Constrained vs unconstrained labor supply: the economics of dual job holding

Abstract

This paper develops a unified model of dual and unitary job holding based on a Stone-Geary utility function. The model incorporates both constrained and unconstrained labor supply. Panel data methods are adapted to accommodate unobserved heterogeneity and multinomial selection into six mutually exclusive labor supply regimes. We estimate the wage and income elasticities arising from selection and unobserved heterogeneity as well as from the Stone-Geary Slutsky equations. The labor supply model is estimated with data from the British Household Panel Survey 1991–2008. Among dual job holders, our study finds that the Stone-Geary income and wage elasticities are much larger for labor supply to the second job compared with the main job. When the effects of selection and unobserved heterogeneity are taken account of, the magnitudes of these elasticities on the second job tend to be significantly reduced.

A Technical Appendix

1.1 A.1 Theoretical Stone-Geary elasticities

Unconstrained dual job holder

The own wage Slutsky equation in elasticity form may be expressed by

$$\eta_{mm}=\eta_{mm}^{c}+\epsilon_{mm\text{I} }, $$

where

$$\begin{array}{@{}rcl@{}} \eta_{mm} & =&\frac{w_{m}}{h_{m}^{\ast}}\frac{\partial h_{m}^{\ast}}{\partial w_{m}}\\ & =&\frac{\alpha_{m}}{w_{m}h_{m}^{\ast}}\left( \gamma_{k}w_{k}+\text{}I\text{} -\gamma_{3}\right) \gtreqless0, \end{array} $$

is the uncompensated own wage elasticity for job m,

$$\begin{array}{@{}rcl@{}} \eta_{mm}^{c} & =&\frac{w_{m}}{h_{m}^{\ast}}S_{mm}\\ & =&\frac{\alpha_{m}}{w_{m}h_{m}^{\ast}}\left( \gamma_{k}w_{k}+w_{m}h_{m}^{\ast}+\text{} I\text{} -\gamma_{3}\right) >0, \end{array} $$

is the compensated own substitution effect elasticity for job m, S _{m m} is the compensated own substitution effect, and

$$\epsilon_{mm\text{I} }=-\alpha_{m\text{} }\,<0, $$

is the own wage income effect elasticity.

The pure income effect elasticity for job m is given by

$$\begin{array}{@{}rcl@{}} \eta_{mI\text{} } & =&\frac{I}{h_{m}}\frac{\partial h_{m}^{\ast}}{\partial I}\\ & =&-\frac{\alpha_{m}I}{w_{m}h_{m}^{\ast}}<0. \end{array} $$

The Slutsky equation for cross wage effects in elasticity form is given by

$$\eta_{mk}=\eta_{mk}^{c}+\epsilon_{mk\text{I}}, $$

where

$$\begin{array}{@{}rcl@{}} \eta_{mk} & =&\frac{w_{k}}{h_{m}^{\ast}}\frac{\partial h_{m}^{\ast}}{\partial w_{k}}\\ & =&\frac{-\alpha_{m}\gamma_{k}w_{k}}{w_{m}h_{m}^{\ast}}<0, \end{array} $$

is the uncompensated cross wage effect elasticity of labor supply to job m from a change in the wage for job k,

$$\begin{array}{@{}rcl@{}} \eta_{mk}^{c} & =&\frac{w_{k}}{h_{m}^{\ast}}S_{mk}\\ & =&\frac{-\alpha_{m}w_{k}}{w_{m}h_{m}^{\ast}}\left( \gamma_{k}-h_{k}^{\ast} \right) <0, \end{array} $$

is the compensated cross substitution effect elasticity, S _{m k} is the compensated cross substitution effect of a change in the wage on job k on labor supply to job m, and

$$\epsilon_{mkI}=\frac{-\alpha_{m}w_{k}h_{k}^{\ast}}{w_{m}h_{m}^{\ast}}<0, $$

is cross-wage income effect elasticity. Observe that both uncompensated and compensated increases in the wage for job k lead to reductions in labor supply to job m.

Unconstrained unitary job holders

The own wage Slutsky equation in elasticity form for job 1 when not working a second job is expressed as

$$\eta_{11}\vert_{h_{2}= 0} =\eta_{11}^{c}\vert_{h_{2} = 0}+\epsilon_{11I}\vert_{h_{2}= 0} , $$

where

$$\begin{array}{@{}rcl@{}} \eta_{11}\vert_{h_{2}= 0} & =&\frac{w_{1}}{h_{1}^{\ast}} \frac{\partial h_{1}^{\ast}}{\partial w_{1}}\vert_{h_{2}= 0} \\ & =&\left( \frac{\alpha_{1}}{1-\alpha_{2}}\right) \left( \frac{1}{w_{1}h_{1}^{\ast}}\right) \left( I-\gamma_{3}\right) \gtreqless0, \end{array} $$

is the uncompensated own wage effect elasticity for job 1,

$$\begin{array}{@{}rcl@{}} \eta_{11}^{c}\vert_{h_{2}= 0} & =&\frac{w_{1}}{h_{1}^{\ast}} S_{11}\vert_{h_{2}= 0} \\ & =&\left( \frac{\alpha_{1}}{1-\alpha_{2}}\right) (\frac{1}{w_{1}h_{1}}) \left( w_{1}h_{1}+I-\gamma_{3}\right) >0, \end{array} $$

is the compensated own substitution effect elasticity for job 1,$S_{11}\vert _{h_{2}= 0}$. is the own compensated substitution effect, and

$$\epsilon_{11I}\vert_{h_{2}= 0} =\frac{-\alpha_{1}}{1-\alpha_{2}} <0, $$

is the income effect elasticity from the own wage.

