On-the-Job Search in a Matching Model with Heterogenous Jobs and Workers* by Juan J. Dolado^a Marcel Jansen^b Juan F. Jimeno DOCUMENTO DE TRABAJO 2003-21

September 2003

This paper has been presented as an Invited Lecture at the annual conference of the European Association of Labour Economics (EALE 2003), Seville, 18-21 september, 2003. We are very grateful to Maite Blazquez for excellent research assistance. We also wish to thank James Albrecht, Barbara Petrongolo, Robert Shimer, Manuel Santos and participants in seminars at UC3M, CORE, Essex, LSE, EEA-ESEM 2003 (Stockholm) and ESSLE 2003 (Ammersee) for helpful comments. The usual disclaimer applies.

a Universidad Carlos III de Madrid and CEPR.

b Universidad Carlos III de Madrid.

c Universidad de Alcalá, FEDEA and CEPR.

ISSN 1696-750X

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On-the-Job Search in a Matching Model with Heterogenous Jobs and Workers ¹

Juan J. Dolado^a, Marcel Jansen^b and Juan F. Jimeno^{c a}Universidad Carlos III de Madrid and CEPR. ^bUniversidad Carlos III de Madrid. ^cUniversidad de Alcalá, FEDEA and CEPR.

September 15, 2003

¹This paper has been presented as an Invited Lecture at the annual conference of the European Association of Labour Economics (EALE 2003), Seville, 18-21 september, 2003. We are very grateful to Maite Blazquez for excellent research assistance. We also wish to thank James Albrecht, Barbara Petrongolo, Robert Shimer, Manuel Santos and participants in seminars at UC3M, CORE, Essex, LSE, EEA-ESEM 2003 (Stockholm) and ESSLE 2003 (Ammersee) for helpful comments. The usual disclaimer applies.

Abstract

This paper considers a matching model with heterogenous jobs (unskilled and skilled) and workers ( low and high- educated) which allows for on-thejob search by mismatched workers. The latter are high-educated workers who transitorily accept unskilled jobs and continue to search for skilled jobs. Our findings show that on-the-job search introduces an additional source of between and within-group wage inequality. Furthermore, the higher quit rate of mismatched workers exerts a negative externality on unskilled jobs and weakens the labour market position of low-educated workers. This last feature changes the efects of skill-biased technological change and it alters the response of the labour market to shifts in the skill distribution.

Keywords: job search, skills, unemployment, wage inequality JEL codes: J63, J64

1 Introduction

This paper develops a matching model with two-sided heterogeneity that allows for skill mismatch and introduces the novelty of on-the-job search. Firms have the choice between creating two types of jobs (skilled and unskilled) and the labour force consists of two kinds of workers (high-educated and low-educated).¹ High-educated workers have the advantage that they can perform both jobs, while low-educated workers can only be employed in unskilled jobs. Furthermore, in contrast to most of the literature on matching models with heterogenous agents, we allow for on-the-job search by higheducated workers performing unskilled jobs. As a result, high-educated workers continue to accept unskilled jobs even when their productivity on skilled jobs is much higher. Moreover, given that high-educated workers will quit these jobs at a higher rate than low-educated workers, on-the-job search exerts a negative externality on firms with unskilled jobs, and this new channel tends to weaken the labour market position of the low-educated.

The job competition between mismatched workers and properly matched ones in the lower segment of the labour market has interesting implications corcerning the issues of “overeducation” and “crowding-out” of low-educated workers by high-educated workers when both apply for the unskilled jobs.² As recently highlighted by Eurostat (2003), these phenomena are particularly relevant in some EU countries like the South Mediterranean ones.³

¹For notational convenience, we associate workers’ skills to their educational attainments. Moreover, the latter are exogenously given. Thus, any imperfect correlation that there might exist between education and skills is ignored in the paper.

²One of the first models of job competition and crowding-out can be found in Thurow (1975). In that model, the marginal product of labour is associated with the characteristics of jobs rather than of individuals. Hence, when there is a fixed amount of jobs and an excess of high-educated workers, low-educated workers get “crowded-out” from their traditional jobs and their unemployment rate will increase. Moreover, if the educational drive is sufiently strong, despite being able to take unskilled jobs, the unemployment rate of higheducated workers may go up as well. A basic problem with this approach is that, by assuming a fixed supply of jobs, the supply and composition of vacancies does not adjust to the composition of workers.

³Using a slight variant of the “objective” procedure proposed by Verdugo and Verdugo (1989) to measure overeducation, whereby overeducated workers are those whose years of education are more than one standard deviation above the average years of education in the socioeconomic group where they belong, Oliver and Raymond (2003) show that the proportion of overeducated workers with a college degree in Spain (Diplomados and Licenciados) increased from 14% in 1985 to 21% in 1998. During this period, the share of

These countries have experienced an intense upgrading of tertiary education over the last fifteen years and this shift in the distribution of educational attainments seems to have outpaced the growth of skilled jobs, giving rise to shares between 15% and 25% of skill-mismatch at the highest educational level (ISCED 5-6 in terms of the educational categories of Eurostat). In this respect, Figure 1a illustrates this fact by showing that there is a positive correlation between the the intensity of the tertiary educational upgrading in thirteen EU countries over the last two decades and the degree of “over-education”.⁴ Likewise, Figure 1b shows that overeducated workers have higher job mobility, as illustrated by the positive correlation between the degree of “over-education” and the percentage- point diferences in the proportions of mismatched and correctly matched workers who are searching for another job. Both stylized facts constitute the main motivations for this paper.

