Compensating for unequal parental investments in schooling

Abstract

This paper investigates how rural families in China use marital and post-marital transfers to compensate their sons for unequal schooling expenditures. Using a common behavioral framework, we derive two methods for estimating the relationship between parental transfers and schooling investments: the log-linear and multiplicative household fixed-effects regression models. Using data from a unique household-level survey, we strongly reject the log-linear specification. Results from the multiplicative model suggest that when a son receives 1 yuan less in schooling investment than his brother, he obtains 0.47 yuan more in transfers as partial compensation. Since our measure of transfers represents a substantial fraction of total parental transfers, sons with more schooling likely enjoy higher lifetime consumption. Redistribution within the household may be limited by either the parents’ desire for consumption equality or bargaining constraints imposed by their children. Controlling for unobserved household heterogeneity and a fuller accounting of lifetime transfers are quantitatively important.

Appendices

Appendix

A. A framework for parental investments and transfers

This appendix provides a general framework of intra-household resource allocation. Decisions are made in two stages. In the first stage, parents choose schooling investments (s _ih ) among different children; in the second stage, consumption is allocated among family members.

We start from the second stage of the problem. Taking s _ih as given, consumption levels are determined by solving (P1):

$$ \underset{c_h,{c}_{1 h},{c}_{2 h}}{ \max }\ {U}_h\left({c}_h,{c}_{1 h},{c}_{2 h}\right)=\left(1 - {k}_{1 h}-{k}_{2 h}\right) u\left({c}_h\right)+{k}_{1 h} u\left({c}_{1 h}\right)+{k}_{2 h} u\left({c}_{2 h}\right) $$

(P1)

subject to the household budget constraint:

$$ {c}_h+{c}_{1 h}+{c}_{2 h}={w}_h\left({s}_{1 h},{s}_{2 h}\right)={m}_h+{\displaystyle \sum_{i=1}^2\left( R\left({s}_{i h}\right)- C\left({s}_{i h},{a}_{i h},{m}_h\right)\right)} $$

(19)

It is convenient to impose a homothetic assumption on U _h (·), say u(c) = ln(c) (see, Becker (1991)). Then optimal consumption levels are proportional to the total family wealth:

$$ {c}_{1 h}^{*}={k}_{1 h}{w}_h $$

(20)

$$ {c}_{2 h}^{*}={k}_{2 h}{w}_h $$

(21)

$$ {c}_h^{*}=\left(1 - {k}_{1 h} - {k}_{2 h}\right){w}_h $$

(22)

This consumption allocation solution can be rationalized through either a unitary or collective model. Their difference lies in the interpretation of k _ih . In the context of the unitary model, equation (P1) represents the utility function of altruistic parents, who allocate consumption among family members. k _ih can be interpreted as the relative weight they put on their children’s welfare relative to their own. Under Chiappori’s (1988, 1992) collective model, family members bargain efficiently over the division of family wealth to obtain their own consumption. The efficient bargaining results can be implemented through maximizing a "household welfare" function as in (P1), where k _ih represents an individual’s bargaining power. In general,

$$ {k}_{ih}={k}_{0 h}+\mu \frac{r{ s}_{ih}}{w_h};\mu \ge 0 $$

(23)

When μ > 0, the parents’ marginal utility from a child’s consumption is increasing in the child’s earnings from schooling expenditure relative to family wealth.

In the first stage, parents make decisions on schooling to maximize their own utility function (assumed to be homothetic as well), while taking into account the allocation decisions (20)–(22):

$$ \begin{array}{l}\underset{c_h,{c}_{1 h},{c}_{2 h}}{ \max}\left(1-{b}_{1 h}-{b}_{2 h}\right) \ln {c}_h+{b}_{1 h} \ln {c}_{1 h}+{b}_{2 h} \ln\ {c}_{2 h}\hfill \\ {}\mathrm{s}.\mathrm{t}.(20)-(22)\hfill \end{array} $$

(P2)

where b _ih are the relative weights they put on their children’s welfare relative to their own.

Notice that our framework is equivalent to Becker’s benchmark unitary model when b _ih  = k _ih and μ = 0.

The value of μ is crucial for the efficiency of schooling investments. When μ = 0, k _ih is independent of schooling attainment. Optimal schooling investments s ^* _ih (a _1h , a _2h , m _h ) satisfy:

$$ \begin{array}{l}\frac{\partial w\left({s}_{1 h},{s}_{2 h}\right)}{\partial {s}_{ih}}=0\hfill \\ {}\mathrm{i}.\mathrm{e}.\frac{\partial R\left({s}_{ih}\right)}{\partial {s}_{ih}}=\frac{\partial C\left({s}_{ih},{a}_{ih},{m}_h\right)}{\partial {s}_{ih}}\hfill \end{array} $$

i.e., the efficient schooling investment level (s ^* _i ) will maximize total family wealth.³⁹ This is true under both the unitary and collective model. Even when household members bargain over the division of family wealth and parents have their own private interests, schooling investment is chosen to maximize total family wealth. Although parental preferences may differ from the bargaining weights, (b _ih  ≠ k _ih ), the best the parents can do given the allocation constraints (20)–(22) is to maximize total family wealth. This result is reminiscent of Becker’s Rotten Kid Theorem.

