Wage shocks, household labor supply, and income instability

Abstract

Do married couples make joint labor supply decisions in response to each other’s wage shocks? The study on this question aids in understanding the link between the rising income instability and household insurance. Existing studies on household insurance either focus on consumption smoothing and take labor supply as a given, or only focus on wife’s labor responses to husband’s unemployment shocks. This article develops an intra-household insurance model that allows for insurance against permanent and transitory wage shocks from both partners. Estimation using the Survey of Income and Program Participation shows that wife increases labor supply in response to husband’s adverse wage shocks when both of them are working, and wife gets more nonlabor income when she is out of work. This intra-household insurance reduces earnings instability by about 2 to 9 %. These results suggest that joint labor supply decisions provide an extra smoothing effect on income instability.

Appendix

Derivation of Eq. 16

Based on the equation of Donni (2003) in his Section 3.2.3 and our functional form, a partial differential equation that identifies the nonparticipation set gives

$$ A_{1} K_{3} =K_{1} A_3 $$

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From Eq. 14, we have A ₁ = a ₁ + s _f b ₁, and A ₃ = a ₃ + s _f b ₃. From Eq. 15, we have K ₃ = k ₃ + q _f b ₃, and K ₁ = k ₁ + q _f b ₁. Plug these into the above Eq. 24 together with the expression of k ₃ and k ₁ derived from Eq. 9, and we have:

$$ \left( {a_{1} +s_{f} b_{1} } \right)\left( {\frac{a_{3} b_{8} }{\Delta }+q_{f} b_{3} } \right)=\left( {\frac{a_{1} b_{8} }{\Delta }+q_{f} b_{1} } \right)\left( {a_{3} +s_{f} b_{3} } \right) $$

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This can be further simplified to Eq. 16: $q_{f} =\frac {s_{f} b_{8} }{\Delta }$.

1.1 Derivation of likelihood function

This section derives log-likelihood function for male and female labor supply when both partners are working and female labor supply when husband is out of work. Assume that preference shocks $u_{it}^{f}$ and $u_{it}^{m}$ in labor supply function (Eq. 22) follow a joint normal distribution with zero mean and the following covariance matrix:

$$ \left( {{\begin{array}{*{20}c} {\sigma_f^2 } & {\rho \sigma_f \sigma_m } \\ {\rho \sigma_f \sigma_m } & {\sigma_m^2 } \\\end{array} }} \right) $$

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The log-likelihood function takes the following form:

$$ \begin{array}{rll} \mathcal{L}&=& \sum\limits_{i\in P} \log \mathcal{L}_i (h^f,h^m)+\mathop \sum\limits_{i\in fNP} \log \mathcal{L}_i (h^f)+\mathop \sum\limits_{i\in mNP} \log \mathcal{L}_i (h^m)\\&&+\sum\limits_{i\in fmNP} \log \mathcal{L}_i (h^f=0,h^m=0) \end{array} $$

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The likelihood function when both partners are working follows a joint normal distribution:

$$ \mathcal{L}_{i} =\frac{1}{\sigma_{f} \sigma_{m} }\varphi \left( {\frac{h^f-a\prime x}{\sigma_{f} },\frac{h^m-b\prime x}{\sigma_{m} },\rho } \right) $$

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where φ is the density function of standard normal distribution.

The likelihood function in the male nonparticipation set is different. First, based on Eq. 14, the error term in female labor supply function when husband does not work is now $u_{it}^{f} +s_{f} u_{it}^{m}$. This suggests that the upper left element in the covariance matrix is $\sigma _{f}^{2} +2s_{f} \rho \sigma _{f} \sigma _{m} +s_{f}^{2} \sigma _{m}^{2}$. The covariance between $(u_{it}^{f} +s_{f} u_{it}^{m} )$ and $u_{it}^{m}$ is $\rho \sigma _{f} \sigma _{m} +s_{f} \sigma _{m}^{2}$. Therefore, the covariance matrix becomes

$$ \left( {{\begin{array}{*{20}c} {\sigma_{f}^{2} +2s_{f} \rho \sigma_{f} \sigma_{m} +s_{f}^{2} \sigma_{m}^{2} } & {\rho \sigma_{f} \sigma_{m} +s_{f} \sigma_{m}^{2} } \\ {\rho \sigma_{f} \sigma_{m} +s_{f} \sigma_{m}^{2} } & {\sigma_{m}^{2} } \\ \end{array} }} \right) $$

