table 6 provides five model specifications to reveal the determinants of under-investment in education. Specification [C1] considers income, gender, rural settings, and dummies for educational achievements. As in previous results, females are less likely to under-invest in education, which is due to lower returns from education caused by the gender pay gap. Individuals in rural settings experience a similar problem as education leads to smaller benefits. The level of education has a positive and significant impact of being under-invested. In addition, the impact diminishes from no formal education to secondary education. Model [C2] tests for financial inclusion and shows that having a bank account reduces the risk of under-investment, whereas access to loans

Financial inclusion and inequality
To test whether investing in education and financial inclusion improves a household's position in terms of its income decile, we focus on a sub-sample of households surveyed in 2011 and 2013. 38.9% of households in the 2013 survey were also included in the 2011 survey. All variables are aggregated on the household level. HH INC refers to the combined household income, HH EDU and HH FIN represent investment in education and financial assets on the household level, respectively. We take the average age of members of the household (HH AGE) to account for cohort effects. We use indicators for account ownership (HH ACCOUNT), having a formal loan (HH LOAN), and having an informal loan (HH LOAN INF). If one member of the household has an account, a formal or informal loan, the household is assigned the value one in the respective indicator variable. Finally, we also include whether households have a time deposit for the purpose of financing education HH DEPOSIT EDU, whether households take informal loans to fund education HH INF EDU and if so the amount of loans taken HH INFDEBT EDU. The position of households in income deciles is determined in 2011 and 2013, resulting in an ordinal variable from 1 (poorest) to 10 (richest) household. To account for the ordinal nature of the dependent variable, we estimate ordered logistic regressions reported in Table 7.
Holding the sample composition constant permits the use of lagged explanatory variables on the household level. Accordingly, we can establish the causal order or relationships, which goes beyond the previous empirical models (Kling, Harvey, and Maclean 2017). All explanatory variables are lagged, i.e. they refer to 2011, whereas the dependent variable is the income decile of the household in 2013. The previous income decile in 2011 is a predictor of the future outcome, and as expected exhibits a positive and significant partial impact. Note: All models refer to ordered logistic regressions with robust standard errors. All specifications control for provinces and years. * p < 0.05, * * p < 0.01, * * * p < 0.001. Note: All models refer to ordered logistic regressions with robust standard errors. All specifications control for provinces and years. * p < 0.05, * * p < 0.01, * * * p < 0.001. In line with Theorem 3.2, we rerun the analyzes for different income deciles, which reveals a very different pattern if we consider the bottom 40% of the income distribution. 6 Table 8 restricts the sample to households below the 40th income percentile, indicating that informal loans HH LOAN INF do lift households in the bottom 40% to higher income deciles. In addition, poorer households seem to be better off in rural than urban settings. Accordingly, splitting the sample in line with theoretical suggestions provides further insides.

Conclusion
Our theory relaxes the assumption of binary choices made in prior studies (Banerjee and Newman 1993;Galor and Zeira 1993;Aghion and Bolton 1997) permitting continuous investment decisions. Intuitively, increasing financial inclusion should lead to more investment in education, building human capital, which translates into higher earnings. However, depending on the initial endowment some households might under-invest as they face higher interest rates on loans. Financial inclusion also constitutes access to financial assets. If returns from financial assets exceed returns from investing in human capital, inequality can increase. Theorem 3.2 demonstrates that not all individuals might benefit from financial inclusion as a 'middle class' might find it unattractive to invest in education using expensive loans compared to their time preference.
Using household data from China, we show that education increases incomes; however, marginal benefits of higher attainments fade. In addition, females and individuals in rural settings receive lower earnings, diminishing their returns from investing in education. This in turn leads to lower optimal levels of investment in education as predicted by theoretical considerations. Quantifying under-investment in education reveals that a segment of the population should invest more as their marginal benefits outweigh marginal costs in terms of interest rates on formal and informal loans. Financial inclusion does mitigate the investment gap in education; however, this effect is predominately driven by having access to savings accounts and not formal bank loans. In addition, informal loans have a positive but smaller effect on closing the gap in education. Finally, we demonstrate that households can move to higher income deciles if they have access to bank accounts. Yet, having formal or informal loans diminishes the prospects of households. Hence, the impact of financial inclusion on inequality is mixed, suggesting that increasing access to loans might not result in better outcomes.
As suggested by Theorem 3.2 and in line with Demirguc-Kunt, Klapper, and Singer (2017, 18), we split the sample into the bottom 40% of households and the top 60% in terms of household income. Our theoretical prediction that only poorer households benefit from loans is confirmed empirically. This is in line with recent research focusing on rural China that poor households benefit more from financial inclusion (Zhang and Posso 2019). Moreover, we show that informal loans are beneficial, while formal loans do not help households to climb the income ladder. This finding underlines that formal loans are not yet as effective as informal channels of finance, requiring further policy responses. These findings are consistent with the observation made by Demirguc-Kunt, Klapper, and Singer (2017, 18-19) that 'not all financial products are equally effective in reaching development goals, such as reductions in poverty and inequality'.
While more research is needed, recent studies and our own findings suggest that the largest impacts come from access to bank accounts and savings -but also informal loans in the case of poorer households (Demirguc-Kunt, Klapper, and Singer 2017). In fact, findings from recent experimental studies conducted in various developing countries indicate that the impact of payment, (micro-)savings, and (micro-)insurance services on the income of poor and low-income households is consistently and significantly more positive than that of (micro-) credit (Brune et al. 2011;Dupas and Robinson 2013;Karlan, Ratan, and Zinman 2014;Banerjee, Karlan, and Zinman 2015;Demirguc-Kunt, Klapper, and Singer 2017). More research is needed to gain a better understanding of how different dimensions of financial inclusion promote inclusive development.
Notes 4. These various dimensions of financial inclusion are measured by: the number of automated teller machines and commercial bank branches per 100,000 adults; the number of borrowers from, and depositors with, commercial banks per 1,000 adults; and the ratio of domestic credit to GDP (Park and Mercado 2018). 5. Specifically, they find that financial inclusion reduces rural-urban income inequality in the long run, but can increase it in the short term, due to initial disparities between rural and urban areas in terms of access to financial infrastructure and education (Huang and Zhang 2019). 6. Splitting the sample into the bottom 40% of households and the top 60% in terms of household income is in line with Demirguc-Kunt, Klapper, and Singer (2017, 18).