The pure income effect elasticity for h1∗is determined by

$$\begin{array}{@{}rcl@{}} \eta_{1I}\vert_{h_{2}= 0} & =&\frac{I}{h_{1}^{\ast}} \frac{\partial h_{1}^{\ast}}{\partial I}\vert_{h_{2}= 0}\\ & =&\left( \frac{-\alpha_{1}}{1-\alpha_{2}}\right) (\frac{I}{w_{1}h_{1}^{\ast}}) <0. \end{array} $$

Constrained dual job holder

The own wage Slutsky equation in elasticity form for job 2 when constrained on job 1 may be expressed as

$$\eta_{22}\vert_{h_{1}=\ddot{h}_{1}}=\eta_{22}^{c}\vert_{h_{1}=\ddot{h}_{1}}+\epsilon_{22I}\vert_{h_{1}=\ddot{h}_{1}} ,$$

where

$$\begin{array}{@{}rcl@{}} \eta_{22}\vert_{h_{1}=\ddot{h}_{1}}& =&\frac{w_{2}} {h_{2}^{\ast}}\frac{\partial h_{2}^{\ast}}{\partial w_{2}}\vert_{h_{1}=\ddot{h}_{1}}\\ & =&\left( \frac{\alpha_{2}}{1-\alpha_{1}}\right) (\frac{1}{w_{2}h_{2}^{\ast}}) \left( w_{1}\ddot{h}_{1}+ I -\gamma_{3}\right) \gtreqless0, \end{array} $$

is the uncompensated wage uncompensated own wage elasticity for job 2,

$$\begin{array}{@{}rcl@{}} &&\eta_{22}^{c}\vert_{h_{1}=\ddot{h}_{1}} =\frac{w_{2}}{h_{2}^{\ast}}S_{22}\vert_{h_{1}=\ddot{h}_{1}}\\ &=&\left( \frac{\alpha_{2}}{1-\alpha_{1}}\right)\left( \frac{1}{w_{2}h_{2}^{\ast}}\right)\left( w_{1}\ddot{h}_{1}+w_{2}{h_{2}^{\ast}}+I-\gamma_{3}\right) >0, \end{array} $$

is the compensated own substitution elasticity, $S_{22}{\vert }_{h_{1}={\ddot {h}}_{1}}$ is the compensated own substitution effect, and

$$\begin{array}{@{}rcl@{}} \epsilon_{22I}\vert_{h_{1}=\ddot{h}_{1}} & =&\eta_{22}\vert_{h_{1}=\ddot{h}_{1}} -\eta_{22}^{c}\vert_{h_{1}=\ddot{h}_{1}} \\ & =&\frac{-\alpha_{2}}{1-\alpha_{1}}<0, \end{array} $$

is the income effect elasticity from the own wage.

The pure income effect elasticity for h2∗is determined by

$$\begin{array}{@{}rcl@{}} \eta_{2I}\vert_{h_{1}=\ddot{h}_{1}} & =&\frac{I}{h_{2}} \frac{\partial h_{2}^{\ast}}{\partial I}\vert_{h_{1}=\ddot{h}_{1}}\\ & =&\left( \frac{-\alpha_{2}}{1-\alpha_{1}}\right) \left( \frac{I}{w_{2}h_{2}^{\ast}}\right) <0. \end{array} $$

Note that the compensated cross-substitution effect of wages on job 1 on labor supply to job 2 is necessarily zero when hours are constrained in job 1 because wages on job 1 can only have income effects. Hence, the uncompensated cross-wage elasticity of w ₁ on h2∗ is the same as the cross-wage income effect elasticity:

$$\begin{array}{@{}rcl@{}} \eta_{21}\vert_{h_{1}=\ddot{h}_{1}} & =&\frac{w_{1}}{h_{2}} \frac{\partial h_{2}^{\ast}}{\partial w_{1}}|_{h_{1}=\ddot{h}_{1}} =\epsilon_{21I}\vert_{h_{1}=\ddot{h}_{1}}\\ & =&\left( \frac{-\alpha_{2}}{1-\alpha_{1}}\right) \left( {w_{1}\ddot{h}_{1}}{w_{2}h_{2}^{\ast}}\right) <0. \end{array} $$

Constrained unitary job holder

For a constrained unitary job holder, the hours worked $(\ddot {h}_{1})$ are treated as exogenous so there is no corresponding labor supply equation.

1.2 A.2 Empirical elasticities based on the effects of key economic variables on the effects of selection and unobserved heterogeneity.