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The likelihood in male nonparticipation set becomes

$$ \mathcal{L}_{i} =\frac{1}{\sigma_{\nu_{f} } }\varphi \left( {\frac{h^f-a\prime x-s_{f} b\prime x}{\sigma_{\nu_{f} } }} \right)\Phi \left( {\frac{-\frac{b\prime x}{\sigma_{m} }-r_{f} \frac{h^f-a\prime x-s_{f} b\prime x}{\sigma_{\nu_{f} } }}{\sqrt{1-r_{f}^{2} }}} \right) $$

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where Φ stands for the cumulative distribution function of standard normal distribution, $\sigma _{\nu _{f} }$ is the abbreviation for the square root of the upper left element in covariance matrix, and r _f is the correlation parameter in the nonparticipation set $\left ( {\frac {\rho \sigma _{f} +s_{f} \sigma _{m} }{\sigma _{\nu _{f} } }} \right )$.

Similarly, the likelihood in female nonparticipation set becomes

$$ \mathcal{L}_{i} =\frac{1}{\sigma_{\nu_{m} } }\varphi \left( {\frac{h^m-b\prime x-s_{m} a\prime x}{\sigma_{\nu_{m} } }} \right)\Phi \left( {\frac{-\frac{a\prime x}{\sigma_{f} }-r_{m} \frac{h^m-b\prime x-s_{m} a\prime x}{\sigma_{\upsilon m} }}{\sqrt{1-r_{m}^{2} }}} \right) $$

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and the likelihood in the set where neither the husband nor wife participates follows a bivariate normal distribution:

$$ \mathcal{L}_{i} =\Phi \left( {\frac{-a\prime x}{\sigma_{f} },\frac{-b\prime x}{\sigma_{m} },\rho } \right) $$

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1.2 Unitary model

In this section, we derive an alternative household decision model, the unitary model, which assumes that a household as a unit, rather than two individuals, makes decision and maximizes utility. The unitary model implies a different set of testable restrictions. In Section 5.4, we test the hypothesis of both collective model and unitary model. There are two restrictions imposed on the unitary model without corner solution: income pooling restriction and the Slutsky restrictions. The income pooling restriction assumes that household members pool income together, which fully insure themselves against all shocks. The other restriction is the Slutsky symmetry of the substitution matrix and positive semi-definiteness of the substitution matrix. The unitary model generates different testable restrictions from the collective model. The unitary model becomes

$$ {\begin{array}{*{20}c} {{\begin{array}{*{20}c} {\max } \\ {h_{it}^{f} ,h_{it}^{m} ,c_{it}^{m} ,c_{it}^{f} } \\ \end{array} } U\left( {1-h_{it}^{f} ,1-h_{it}^{m} ,c_{it}^{f} ,c_{it}^{m} } \right)} \\ {s.t. c_{it}^{f} +c_{it}^{m} \le w_{it}^{f} h_{it}^{f} +w_{it}^{m} h_{it}^{m} +y_{it} } \\ \end{array} } $$

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Labor supply functions will be derived the same as in Eq. 8. The Slutsky symmetry of compensated cross wage effects can be expressed as follows:

$$ S_{fm} =S_{mf} $$

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where $S_{ij} =\frac {\partial h^{i}}{\partial w^{j}}-h^{j}\frac {\partial h^{i}}{\partial y},i,j=f,m$

