Gender wage differentials in Italy: a structural estimation approach

Abstract

This paper studies gender wage differentials by providing a maximum likelihood structural estimation of the frictional parameters of an equilibrium search model with on-the-job search and firm heterogeneity. In a second step, I also consider the role of discrimination. Results indicate higher level of search frictions for women; this result is confirmed by various robustness checks and by different specification and estimation strategies. I also find that the resulting mapping from productivity to wages for men is highly non-linear, while for women it is almost linear. Search, productivity and discrimination play different roles in shaping the gender differential depending on the specification and estimation of the model.

Appendices

Appendix 1: Estimating discrimination parameters

In this Appendix, I briefly present the method used to estimate discrimination parameters and report equations used to predict average earnings as proposed by Bowlus and Eckstein (2002). After estimating transition and productivity parameters in the first step, the ratio between estimated arrival rates for women λ _W and men λ _M can be used to estimate z.³⁹ The two remaining parameters are estimated from mean wage offers and earnings, respectively. As in the rest of the paper, M is for men, W is for women, G(w) denotes the earnings distribution, F(w) is the wage offer distribution, E denotes expectations, P is productivity, R is the reservation wage, d is the disutility parameter, γ _d is the proportion of discriminating firms, while z is the search intensity parameter. Discrimination parameters are obtained by solving the system of three equations below to match moments in the data:

$$ \frac{\lambda _{\rm W}}{\lambda _{\rm M}}=(1-\gamma _{\rm d}+z\gamma _{\rm d}), \label{eq: link} $$

(13)

$$ \begin{array}{rll} E_{F(w)}^{\rm W}&=&\left( \frac{ (k_{\rm eW})^{z}(3+2(k_{\rm eW})^{z})P_{\rm W}+(3+3(k_{\rm eW})^{z}+((k_{\rm eW})^{z})^{2})R_{\rm W} }{3(1+(k_{\rm eW})^{z})^{2}}\right) \\ &&+\,\left( \frac{z(k_{\rm eW})\gamma _{\rm d}d(2(1+(k_{\rm eW})(1-\gamma _{\rm d}))+(zk_{\rm eW}\gamma _{\rm d})}{3(k_{\rm eW})^{z}(1+k_{\rm eW}(1-\gamma _{\rm d}))^{2}(1+(k_{\rm eW})^{z})^{2}}\right. \\ &&{\kern18pt} \left. -\frac{2(1+k_{\rm eW}(1-\gamma _{\rm d}))^{2}(1+(k_{\rm eW})^{z})^{2}}{ 3(k_{\rm eW})^{z}(1+k_{\rm eW}(1-\gamma _{\rm d}))^{2}(1+(k_{\rm eW})^{z})^{2}}\right) ,\label{eq: EFwW} \end{array} $$

(14)

$$ \begin{array}{lll} &&{\kern-6pt} E_{G(w)}^{\rm W}\\ &&=(1-\gamma _{\rm d})\frac{(1+(k_{\rm eW})^{z})}{(1-\gamma _{\rm d}+z\gamma _{\rm d})}\left( \frac{k_{\rm eW}(1-\gamma _{\rm d})P_{\rm W}}{(1+k_{\rm eW}(1-\gamma _{\rm d}))^{2}} +\frac{R_{\rm W}}{(1+(k_{\rm eW})^{z})^{2}}\right) \label{eq: EGwW} \\ &&\phantom{=}+\left( \frac{\gamma _{\rm d}z}{(1-\gamma _{\rm d}+z\gamma _{\rm d})(1+(k_{\rm eW})^{z})} \right) \left( \frac{(zk_{\rm eW}\gamma _{\rm d})(P_{\rm W}-d)}{(1+k_{\rm eW}(1-\gamma _{\rm d})) }+R_{\rm W}\right) \\ &&\phantom{=}+\left( \frac{(\gamma _{\rm d})(1-\gamma _{\rm d})zk_{\rm eW}(1+(k_{\rm eW})^{z})(2+2k_{\rm eW}(1-\gamma _{\rm d})+zk_{\rm eW}\gamma _{\rm d})(P_{\rm W}-d)}{(1-\gamma _{\rm d}+z\gamma _{\rm d})(1+(k_{\rm eW})^{z})^{2}(1+k_{\rm eW}(1-\gamma _{\rm d}))^{2}}\right) .\\ \end{array} $$

(15)

The two other relevant equations used to predict averages of the earnings and offer distribution for men in the wage decomposition exercise read as

$$ E_{F(w)}^{\rm M}=\frac{k_{\rm eM}(3+2k_{\rm eM})P_{\rm M}+(3+3k_{\rm eM}+(k_{\rm eM})^{2})R_{\rm M}}{ 3(1+k_{\rm eM})^{2}} \label{eq: EFwM} $$

(16)

$$ E_{G(w)}^{\rm M}=\frac{k_{\rm eM}P_{\rm M}+R_{\rm M}}{(1+k_{\rm eM})} \label{eq: EGwM} $$

(17)

Note that the estimates of discrimination parameters are obtained by numerically solving the system of Eqs. 13, 14 and 15 reported above; hence, they can be very sensitive to the choice of previously estimated transition and productivity parameters, as little variations in starting values could result in implausible results or non-existence of a solution. What is more, no standard errors for these estimates are available.