Unconstrained dual job holder

We express the total labor supply effect of w _{1i t} as

$$\left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}}\right)^{T}=\left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}}\right)^{SG}+\hat{\theta}_{11}\left[ \frac{1}{w_{1it}}\frac{\partial \hat{\lambda}_{1it}}{\partial w_{1it}}-\frac{\hat{\lambda}_{1it}}{\left( w_{1it}\right)^{2}}\right] - \frac{\bar{Z}_{1i}\hat{\pi}_{11}}{\left( w_{1it}\right)^{2}} $$

where $\left (\frac {\partial \hat {h}_{1it}}{\partial w_{1it}}\right )^{SG}$ is the slope of the uncompensated labor supply curve obtained from the Stone-Geary utility function and $\frac {\partial \hat {\lambda }_{1it}}{\partial w_{1it}}$ $=\left [ \dfrac {-\hat {\lambda }_{1it}\left (z_{1it}+\hat {\lambda }_{1it}\right )} {\phi \left (z_{1it}\right )} \right ] \left [ \frac {\partial \hat {P}_{1it}}{\partial w_{1it}}\right ] ,$ $z_{1it}={\Phi }^{-1}\left (\hat {P}_{1it}\right )$, and $\hat {P}_{1it}={\Lambda }\left (x_{it},\bar {\omega }_{i},\tilde {\beta }_{1}\right )$. Accordingly, the total labor supply elasticity with respect to w _{1i t} is obtained from

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}{{~}_{_{11it}}^{T}} =\left( \frac{w_{1it}}{\hat{h}_{1it}}\right) \left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}}\right)^{T} \\ & =&\hat{\eta}_{_{11it}}+\hat{\theta}_{11}\left[ \frac{1}{\hat{h}_{1it}}\frac{ \partial \hat{\lambda}_{1it}}{\partial w_{1it}}-\frac{\hat{\lambda}_{1it}}{ w_{1it}\hat{h}_{1it}}\right] -\frac{\bar{Z}_{1i}\hat{\pi}_{11}}{w_{1it}\hat{h}_{1it}} \end{array} $$

where $\hat {\eta }_{_{11it}}=\hat {\eta }{{~}_{_{11it}}^{c}}+\hat {\epsilon }_{11\text {I}}$ is the estimated Stone-Geary uncompensated labor supply elasticity with respect to w _{1i t} , $\hat {\eta }{_{_{11it}}^{c}}$ is the compensated labor supply elasticity with respect to w _{1i t} (evaluated at $\hat {h}_{1it}$) , and $\hat {\epsilon }_{11\text {I} }$ is the own wage income effect elasticity. The term $\hat {\theta }_{11}\left [ \frac {1}{\hat {h}_{1it}}\frac {\partial \hat {\lambda }_{1it}}{\partial w_{1it}}-\frac {\hat {\lambda }_{1it}}{w_{1it} \hat {h}_{1it}}\right ]$ captures the job 1 labor supply elasticity effects of w _{1i t} on the probability of being an unconstrained dual job holder. Although w _{1i t} does not directly affect the controls for unobserved heterogeneity, it does impact the effect of unobserved heterogeneity on job 1 labor supply via the term $-\frac {\bar {Z}_{1i}\hat {\pi }_{11}}{w_{1it}\hat {h}_{1it}}$.

The total cross labor supply effect of w _{2i t} is identical to the uncompensated cross wage effect of w _{2i t} on labor supply to job 1 from the Stone-Geary utility function:

$$\left( \frac{\partial \hat{h}_{1it}}{\partial w_{2it}}\right)^{T}=\left( \frac{\partial \hat{h}_{1it}}{\partial w_{2it}}\right)^{SG}. $$

Therefore, the total labor supply elasticity for job 1 with respect to w _{2i t} is the same as the estimated Stone-Geary uncompensated cross wage elasticity:

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}_{_{12it}}^{T}=\left( \frac{w_{2it}}{\hat{h}_{1it}}\right) \left( \frac{\partial \hat{h}_{1it}}{\partial w_{2it}}\right)^{T} \\ &=&\hat{\eta}_{_{12it}}, \end{array} $$

where $\hat {\eta }_{_{12it}}=\hat {\eta }{~}_{_{12it}}^{c}+\hat {\epsilon }_{12\text {I} }$, $\hat {\eta }_{_{12it}}^{c}$ is the compensated cross substitution effect elasticity, and $\hat {\epsilon }_{12\text {I} }$ is the cross wage income effect elasticity.

The total pure income effect of non-labor income on labor supply to job 1 is obtained as

$$\left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}}\right)^{T}=\left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}}\right)^{SG}+\left( \frac{\hat{\theta}_{11}}{w_{1it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}} {\partial I_{it}}\right) $$

where $\left (\frac {\partial \hat {h}_{1it}}{\partial I_{it}}\right )^{SG}$ is the Stone-Geary pure income effect for labor supply to job 1 and $\frac {\partial \hat {\lambda }_{1it}}{\partial I_{it}}=\left [ \frac {-\hat {\lambda }_{1it}\left (z_{1it}+\hat {\lambda }_{1it}\right )} {\phi \left (z_{1it}\right )} \right ] \left [ \frac {\partial \hat {P}_{1it}}{\partial I_{it}}\right ]$. Accordingly, the total labor supply pure income effect elasticity is obtained from

$$\begin{array}{@{}rcl@{}} && \hat{\eta}{_{_{1Iit}}^{T}}=\left( \frac{I_{it}}{\hat{h}_{1it}}\right) \left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}}\right)^{T} \\ & =&\hat{\eta}_{_{1Iit}}+\left( \frac{\hat{\theta}_{11}I_{it}}{w_{1it}\hat{h}_{1it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}}{\partial I_{it}} \right) \end{array} $$

where $\hat {\eta }_{_{1Iit}}$ is the Stone-Geary pure income effect elasticity for job 1. The term $\left (\frac {\hat {\theta }_{11}I_{it}}{w_{1it}\hat {h}_{1it}}\right ) \left (\frac {\partial \hat {\lambda }_{1it}}{\partial I_{it}}\right )$ captures the job 1 labor supply elasticity effects of I _{i t} on the probability of being an unconstrained dual job holder.

