Complementarities Between Native and Immigrant Workers in Italy by Sector

Abstract

This chapter investigates the existence of complementarities between immigrant and native workers across sectors in Italy and the effects on wages due to immigration during the period 2011–2016. The analysis is based on a production function framework, where the aggregate labor is the result of a nested-CES function, and workers are differentiated according to their educational attainment, job experience, and nationality. This approach allows the analysis to estimate the elasticity of substitution between immigrant and native workers with the same education-experience level by sector. The contribution is twofold. First, it provides an estimate of the elasticity of substitution between native and immigrant workers by sector. Second, by considering explicitly the different degrees of substitutability between immigrant and native workers, the analysis provides an estimate of the wage impact for the two groups of workers at sectoral level. We find noticeable differences in the elasticity parameters across sectors. Similarly, the wage impact of immigrant is remarkably different both across sectors and between immigrant and native workers.

Appendix

A. Intermediate Formulae to Get Eq. (14.5)

In the competitive equilibrium, in each sector and for each type of workers, firms equalize wages to the marginal productivity of labor. Accordingly, we compute the marginal product of a native worker for the specific education-experience group from Eq. (14.1), then we take the natural logarithm such that

$$\begin{aligned} ln\left( {w_{m,j,s,t}^{N} } \right) = & ln\left( {\alpha A_{s,t} k_{s,t}^{1 - \alpha } } \right) + \frac{1}{{\sigma_{ED} }}lnL_{s,t} + ln\theta_{j,s} \\ & - \left( {\frac{1}{{\sigma_{ED} }} - \frac{1}{{\sigma_{EX} }}} \right)lnL_{j,s,t} + ln\theta_{m,s} - \left( {\frac{1}{{\sigma_{EX} }} - \frac{1}{{\sigma_{IM} }}} \right)lnL_{m,j,s,t} \\ & + ln\theta_{N,m,j,s} - \frac{1}{{\sigma_{IM} }}lnL_{N,m,j,s,t} \\ \end{aligned}$$

(A.1)

where k_s,t = K_s,t/L_s,t. Similarly, for immigrant workers we obtain

$$\begin{aligned} ln\left( {w_{j,m,s,t}^{I} } \right) =\, & ln\left( {\alpha A_{s,t} k_{s,t}^{1 - \alpha } } \right) \\ & + \frac{1}{{\sigma_{ED} }}lnL_{t,s} + ln\theta_{j,s} - \left( {\frac{1}{{\sigma_{ED} }} - \frac{1}{{\sigma_{EX} }}} \right)lnL_{j,s,t} + ln\theta_{m,s} - \left( {\frac{1}{{\sigma_{EX} }} - \frac{1}{{\sigma_{IM} }}} \right)lnL_{m,j,s,t} \\ & + ln\theta_{I,m,j,s} - \frac{1}{{\sigma_{IM} }}lnL_{I,m,j,s,t} \\ \end{aligned}$$

(A.2)

Subtracting Eq. (A.2) from Eq. (A.1) brings to the following equation:

$$ln\left( {\frac{{w_{m,j,s,t}^{I} }}{{w_{m,j,s,t}^{N} }}} \right) = ln\left( {\frac{{\theta_{I,m,j,s} }}{{\theta_{N,m,j,s} }}} \right) - \frac{1}{{\sigma_{IM} }}ln\left( {\frac{{L_{I,m, j,s,t} }}{{L_{N,m,j,s,t} }}} \right)$$

(A.3)

Therefore, we can estimate $\sigma_{IM}$ from the following regression model:

$$ln\left( {\frac{{w_{m,j,s,t}^{I} }}{{w_{m,j,s,t}^{N} }}} \right) = \mu_{m,j,s} - \frac{1}{{\sigma_{IM} }}ln\left( {\frac{{L_{I,m, j,s,t} }}{{L_{N,m,j,s,t} }}} \right) + \varepsilon_{m,j,s, t}$$

(A.4)

B. Analytical Derivation of Eqs. (14.6) and (14.7)

By aggregating the profit-maximizing conditions of Eqs. (A.1) and (A.2), it is possible to obtain the following optimal condition for the labor supply of workers with the same education and experience at sector level:

$$\begin{aligned} ln\bar{w}_{m,j,s,t} = & ln\left( {\alpha A_{t,s}^{{\frac{1}{\alpha }}} k_{t,s}^{{\frac{1 - \alpha }{\alpha }}} } \right) + \frac{1}{{\sigma_{ED} }}ln\left( {L_{s,t} } \right) \\ & + ln\theta_{j,s,t} - \left( {\frac{1}{{\sigma_{ED} }} - \frac{1}{{\sigma_{EX} }}} \right)ln\left( {L_{j,s,t} } \right) + ln\theta_{m,j,s} \\ & - \frac{1}{{\sigma_{EX} }}ln\left( {L_{m,j,s,t} } \right) \\ \end{aligned}$$

(B.1)

The elasticity of substitution between workers with identical education and different experience levels, that is $\sigma_{EX }$, can be estimated using Eq. (14.6). In this regard, $\bar{w}_{m,j,s,t}$ equals to the average weighted wage paid to immigrant and native workers with same education and experience, where the weights are the share of hours worked by each group of workers. The related empirical model can be specified as follows:

$$ln\bar{w}_{m,j,s,t} = {\text{E}}_{s,t} + {\text{E}}_{j,s,t} + {\text{E}}_{m,j,s} - \frac{1}{{\sigma_{EX} }}ln\left( {L_{m,j,s,t} } \right) + e_{ m,j,s,t}$$

(B.2)

A similar approach is followed to estimate the elasticity of substitution between workers with different education levels. Accordingly, the optimizing relationship is

$$ln\bar{w}_{j,s,t} = ln\left( {\alpha A_{s,t}^{{\frac{1}{\alpha }}} k_{s,t}^{{\frac{1 - \alpha }{\alpha }}} } \right) + \frac{1}{{\sigma_{ED} }}ln\left( {L_{s,t} } \right) + ln\theta_{j,s,t} - \frac{1}{{\sigma_{ED} }}ln\left( {L_{j,s,t} } \right)$$

(B.3)

And the corresponding empirical model is specified as follows:

$$ln\bar{w}_{j,s,t} = {\text{E}}_{s,t} + {\text{E}}_{j,s,t} - \frac{1}{{\sigma_{ED} }}ln\left( {L_{j,s,t} } \right) + u_{ j,s,t}$$

(B.4)

where the sector by time common effects (${\text{\rm E}}_{s,t}$) control for the variation of $ln\left( {\alpha A_{t,s}^{{\frac{1}{\alpha }}} k_{t,s}^{{\frac{1 - \alpha }{\alpha }}} } \right) + \frac{1}{{\sigma_{ED} }}ln\left( {L_{s,t} } \right)$, the education by sector by time effects (${\text{\rm E}}_{j,s,t}$) control for the variation of $ln\theta_{j,s,t} - \left( {\frac{1}{{\sigma_{ED} }} - \frac{1}{{\sigma_{EX} }}} \right)ln\left( {L_{j,s,t} } \right)$, and the education by experience by sector fixed effects (${\text{\rm E}}_{m,j,s}$) control for the variation of $ln\theta_{m,j,s}$.