Based on the Slutsky symmetry (34) and the labor supply functions (Eq. 8), we derive the following testable restrictions:

$$ {\begin{array}{l} {a_{3} -a_{1} b_{0} =b_{2} -b_{1} a_{0} } \\ {a_{1} b_{2} =a_{2} b_{1} } \\ {a_{1} b_{3} =a_{3} b_{1} } \\ {a_{1} b_{4} =a_{4} b_{1} } \\ {a_{1} b_{5} =a_{5} b_{1} } \\ {a_{1} b_{6} =a_{6} b_{1} } \\ {a_{1} b_{7} =a_{7} b_{1} } \\ \end{array} } $$

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In addition, nonparticipation generates another set of restrictions for the unitary model. In the collective model, when the husband is out of work, his potential wage still affects the sharing rule, thus labor supply. In the unitary model, his potential wage no longer affects labor supply. This implies that the effect of male potential wage on female labor supply is zero, when the husband is out of work:

$$ A_{3} =0\Rightarrow a_{3} +s_{f} b_{3} =0 $$

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Similarly, when the wife is out of work, we have

$$ b_{3} +s_{m} a_{3} =0 $$

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1.3 Derivation of log earnings as a function of both partners’ wage shocks

In this section, we use Taylor expansions to derive earnings as a function of the partial response to both partners’ wage shocks. Based on the specification of the sharing rule in our model, the higher-order derivatives of labor supply with respect to wage shocks all become zeros. The first-order Taylor expansion for log male earnings is as follows:

$$\begin{array}{@{}rcl@{}} \log \left( {h^{m}\epsilon^{m}} \right)&=&f\left( {\text{log}\epsilon^{m},\text{log}\epsilon^{f}} \right)\notag \\ &=&\log \left( {h_{0}^{m} \epsilon_{0}^{m} } \right)+\frac{\partial \text{log}h^{m}\epsilon^{m}}{\partial \text{log}\epsilon^{m}}\left( {\text{log}\epsilon^m-\text{log}\epsilon_{0}^{m} } \right)\notag\\ &&+\frac{\partial \text{log}h^{m}\epsilon^{m}}{\partial \text{log}\epsilon^{f}}\left( {\text{log}\epsilon^f-\text{log}\epsilon_{0}^{f} } \right) \end{array} $$

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where 𝜖 ^f and 𝜖 ^m are transitory components of female and male unlogged wage, respectively; thus, log𝜖 ^m is equivalent to v ^m in Eq. 21. Take the variance of Eq. 38, and the constant term on the right-hand side drops out.

$$ \begin{array}{rll} var\left[ {\text{log}h^{m}\epsilon^{m}} \right]&=&0+var\left[ {\frac{\partial \text{log}h^{m}\epsilon^{m}}{\partial \text{log}\epsilon^{m}}\left( {log\epsilon^{m}} \right)} \right]+\text{var}\left[ {\frac{\partial \text{log}h^{m}\epsilon^{m}}{\partial \text{log}\epsilon^{f}}\left( {\text{log}\epsilon^{f}} \right)} \right] \\ &=&\left[ {\left( {\frac{\partial \text{log}h^{m}}{\partial \text{log}\epsilon^{m}}+1} \right)^2var\left( {\log \epsilon^{m}} \right)-\frac{\partial \text{log}h^{m}}{\partial \text{log}\epsilon^{f}}var\left( {\log \epsilon^{f}} \right)} \right]\\ &=&\left( {\beta_{1} -\beta_{2} k_{7} +1} \right)^2v^m-\left( {\beta_{2} k_{6} } \right)^2v^{f} \end{array}$$

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where $\frac {\partial \text {log}h^{m}}{\partial \text {log}\epsilon ^{m}}$ and $\frac {\partial \text {log}h^{m}}{\partial \text {log}\epsilon ^{f}}$ are directly derived from labor supply functions (Eq. 5) and sharing rule (Eq. 6). All items on the right-hand side can be computed using estimates from our model.

