## 1. INTRODUCTION

Over the last few years, search-matching models have become the subject of renewed scrutiny. This spurt of interest is largely motivated by the difficulty to reconcile both micro and macro features of the data with the theory. Central to the issue is the difference between market productivity and the value of leisure. Costain and Reiter (2008) or Hagedorn and Manovskii (2008) show that, by setting it close enough to zero, one can ensure that unemployment and posted vacancies fluctuate as much in the Mortensen-Pissarides model as in the data. This solution, however, leads to a large discrepancy between predicted and observed wage inequality. Hornstein *et al.* (2007) illustrate this tension in a calibration exercise, finding that parameter values matching business cycle fluctuations greatly underestimate the degree of wage dispersion. Accordingly, recent research highlights the need to carefully analyze the micro predictions of the Mortensen–Pissarides framework.

This paper contributes to this project by estimating a search-matching model where workers accumulate job-specific skills through learning-by-doing (hereafter LBD). This feature allows us to control for the effect of job tenure on the wage distribution; hence it lowers the amount of residual wage dispersion and consequently the cost of being unemployed. We find that LBD reduces the gap between the value of non-market activity derived from our structural estimation and the one needed to fit business cycle fluctuations, but only marginally so. Our analysis therefore concurs with the findings in Hornstein *et al.* (2007).

The rate of LBD cannot be estimated in a purely deterministic set-up because the lower bound of the wage distribution increases with tenure when workers regularly progress along the learning curve. Given that some highly senior workers earn a wage close to their reservation wage, the estimated rate of LBD necessarily collapses to zero in a deterministic set-up. This is why one needs to introduce some noise into the learning process. A possibility is to allow for measurement errors. The drawback of this approach is that it ignores the interaction between job destruction and the rate of LBD. We assume instead that jobs are affected by idiosyncratic productivity shocks. The addition of this realistic feature allows us to endogenize job separations so that our structural model takes into account the positive influence of LBD on job retention.1

Our analytical framework is therefore closely related to the canonical Mortensen and Pissarides (1994) model (hereafter MP94) with endogenous separations. We assume that productivity and thus human capital are purely match-specific. We also assume that firms and workers cannot commit so that wages are set by Nash bargaining. We extend MP94 by letting initial productivities differ across jobs and by allowing workers to accumulate specific human capital. We also replace Poisson processes with Brownian motions in order to capture the high persistence of earnings shocks. Our model therefore builds on the framework described in Prat (2006). The contribution of this paper consists in structurally estimating the model on micro data. The analysis shows that the proposed extension of the MP94 model is able to match the joint distribution of wages and job spells, and identifies the parameter values required to achieve a good fit.

Given the considerable influence of the MP94 model, it may seem surprising that it has not yet been structurally estimated. A likely explanation is that doing so raises several challenges. First of all, endogenous separations greatly complicate the derivation of the likelihood function because one has to deduce all the sample paths that breach the reservation productivity. We show how this problem can be solved through the introduction of geometric Brownian motions. We therefore assume that log-wages follow a random walk with a deterministic trend. This specification accords well with the high persistence of earning shocks. Yet it substantially simplifies the canonical ARMA decomposition of the individual earnings process since it only retains its martingale component, disregarding transitory shocks.2 This simplification yields substantial analytical gains, allowing us to solve for equilibrium in closed-form, and thus to estimate the model by full-information methods.3 Hence, although earnings processes of the ARMA-type are more accurate, we propose geometric Brownian motions as a useful first-order approximation.

The other main technical difficulty is due to the non-standard properties of the likelihood function: the reservation wage and consequently the range of the data is a function of the estimated parameters. This peculiarity of job search models is well known (see, for example, Flinn and Heckman, 1982). In order to circumvent this problem, they proposed to evaluate the likelihood function in two steps. Unfortunately, we cannot use their methodology because endogenous job destruction implies that workers and firms separate as soon as the match productivity breaches the reservation threshold. As a result, the lowest reported wage is no longer a super-consistent estimator because the density at the reservation wage is equal to zero. We are nonetheless able to establish the asymptotic property of the estimators. We adapt to our set-up the proof by Greene (1980) that, in order to establish asymptotic normality, standard regularity conditions need not be satisfied when the likelihood function is equal to zero at the parameter-dependent boundary. Hence endogenous separation actually simplifies the analysis since it enables us to estimate the likelihood function as if it were standard.