The literature on matching models with heterogeneous agents is fairly recent, dating back to the influential contributions by Acemoglu (1999) and Mortensen and Pissarides (1999).⁵ Assuming random search and a constant contact rate between unemployed workers and vacancies, Acemoglu (1999) ofers a theory of how a pooling equilibrium (defined as a situation in which firms only create “middling” jobs in order to avoid screening costs) is replaced by a separating equilibrium when either the proportion of skilled workers or the skill diferential is high enough. In the new equilibrium firms find it profitable to create skilled and unskilled jobs, and search for the appropriate candidates. This leads endogenously to changes in the unemployment rates of both types of workers and to an increase in between-group wage dispersion. Mortensen and Pissarides (1999), in turn, study a model with directed search and endogeneous job destruction. In their model there is a perfect match between workers’ skills and firms’ skill requirements, and between and withingroup wage dispersion arise from shocks to the productivities of the two types of jobs. Hence, although these papers set up the foundations of the role of search frictions in models with heterogeneity, they do not deal with the spillover efects of high-educated workers onto the creation and filling of unskilled job vacancies.

the Spanish population (25-64) with a tertiary educational attainment raised from 15% to 23%. Alba-Ramírez (1993) and Dolado et al. (2000) have documented some of the stylised facts of overeducation and crowding-out in Spain since the mid-1980s.

⁴The data for Figures 1 a and b come from an ad hoc module carried out by Eurostat in the EU LFS 2000 designed to collect specific information on the transition from school to working life in EU countries. A job mismatch is defined as a job outside the field of education of school leavers which is considered to have lower skill requirements than those corresponding to the workers’ educational attainments. The educational upgrading is measured by the ratio of the fraction of the population aged 25-34 with tertiary education and the corresponding proportion of the population aged 45-54. Data correspond to 1999 (see Education at the Glance, OECD 2001).

⁵Other relevant contributions in this line of research are Burgess (1993), Pissarides (1994) and McKenna (1996).

Figure 1a: Relationship between “educational upgrading” and mismatch

Figure 1b: Relationship between “over-education” and job search intensity

More recently, however, contributions by Gautier (2002) and Albrecht and Vroman (2002) have started to address directly this issue. In Gautier (2002), badly matched workers (high-skilled workers in simple jobs) continue to search on the job. Thus, his model is similar to ours, yet with two notable exceptions. First, rather than assuming that wages are determined by Nash bargaining, he adopts the simplifying assumption that firms and workers share their output in some fixed proportion. As a result, wages are unafected by the overall labour market conditions. Second, in his model, high-skilled workers may be more productive on simple jobs than low-skilled workers. This second assumption implies that low-skilled workers may benefit from job competition by high-skilled workers if the latter are suficiently more productive on unskilled jobs.⁶

Although there is anecdotal evidence about simple tasks that could be better performed by low-skilled workers (say, hamburger flipping or garbage collection) and viceversa (say, some clerical or services jobs which may require fluency in a foreign language), on average it seems sensible to assume that both types of workers are equally productive on simple jobs.⁷ This is precisely the assumption made by Albrecht and Vroman (2002) (AV, henceforth), who analyse a model with endogenous skill requirements. They show that the labour market equilibrium may switch from one in which high-skilled workers match with both types of jobs (a cross-skill matching equilibrium) to another where they refuse unskilled jobs (an ex-post segmentation equilibrium). Like in Acemoglu (1999), when either the productivity diferential in the two jobs or the proportion of high-educated workers in the population is suficiently large, the equilibrium switches from the first to the second type with important implications for unemployment and wage dispersion.

In our paper, we adopt the basic structure of their model (skill diferences across workers and job requirements, Nash bargaining approach and undirected search) but we introduce the novel feature of allowing for on-the-job search by mismatched workers. By incorporating on-the-job search, we show that the properties of this type of models change considerably.

First, with on-the-job search, the equilibrium always exhibits cross-skill matching. The key diference with AV’s model is that, by allowing mismatched workers to perform on-the-job search, they retain the option value from employment in a skilled job. It is therefore optimal for high-educated workers to accept an unskilled job if it pays more than their income during unemployment.

⁶A third, less relevant, diference with our model is the assumption that simple and complex jobs are ofered in diferent markets. Hence, unlike our model, high-skilled workers congest the market for low-skilled workers but not the other way around.

⁷Another relevant aspect which is not considered here is that of internal promotion, whereby “over-educated” workers may accept unskilled jobs as “stepping stones” for better jobs within the firm in the future (see, e.g. Brunello, 1996).

Second, on-the-job search introduces an interesting, additional source of within-group wage inequality. In AV’s model, high-educated workers earn diferent wages on the two types of jobs, but they always earn a higher wage than low-educated workers. In our model, by contrast, over-qualified workers end up earning a lower wage than low-educated workers, despite having a higher outside option value. The explanation for this result lies in their higher quit rate. On-the-job search reduces the surplus value of jobs performed by mismatched workers and as a result they earn a lower wage than those low-educated workers who are correctly matched. The empirical evidence on this last issue is mixed. For instance, on the one hand, there is some evidence for the Netherlands (see, e.g. Hartog and Oosterbeek, 1988, and Gautier et al., 2002, ) and for the U.S. (see, e.g., Sicherman, 1991) that over-educated workers earn more than correctly allocated workers but, on the other hand, Groot (1993, 1996), Verdugo and Verdugo (1989) and Alba-Ramírez and Blázquez (2003) obtain the opposite result for the U.K, the U.S and Spain, respectively.⁸ Furthermore, the fact that this type of workers are being hired for simple jobs may be an indication of the existence of unobserved characteristics that make them more productive, as in Gautier (2002). Our model excludes this type of non-observable heterogeneity.