The implied hours equation for job 2 obtained from the estimated earnings (12) is given by

$$ \hat{h}_{2it}=\left( 1-\hat{\alpha}_{2}\right) \tilde{\gamma}_{2} +\ \hat{\alpha}_{2} \left( \frac{\tilde{\gamma}_{3}-\tilde{\gamma}_{1}w_{1it} -I_{it}}{w_{2it}}\right) +\hat{\theta}_{21}\frac{\hat{\lambda}_{1it}}{w_{2it}}+\frac{\bar{Z}_{1i} \hat{\pi}_{21}}{w_{2it}}. $$

(20)

Similarly as in job 1, we express the total labor supply effect of w _{2i t} on job 2 as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\right)^{SG}-\frac{\hat{\theta}_{21}\hat{\lambda}_{1it}}{\left( w_{2it}\right)^{2}}-\frac{\bar{Z}_{1i}\hat{\pi}_{21}}{\left( w_{2it}\right)^{2}}, $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial w_{2it}}\right )^{SG}$ is the slope of the uncompensated labor supply curve for job 2 obtained from the Stone-Geary utility function. Accordingly, the total job 2 labor supply elasticity with respect to w _{2i t} is obtained from

$$\begin{array}{@{}rcl@{}} & & \hat{\eta}{_{_{22it}}^{T}}=\left( \frac{w_{2it}}{\hat{h}_{2it}}\right) \left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\right)^{T} \\ & =&\hat{\eta}_{_{22it}}-\frac{\hat{\theta}_{21}\hat{\lambda}_{1it}}{w_{2it} \hat{h}_{2it}} -\frac{\bar{Z}_{1i}\hat{\pi}_{21}}{w_{2it}\hat{h}_{2it}} \end{array} $$

where $\hat {\eta }_{_{22it}}=\hat {\eta }{~}_{_{22it}}^{c}+\hat {\epsilon }_{22\text {I} }$ is the estimated Stone-Geary uncompensated job 2 labor supply elasticity with respect to w _{2i t} , $\hat {\eta }{_{_{22it}}^{c}}$ is the compensated labor supply elasticity with respect to w _{2i t} (evaluated at $\hat {h}_{2it}$) , and $\hat {\epsilon }_{22\text {I} }$ is the own wage income effect elasticity. Although w _{2i t} does not affect the probability of a worker being an unconstrained dual job holder, it does affect the impact of selection on job 2 labor supply via the term $-\frac {\hat {\theta }_{21}\hat {\lambda }_{1it}}{w_{2it}\hat {h}_{2it}}$. Similarly, w _{2i t} impacts the effect of unobserved heterogeneity on job 2 labor supply via the term $\frac {\bar {Z}_{1i}\hat {\pi }_{21}}{w_{2it}\hat {h}_{2it}}$.

The total cross-labor supply effect of w _{1i t} on job 2 labor supply is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\right)^{SG}+\left( \frac{ \hat{\theta}_{21}}{w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}} {\partial w_{1it}}\right) , $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial w_{1it}}\right )^{SG}$ is the cross-wage effect for the uncompensated job 2 labor supply curve obtained from the Stone-Geary utility function. Therefore, the total cross-labor supply elasticity of w _{1i t} on job 2 labor supply is obtained as

$$\begin{array}{@{}rcl@{}} \hat{\eta}{_{_{21it}}^{T}}& =&\left( \frac{w_{1it}}{\hat{h}_{2it}}\right) \left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\right)^{T} \\ & =&\hat{\eta}_{_{21it}}+\left( \frac{\hat{\theta}_{21}w_{1it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}}{\partial w_{1it}} \right) , \end{array} $$

where $\hat {\eta }_{_{21it}}=\hat {\eta }{{~}_{_{21it}}^{c}}+\hat {\epsilon }_{21\text {I} }$, $\hat {\eta }{_{_{21it}}^{c}}$ is the compensated cross substitution effect elasticity, and $\hat {\epsilon }_{21\text {I} }$ is the cross wage income effect elasticity. The term $\left (\frac {\hat {\theta }_{21}w_{1it}}{w_{2it}\hat {h}_{2it}}\right ) \left (\frac {\partial \hat {\lambda }_{1it}}{\partial w_{1it}}\right )$ captures the job 2 labor supply elasticity effects of w _{1i t} on the probability of being an unconstrained dual job holder.