When μ > 0, schooling investment will not be wealth maximizing. Under the unitary model, b _i (s _i ) = k _i (s _i ). The optimal solution for child i’s schooling s ^′ _ih satisfies the first-order condition of (P2):

$$ \frac{\partial w\left({s}_{1 h},{s}_{2 h}\right)}{\partial {s}_{ih}}=-\mu r\cdot \ln \frac{k_{ih}}{1-{k}_{1 h}-{k}_{2 h}} $$

When ${k}_{ih}>1-{k}_{1 h}-{k}_{2 h},\frac{\partial {w}_h}{\partial {s}_{ih}}<0$, hence s ^′ _ih  > s ^* _ih ; when ${k}_{ih}<1-{k}_{1 h}-{k}_{2 h},\frac{\partial {w}_h}{\partial {s}_{ih}}>0$, hence s ^′ _ih  < s ^* _ih , i.e., parents will over/under invest (relative to the wealth maximizing level) in one child’s schooling if he receives a higher/lower weight in the parental utility function than do the parents.⁴⁰

Under the collective model, b _ih (s _ih ) ≠ k _ih (s _ih ). The optimal solution for child i’s schooling s ^′ ′ _ih satisfies:

$$ \frac{\partial w\left({s}_{1 h},{s}_{2 h}\right)}{\partial {s}_{ih}}=-\mu r\cdot \frac{b_{ih}\left(1-{k}_{jh}\right)-{k}_{ih}\left(1-{b}_{jh}\right)}{k_{ih}\left(1-{k}_{1 h}-{k}_{2 h}\right)} $$

In particular, if b _ih  < k _ih (s ^′ ′ _ih ) for i = 1, 2, then $\frac{\partial {w}_h}{\partial {s}_{ih}}>0$ and therefore s ^′ ′ _ih  < s ^* _ih . This means that, if at the wealth maximizing schooling level, children’s bargaining powers are greater than their parents would like them to be, the parents will invest less in children’s schooling relative to the efficient level.

B. Estimation of the multiplicative model

Given a random sample of households h = 1,…, H, each with 2 children i = 1, 2, consider the regression:

$$ {y}_{ih}=\left(1+{D}_{ih}\delta \right) \cdot {\omega}_h+{X}_{ih} b+{\varepsilon}_{ih} $$

(24)

where y _ih is the dependent variable; D _ih and X _ih are vectors of children’s characteristics; and ω _h is the household fixed effect.⁴¹ The parameters to be identified are θ = (δ, {ω _h } ^H _h = 1 , b).

A nonlinear least squares (NLS) estimator of $\theta, \widehat{\theta}$, solves:

$$ \underset{\theta}{ \min }{\displaystyle \sum_{h=1}^H{\displaystyle \sum_{i=1}^2{\left[{y}_{i h}-\left(1+{D}_{i h}\delta \right)\cdot {W}_h-{X}_{i h} b\right]}^2}} $$

To reduce the computational burden of a global search for all parameters, we take advantage of the partial linearity in (24): given a value of δ, δ ₀, equation (24) is a linear model. {ω _h } ^H _h = 1 and b can be estimated through OLS. Denoting these estimators as {ω ^ols _h (δ ₀|X _ih , D _ih )} ^H _h = 1 and b ^ols(δ ₀|X _ih , D _ih ), our problem is to find:

$$ \widehat{\delta}= \arg \underset{\delta_0}{ \min }{\displaystyle \sum_{h=1}^H{\displaystyle \sum_{i=1}^2{\left({y}_{i h}-\left(1+{D}_{i h}{\delta}_0\right)\cdot {\omega}_h^{ols}\left({\delta}_0\left|{X}_{i h},{D}_{i h}\right.\right)-{X}_{i h}{b}^{ols}\left({\delta}_0\left|{X}_{i h},{D}_{i h}\right.\right)\right)}^2}} $$

Given $\widehat{\delta}$, the final estimator $\widehat{\theta}$ is:

$$ \widehat{\theta}=\left(\widehat{\delta},{\left\{{\omega}_h^{ols}\left(\widehat{\delta}\left|{X}_{ih},{D}_{ih}\right.\right)\right\}}_{h=1}^H,{b}^{ols}\left(\widehat{\delta}\left|{X}_{ih},{D}_{ih}\right.\right)\right) $$

C. The incidental parameter problem

The least square estimators outlined above are subjected to a so-called "incidental parameter" problem. As first observed by Neyman and Scott (1948), standard estimators of nonlinear panel data models are usually inconsistent if the length of the panel is small relative to the number of observations. In this case, the finite sample bias in the fixed effects parameters ({ω _h } ^H _h = 1  in our context) will contaminate estimates of other parameters (δ and b in (24)).