Third, in this respect, one of the striking implications of our model is that on-the-job search provides a novel mechanism to explain the widening of the wage distribution which has been observed notably in the US but also in many OECD countries since the 1980s, despite the rise in the relative supply of college graduates. In efect, even abstracting from the efects of a secular increase in the demand for skills induced by “skill-biased” technical change (e.g, Autor et al., 1998) or in trade with LDCs (e.g. Borjas and Ramey, 1995), or a combination of both phenomena (e.g., Acemoglu, 2003), “skillupgrading” gives rise to on-the-job search which, in turn, would produce an increase in both the average wage premium and within-group wage inequality for workers with high educational attainments. This is so because a larger supply of high-educated workers end up being mismatched in unskilled jobs and since they are paid less than correctly-matched workers, both measures of wage inequality increase.

⁸For example, in the influential study of Verdugo and Verdugo (1989) it is found that, while the wage return to an extra year of required education is 6.2%, the return to a year of overeducation is -8.0% in the U.S.

Fourth, while “skill-biased ” technological change (an increase in the productivity of skilled jobs) always increases the unemployment rate of loweducated workers in models without on-the job search, this is not necessarily the case in our framework. In general, when the cost of opening both types of jobs is the same, we find that the efect is ambiguous although for reasonable parameter values the tradional result holds. However, if skilled jobs are relatively scarce to start with , say, because the cost of opening skill vacancies is higher than the cost of opening unskilled vacancies, then the increase in the supply of skilled jobs due to “skill-biased” technological progress will lead to relatively more high-educated workers in skilled jobs and to a lower share of over-qualified job seekers. Given that firms opening unskilled job vacancies prefer to match with low-educated workers rather than with high-educated ones (who are equally productive but who with a higher quit probability), a higher productivity in skilled jobs may therefore help those firms in finding stable and appropriately-qualified workers, leading to a decrease of the unemployment rate of low-educated workers.Likewise, for a given productivity diferential, the unemployment rate of low-educated workers will be lower the smaller is the (exogenous) job-destruction rate of skilled jobs relative to that of unskilled jobs. The explanation is again that the higher job tenure of adequately matched high-educated workers weakens the negatively externality that mismatched workers impose on firms creating unskilled vacancies. These properties suggest that there may be scope for diferentiated labour market policies that reduce the turnover of high-educated workers. A full treatment of this issue is beyond the scope of this paper. However, it is easy to construct examples in which the combined efect of a reduction in the separation rate of skilled jobs and an increase in the cost of those jobs (an approximation of the efects of firing costs on skilled jobs) reduces the unemployment rate for both types of workers.

The above results contrasts with the ones obtained from of an increase in the share of high-educated workers in the population. In this case, firms will also open more skilled jobs but, due to the “skill-upgrading”, mismatched workers represent a larger fraction of job seekers for unskilled vacancies, leading to lower profits for firms creating those jobs and a higher unemployment rate among the low-educated.

Finally, as discussed above, both AV’s and our model provide a useful analytical set-up to analyse the consequences of “overeducation” and “crowding-out”. Matching models with and without on-the-job search share the prediction that “skill- upgrading” will increase the unemployment rate of less-educated workers. However, while the unemployment rate of higheducated workers is invariant to changes in the skill distribution in AV´s model, we tend to find a negative correlation between the unemployment rate and the cohort size of high-educated workers. Hence, although a larger number of high-educated workers may end up accepting unskilled jobs, these workers do experience an improvement in their labour market position due to both the increase in the exit rate out of unemployment and the higher share of skilled job vacancies.⁹ Thus, despite the sometimes popular view that “over-education”, under on a fixed supply of jobs like in Thurow (1975), may lead to an increase in the unemployment rates of both low and high-educated workers, our model yields the opposite result for the latter.

The plan for the rest of the paper is as follows. Section 2 presents the main characteristics of the model while section 3 discusses the properties of the steady-state equilibrium in terms of its existence, uniqueness, and the implications for the wage distribution. Section 4 focusses on the comparative statics stemming from two important changes in the parameters of the model: (i) an increase in the proportion of high-educated workers reflecting the “skillupgrading” that has taken place in many countries, and (ii) an increase in the productivity of skilled jobs reflecting “skill-biased” technological progress. Section 5 discusses a few simulations of the proposed model which shed light on the size of the efects derived in the analytical sections. Finally, Section 6 concludes. Proofs of the main propositions and a slight generalization of the model entailing diferent separation rates are gathered in Appendices A and B, respectively.

⁹According to our knowledge the only other paper that obtains this type of cohort-size efects is Shimer (2001). He analyses a model with young and old workers, where young workers change jobs more frequently. As a result, an inflow of young workers will be matched with a creation of additional jobs and this reduces the cyclical unemployment of both the young and old cohorts in the labour force.

2 The Model

2.1 Main assumptions

We consider an economy populated by a continuum of risk-neutral workers with measure normalised to unity. An exogenous fraction of the workers is low-educated (l) while the remaining fraction, , is higheducated (h). All workers are infinitely-lived and time is continuous.