The total pure income effect of non-labor income on labor supply to job 2 is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\right)^{SG}+\left( \frac{ \hat{\theta}_{21}}{w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}} {\partial I_{it}}\right) $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial I_{it}}\right )^{SG}$ is the Stone-Geary pure income effect for labor supply to job 2. Accordingly, the total labor supply pure income effect elasticity is obtained from

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}{{~}_{_{2Iit}}^{T}} =\left( \frac{I_{it}}{\hat{h}_{2it}}\right) \left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\right)^{T} \\ & =&\hat{\eta}{_{_{2Iit}}}+\left( \frac{\hat{\theta}_{21}I_{it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{1it}}{\partial I_{it}} \right) \end{array} $$

where $\hat {\eta }_{_{2Iit}}$ is the Stone-Geary pure income effect elasticity for job 2. The term $\left (\frac {\hat {\theta }_{21}I_{it}}{w_{2it}\hat {h}_{2it}}\right ) \left (\frac {\partial \hat {\lambda }_{1it}}{\partial I_{it}}\right )$ captures the job 2 labor supply elasticity effects of I _{i t} on the probability of being an unconstrained dual job holder.

Unconstrained unitary job holders

We express the total labor supply effect of w _{1i t} as

$$\left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}}\vert_{h_{2}= 0}\right)^{T}=\left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}}\vert_{h_{2}= 0}\right)^{SG}+\hat{\theta}_{12}\left[ \frac{1}{w_{1it}}\frac{\partial \hat{\lambda}_{2it}}{\partial w_{1it}}-\frac{\hat{\lambda}_{2it}}{\left( w_{1it}\right)^{2}}\right] -\frac{\bar{Z}_{2i}\hat{\pi}_{12}}{\left( w_{1it}\right)^{2}}, $$

where $\left (\frac {\partial \hat {h}_{1it}}{\partial w_{1it}}\vert _{h_{2}= 0}\right )^{SG}$ is the slope of the uncompensated labor supply curve obtained from the Stone-Geary utility function and $\frac {\partial \hat {\lambda }_{2it}} {\partial w_{1it}}=\left [ \frac {-\hat {\lambda }_{2it}\left (z_{2it}+\hat {\lambda }_{2it}\right )} {\phi \left (z_{2it}\right )} \right ] \left [\frac {\partial \hat {P}_{2it}}{\partial w_{1it}}\right ]$, $z_{2it}={\Phi }^{-1}\left (\hat {P}_{2it}\right )$, and $\hat {P}_{2it}={\Lambda }\left (x_{it},\bar {\omega }_{i},\tilde {\beta }_{2}\right )$. Accordingly, the total labor supply elasticity with respect to w _{1i t} is obtained from

$$\begin{array}{@{}rcl@{}} \hat{\eta}_{11it}^{T}\vert_{h_{2}= 0} & =&\left( \frac{w_{1it}}{ \hat{h}_{1it}}\right) \left( \frac{\partial \hat{h}_{1it}}{\partial w_{1it}} \vert_{h_{2}= 0}\right)^{T} \\ & =&\hat{\eta}_{_{11it}}\vert_{h_{2}= 0}+\hat{\theta}_{12}\left[ \frac{1}{\hat{h}_{1it}}\frac{\partial \hat{\lambda}_{2it}}{\partial w_{1it}}- \frac{\hat{\lambda}_{2it}}{w_{1it}\hat{h}_{1it}}\right] -\frac{\bar{Z}_{2i} \hat{\pi}_{12}}{w_{1it}\hat{h}_{1it}}, \end{array} $$

where $\hat {\eta }_{_{11it}}\vert _{h_{2}= 0}=\hat {\eta }{{~}_{_{11it}}^{c}}\vert _{h_{2}= 0} +\hat {\epsilon }_{11\text {I}} \vert _{h_{2}= 0}$ is the estimated Stone-Geary uncompensated labor supply elasticity with respect to w _{1i t} , $\hat {\eta }{{~}_{_{11it}}^{c}}\vert _{h_{2}= 0}$ is the compensated labor supply elasticity with respect to w _{1i t} (evaluated at $\hat {h}_{1it}$) , $\hat {\epsilon }_{11\text {I} }\vert _{h_{2}= 0}$ is the own wage income effect elasticity. The term $\hat {\theta }_{12}\left [ \frac {1}{\hat {h}_{1it}}\frac {\partial \hat {\lambda }_{2it}}{\partial w_{1it}}-\frac {\hat {\lambda }_{2it}}{w_{1it}\hat {h}_{1it}}\right ]$ captures the job 1 labor supply elasticity effects of w _{1i t} on the probability of being an unconstrained dual job holder. w _{1i t} impacts the effect of unobserved heterogeneity on job 1 labor supply via the term $-\frac {\bar {Z}_{2i}\hat {\pi }_{12}}{w_{1it}\hat {h}_{1it}}$.

