Welfare Sharing Within Households: Identification from Subjective Well-being Data and the Collective Model of Labor Supply

Abstract

The paper proposed a novel approach to identifying intra-household allocations in households with two decision-makers. We addressed the issue by using the collective model of labor supply. Most empirical studies based on the collective approach were restricted to the identification of the sharing rule guiding individual allocations up to a constant. We suggested using individual welfare satisfaction data as an additional source of identification. An empirical example was given using the Russian Longitudinal Monitoring Survey. The sharing rule was found to be related to the spouses’ wage and age differences and number of children. Sharing varied with the level of household income: In low-income households, sharing was equal; in middle (high) income households men held a slight (strong) advantage.

Appendix

The parameters derived in this section are related to the distribution of $\varepsilon$ from

$$\begin{aligned} I= \left\{ \begin{array}{rl} 0 &{} \text{ if } \varepsilon \le z_1 \\ 1 &{} \text{ if } z _1 < \varepsilon \le z _2\\ 2 &{} \text{ if } \varepsilon > z_2\\ \end{array} \right. \end{aligned}$$

(A1)

where $z_1 =\kappa _1 - \mathbf{\gamma }'\mathbf{Z}$ and $z_2 =\kappa _2 - \mathbf{\gamma }'\mathrm{\mathbf{Z}}$.

The Mean and Variance of the $\mathbf u _\mathbf eq$ Distribution

The mean of the truncated standard normal distribution is found using the following property of the standard normal density function $f(\varepsilon ) = \frac{1}{\sqrt{2\pi } }\exp ( - \frac{\varepsilon ^2}{2})$: $\frac{\partial f(\varepsilon )}{\partial \varepsilon } = - \varepsilon f(\varepsilon )$.

$$\begin{aligned} \begin{array}{lll} E(u_{eq})= E (\varepsilon \vert I = 1) = E (\varepsilon \vert z_1 < \varepsilon < z_2 ) = \int \limits _{z_1 }^{z_2 } {\varepsilon f(\varepsilon \left| {z_1 < \varepsilon < z_2 } \right. )d\varepsilon = } \\ = \int \limits _{z_1 }^{z_2 } {\varepsilon \frac{f(\varepsilon )}{P(z_1 < \varepsilon < z_2 )}d\varepsilon = \frac{1}{F(z_2 ) - F(z_1 )}} \int \limits _{z_1 }^{z_2 } {\varepsilon f(\varepsilon )d\varepsilon } = \\ = \frac{1}{F(z_2 ) - F(z_1 )}\int \limits _{z_1 }^{z_2 } {\left( { -f'(\varepsilon )} \right) d\varepsilon } = \frac{1}{F(z_2 ) - F(z_1 )}\left[ { - f(\varepsilon )} \right] _{z_1 }^{z_2 } = \frac{f(z_1 ) - f(z_2 )}{F(z_2 ) - F(z_1 )} \end{array} \end{aligned}$$

(A2)

where $F$(.) is the standard normal cumulative density function.

The variance is calculated⁵ Using the same approach:

$$\begin{aligned} Var(u_{eq})= Var(\varepsilon \vert I = 1) = Var (\varepsilon \vert z_1 < \varepsilon < z_2 ) = E (\varepsilon ^2\vert z_1 < \varepsilon < z_2 ) - \left( {E (\varepsilon \vert z_1 < \varepsilon < z_2 )} \right) ^2 \end{aligned}$$

(A3)

$( {E (\varepsilon \vert z_1 < \varepsilon < z_2 )}$ is defined by (A2). $E (\varepsilon ^2\vert z_1 < \varepsilon < z_2 )$ is found by integrating by parts:

$$E({\varepsilon ^2}|{z_1} < \varepsilon < {z_2}) = {1 \over {F({z_2}) - F({z_1})}}\int\limits_{{z_1}}^{{z_2}} {\left( { - \varepsilon f'(\varepsilon )} \right)d\varepsilon } = {1 \over {F({z_2}) - F({z_1})}}\left( {\left[ { - \varepsilon f(\varepsilon )} \right]_{{z_1}}^{{z_2}} + \int\limits_{{z_1}}^{{z_2}} {\left( {f(\varepsilon )} \right)d\varepsilon } } \right) = {1 \over {F({z_2}) - F({z_1})}}\left( { - {z_2}f({z_2}) + {z_1}f({z_1}) + F({z_2}) - F({z_1})} \right) = 1 + {{{z_1}f({z_1}) - {z_2}f({z_2})} \over {F({z_2}) - F({z_1})}}$$

(A4)

Finally, using (A2–A4) the variance of the truncated low is found as

$$\begin{aligned} Var(u_{eq}) = 1 + \frac{z_1 f(z_1 ) - z_2 f(z_2 )}{F(z_2 ) - F(z_1 )} - \left( {\frac{f(z_1 ) - f(z_2 )}{F(z_2 ) - F(z_1 )}} \right) ^2 \end{aligned}$$

Selection Bias Correction

Given selection of the households whose members report the same level of satisfaction, the expected value of the error terms of the labor supply equations are not zero:

$$\begin{aligned} E (u_i \vert I = 1) = E(E (u_i\vert \varepsilon ) \vert I =1) = M(\sigma _{u_i\varepsilon })E(\varepsilon \vert I=1) \end{aligned}$$

where $M(\sigma _{u_i\varepsilon })$ is an unknown coefficient depending on the covariance between $\varepsilon$ and $u_i$ .

The correction term $E(\varepsilon \vert I=1)$ is defined for each individual in the same way as $E(u_{eq} \vert I=1)$ shown by (A2):

$$\begin{aligned} E(\varepsilon \vert I=1) = \frac{f(z_1 ) - f(z_2 )}{F(z_2 ) - F(z_1 )} \end{aligned}$$

where $f(\cdot )$ and $F(\cdot )$ are the standard normal density and cumulative density functions respectively.

Constraint on the Cholesky Parameters

The constraint on $\sigma _{u_{eq}} ^2 = Var (\varepsilon \vert z_1 < \varepsilon < z_2 )$ is respected by normalizing the error term of the third equation by $u^{*}_{eq} = \frac{u_{eq} }{\sqrt{Var (\varepsilon \vert z_1 < \varepsilon < z_2 )} }$. Constrained (and normalized to 1) variance of $u^{*}_{eq}$ implies a constraint on Cholesky matrix parameters $L$. This constraint is found setting the relationship between $Var(u^{*}_{eq})$ and matrix $L~$elements:

$$\begin{aligned} Var({u_{eq} }\bigg /{\sqrt{Var (\varepsilon \vert z_1 < \varepsilon < z_2 )} }) = (l_{31} )^2 + (l_{32} )^2 + (l_{33} )^2, \end{aligned}$$

which implies the following restriction on Cholesky matrix parameters

$$\begin{aligned} (l_{31} )^2 + (l_{32} )^2 + (l_{33} )^2 = 1 \end{aligned}$$

A constraint on $l_{33}$ is then $l_{33} = \sqrt{1 - (l_{31} )^2 - (l_{32} )^2}$.