There are two types of jobs: skilled jobs (s), and unskilled jobs (n). Unskilled jobs can be performed by both types of workers, while a skilled job requires a high-educated worker. Furthermore, we assume that both types of workers are equally productive on unskilled jobs, while high-educated workers are more productive when matched to a skilled job. Formally, let denote the flow output of a job of type i that is filled by a worker of type . Our assumptions on the production technology can then be summarised as follows

\[y (s, h) = y (s) > y (n, h) = y (n, l) = y (n) > y (s, l) = 0.\]

For convenience, we assume that firms can open at most one job. The choice of the type of job is irreversible and the mass of each type of job is determined by a free-entry condition. Finally, job destruction is exogeneous and follows a Poisson process with arrival rate s that is common to both jobs.¹⁰ Whenever a job is destroyed, the worker becomes unemployed while the job becomes vacant. During unemployment workers receive a flow income which is to be interpreted as home production or leisure. The flow cost of maintaining a vacant job is denoted by c and is common to both types of jobs.

2.2 Matching

The labour market is characterised by matching frictions and on-the-job search is allowed for mismatched workers, namely, high-educated workers in unskilled jobs. As in AV, the meeting process between workers and vacancies is assumed to be undirected. This assumption allows us to capture the idea that, given the overall labour market conditions, low-educated workers are better of the greater the fraction of unskilled vacancies, and firms ofering unskilled vacancies benefit from a greater fraction of low-educated job seekers. A similar situation applies to both high-educated job seekers and firms opening skilled jobs. Specifically, the total number of matches between a worker and a firm is determined by a constant returns to scale matching function

¹⁰Despite the common rate of job destruction rate, s, the efective job separation rate of unskilled jobs will be partially endogenous due to the on-the-job search by mismatched workers. On average workers in skilled jobs will therefore enjoy more stable employment relationships than workers on unskilled jobs, in line with the available empirical evidence.

\[m [ v (n) + v (s), u (l) + u (h) + e (n, h) ],\]

where is the mass of unemployed workers of type denotes the mass of vacancies of and is the mass of high educated workers performing unskilled jobs. The total mass of unemployed workers is denoted as u while and denote the unemployment rates of low e eand high-educated workers, respectively. We assume that is strictly increasing in both arguments and denote the “labour market tightness” by

Since the mass of job seekers includes the mass of “mismatched” workers, , in the analysis of the transitions between job vacancies and workers it is important to distinguish beween the shares of and in the pool of unemployed and in the pool of job seekers. For this purpose, we define the following two shares: which denotes the proportion of low-educated unemployed workers in the mass of unemployed, and (ii) which represents the fraction of all unemployed workers in the mass of job seekers. With this notation, the share of low-educated unemployed workers in the total mass of job seekers becomes φψ . Likewise, the shares of high-educated unemployed workers in the mass of unemployed and in the mass of job seekers are and , respectively.

For the sake of realism we assume in the sequel that mismatched workers only change employer if they find a skilled job. Accordingly, the rate at which firms meet a job-seeker is equal to , but some unskilled jobs will meet a mismatched worker who will refuse to match. The efective matching rate of an unskilled job with a low-educated worker is therefore , while the corresponding rate with a high-educated worker is Similarly, skilled jobs may meet a low-educated worker who is not qualified for the job. Thus, the matching rate of a skilled job can be written as . To define the matching rates of workers we introduce the share which denotes the share of unskilled vacancies. Low-educated workers therefore exit unemployment at rate , while mismatched workers change employers at rate

¹¹Had we assumed mismatched workers to be part of the mass of high-educated workers

Finally, the properties of the matching function imply that the matching rate of workers (firms) is increasing (decreasing) in θ, and, conventionally, we assume that lim_θ and lim_θ

2.3 Bargaining

In equilibrium, our model considers three types of matches: (i) high-educated workers on skilled jobs, (ii) high-educated workers on unskilled jobs, and (iii) low-educated workers on unskilled jobs. In each of these matches, the firmworker pair divides the surplus of the match according to the asymmetric Nash bargaining solution. The exogenous surplus share of workers is denoted by . Moreover, we adopt the following standard notation: denotes the value of unemployment for a worker of type j, denotes the value of a vacant job of type i, denotes the value of employment for a worker of type j on a job of type i and denotes the value to the firm of filling a job of type i with a worker of type j. Accordingly, the surplus of a match between a job of type i and a worker of type j can be expressed as and, when a match is consumated, the wage satisfies the standard Nash bargaining condition¹²

φ)q(θ).

(1-ψφ)q(θ)

seeking an unskilled job, then the matching rate would become instead of ψ(1- The latter would imply an even lower stability of unskilled jobs and this would exacerbate the negative search externality of mismatched workers on low-educated workers.

(w∗)

¹²We are grateful to Robert Shimer for pointing our to us in a note (see Shimer, 2003) that in models with on-the-job search, the equivalence between the Nash bargaining solution and the linear sharing rule in (1) may not be valid since firms may prefer to pay a higher eficiency wage in order to reduce worker turnover. However, his result obtains when employed workers pay a search cost (σ) which allows them to search with the same eficiency as unemployed workers. Nonetheless, our assumption that implies that σ = 0 the eficiency wage would be and, since in equilibrium, w∗ = w(s, h) w(s, h) > y(n) firms will not find profitable to do so. In reality, σ = 0 can be justified by the fact that σ = 0 on-the-job seekers, despite having less time to search than unemployed workers, may have more contacts to find an alternative job.

\[(1 - \beta) [ W (i, j) - U (j) ] = \beta [ J (i, j) - V (i) ].\tag{1}\]

Below we concentrate on steady states and we assume that high-educated workers accept both types of jobs. The proof that this strategy is optimal is provided in Section 3. Formally, a cross-skill matching equilibrium can be summarised by a vector that satisfies the following conditions: (i) match formation is voluntary, (ii) the expected profit of each type of job is equal to zero, and (iii) the state variables and satisfy the appropriate steady-state conditions defined below.