The total pure income effect of non-labor income on labor supply to job 1 is obtained as

$$\left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}}\vert_{h_{2}= 0} \right)^{T}=\left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}}\vert_{h_{2}= 0} \right)^{SG}+\left( \frac{\hat{\theta}_{12}}{w_{1it}}\right) \left( \frac{\partial \hat{\lambda}_{2it}}{\partial I_{it}}\right) $$

where $\left (\frac {\partial \hat {h}_{1it}}{\partial I_{it}}\vert _{h_{2}= 0} \right )^{SG}$ is the Stone-Geary pure income effect for labor supply and $\frac {\partial \hat {\lambda }_{2it}} {\partial I_{it}}=\left [ \frac {-\hat {\lambda }_{2it}\left (z_{2it}+\hat {\lambda }_{2it}\right )} {\phi \left (z_{2it}\right )} \right ] \left [ \frac {\partial \hat {P}_{2it}}{\partial I_{it}}\right ]$. Accordingly, the total labor supply pure income effect elasticity is obtained from

$$\begin{array}{@{}rcl@{}} \hat{\eta}_{_{1Iit}}^{T}\left|{~}_{h_{2}= 0}\right. & =&\left( \frac{I_{it}} {\hat{h}_{1it}}\right) \left( \frac{\partial \hat{h}_{1it}}{\partial I_{it}} \left|{~}_{h_{2}= 0}\right. \right)^{T} \\ & =&\hat{\eta}_{_{1Iit}}\left|{~}_{h_{2}= 0}\right. +\left( \frac{\hat{\theta} _{12}I_{it}}{w_{1it}\hat{h}_{1it}}\right) \left( \frac{\partial \hat{\lambda} _{2it}}{\partial I_{it}}\right) \end{array} $$

where $\hat{\eta}_{_{1Iit}}\left\vert _{h_{2}=0}\right.$ is the Stone-Geary pure income effect elasticity. The term $\left (\frac {\hat {\theta }_{12}I_{it}}{w_{1it}\hat {h}_{1it}}\right ) \left (\frac {\partial \hat {\lambda }_{2it}}{\partial I_{it}}\right )$ captures the labor supply elasticity effects of I _{i t} on the probability of being an unconstrained unitary job holder.

Constrained dual job holders

For over-employed dual job holders:

We express the total labor supply effect of w _{2i t} on job 2 as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right) ^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right) ^{SG}-\frac{\hat{\theta}_{23}\hat{\lambda}_{23it}}{\left( w_{2it}\right) ^{2}}-\frac{\bar{Z}_{3i}\hat{\pi}_{23}}{\left( w_{2it}\right) ^{2}}, $$

where $\left( \dfrac{\partial \hat{h}_{2it}}{\partial w_{2it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right) ^{SG}$ is the slope of the uncompensated labor supply curve for job 2 obtained from the Stone-Geary utility function and $\frac {\partial \hat {\lambda }_{23it}} {\partial w_{1it}}=\left [\frac {-\hat {\lambda }_{23it}\left (z_{3it}+\hat {\lambda }_{3it}\right )}{\phi \left (z_{3it}\right )} \right ] \left [ \frac {\partial \hat {P}_{3it}}{\partial w_{1it}}\right ] ,z_{3it}={\Phi }^{-1}\left (\hat {P}_{3it}\right )$, and $\hat {P}_{3it}={\Lambda }\left (x_{it},\bar {\omega }_{i},\tilde {\beta }_{3}\right )$. Accordingly, the total job 2 labor supply elasticity with respect to w _{2i t} is obtained from

$$\begin{array}{@{}rcl@{}} \hat{\eta}_{_{22it}}^{T}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. & =&\left( \frac{w_{2it}}{\hat{h}_{2it}}\right) \left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right)^{T} \\ & =&\hat{\eta}_{_{22it}}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. -\frac{\hat{\theta}_{23}\hat{\lambda}_{23it}}{w_{2it}\hat{h}_{2it}}\ -\frac{\bar{Z}_{3i}\hat{\pi}_{23}}{w_{2it}\hat{h}_{2it}} \end{array} $$

where $\hat {\eta }_{_{22it}}\left |{~}_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast }}\right . =\hat {\eta }_{_{22it}}^{c}\left |{~}_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } }\right . +\hat {\epsilon }_{22\text {I} }\left |{~}_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } }\right .$ is the estimated Stone-Geary uncompensated job 2 labor supply elasticity with respect to w _{2i t} , $\hat {\eta }{{~}_{_{22it}}^{c}}_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast }}$ is the compensated labor supply elasticity with respect to w _{2i t} (evaluated at $\hat {h}_{2it}\left |{~}_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast }}\right .$), and $\hat {\epsilon }_{22\mathrm {I} }\left |_{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } }\right .$ is the own wage income effect elasticity. The effects of selection and unobserved heterogeneity on job 2 labor supply are affected by changes in w _{2i t} because the selection and unobserved heterogeneity terms in the job 2 labor supply function include w _{2i t} in the denominator. After differentiating job 2 labor supply with respect to w _{2i t} and converting to elasticities, the selection and unobserved heterogeneity effects are captured by $-\frac {\hat {\theta }_{23}\hat {\lambda }_{23it}}{w_{2it}\hat {h}_{2it}}\$and $-\frac {\bar {Z}_{3i}\hat {\pi }_{23}}{w_{2it}\hat {h}_{2it}}$, respectively.

