A tale of two decades: Relative intra-family earning capacity and changes in family welfare over time

Abstract

The purpose of this paper is to assess the impact of economic changes in the 1990s and 2000s on the welfare of married households, taking into account the relative earnings structure of husband and wife. Modeling the household members’ joint labor supply, we find that families in which the wife is the higher wage earner experienced as much welfare gain in the 1990s and significantly higher welfare gains in the 2000s as families in which the husband is the higher wage earner.

Appendices

Appendix 1

1.1 This appendix presents the first order conditions of families' utility maximization problem, derivation of the labor supply equations, and the liklihood function estimated

The quadratic functional form as presented in Eq. (5) in the text can also be written in the following form:

$$\begin{array}{ccccc}\ U\left( Z \right) = {a_1}\left( {{L_1}} \right) + {a_2}\left( {{L_2}} \right) + {a_3}\left( C \right) - \frac{1}{2}{b_{11}}{\left( {{L_1}} \right)^2} - \frac{1}{2}{b_{22}}{\left( {{L_2}} \right)^2} - \frac{1}{2}{b_{33}}{\left( C \right)^2} - {b_{12}}{L_1}{L_2}\\ \\ - {b_{13}}{L_1}C - {b_{23}}{L_2}C\\ \end{array}$$

(13)

where ${L_1} = T - {h_1};{L_2} = T - {h_2};\;{\rm{and}}\;C = {w_1}{h_1} + {w_2}{h_2} + Y$

This becomes an unconstrained utility maximization problem which depends on the working hours h ₁ and h ₂, assuming that Y (non-labor income) is exogenous. The corresponding first order conditions become:

$$\begin{array}{ccccc}\ \frac{{\partial u}}{{\partial {h_1}}} = a_1^* + a_3^*{w_1} - {b_{11}}{h_1} - {b_{33}}{w_1}\left( {{w_1}{h_1} + {w_2}{h_2} + Y} \right)\\ \\ - {b_{12}}{h_2} + {b_{13}}\left( {2{w_1}{h_1} + {w_2}{h_2} + Y} \right) + {b_{23}}{w_1}{h_2} = 0\\ \end{array}$$

(14)

$$\begin{array}{ccccc}\ \frac{{\partial u}}{{\partial {h_2}}} = a_2^* + a_3^*{w_2} - {b_{22}}{h_2} - {b_{33}}{w_2}\left( {{w_1}{h_1} + {w_2}{h_2} + Y} \right)\\ \\ - {b_{12}}{h_1} + {b_{23}}\left( {{w_1}{h_1} + 2{w_2}{h_2} + Y} \right) + {b_{13}}{w_2}{h_1} = 0\\ \end{array}$$

(15)

There is no need to specify a time endowment (T) in order to estimate the labor supply functions because $a_1^*$, $a_2^*$, and $a_3^*$ are re-parameterized functions of T and Y. This re-parameterization is necessary for identification of the labor supply equations. It is through these starred parameters that differences in tastes across families are allowed to enter. Specifically,

$$a_1^* = {X_1}{{\rm{\Gamma }}_1}\,{\rm{and}}\,a_2^* = {X_2}{{\rm{\Gamma }}_2}$$

where X ₁ and X ₂ are vectors of individual and family characteristics and Γ₁ and Γ₂ are parameters to be estimated. Individual characteristics included in X are age, age squared, race, and education. Family characteristics include number of children. Non-labor family income enters separately.

Using Eqs. (14) and (15), we can solve the system obtaining the values of h ₁ and h ₂ that maximize the utility function, in the following way:

$${{\rm{\Omega }}_1}h_1^* + {{\rm{\Omega }}_2}h_2^* + {{\rm{\Omega }}_3} = 0$$

(16)

$${{\rm{\Omega }}_2}h_1^* + {{\rm{\Omega }}_4}h_2^* + {{\rm{\Omega }}_5} = 0$$

(17)

where:

$${{\rm{\Omega }}_1} = 2{b_{13}}{w_1} - {b_{11}} - {b_{33}}w_1^2;$$

(18)

$${{\rm{\Omega }}_2} = {b_{23}}{w_1} + {b_{33}}{w_1}{w_2} - {b_{12}} + {b_{13}}{w_2};$$

(19)

$${{\rm{\Omega }}_3} = {a^*}_1 + {a^*}_3{w_1} + \left( {{b_{33}}{w_1} + {b_{13}}} \right)Y;$$

(20)

$${{\rm{\Omega }}_4} = 2{b_{23}}{w_2} - {b_{22}} - {b_{33}}w_2^2;{\rm{and}}$$

(21)

$${{\rm{\Omega }}_5} = {a^*}_2 + {a^*}_3{w_2} + \left( {{b_{33}}{w_2} + {b_{23}}} \right)Y.$$

(22)

From Eqs. (16) and (17), the solutions for $h_1^*$ and $h_2^*$ become:

$$h_1^* = \frac{{{{\rm{\Omega }}_3}{{\rm{\Omega }}_4} - {{\rm{\Omega }}_2}{{\rm{\Omega }}_5}}}{{{\rm{\Omega }}_2^2 - {{\rm{\Omega }}_1}{{\rm{\Omega }}_4}}}$$

(23)

$$h_2^* = \frac{{{{\rm{\Omega }}_1}{{\rm{\Omega }}_5} - {{\rm{\Omega }}_2}{{\rm{\Omega }}_3}}}{{{\rm{\Omega }}_2^2 - {{\rm{\Omega }}_1}{{\rm{\Omega }}_4}}}$$

(24)

These derivatives are obtained with the help of Mathematica® (Wolfram Research, Inc. 2010). We calculate expected hours conditional on being positive according to (Muthen 1990).

Table 5 Maximum likelihood utility parameter estimates

Appendix 2