2.4 Flow equations

Figure 2 illustrates the flows of workers between the three possible states. At each moment in time a flow of low-educated unemployed find employment and this flow is equal in steady state to the flow of low-educated workers into unemployment, . Similarly, the flow of high-educated workers exiting unemployment equals the flow into unemployment. A share of these workers encounters a skilled job, while the remaining fraction η is employed in an unskilled job and will continue to search on the job. Finally, the flows into and out of unemployment of correctly-matched workers and of mismatched workers are the same.

Accordingly, the steady state condition for is

\[\eta \theta q (\theta) \phi u = s (\mu - \phi u).\tag{2}\]

Similarly, for , we have

\[\theta q (\theta) (1 - \phi) u = s [ 1 - \mu - (1 - \phi) u ],\tag{3}\]

whereas the mass of mismatched workers satisfies

\[\eta \theta q (\theta) (1 - \phi) u = [ s + \theta q (\theta) (1 - \eta) ] e (n, h).\tag{4}\]

2.5 Asset values

2.5.1 Workers

The asset value of a low-educated unemployed, , satisfies

\[r U (l) = b + \theta q (\theta) \eta [ W (n, l) - U (l) ].\tag{5}\]

Similarly, given the assumption that high-educated workers accept both types of jobs, the asset value of a high-skill unemployed, , verifies

\[r U (h) = b + \theta q (\theta) [ \eta (W (n, h) - U (h)) + (1 - \eta) (W (s, h) - U (h)) ],\tag{6}\]

while the asset values of low-educated and high-educated workers in unskilled and skilled jobs, respectively, satisfy

\[r W (n, l) = w (n, l) + s (U (l) - W (n, l)),\tag{7}\]

\[r W (s, h) = w (s, h) + s (U (h) - W (s, h)).\tag{8}\]

Finally, the asset value of employment for mismatched workers verifies

\[r W (n, h) = w (n, h) + s [ U (h) - W (n, h) ] + \theta q (\theta) (1 - \eta) [ W (s, h) - W (n, h) ],\tag{9}\]

where the term corresponds to the expected return from successful on-the-job search.

2.5.2 Firms

As regards firms, the values of opening unskilled and skilled vacancies are given, respectively, by

\[r V (n) = - c + \psi q (\theta) \left[ \phi (J (n, l) - V (n)) + (1 - \phi) (J (n, h) - V (n)) \right],\tag{10}\]

\[r V (s) = - c + q (\theta) (1 - \psi \phi) [ J (s, h) - V (s) ],\tag{11}\]

whereas the values to the employer of filling those vacancies with workers of the required type, verify

\[r J (n, l) = y (n) - w (n, l) + s [ V (n) - J (n, l) ],\tag{12}\]

\[r J (s, h) = y (s) - w (s, h) + s [ V (s) - J (s, h) ].\tag{13}\]

Lastly, the value to a firm that fills an unskilled vacancy with a higheducated (mismatched) worker is

\[r J (n, h) = y (n) - w (n, h) + s [ V (n) - J (n, h) ] + \theta q (\theta) (1 - \eta) [ (V (n) - J (n, h) ].\tag{14}\]

3 Equilibrium

Below we derive the equilibrium. Following the strategy in AV we express all equilibrium relations in terms of the labour market tightness, and the share of unemployed workers with low education,

3.1 Worker flows

We start with the equilibrium flow equations. From equations (2) and (3) we can solve for u and as a a function of θ and (plus the exogenous variable . This yields

\[u (\theta , \phi ; \mu) = \frac {s}{s + \theta q (\theta)} \frac {1 - \mu}{1 - \phi},\tag{15}\]

\[\eta (\theta , \phi ; \mu) = \frac {(1 - \phi) \theta q (\theta) \mu + s (\mu - \phi)}{\theta q (\theta) \phi (1 - \mu)}.\tag{16}\]

For given and ,the unemployment rate of high-educated workers is decreasing in ewhilst the unemployment rate of low-educated wokers is decreasing in both and eIt is also straightforward to verify that is decreasing in , and increasing in as in a cross-skill matching equilibrium.¹³

Next, since is equal to , equation (4) yields the equilibrium value of as a function of θ, φ and

\[\psi (\theta , \eta ; \mu) = \frac {1}{1 + \frac {\eta (1 - \phi) \theta q (\theta)}{s + (1 - \eta) \theta q (\theta)}}\tag{17}\]

with . Intuitively, at a higher value of the labour market tightness there will be less unemployed workers and more mismatched workers, reducing . Similarly, an increase in the fraction of unskilled jobs reduces and increases , resulting in a drop of . Finally, substituting equation (16) into the above expression yields a new function with the same arguments as and . This is our third equilibrium expression which yields and

>From the above results we can conclude that the efective matching rate of unskilled jobs, decreases unambiguously in (for given values of . By contrast, in the case of skilled jobs, the changes in and have an opposite efect on the efective matching rate On the one hand, there is the well-known congestion efect, captured by , which makes it more dificult to find a worker and, on the other, there is composition efect of opposite sign, captured by 2 as the proportion of high-educated workers searching for a good job increases. Nonetheless, it can be shown that for suficiently high values of the change in the indirect composition efect turns out to be dominated by the direct congestion efect. In what follows we shall therefore assume that, for given values of an increase in θ reduces 15

¹³Formally, since low-educated workers have a lower exit rate out of unemployment than high-educated workers, they must be relatively over-represented in the mass of unemployed as the inflow into unemployment is the same for both types of workers.