Similarly, the Taylor expansion of log household earnings becomes

$$ \begin{array} {rll}\log \left( {h^{m}\epsilon^m+h^{f}\epsilon^{f}} \right)&=&f\left( {\text{log}\epsilon^{m},\text{log}\epsilon^{f}} \right)=\log \left( {h_{0}^{m} \epsilon_{0}^{m} +h_{0}^{f} \epsilon_{0}^{f} } \right)\\ &&+\frac{\partial \text{log}(h^{m}\epsilon^m+h^{f}\epsilon^f)}{\partial log\epsilon^{m}}\left( {\text{log}\epsilon^m-\text{log}\epsilon_{0}^{m} } \right) \\ &&+\frac{\partial \text{log}(h^{m}\epsilon^m+h^{f}\epsilon^f)}{\partial \text{log}\epsilon^{f}}\left( {\text{log}\epsilon^f-\text{log}\epsilon_{0}^{f} } \right) \end{array}$$

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Taking the variance of Eq. 40, the variance of the constant terms becomes zero. Again substitute partial derivatives using the estimated coefficients from Eqs. 5 and 6, and the variance becomes

$${} \begin{array}{rll} var\!\left[\! {\log\! \left(\! {h^{m}\epsilon^{m}\!\,+\,h^{f}\!\epsilon^{f}} \!\right)}\! \right]\!&=\!var\left( \!{\frac{\partial \log (h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^f)}{\partial \log \epsilon^{m}}\log \epsilon^{m}}\! \right)\\ &\quad\!+var\!\left(\! {\frac{\partial \log (h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^f)}{\partial \log \epsilon^{f}}\log \epsilon^{f}}\! \right) \\ &=\!var\!\left[ \!{\frac{1}{h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^{f}}} \right.\!\left( \frac{\partial (h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^f)}{\partial \text{log}\epsilon^{m}}\text{log}\epsilon^{m}\right.\\ &\qquad\qquad\qquad\qquad\qquad+\!\left.\left. {\frac{\partial (h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^f)}{\partial \text{log}\epsilon^{f}}\text{log}\epsilon^{f}}\! \right)\! \right] \\ &=\!var\!\left[\! {\frac{1}{h^{m}\epsilon^{m}\!\,+\,h^{\!f}\!\epsilon^{\!f}}\!\left(\! {\frac{\partial \exp (\log h^{m}\!\epsilon^m)\,+\,\partial \exp (\log h^{\!f}\!\epsilon^{\!f}\!)}{\partial \log \epsilon^{m}}\!\log\! \epsilon^{m}} \right.} \right. \\ &\quad\left. {\left. {+\frac{\partial \exp (\log h^{m}\epsilon^m)\,+\,\partial \exp (\log h^{\!f}\epsilon^{\!f}\!)}{\partial \log \epsilon^{f}}\log \epsilon^{f}} \!\right)}\! \right] \\ &=\!var\!\left[\! {\frac{1}{h^{m}\epsilon^{m}\,+\,h^{f}\epsilon^{f}}\!\left(\! {h^{m}\!\epsilon^{m}\!\left( {\beta_{1} \,-\,\beta_{2} k_{7} \,+\,1} \right)\,+\,h^{f}\!\epsilon^{f}\!\alpha_{2} k_{7} }\! \right)\!\log \epsilon^{m}} \right. \\ &\quad+\left( {-h^{m}\epsilon^{m}\beta_{2} k_{6} +h^{f}\epsilon^{f}\left( {1+\alpha_{1} +\alpha_{2} k_{6} } \right)\log \epsilon^{f}} \right] \\ &=\!var\!\left[\! {\frac{1}{h^{m}\exp (v^m)\,+\,h^{f}\exp (v^f)}\!\left({\vphantom{v^{f}}} \!{h^{m}\exp \left( {v^{m}} \right)\!\left( {\beta_{1} \,-\,\beta_{2} k_{7} \,+\,1} \right)} \right.} \right.\\ &\left.\quad + \,h^{f}\exp \left( {v^{f}} \right)\alpha_{2} k_{7} \right)v^m+\left( -h^{m}\exp \left( {v^{m}} \right)\beta_{2} k_{6}\right.\\ &\left.\quad+h^{f}\text{exp}(v^f)(1+\alpha_{1} +\alpha_{2} k_{6} )v^{f} {\vphantom{\frac{1}{2}}}\right] \end{array} $$

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