Do coresidency and financial transfers from the children reduce the need for elderly parents to works in developing countries?

Abstract

Do elderly parents use coresidence with or financial transfers from children to reduce their own labour supply in old age? This paper is one of only a few studies that seeks to formally model elderly labour supply in the context of a developing country while taking into account coresidency with and financial transfers from children. We find little evidence that support from children—either through transfers or coresidency—substitutes for elderly parents’ need to work. Thus, as in developed countries, there is a role for public policy to enhance the welfare of the elderly population.

Appendices

Appendix 1

Table 6 Variable definitions

Table 7 Mean parental and child characteristics by gender and coresidency

Table 8 Transfers from non-coresiding children based on single-equation model (Tobit marginal effects and t statistics)

Table 9 Determinants of weekly normal hours of work based on single-equation model (marginal effects and t statistics)

Appendix 2 Joint maximum likelihood estimation of the coresidency, transfers and labour supply equations

The estimation is conducted separately by coresidency status and gender. We present the likelihood function for a non-coresiding mother or father below. The procedure for coresiding parents is analogous to this.

We begin by simplifying the notation used in Eqs. 1–6 in the text. This will make it easier to write out the likelihood expressions. The simplified coresidency equation is:

$$\begin{array}{*{20}c} {{C_{i} = 1{\left( {\eta _{0} + \eta _{1} Z^{P}_{i} + \eta _{2} Z^{C}_{i} + \eta _{3} H_{i} + \nu _{i} > 0} \right)}}} \\ {{ = 1{\left( {\eta Z_{i} + \nu _{i} > 0} \right)}}} \\ \end{array} $$

(7)

where 1( ) is an indicator function that equals 1 if the expression in the brackets is true and 0 otherwise, and Z _i is the column vector ${\left[ {1,{\mathbf{Z}}_{i} ^{{\text{P}}} ,{\mathbf{Z}}_{i} ^{{\text{C}}} ,H_{i} } \right]}$.

For transfers, we have:

$$ \begin{array}{*{20}c} {{\text{TR}}_{i} }{ = \max {\left( {\pi _{{0n}} + \pi _{{1n}} {\mathbf{Z}}_{i} ^{{\text{P}}} + \pi _{{2n}} {\mathbf{Z}}_{i} ^{{{\text{NC}}}} + u_{{1i}} ,0} \right)}} \\ {}{ = \max {\left( {\pi {\mathbf{X}}_{i} + u_{{1i}} ,0} \right)}} \\ \end{array} , $$

(8)

where X _i is the column vector $[1,{\text{ }}{\mathbf{Z}}_{i} ^{{\text{P}}} {\text{, }}{\mathbf{Z}}_{i} ^{{{\text{NC}}}} ].$ Finally, labour supply is given by:

$$ \begin{array}{*{20}c} {LS_{i} ^{P} }{ = \max {\left( {\beta _{{0n}} + \beta _{{1n}} {\mathbf{Z}}_{i} ^{{\text{P}}} + \gamma _{{1n}} {\text{TR}}_{i} + \varepsilon _{{1i}} ,0} \right)}} \\ {}{ = \max {\left( {\beta {\mathbf{W}}_{i} + \gamma {\text{TR}}_{i} + \varepsilon _{{1i}} ,0} \right)}} \\ \end{array} , $$

(9)

where W _i is the column vector ${\left[ {1,{\mathbf{Z}}_{i} ^{{\text{P}}} } \right]}.$

We assume that the error terms are jointly normally distributed such that:

$${\left( {\begin{array}{*{20}c} {{\nu _{i} }} \\ {{u_{{1i}} }} \\ {{\varepsilon _{{1i}} }} \\ \end{array} } \right)} \sim N{\left( {\begin{array}{*{20}c} {0\;1}{\rho _{{\nu u_{1} }} \sigma _{{u_{1} }} }{\rho _{{\nu \varepsilon _{1} }} \sigma _{{\varepsilon _{1} }} } \\ 0{}{\rho _{{u_{1} \varepsilon _{1} }} \sigma _{{u_{1} }} \sigma _{{\varepsilon _{1} }} } \\ {0,}{}{\sigma ^{2}_{{\varepsilon _{1} }} } \\ \end{array} } \right)}.$$

2.1 Deriving the likelihood function

The elderly non-coresiding individual can be in one of four states. Below, we list the states and the expression for the associated probability of being in each state.

2.1.1 Coresiding (C _i = 1)

The probability associated with being in this state is written as:

$$ \begin{array}{*{20}c} {L_{{1i}} }{ = \Pr {\left( {C_{i} = 1} \right)}} \\ {}{ = \Pr {\left( {\nu _{i} > - \eta z_{i} } \right)}} \\ {}{ = 1 - \Phi {\left( { - \eta z_{i} } \right)}} \\ \end{array} , $$

where Φ is the normal cumulative distribution function. Note that we are following the convention of using uppercase letters to represent the random variables and lowercase to represent the realization of the variables.