Given the partial nonlinearity feature of our model, an alternative approach that can help get around the incidental parameter problem is quasi-differencing. To see this, divide (24) by 1 + D _ih δ on both sides, and then take the sibling difference within the same household to obtain:

$$ \varDelta \frac{y}{1+ D\delta} h=\varDelta \frac{X}{1+ D{\delta}_h} b+{\upsilon}_h $$

(25)

where $\varDelta \frac{x}{1+ D{\delta}_h}=\frac{x_{1 h}}{1+{D}_{1 h}\delta}-\frac{x_{2 h}}{1+{D}_{2 h}\delta}$ and ${\upsilon}_h=\frac{\varepsilon_{1 h}}{1+{D}_{1 h}\delta}-\frac{\varepsilon_{2 h}}{1+{D}_{2 h}\delta}$. Notice that household fixed effects are eliminated in (25). The parameters to be determined are (δ, b), which can be consistently estimated through NLS.⁴²

In order to compare the accuracy and efficiency between our least-square estimate and the above quasi-differencing estimate, we conduct Monte Carlo experiments. The number of households is 140, consistent with our multiple-sons sample. In each simulation, each household is endowed with a k ₀ w _h , which is assumed to be log normal distributed. Its distribution parameters are calibrated from the moments of the household fixed effects {w _h } ^H _h = 1 that we estimate from the data.

Each household has two children, one of whom is the "first son". We randomly generate two other attributes of the children, "indicator of the taller son" (D ^taller _ih ) and "indicator of living with parents post marriage" (D ^livep _ih ), such that their variance-covariance matrices with the first son indicator (D ^1stson _ih ) replicate the ones in the actual data.

We regress schooling expenditure on k ₀ w _h and children’s attributes using the actual dataset. We then use the coefficients and the distribution parameters of the error terms to generate the simulated schooling investments. Note that the error terms here serve as the ability endowments of the children. The intervivos transfers are simulated using equation (17) with error terms randomly generated to match the sample moments of the marital transfers (t _ih ). The resulting data set is {s _ih , t _ih , D ^taller _ih , D ^livep _ih , D ^1stson _ih }. We estimate the model:

$$ {t}_{ih}=\left(1+{\delta}_1{D}_{ih}^{t aller}+{\delta}_2{D}_{ih}^{livep}+{\delta}_3{D}_{ih}^{1 stson}\right)\cdot {\omega}_h+\beta {s}_{ih}+{\varepsilon}_{ih} $$

with the "true" parameters being set as (β,δ _1, δ ₂,δ ₃) = (−0.3, 0.2, 0.4, −0.2). Note that we allow cov(s _ih , ω _h ) > 0 and cov(s _ih , D _ih ) > 0, but keep cov(s _ih , ε _ih ) = 0 in the simulated data, which are the basic identification assumptions maintained throughout our paper.

We conduct a Monte Carlo analysis with 500 simulations. The results are reported in an appendix table available online. Our results show that, the least square approach used in our paper delivers highly precise estimates. By contrast, the quasi-differencing results exhibit larger bias and larger standard errors, especially when all three attributes of the children are controlled for simultaneously. ⁴³We conclude that, even though our least square estimates may be theoretically inconsistent, their finite sample bias is negligible and they are more efficient than the consistent quasi-differencing estimates. We therefore base our inferences on the least squared results in our paper.

D. An illustrative explanation for why the first son can have both smaller k _ih and smaller (1–μ _ih )r _ih at the same time

In an appendix figure available online, we show, in an illustrative way, the fitted linear relationship between educational expenditure and the transfers for the first son and his sibling (denoted as the second son).⁴⁴ The readers are referred to that figure for the discussion below. By definition, the two lines in the figure should pass through the sample mean for the two sons, denoted by point A and B, respectively. Their relative positions in the figure are due to the fact that, on average, eldest sons receive less schooling investments and intervivos transfers than their siblings in the data.⁴⁵

Subfigure (1) illustrates the case of Table 4, where the slopes of the two fitted lines, i.e. the marginal compensation coefficients, are assumed to be the same. But intercepts of the lines can be different, which identifies δ in equation (17) for the first son. The assumption that the two lines are parallel, along with the relative position of point A and B, determines that l ₁ lies below l ₂, which is consistent with the negative value for δ for the first son in Table 4.

Subfigure (2) illustrates the case of Table 6, where the slopes of the two lines are allowed to be different. Notice that the coefficient on the interaction between the first son dummy with the educational expenditure in Table 6 is positive, which means that the marginal coefficient is smaller for the first son. This result, along with the relative position of A and B, determines that magnitude of δ for the first son should be larger in Table 6 than in Table 4, which is consistent with our results.