∂ψφ ∂ψφ > 0 ∂φ ∂φ

∂ψ(1 φ) ∂ψ(1−φ) ∂θ ∂θ < 0.

∂ψφ_∂θ ∂ψφ ∂θ <0, <0,

∂ψ(1 φ) ∂φ ∂φ < 0.

¹⁴>From the signs of those derivatives, it is straightforward to obtain that > 0 and Morever, we assume that − < 0. The sign of those derivatives is used to prove equilibrium uniqueness in Appendix A.

3.2 Equilibrium wages

We now proceed with a derivation of the equilibrium wages. These solutions are needed to obtain the two free entry conditions.

3.2.1 The wage of low-educated workers

Substituting (5), (7), (10) and (12) into equation (1) and imposing the freeentry condition for unskilled vacancies, , we obtain that the match surplus of a low-educated worker, 1， satisfies

\[(r + s) S (n, l) = y (n) - r U (l),\]

implying that is given by

\[w (n, l) = r U (l) + \beta [ y (n) - r U (l) ].\tag{18}\]

3.2.2 The wage of high-educated workers on skilled jobs

Likewise, substitution of equations (6), (8), (11) and (13) into equation (1) and the free-entry condition for skilled vacancies, , yields the match surplus of a high-educated worker, which verifies

^∂(1 − ^{ψφ)q(θ)/∂θ < 0}

µ 0.5.

¹⁵Compared to a standard matching model this is the only additional restriction that we impose on the matching technology. Moreover, although a precise suficient condition for is cumbersome to obtain, in our numerical simulations we find that the above derivative is always negative for values of

\[(r + s) S (s, h) = y (s) - r U (h),\]

so that the wage of a high-educated worker in a skilled job is given by

\[w (s, h) = r U (h) + \beta [ y (s) - r U (h) ].\tag{19}\]

3.2.3 The wage of mismatched workers

The derivation of the wage of mismatched workers is slightly more complicated and can be obtained from substitution of equations (6), (9), (10) and (14) into equation (1) which, together with , implies that the match surplus of a mismatched worker, satisfies

\[[ r + s + \theta q (\theta) (1 - \eta) ] S (n, h) = y (n) - r U (h) + \theta q (\theta) (1 - \eta) \beta \left[ \frac {y (s) - r U (h)}{r + s} \right],\]

leading to the following expression for their wage

\[w (n, h) = r U (h) + \beta [ y (n) - r U (h) ] - (1 - \beta) \theta q (\theta) (1 - \eta) \beta \left[ \frac {y (s) - r U (h)}{r + s} \right].\tag{20}\]

According to (20), mismatched workers earn less than a share of the flow surplus . The reason is that their wages are reduced by the amount times the capital gain from successful on-the-job search, namely, the firm’s share of the surplus that the worker creates by searching on the job.¹⁶

¹⁶This result was previously obtained by Pissarides (1994) in a model with two types of jobs and a homogenous workers where on-the-job search by mismatched workers takes place only at short tenure in the bad jobs. However, that paper does not yield a closed-form comparison of both wages and we shall do in section 3.5.3.

3.3 Match surpluses

The equilibrium expressions for the match surpluses can now be obtained in two steps. First, we obtain the equilibrium asset value of unemployed workers by substituting the expressions for and into (5) and (6). This yields

\[r U (l) = \frac {(r + s) b + \theta q (\theta) \eta \beta y (n)}{r + s + \theta q (\theta) \eta \beta}\tag{21}\]

\[r U (h) = \frac {(r + s) \lambda_ {3} b + \theta q (\theta) \beta [ \eta (r + s) y (n) + (1 - \eta) \lambda_ {2} y (s) ]}{\lambda_ {1} \lambda_ {2}},\tag{22}\]

where ,and

Next, subtituting (21) and (22) back into the surplus expressions yields the closed- form solutions

\[S (n, l) = \frac {y (n) - b}{r + s + \theta q (\theta) \eta \beta}, S (n, h) = \frac {y (n) - b}{r + s + \theta q (\theta) (1 - \eta + \eta \beta)}.\]

from which it follows that . Furthermore, since mismatched workers retain the option value of employment in skilled jobs, is independent of 17

3.4 Free-entry conditions

Finally, using expression (1) for the Nash bargaining solution, we can write the free-entry conditions for the two types of jobs as follows

\[r V (n) = - c + \psi q (\theta) (1 - \beta) [ \phi S (n, l) + (1 - \phi) S (n, h) ] = 0,\]

rU(h),

¹⁷Formally, with on-the job-search the increase in the wage due to the rise in w(n, h) the outside option of high-educated workers, is exactly ofset by the the increase in the future surplus in a skilled job, which leads to a fall in Without on-the job w(n, h). search, the second efect is absent leading to a negative relation between and S(n, h) y(s).

