Beyond Wage Gap, Towards Job Quality Gap: The Role of Inter-Group Differences in Wages, Non-Wage Job Dimensions, and Preferences

Abstract

Wage is not the only thing people care about when assessing the quality of their jobs. Non-wage job dimensions, such as autonomy at work and work-life balance, are important as well. Nevertheless, there is vast literature comparing groups of employed people that focuses on the inter-group wage gaps only. We go beyond the wage gap by proposing a framework for analysing inter-group gaps in multidimensional job quality. Job quality is measured by the so-called equivalent wage, a measure combining wage and multiple non-wage job dimensions in accordance with preferences over jobs as combinations of job dimensions. We derive a decomposition of the inter-group equivalent wage gap into three components: (1) the standard wage gap, (2) the gap in non-wage dimensions, and (3) inter-group preference heterogeneity. In an illustrative empirical application, we focus on the gender gap for recent university graduates using survey data from 19 countries. Men’s equivalent wages are substantially higher than women’s, and the equivalent wage gaps are significantly larger than the wage gaps. This is because the non-wage job dimensions are on average to men's advantage, and the preference heterogeneity is such that men care about the non-wage dimensions less than women do, and thus suffer less from having the non-wage dimensions at levels below the perfect level. This type of decompositions broadens information about labour market inequalities available to policy makers, but it is up to them to decide which of the three components of the equivalent wage gap are normatively relevant for them and whether they should aim to eliminate them.

Appendix

1.1 Derivation of expression (10) in Sect. 2.4

As stated in Sect. 2.4, upon allowing for preferences to depend on $Z$, instead of $\varphi_{k}^{g}$ we now have $\varphi_{k}^{g} \left( Z \right)$. By the same token, in Eq. (5) in Sect. 2.3, $\varphi_{k}$ is replaced by $\varphi_{k} \left( Z \right)$. Starting from (5) so amended, we have:

$${\Delta }{\mathbb{E}}\left[ {\ln W^{*} } \right] = {\Delta }{\mathbb{E}}\left[ {\ln W} \right] + {\Delta }{\mathbb{E}}\left[ {\mathop \sum \limits_{k = 1}^{K} \varphi_{k} \left( Z \right)\left( {D_{k} - D_{k}^{r} } \right)} \right]$$

$$= {\Delta }{\mathbb{E}}\left[ {\ln W} \right] + \mathop \sum \limits_{k = 1}^{K} {\Delta }{\mathbb{E}}\left[ {\varphi_{k} \left( Z \right)\left( {D_{k} - D_{k}^{r} } \right)} \right]$$

$$= {\Delta }{\mathbb{E}}\left[ {\ln W} \right] + \mathop \sum \limits_{k = 1}^{K} {\Delta }\left( {{\mathbb{E}}\left[ {\varphi_{k} \left( Z \right)} \right]{\mathbb{E}}\left[ {D_{k} - D_{k}^{r} } \right] + {\text{Cov}}\left[ {\varphi_{k} \left( Z \right),D_{k} - D_{k}^{r} } \right]} \right)$$

$$= \Delta {\mathbb{E}}\left[ {\ln W} \right] + \mathop \sum \limits_{{k = 1}}^{K} \Delta \left( {{\mathbb{E}}\left[ {\varphi _{k} \left( Z \right)} \right]{\mathbb{E}}\left[ {D_{k} - D_{k}^{r} } \right]} \right) + \mathop \sum \limits_{{k = 1}}^{K} \Delta {\text{Cov}}\left[ {\varphi _{k} \left( Z \right),D_{k} } \right]$$

$$= \Delta {\mathbb{E}}\left[ {\ln W} \right] + \mathop \sum \limits_{{k = 1}}^{K} \overline{{\varphi _{k} \left( Z \right)}} \Delta {\mathbb{E}}\left[ {D_{k} } \right] + \mathop \sum \limits_{{k = 1}}^{K} \Delta{\mathbb{E}}\left[ {\varphi _{k} \left( Z \right)} \right]\overline{{{\mathbb{E}}\left[ {D_{k} - D_{k}^{r} } \right]}} + \mathop \sum \limits_{{k = 1}}^{K} \Delta Cov\left[ {\varphi _{k} \left( Z \right),D_{k} } \right]$$