Appendix 2: Likelihood function with homogeneous productivity

In what follows, I briefly present the method used to estimate the transition and productivity parameters in the Burdett and Mortensen (1998) model with homogeneous workers and firms. The model is estimated using duration and wage data; however, it is not based on the nonparametric estimation of the wage distribution used in the estimation of the model with productive heterogeneity, and it has the counterfactual implication of increasing wage densities. Data requirement is very similar to that needed for the model with heterogeneous productivity: the duration of the spell of unemployment or employment, the wage earned in each state and the destination after the employment spell.⁴⁰ The likelihood is obtained by multiplication of the probability distributions of the observables.

The probabilities of being in each state are

$$ \Pr (\rm u) = \delta /(\delta +\lambda _{\rm u}), $$

(18a)

$$ \Pr (\rm e) = \lambda _{\rm u}/(\delta +\lambda _{\rm u}). $$

(18b)

The duration of unemployment has an exponential distribution with parameter λ _u, and the marginal distribution of t ₀ is

$$ f(t_{0})=\lambda _{\rm u}\exp (-\lambda _{\rm u}t_{0}). $$

(19)

The marginal distribution of wages of the re-employed is a drawing from the equilibrium wage distribution, and the density function is

$$ f(w)=\left[ \frac{\delta +\lambda _{\rm u}}{2\lambda _{\rm e}}\right] \left[ \left( P-w\right) (P-R)\right] ^{-\frac{1}{2}}. $$

(20)

The conditional distribution of the job length, conditional on the wage received, is

$$ f(t_{1}|w)=\left\{ \delta +\lambda _{\rm e}\left[ 1-F(w)\right] \right\} \exp \left\{ -(\delta +\lambda _{\rm e}\left[ 1-F(w)\right] t_{1})\right\} , $$

(21)

where

$$ F(w)=\frac{\delta +\lambda _{\rm e}}{\lambda _{\rm e}}\left[ 1-\left( \frac{P-w}{P-R} \right) ^{\frac{1}{2}}\right] . $$

(22)

The marginal distribution of wages of the employed is a drawing from the equilibrium earnings distribution, and the density function is

$$ g(w)=\frac{\delta (P-R)^{\frac{1}{2}}}{2\lambda _{\rm e}}(P-w)^{-\frac{3}{2}}. $$

(23)

Transition probabilities from employment to other states, conditional on the wage, read as follows

$$ \Pr ({\rm jtu}|w) = \frac{\delta }{\delta +\lambda _{\rm e}\left[ 1-F(w)\right] },$$

(24a)

$$ \Pr ({\rm jtj}|w) = \frac{\lambda _{\rm e}\left[ 1-F(w)\right] }{\delta +\lambda _{\rm e} \left[ 1-F(w)\right] }. $$

(24b)

After appropriately dealing with right and left censoring, the likelihood can be written as the multiplication of above terms

$$ \begin{array}{rll} L(\theta )&=&f(w)\lambda _{\rm u}\exp \left[ -\lambda _{\rm u}(t_{0})\right] g(w)\left[ \delta +\lambda _{\rm e}(1-F(w))\right] \\ &&\times \exp \left\{ -\left[ \delta +\lambda _{\rm e}(1-F(w_{1}))\right] (t_{1})\right\} \delta ^{v}\left[ \lambda _{\rm e}(1-F(w))\right] ^{1-v} \end{array} $$

(25)

where v is equal to 0 if the employment terminates in a voluntary quit and 1 if there is an involuntary layoff.

In this setting, the reservation wage and the productivity parameter are estimated as follows:

$$ \widehat{R}=\min (w), $$

(26)

$$ \widehat{P}=\left[ \frac{(1+k_{\rm e})^{2}}{(1+k_{\rm e})^{2}-1}\right] \max (w)- \left[ \frac{1}{(1+k_{\rm e})^{2}-1}\right] \min (w). $$

(27)

To simplify notation, k _e = λ _e/δ has been used to estimate the productivity parameter, where the latter is estimated using wage and transition data without using non-parametric estimates of earnings. However, note that using the minimum and the maximum observed wage is sensitive to data manipulation and trimming; hence, estimates of the productivity parameters have been conducted using the 99th percentile of the wage distribution after some sensitivity tests.

Appendix 3: Method of moments

In this last part of the Appendix, I follow computation procedures proposed by Bowlus and Eckstein (2002) to estimate δ, λ _u, λ _e and P by matching first moments observed in the data.⁴¹ The reservation wage R is assumed to be gender specific, and it is estimated as the lowest wage observed in the sample for men and women. The basic system of equations, based on the standard Burdett and Mortensen (1998) theoretical model, is described below.

Unemployment duration identifies the arrival rate of offers when unemployed λ _u:

$$ u_{\rm dur}=\frac{1}{\lambda _{\rm u}}. \label{eq: udur} $$

(28)

Having estimated λ _u, the unemployment rate identifies the job destruction rate δ:

$$ u_{\rm rate}=\frac{\delta }{\delta +\lambda _{\rm u}}, \label{eq: urate} $$

(29)

while the proportion of jobs terminating into unemployment identifies the arrival rate of offers when employment λ _e:

$${\rm jtu}=\frac{\lambda _{\rm e}}{(\delta +\lambda _{\rm e})\ln (1+\frac{\lambda _{\rm e}}{ \delta })}. \label{eq: jtu} $$

(30)

Finally, the average wage of the cross-section earnings distribution identifies average productivity:

$$ E_{G(w)}=\frac{\lambda _{\rm e}P+\delta R}{\lambda _{\rm e}+\delta }. \label{eq: EGw} $$

(31)

The above model is separately estimated for men and women; hence, all parameters are gender specific. Note that when discrimination is considered, the average of the earnings distribution E _G(w) is used to estimate discrimination parameters; hence, productivity can be estimated as in the model with homogeneous productivity discussed in “Appendix 2” using Eq. 27.