\[r V (s) = - c + q (\theta) (1 - \psi \phi) (1 - \beta) S (s, h) = 0,\]

and substituting in the closed-form solutions for and derived above, yields

\[\frac {c}{q (\theta)} = \psi (1 - \beta) \left[ \phi \frac {y (n) - b}{r + s + \theta q (\theta) \eta \beta} + (1 - \phi) \frac {y (n) - b}{\lambda_ {2}} \right],\tag{23}\]

\[\frac {c}{q (\theta)} = (1 - \beta) (1 - \psi \phi) \left[ \frac {y (s) - b}{\lambda_ {1}} - \theta q (\theta) \eta \beta \frac {y (n) - b}{\lambda_ {1} \lambda_ {2}} \right].\tag{24}\]

Equations (23) and (24) are the free-entry conditions for unskilled and skilled jobs, respectively, which henceforth, for given η and , will be denoted in implicit form as and . Note that the right-hand side of these equations defines the expected future profits when the job is filled whereas the left-hand side represents the expected cost of keeping a vacancy unfilled. Thus, equations (23) and (24), together with the expressions in (15), (16) and (17) for and , defines the system of equations determining the equilibrium of our model.

3.5 Properties of the Equilibrium

3.5.1 Existence

We start the analysis by deriving the conditions under which a a cross-skill matching equilibrium exists. First of all, as in AV, we need to rule out the corner solution in which firms only create unskilled jobs. A suficient condition to ensure that firms are willing to create skilled jobs is that 0 if . This condition can be easily derived.

When , the outside option value of workers (common to both types) would simplify to

\[r U = \frac {(r + s) b + \beta \theta q (\theta) y (n)}{r + s + \beta \theta q (\theta)}.\]

Substituting this value into the free-entry condition when no skilled jobs are available, i.e. , we obtain

\[{\frac {c}{q (\theta)}} = (1 - \beta) {\frac {y (n) - b}{r + s + \beta \theta q (\theta)}}.\]

The above equation yields a unique solution for ,denoted as . The necessary condition to rule out the corner solution with , can thus be written as which is equivalent to the condition that which gives ¹⁸

\[(1 - \mu) \frac {(r + s) (y (s) - b) + \theta^ {*} q (\theta^ {*}) \beta (y (s) - y (n))}{(r + s) (r + s + \theta^ {*} q (\theta^ {*}) \beta)} > \frac {y (n) - b}{r + s + \theta^ {*} q (\theta^ {*}) \beta}.\]

Finally, rearranging terms, we obtain

\[y (s) - b > \left[ 1 + \frac {\mu (r + s)}{(1 - \mu) (r + s + \beta \theta^ {*} q (\theta^ {*}))} \right] (y (n) - b),\tag{25}\]

which is equivalent to the existence condition given in AV. Hence, according to equation (25), skilled jobs need to be more productive than unskilled jobs and the required productivity diferential increases with

Second, to conclude the proof of existence, we need to show that higheducated workers accept unskilled jobs when firms create both types of jobs. This result requires that the match surplus is positive when (25) is satisfied. Since , this is equivalent to assuming that . Hence, as anticipated in Section 2, in our economy higheducated workers never find it optimal to refuse unskilled jobs.

Proposition 1 In any equilibrium high-educated workers accept unskilled jobs.

η = 1

φ = µ.

¹⁸Notice that this condition is derived under the assumption that workers do not engage in on-the-job search unless there is some strictly positive mass of skilled jobs. This assumption is natural given our assumption of purely random search. Furthermore, when it is necessarily true that

The intuition behind the above result is rather simple. Since workers with high education can search as eficiently during employment as during unemployment, they will accept any job that ofers a wage above A cross-skill matching equilibrium is therefore the only possible type of non-trivial equilibrium with two types of jobs. Furthermore, given Proposition 1, equation (25) is both a necessary and a suficient condition to ensure existence of equilibrium.

3.5.2 Uniqueness

Since we are interested in the comparative statics properties of the model, a necessary preliminary step in the analysis should focus on the conditions for uniqueness. As shown in Appendix A, uniqueness is guaranteed under the following three suficient conditions: (i) low-educated workers are a majority of the population, (ii) workers obtain at least half of the surplus of any match, and (iii) the productivity diferential exceeds a certain threshold value, (defined in Appendix A). Formally, these restrictions on the parameter space can be stated as follows

\[\mu \geq 0. 5, \beta \geq 0. 5 \text { and } \{y (s) - y (n) \} \geq y ^ {*}.\]

To obtain uniqueness, we substitute the solutions in equations (16) and (17) for and into the free-entry conditions (23) and (24). This yields a system of two equations in two unknowns, namely, the labour market tightness (θ) and the fraction of low-educated unemployed workers (φ). Furthermore, when and , we show in Appendix A that the profits of unskilled jobs increase with φ and decrease with θ. The free- entry condition for unskilled jobs is therefore associated with an upward-sloping locus in the space . Intuitively, an increase in the labour market tightness makes it more dificult to fill any job. Thus, the profits of unskilled jobs will decrease unless the fraction of low-educated workers in the mass of unemployed workers and in the mass of job seekers increases enough.²⁰ Likewise, when the productivity diferential exceeds , the profits of skilled jobs decrease both with and . This results in a downward-sloping curve that intersects the free entry curve of unskilled jobs at most once.

y(s),

¹⁹By contrast, in AV mismatched workers do not engage in on-the-job search. By accepting an unskilled job high-educated workers therefore forego the option of employment in a skilled job. Consequently, beyond some threshold level of these workers will prefer to refuse unskilled jobs and continue to search, giving rise to an ex-post segmentation matching equilibrium.

∂(φψ)/∂φ = ψ + φ∂ψ/∂φ > 0.

