## 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.

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## Notes

Standard wage regression control for most observable characteristics that can account for productivity differentials, i.e. human capita, experience, industry, occupation and unobserved heterogeeity.

Flabbi (2010b) explicitly addresses the issue of stable gender wage differentials over time by considering the role of discrimination. He concludes that the proportion of prejudiced employers in the USA drops from 70% in 1985 to 32% in 2005. Flabbi and Moro (2010) and Usui (2007) explicitly consider the importance of preferences towards more flexible jobs in shaping different labour market outcomes for men and women.

Bowlus and Grogan (2009) estimate an equilibrium search model on UK data, while Bartolucci (2009) uses data on productivity for German firms to disentangle more adequately the role of productivity and discrimination. His paper is the only example I found of structural estimation of search models to analyse gender or racial differences in continental Europe. See Sulis (2007) for comparison between structural and reduced form methods to analyse gender differentials.

Although this has not been done explicitly before, this intuition was already discussed by footnote 32, p. 1327 Bowlus and Eckstein (2002). In fact, they claim that mean earnings can identify the average of the productivity distribution in a model of heterogeneous firm productivity. In my model, average productivity is already estimated in the previous step; hence, first moments of the offer and earnings distributions are jointly used with differences in transition rates to estimate discrimination parameters.

To the best of my knowledge, no attempt has been made to jointly model firms’ heterogeneity, discrimination and on-the-job search. Bowlus (1997) estimates a model with discrete productive heterogeneity and on-the-job search but no discrimination.

Similarly, the equilibrium unemployment rate is obtained by equating flows into and out of this state and reads as \(\frac{\delta }{\delta +\lambda _{0}} .\)

This function is continuous and monotone. See Bontemps et al. (2000) for proofs regarding uniqueness and existence of the function. Notice that, given continuity of this function, the mapping from productivity to offered wages determines a continuous distribution for

*F*(*w*).Note also that assuming

*ex-ante*heterogeneity in ability across workers guarantees that the resulting distribution of wages has the correct shape.I offer some evidence on these issues using Italian Labour Force data, which does not have any information on wages, but it has standard information regarding labour mobility. Available data for a sample of the population aged 15–50 for the period 1993–1994 show that for men, the probability of staying in the same macro-sector is equal to 97% for private sector employees and 90% for public sector ones. For women, corresponding figures are 96% and 94%, respectively. Possible transitions to other states are discussed next.

I further discuss this important point when commenting on the results obtained for structural transition parameters.

The overall sample selection procedure is available upon request. Note that in the empirical section of the paper I present various sensitivity checks to control the robustness of my estimates to different definitions of the main variables of interest as unemployment, job-to-job transitions and reservation wages.

Flabbi (2010a) identifies the distribution of match-specific values of productivity using parametric assumptions on the observed wage distribution.

When productivity is homogeneous but differs between men and women, it is also possible to identify productivity parameters using the minimum and the maximum wage observed in the sample jointly with estimated transition parameters (see Van den Berg 1999).

In Sulis (2008), I use this estimation procedure to analyse regional labour market differentials. Results for transition parameters for men are also reported in that paper.

See also Postel-Vinay and Robin (2002) for similar choice on French data. In the next sections, I conduct some robustness analysis on the definition of job-to-job transitions.

I consider both left and right censored observations for those spells in progress in January 1985 and December 1996, which are the two bounds of the observation period.

Bontemps et al. (2000) also derive a closed form solution for the density of the productivity of firms that are active in the market equilibrium. See next sections for further details about the recoverability of the density of the productivity distribution.

This parameter suggests the possibility that arrival rates are influenced by preferences of employers towards workers’ types. This proportional factor 0 ≤

*z*≤ 1 is added to the model. If*z*= 0, prejudiced firms do not search for women, while if*z*= 1 (*d*= 0 and*γ*_{d}= 0) arrival rates are not influenced by discrimination.See “Appendix 1” for a detailed description of the estimation procedure.

As adequate data on wage offers are not available, this figure is a first necessary step to check the relationship between offers and earnings and before estimating frictional parameters.

However, increasing the bandwidth of the kernel estimator, the bimodality tends to disappear. Given a baseline value of 80, the second mode disappears at 200.

Further information is also hardly available in Labour Force Survey data. However, it is interesting to compare my results to those obtained from raw transition probabilities for a sample of the population aged 15–50 for the period 1993–1994 extracted from that dataset. Expected differences between men and women in transition rates are detected. In a given year, about 5% of employed men lose their jobs, half of those go to unemployment, while the rest goes out of the labour force; 24% of the unemployed find a job, while a similar proportion goes out of the labour force; finally, about 10% of those that were out of the labour force transit directly to employment. For women, the destruction rate is equal to 8%, 3% goes to unemployment and the rest moves to non-participation. The probability of finding a job is much lower (17% of unemployed find a job), while the probability of moving out of the labour force is very high (34%). Finally, less than 5% of women out of the labour force begin working in a given year.

Although the parameter estimates are reliable and make economic sense, the model can have difficulties in fitting the data, and an admissible underlying distribution of productivity could not exist. See next sections and Sulis (2008) for more discussion on this point.

Allowing for different search intensities can have interesting implications for the shape of the wage–productivity profile. This is left for future research.

Further details regarding the estimation procedure are in “Appendix 1”.

Technical details concerning the likelihood function and a brief summary of the identification strategy are in “Appendix 2”.