2.1.2 Non-coresiding, receiving positive transfers and having positive labour supply

$$ {\left( {C_{i} = 0,{\text{TR}}_{i} > 0,LS^{P} _{i} > 0} \right)} $$

$$ \begin{array}{*{20}c} {L_{{2i}} = \Pr {\left( {C_{i} = 0,{\text{TR}}_{i} = tr_{i} ,LS_{i} ^{P} = ls_{i} } \right)}} \\ { = \Pr {\left( {{\text{TR}}_{i} = tr_{i} ,LS_{i} ^{P} = ls_{i} } \right)} \times \Pr {\left( {C_{i} = 0\left| {{\text{TR}}_{i} } \right. = tr_{i} ,LS_{i} ^{P} = ls_{i} } \right)}} \\ { = \Pr {\left( {u_{{1i}} = tr_{i} - \pi x_{i} ,\varepsilon _{{1i}} = ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)} \times \Pr {\left( {\nu _{i} < - \eta z_{i} \left| {u_{{1i}} } \right. = tr_{i} - \pi x_{i} ,\varepsilon _{{1i}} = ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)}} \\ {\varphi _{2} {\left( {tr_{i} - \pi x_{i} ,ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)} \times \Phi {\left( { - \eta z_{i} \left| {tr_{i} } \right. - \pi x_{i} ,ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)}} \\ \end{array} , $$

where ϕ ₂ is the bivariate normal density function.

2.1.3 Non-coresiding, receiving positive transfers and not working

$$ {\left( {C_{i} = 0,\,{\text{TR}}_{i} > 0,\,LS^{P} _{i} = 0} \right)} $$

$$ \begin{array}{*{20}c} {L_{{3i}} = \Pr {\left( {C_{i} = 0,LS_{i} ^{P} = 0,{\text{TR}}_{i} = tr_{i} } \right)}} \\ { = \Pr {\left( {{\text{TR}}_{i} = tr_{i} } \right)} \times \Pr {\left( {LS_{i} ^{P} = 0,C_{i} = 0\left| {{\text{TR}}_{i} } \right. = tr_{i} } \right)}} \\ { = \Pr {\left( {u_{{1i}} = tr_{i} - \pi x_{i} } \right)} \times \Pr {\left( {\nu _{i} < - \eta z_{i} ,\varepsilon _{{1i}} < - \beta w_{i} - \gamma tr\left| u \right._{{1i}} = tr_{i} - \pi x_{i} } \right)}} \\ {\varphi _{2} {\left( {tr_{i} - \pi x_{i} } \right)} \times \Phi _{2} {\left( { - \eta z_{i} , - \beta w_{i} - \gamma tr_{i} \left| {tr_{i} - \pi x_{i} } \right.} \right)}} \\ \end{array} , $$

where φ is the normal probability density function, and Φ ₂ is the bivariate normal cumulative distribution function.

2.1.4 Non-coresiding, receiving no transfers and working

$$ {\left( {C_{i} = 0,\;{\text{TR}}_{i} = 0,LS^{P} _{i} > 0} \right)} $$

$$\begin{array}{*{20}c} {L_{{4i}} = \Pr {\left( {C_{i} = 0,\operatorname{TR} _{i} = 0,LS_{i} ^{P} = ls_{i} } \right)}} \\ { = \Pr {\left( {LS_{i} ^{P} = ls_{i} } \right)}.\Pr {\left( {{\text{TR}}_{i} = 0,C_{i} = 0\left| {LS_{i} ^{P} = Is_{i} } \right.} \right)}} \\ { = \Pr {\left( {u_{i} = ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)} \times \Pr {\left( {u_{{1i}} < - \pi x_{i} ,\nu _{i} < - \eta z_{i} ,\left| {\varepsilon _{{1i}} = ls_{i} - \beta w_{i} - \gamma tr} \right._{i} } \right)}} \\ { = \phi {\left( {ls_{i} - \beta w_{i} - \gamma tr_{i} } \right)} \times \Phi _{B} {\left( { - \pi x_{i} , - \eta z_{i} \left| {ls_{i} - \beta w_{i} - \gamma tr_{i} } \right.} \right)}} \\ \end{array} .$$

2.1.5 Non-coresiding, receiving no transfers and not working

$$ {\left( {C_{i} = 0,\,LS^{P} _{i} = 0,\,{\text{TR}}_{i} = 0} \right)} $$

$$ \begin{array}{*{20}c} {L_{{5i}} = \Pr {\left( {C_{i} = 0,LS_{i} ^{P} = 0,{\text{TR}}_{i} = 0} \right)}} \\ { = \Pr {\left( {\nu _{i} < - \eta z_{i} ,\varepsilon _{{1i}} < - \beta w_{i} - \gamma tr_{i} ,u_{{1i}} < - \pi x_{i} } \right)}} \\ { = \Phi _{3} {\left( { - \eta z_{i} , - \beta w_{i} - \gamma tr_{i} , - \pi x_{i} } \right)}} \\ \end{array} , $$

where Φ ₃ is the trivariate normal cumulative distribution function.

2.2 Log likelihood function

The log likelihood is thus written as:

$$\log L_{i} = 1{\left( {C_{i} = 1} \right)} \times \log L_{{1i}} + 1{\left( {C_{i} = 0,TR_{i} > LS_{i} > 0} \right)} \times \log L_{{2i}} + 1{\left( {C_{i} = 0,TR_{i} > 0,LS_{i} = 0} \right)} \times \log L_{{3i}} + 1{\left( {C_{i} = 0,TR_{i} = 0,LS_{i} > 0} \right)}.\log L_{{4i}} + 1{\left( {C_{i} = 0,TR_{i} = 0,LS_{i} = 0} \right)}.\log L_{{5i}} .$$

The analogous states for the coresiding case are the probabilities associated with (1) non-coresiding; (2) coresiding, receiving positive transfers and working; (3) coresiding, receiving positive transfers and not working; (4) coresiding, receiving no transfers and working; and (5) coresiding, receiving no transfers and not working. The set of explanatory variables in the transfers and labour supply equations differ in the coresiding case, as shown in Eqs. 1–4.