Because hours are constrained for job 1, the wage rate for job 1 can only have income and selection effects on labor supply to job 2 but not substitution effects. The total cross-labor supply effect of w _{1i t} on job 2 labor supply is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right) ^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast}}\right. \right) ^{SG}+\left( \frac{\hat{\theta}_{23}} {w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{23it}}{\partial w_{1it}}\right) \ ,$$

where $\left( \dfrac{\partial \hat{h}_{2it}}{\partial w_{1it}}\left\vert _{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right) ^{SG}$ is the cross-wage effect for the uncompensated job 2 labor supply curve obtained from the Stone-Geary utility function. Therefore, the total cross-labor supply elasticity of w _{1i t} on job 2 labor supply is obtained as

$$\begin{array}{@{}rcl@{}} \hat{\eta}_{_{21it}}^{T}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. & =&\left( \frac{w_{1it}}{\hat{h}_{2it}}\right) \left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. \right)^{T} \\ & =&\hat{\eta}_{_{21it}}\left|{~}_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right. +\left( \frac{\hat{\theta}_{23}\ w_{1it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{23it}}{\partial w_{1it}}\right) , \end{array} $$

where $\hat {\eta }_{_{21it}}\vert _{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } } =\hat {\epsilon }_{21\text {I} }\vert _{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } }$ is the estimated Stone-Geary uncompensated job 2 cross-labor supply elasticity, and $\left (\frac {\hat {\theta }_{23}\ w_{1it}}{w_{2it}\hat {h}_{2it}}\right ) \left (\frac {\partial \hat {\lambda }_{23it}}{\partial w_{1it}}\right )$ is the job 2 labor supply elasticity effects of w _{1i t} on the probability of being a constrained over-employed dual job holder.

The total pure income effect of non-labor income on labor supply to job 2 is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} }\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast}}\right)^{SG}+\left( \frac{\hat{\theta}_{23}}{w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{23it}}{\partial I_{it}}\right) $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial I_{it}}\vert _{h_{1}=\ddot {h}_{1}>h_{1}^{\ast } }\right )^{SG}$ is the Stone-Geary pure income effect for labor supply to job 2 and $\frac {\partial \hat {\lambda }_{23it}} {\partial I_{it}}=\left [ \dfrac {-\hat {\lambda }_{23it}\left (z_{3it}+\hat {\lambda }_{3it}\right )} {\phi \left (z_{3it}\right )} \right ] \left [ \frac {\partial \hat {P}_{3it}} {\partial I_{it}}\right ]$. Accordingly, the total labor supply pure income effect elasticity is obtained from

$$\begin{array}{@{}rcl@{}} & & \hat{\eta}{{~}_{_{2Iit}}^{T}}\vert_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} } =\left( \frac{I_{it}}{\hat{h}_{2it}}\right) \left( \frac{ \partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast} } \right)^{T} \\ & =&\hat{\eta}_{_{2Iit}}\vert_{h_{1}=\ddot{h}_{1}>h_{1}^{\ast}} +\left( \frac{\hat{\theta}_{23}I_{it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{23it}}{\partial I_{it}}\right) \end{array} $$

where $\hat {\eta }_{_{2Iit}}\vert _{h_{1}=\ddot {h}_{1}>h_{1}^{\ast }}$ is the Stone-Geary pure income effect elasticity for job 2. The term $\left (\frac {\hat {\theta }_{23}I_{it}}{w_{2it}\hat {h}_{2it}}\right )\left (\frac {\partial \hat {\lambda }_{23it}}{\partial I_{it}}\right )$ captures the job 2 labor supply elasticity effects of I _{i t} on the probability of being a constrained over-employed dual job holder.

For under-employed dual job holders:

We express the total labor supply effect of w _{2i t} on job 2 as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\vert_{h_{1}= \ddot{h}_{1}<h_{1}^{\ast} }\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{2it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } \right)^{SG}-\frac{\hat{\theta}_{24}\hat{\lambda}_{24it}}{\left( w_{2it}\right)^{2}}-\frac{\bar{Z}_{4i}\hat{\pi}_{24}}{\left( w_{2it}\right)^{2}}, $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial w_{2it}} \vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } } \right )^{SG}$ is the slope of the uncompensated labor supply curve for job 2 obtained from the Stone-Geary utility function and $\frac {\partial \hat {\lambda }_{24it}} {\partial w_{1it}}=\left [ \dfrac {-\hat {\lambda }_{24it}\left (z_{4it}+\hat {\lambda }_{4it}\right )} {\phi \left (z_{4it}\right )} \right ] \left [ \frac {\partial \hat {P}_{4it}} {\partial w_{1it}}\right ] ,z_{4it}={\Phi }^{-1}\left (\hat {P}_{4it}\right )$, and $\hat {P}_{4it}={\Lambda }\left (x_{it},\bar {\omega }_{i},\tilde {\beta }_{4}\right )$. Accordingly, the total job 2 labor supply elasticity with respect to w _{2i t} is obtained from