Using the highest and lowest wages observed in the sample is sensitive to data cleaning and trimming. After some sensitivity checks, I decide to use the minimum wage and the 99th percentile of the distribution of earnings. Estimates are not very different from the ones obtained using the average of the earnings distribution discussed above.

See “Appendix 3” for details regarding the methodology.

Note that estimates of productivity obtained using average earnings of men and women are

*P*_{M}= 11,110 and*P*_{W}= 10,191, respectively, resulting in a differential of 0.9 against 0.8 obtained using the minimum and maximum wage observed in the sample.Note that these estimates still respect the condition

*P*>*d*+*R*.It is important to note that these exercises do not take into account the equilibrium change in the reservation wage of workers.

Flabbi (2010a) shows that implementation of equal pay policies may not be possible in contexts in which the match-specific value of productivity is unknown to the policy maker.

Note that the disutility parameter is estimated lower than in previous contexts, while the proportion of discriminating firms is higher. This could be due to the fact that here the parameter

*z*is not considered. Note also that these are not directly comparable to estimates reported in panel C of Table 5 as productivity is estimated in a different way.Equations used to estimate these effects are taken from Bowlus and Eckstein (2002) and read as follows:

$$ F(w)=\left\{ \begin{tabular}{ll} $\frac{1+k_{\rm e}}{k_{\rm e}}-\left( \frac{1+k_{\rm e}}{k_{\rm e}}\right) \left( \frac{ P-(\theta \times d)-w}{P-(\theta \times d)-R}\right) ^{0.5}$ & if $R\leq w\leq w_{\rm hd}$ \\ $\frac{1+k_{\rm e}}{k_{\rm e}}-\left( \frac{1+k_{\rm e}\times (1-\gamma _{\rm d})}{k_{\rm e}} \right) \left( \frac{P-w}{P-w_{\rm hd}}\right) ^{0.5}$ & if $w_{\rm hd}\leq w\leq w_{\rm h}$ \end{tabular} \right. $$where

$$ \begin{array}{lll} w_{\rm hd} &=&P-(\theta d)-\left( \frac{1+k_{\rm e}\times (1-\gamma _{\rm d})}{1+k_{\rm e}} \right) ^{2}(P-(\theta d)-R) \\ w_{\rm h} &=&P-\left( \frac{1}{1+k_{\rm e}(1-\gamma _{\rm d})}\right) ^{2}(P-w_{\rm hd}), \end{array} $$and

*θ*= 0.36 is the proportion of women in the sample. It is important to stress that, in this setting, I do not consider the equilibrium effects of these policy changes on the reservation wage. To calculate those effects, an estimate of the value of leisure is needed. However, such estimates returned implausibly high negative values and are not reported. Hence, it is important to stress the fact that not considering these changes can miss an important channel of transmission that are left for future research.The ratio of arrival rates is calculated as an average between the arrival rate of offers when unemployed and the one when employed.

To ease exposition, censoring is not considered, but information on it was used in the estimation routines.

As for discrimination parameters, I solve the systems below using numerical methods with Maple. As mentioned above, it should made also clear that estimation methods used in this section do not take into account the counterfactual increasing density distributions generated by the model. However, as discussed in Bowlus and Eckstein (2002), the model is identified.

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## Acknowledgements

This paper is based on Chapter 2 of my Ph.D. dissertation at the University of Essex, UK. Thanks to Melvyn Coles and Amanda Gosling for the support and advice and to Barbara Petrongolo and Michele Belot for their comments. Particular thanks also to Dale Mortensen, Bruno Contini, the editor of this journal, Jim Albrecht, and two anonymous referees for comments and suggestions that substantially improved the paper. I also thank participants at the Georgetown University Econometrics Seminar; at the Labour Market Dynamic Growth Conference in Sandbjerg, Denmark; at the Labor Market Flows, Productivity and Wage Dynamics Workshop in Turin; at the Evolution of Inequalities in Italy Workshop in Rome; at the 3rd ICEEE Conference in Ancona and at a seminar at the University of Salerno for their comments and suggestions. Financial help from the Italian Ministry of Education, PRIN Project 2005132317 “L’evoluzione delle disuguaglianze nel mercato del lavoro in Italia tra cambiamento tecnologico e modifiche istituzionali” is gratefully acknowledged. Part of the revision process of this paper was conducted while I was visiting the Department of Economics at Georgetown University; I thank that institution and the academic staff for their hospitality and the very stimulating environment. Of course, I am solely responsible for all remaining errors and misinterpretations.

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## 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*.^{Footnote 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:

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

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.^{Footnote 40} The likelihood is obtained by multiplication of the probability distributions of the observables.

The probabilities of being in each state are

The duration of unemployment has an exponential distribution with parameter *λ*
_{u}, and the marginal distribution of *t*
_{0} is

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

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

where

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

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

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

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:

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.^{Footnote 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}:

Having estimated *λ*
_{u}, the unemployment rate identifies the job destruction rate *δ*:

while the proportion of jobs terminating into unemployment identifies the arrival rate of offers when employment *λ*
_{e}:

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

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.

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Sulis, G. Gender wage differentials in Italy: a structural estimation approach.
*J Popul Econ* **25**, 53–87 (2012). https://doi.org/10.1007/s00148-010-0351-7

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DOI: https://doi.org/10.1007/s00148-010-0351-7