The third equality holds because ${\mathbb{E}}\left[ {PQ} \right] = {\mathbb{E}}\left[ P \right]{\mathbb{E}}\left[ Q \right] + {\text{Cov}}\left[ {P,Q} \right]$ for any two random variables $P$ and $Q$. The last equality holds due to the Shapely value-based decomposition

$${\Delta }\left( {{\mathbb{E}}\left[ {\varphi_{k} \left( Z \right)} \right]{\mathbb{E}}\left[ {D_{k} - D_{k}^{r} } \right]} \right) = \mathop \sum \limits_{k = 1}^{K} \overline{{\varphi_{k} \left( Z \right)}} {\Delta }{\mathbb{E}}\left[ {D_{k} } \right] + \mathop \sum \limits_{k = 1}^{K} {\Delta }{\mathbb{E}}\left[ {\varphi_{k} \left( Z \right)} \right]\overline{{{\mathbb{E}}\left[ {D_{k} - D_{k}^{r} } \right]}}$$

1.2 Derivation of expression (12) in Sect. 2.4

Combining the definition of equivalent wage given by expression (1) with the Box-Cox utility function (12), and solving for $W^{*} \left( {\lambda_{W} } \right)$, yields the following expression

$$W^{*} \left( {\lambda_{W}^{g} } \right) = W\left( {\lambda_{W}^{g} } \right) + \mathop \sum \limits_{k = 1}^{K} \varphi_{k}^{g} \left( {D_{k} \left( {\lambda_{k}^{g} } \right) - D_{k}^{r} \left( {\lambda_{k}^{g} } \right)} \right)$$

Substituting

$$W^{*} \left( {\lambda_{W}^{g} } \right) \equiv \left( {\left( {W^{*} } \right)^{{\lambda_{W}^{g} }} - 1} \right)/\lambda_{W}^{g}$$

$$W\left( {\lambda_{W}^{g} } \right) \equiv \left( {W^{{\lambda_{W}^{g} }} - 1} \right)/\lambda_{W}^{g}$$

$$D_{k} \left( {\lambda_{k}^{g} } \right) \equiv \left( {\left( {D_{k} } \right)^{{\lambda_{k}^{g} }} - 1} \right)/\lambda_{k}^{g}$$

$$D_{k}^{r} \left( {\lambda_{k}^{g} } \right) \equiv \left( {\left( {D_{k}^{r} } \right)^{{\lambda_{k}^{g} }} - 1} \right)/\lambda_{k}^{g}$$

,

and solving for ${W}^{*}$, gives

$$W^{*} = \left\{ {W^{{\lambda_{W}^{g} }} + \lambda_{W}^{g} \mathop \sum \limits_{k = 1}^{K} \frac{{\varphi_{k}^{g} }}{{\lambda_{k}^{g} }}\left[ {\left( {D_{k} } \right)^{{\lambda_{k}^{g} }} - \left( {D_{k}^{r} } \right)^{{\lambda_{k}^{g} }} } \right]} \right\}^{{\frac{1}{{\lambda_{W}^{g} }}}}$$

Taking the natural logarithm of the above equation and averaging it over all members of group $g$ leads to

$${\mathbb{E}}^{g} \left[ {\ln W^{*} } \right] = \frac{1}{{\lambda_{W}^{g} }}{\mathbb{E}}^{g} \left[ {\ln \left\{ {W^{{\lambda_{W}^{g} }} + \lambda_{W}^{g} \mathop \sum \limits_{k = 1}^{K} \frac{{\varphi_{k}^{g} }}{{\lambda_{k}^{g} }}\left[ {\left( {D_{k} } \right)^{{\lambda_{k}^{g} }} - \left( {D_{k}^{r} } \right)^{{\lambda_{k}^{g} }} } \right]} \right\}} \right]$$

Finally, subtracting the above expression for group $B$ from that for group $A$, we get the EW gap as in expression (12) in Sect. 2.4:

$${\Delta }{\mathbb{E}}\left[ {\ln W^{*} } \right] = {\Delta }\left\{ {\frac{1}{{\lambda_{W} }}{\mathbb{E}}\left[ {\ln \left\{ {W^{{\lambda_{W} }} + \lambda_{W} \mathop \sum \limits_{k = 1}^{K} \frac{{\varphi_{k} }}{{\lambda_{k} }}\left[ {\left( {D_{k} } \right)^{{\lambda_{k} }} - \left( {D_{k}^{r} } \right)^{{\lambda_{k} }} } \right]} \right\}} \right]} \right\}$$