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}{{~}_{_{22it}}^{T}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } =\left( \frac{w_{2it}}{\hat{h}_{2it}}\right) \left( \frac{ \partial \hat{h}_{2it}}{\partial w_{2it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} }\right)^{T} \\ & =&\hat{\eta}_{_{22it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } -\frac{\hat{\theta}_{24}\hat{\lambda}_{24it}}{w_{2it}\hat{h}_{2it}}\ -\frac{\bar{Z}_{4i}\hat{\pi}_{24}}{w_{2it}\hat{h}_{2it}} \end{array} $$

where $\hat {\eta }_{_{22it}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } } =\hat {\eta }{_{_{22it}}^{c}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } } +\hat {\epsilon }_{22\text {I} }\vert _{h_{1}= \ddot {h}_{1}<h_{1}^{\ast } }$ is the estimated Stone-Geary uncompensated job 2 labor supply elasticity with respect to w _{2i t} , $\hat {\eta }{_{_{22it}}^{c}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }$ is the compensated labor supply elasticity with respect to w _{2i t} (evaluated at $\hat {h}_{2it}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast }}$), and $\hat {\epsilon }_{22\text {I} }\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }$ is the own wage income effect elasticity. The effects of selection and unobserved heterogeneity on job 2 labor supply are affected by changes in w _{2i t} because the selection and unobserved heterogeneity terms in the job 2 labor supply function include w _{2i t} in the denominator. After differentiating job 2 labor supply with respect to w _{2i t} and converting to elasticities, the selection and unobserved heterogeneity effects are captured by $-\frac {\hat {\theta }_{24}\hat {\lambda }_{24it}}{w_{2it}\hat {h}_{2it}}$ and $-\frac {\bar {Z}_{4i}\hat {\pi }_{24}}{w_{2it}\hat {h}_{2it}}$, respectively.

Because hours are constrained for job 1, the wage rate for job 1 can only have income and selection effects on labor supply to job 2 but not substitution effects. The total cross-labor supply effect of w _{1i t} on job 2 labor supply is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\vert_{h_{1}= \ddot{h}_{1}<h_{1}^{\ast} }\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial w_{1it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } \right)^{SG}+\left( \frac{\hat{\theta}_{24}} {w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{24it}}{\partial w_{1it}}\right) \ , $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial w_{1it}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }\right )^{SG}$ is the cross-wage effect for the uncompensated job 2 labor supply curve obtained from the Stone-Geary utility function. Therefore, the total cross-labor supply elasticity of w _{1i t} on job 2 labor supply is obtained as

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}{{~}_{_{21it}}^{T}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } =\left( \frac{w_{1it}}{\hat{h}_{2it}}\right) \left( \frac{ \partial \hat{h}_{2it}}{\partial w_{1it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } \right)^{T} \\ & =&\hat{\eta}_{_{21it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } +\left( \frac{\hat{\theta}_{24}\ w_{1it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{24it}}{\partial w_{1it}}\right) , \end{array} $$

where $\hat {\eta }_{_{21it}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast }} =\hat {\epsilon }_{21\text {I} }\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }$ is the estimated Stone-Geary uncompensated job 2 cross-labor supply elasticity, and $\left (\frac {\hat {\theta }_{24}\ w_{1it}}{w_{2it}\hat {h}_{2it}}\right ) \left (\frac {\partial \hat {\lambda }_{24it}}{\partial w_{1it}}\right )$ is the job 2 labor supply elasticity effects of w _{1i t} on the probability of being a constrained under-employed dual job holder.

The total pure income effect of non-labor income on labor supply to job 2 is obtained as

$$\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}= \ddot{h}_{1}<h_{1}^{\ast} }\right)^{T}=\left( \frac{\partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } \right)^{SG}+\left( \frac{\hat{\theta}_{24}}{w_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{24it}}{\partial I_{it}}\right) $$

where $\left (\frac {\partial \hat {h}_{2it}}{\partial I_{it}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }\right )^{SG}$ is the Stone-Geary pure income effect for labor supply to job 2and $\frac {\partial \hat {\lambda }_{24it}} {\partial I_{it}}=\left [ \dfrac {-\hat {\lambda }_{24it}\left (z_{4it}+\hat {\lambda }_{4it}\right )} {\phi \left (z_{4it}\right )} \right ] \left [ \frac {\partial \hat {P}_{4it}} {\partial I_{it}}\right ]$. Accordingly, the total labor supply pure income effect elasticity is obtained from

$$\begin{array}{@{}rcl@{}} &&\hat{\eta}{{~}_{_{2Iit}}^{T}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } =\left( \frac{I_{it}}{\hat{h}_{2it}}\right) \left( \frac{ \partial \hat{h}_{2it}}{\partial I_{it}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} }\right)^{T} \\ & =&\hat{\eta}_{_{2Iit}}\vert_{h_{1}=\ddot{h}_{1}<h_{1}^{\ast} } +\left( \frac{\hat{\theta}_{24}I_{it}}{w_{2it}\hat{h}_{2it}}\right) \left( \frac{\partial \hat{\lambda}_{24it}}{\partial I_{it}}\right) \end{array} $$

where $\hat {\eta }_{_{2Iit}}\vert _{h_{1}=\ddot {h}_{1}<h_{1}^{\ast } }$ is the Stone-Geary pure income effect elasticity for job 2. The term $\left (\frac {\hat {\theta }_{24}I_{it}}{w_{2it}\hat {h}_{2it}}\right ) \left (\frac {\partial \hat {\lambda }_{24it}}{\partial I_{it}}\right )$ captures the job 2 labor supply elasticity effects of I _{i t} on the probability of being a constrained under-employed dual job holder.