After having derived the equilibrium of the economy and characterized the likelihood function, we estimate the model using data from the January 2004 supplement of the Current Population Survey. The supplement contains data on accepted wages and employment durations for a supposedly random sample of the US labor force. Its representative dimension accords well with the macro orientation of our model and especially with our focus on the aggregate wage distribution.

We do not estimate the model on panel data for the three following reasons. First, given the relatively stylized structure of the model, we prefer to restrict our analysis to cross-sectional patterns and leave a more thorough inspection of wage dynamics to extensions with general human capital and on-the-job search. Second, macro panel data for the US economy are not readily available. The relatively small size of the PSID makes it difficult to accurately estimate the wage distribution among job entrants,4 while linking households over time using the CPS data leads to a significant proportion of mismatches with potential self-selection issues.5 Lastly, deriving the likelihood function for cross-sectional data turns out to be a comprehensive task since we have to characterize the wage distribution conditional on job tenure. Thus our estimation procedure lays the ground for future empirical research on either panel or cross-sectional data.

We restrict our attention to workers without tertiary education because the estimates do not capture the accumulation of general human capital, which is known to be much more significant for skilled workers.6 The estimation procedure returns estimates for the rate of LBD of around 2% per year. We assess the ability of the model to fit the joint distribution of wages and job spells and find that it reproduces the data surprisingly well given its parsimonious specification. Then we use the estimates to characterize the impact of the rate of LBD. We show that it shifts to the right the wage distribution and significantly increases its dispersion.

### 1.1. Related Literature

To the best of our knowledge, this paper is the first to structurally estimate the MP94 model. This gap in the literature is explained by the fact that jobs' outputs follow stochastic paths in MP94, while estimable search models are typically based on the premise that productivities remain constant through time. As a result, early structural models7 generated flat wage profiles. Only recently has the empirical literature begun to address the observed pattern of wage dynamics.

An influential approach proposed by Postel-Vinay and Robin (2002) consider that workers can bring potential employers into Bertrand competition. Employer competition generates upward-sloping wage profiles because it enables workers to gradually appropriate the output of their jobs. Two recent papers build on this wage-setting rule and combine it with human capital accumulation. Bagger *et al.* (2006) assume that wages are defined as piece rate contracts whose values are determined using the sequential auction model of Postel-Vinay and Robin (2002). Yamaguchi (2006) augments the sequential auction framework with bargaining as in Cahuc *et al.* (2006).

We focus instead on the Nash bargaining rule prevailing in the standard theory of unemployment so that wages follow changes in productivity. In that respect, our approach is more closely related to the paper by Nagypál (2007), which studies a model with both LBD and learning about match quality. Her analysis aims at disentangling the contributions of these two mechanisms. Her focus is quite different from ours: whereas Nagypál (2007) analyzes in great detail the hazard rate of job separation and the wage profile, we put greater emphasis on the aggregate wage distribution.

The additional features included in these three papers greatly complicate the analysis. This is why they all rely on simulation techniques to estimate their structural models. To the contrary, the framework proposed in this paper can be solved analytically and estimated by maximum likelihood. This reflects our focus on the relatively stylized but nevertheless very influential MP94 model.

Lastly, this paper is naturally connected to the large body of empirical research using Mincer equations to evaluate the rate of LBD.8 On the one hand, our model is too stylized to contribute to the debate about the relative importance of job tenure versus experience since it does not include general human capital. On the other hand, our structural framework allows us to quantify the aggregate impact of LBD. We find that it has a significantly positive effect on aggregate output but a small effect on employment.

### 1.2. Structure of the Paper

The rest of the paper is organized as follows. Section 2 discusses the model set-up and characterizes the equilibrium. The econometric procedure and the asymptotic properties of the estimates are detailed in Section 3. Section 4 describes the data and discusses the estimation results. Section 5 assesses the robustness of the estimates. In section 6, we introduce an aggregate matching function to close the model and evaluate the impact of LBD on the equilibrium. Section 7 concludes and the Appendix contains the proofs of the propositions.