²⁰Formally, for a given value of θ an increase in the share of low-educated job seekers, φψ, requires an increase in φ as

The unique equilibrium is depicted in Figure 3. The reason why we need restrictions on and is related to the opposite efect of changes in the share of unskilled vacancies, , on the outside option values of skilled and unskilled workers. While raises with since low-educated workers will match more often with their suitable jobs, decreases with since higheducated workers have a higher chance of being mismatched. Thus, consider for instance an increase in . For a given value of a lower higher value of requires a reduction in the share of skilled vacancies, and as a result will increase while will decrease. These two changes have opposite efects on the profits of unskilled jobs but, under ssumption , the net efect on profits from the fall in and the rise in is always positive.

3.5.3 Wage distribution

Having derived the equilibrium, we can now analyze the efects of on-the-job search on the equilibrium wage distribution.

In our economy all workers are equally productive on unskilled jobs. Nonetheless, firms with unskilled jobs prefer to hire low-educated workers.

The reasons for this outcome are twofold. First, low-educated workers have a lower outside option than high-educated workers because they cannot perform skilled jobs. Second, firms anticipate that mismatched workers will quit an unskilled job whenever they have located a firm with a skilled vacancy. From equation (22), it follows that the first efect tends to raise the wage of mismatched workers, while the second efect tends to reduce it. Thus, the feature that high-educated workers have access to better jobs introduces two opposite efects on the wage of mismatched workers.

Nonetheless, in Appendix A we show that the negative efect always dominates. The explanation is that the wage diferential, , has the same sign as the surplus diferential, , which is negative as shown in section 3.3.

Proposition 2 In any cross-skill matching equilibrium,

Proposition 2 contrasts with the findings of AV who obtain the opposite result. Without on-the-job search, the efective separation rate of higheducated workers is the same as the one of low-educated workers, so that the inequality always holds because . Finally, it is easy to show that in both models is always larger than as high-educated workers have a better outside option and because

In sum, our results clearly indicate that on-the-job search leads to a widening of the within-group wage inequality for high-educated workers relative to the case in which mismatched workers refrain from searching. In Section 5 we shall evaluate the contribution of this additional channel for the overall wage inequality using a calibrated version of the model.

4 Comparative statics

In this section we present some interesting comparative statics on the efects an increase in the fraction of high-educated individuals in the population, referred to as “skill upgrading”, and (ii) an increase in the productivity of high-educated workers, , for given , referred to as “skill-biased” technological change. In the sequel, we will concentrate on the efects of those two changes on both the cohort-specific unemployment rates and the overall degree of labour market tightness in the economy.

4.1 “Skill-upgrading”

What happens if the share of high-educated workers increases? To answer this question, we consider the efects of an inflow of high-educated workers, raising the share of these workers in the population to some value

Immediately after this change, the share of high-educated workers in the pool of job seekers , increases. Skilled jobs will therefore match more frequently with appropriately qualified workers and firms will respond to this change by creating more skilled vacancies. By contrast, unskilled jobs will now meet more frequently with over-qualified workers. Some of these workers are unemployed and will accept the job ofer while others are already employed on unskilled jobs. Yet, in both cases the increase in tends to exert a negative efect on the profits of unskilled jobs as Firms will therefore respond to the increase in the share of high-educated workers by creating less unskilled jobs. In the sequel, we shall refer to this negative search externality as NSE.

Any increase in the proportion of high-educated workers is thus accompanied by an increase in the mass of skilled vacancies, , and a reduction in the mass of unskilled vacancies, Notice as well that these changes give rise to an unambiguous fall in the proportion of unskilled vancies, η, while the efect on the labour market tightness, is unclear. The total number of jobs will increase when the rise in exceeds the fall in . But this need not translate into a higher value of θ, since the distributions of educational levels and jobs also change the elements in the denominator of i.e. and . First, given the shift towards skilled jobs, the unemployment rate of low-educated workers, u(l), tends to increase. Second, in the new equilibrium ewe have a larger share of high-educated workers and, since , this e etends to reduce the overall unemployment rate. Hence, in general it is thus impossible to predict the change in whereas unambiguously decreases.

²¹This statement is true for the absolute number of unskilled jobs, v(s), and for the v(s) ratio between the number of unskilled jobs and the number of low-educated job seekers. The latter is obviously a more meaningful statistic as the mass of low-educated workers falls.

Figure 4 ofers an illustration of the previous efects for parameter configurations that satisfy A.1, where increases. >From (16) we obtain that the increase in reduces for any given value of . According to equations (19) and (20), this shift in the distribution of jobs tends to reduce the outside option value of low-educated workers, , while it improves the corresponding option value of high-educated workers . Under Assumption A.1 the net efect on the profits of unskilled jobs is positive. Furthermore, the reduction in goes in parallel with an increase in the fraction of unemployed in the pool of job seekers, , which exerts further positive efect on the profits of unskilled jobs. Hence, in order to restore zero profits, the locus needs to shift to the right. Similarly, the increase both in and tends to reduce the profits of skilled jobs for given values of . Thus, the locus will shift to the left.

Accordingly, the overall efect of an increase in the share of high-skill workers is thus a fall in and an ambiguous efect on and , given the opposite move of and . From this we cannot draw unambiguous conclusions for the changes in the unemployment rates of high- and low-educated workers, and . Nonetheless, in our numerical experiments we always e efind an increase in ,as in Figure and a fall in . Thus, an increase in the cohort-size of high-educated workers tends to reduce their cohort-specific unemployment rate . That is, each high-educated job seeker enjoys a higher exit rate out of unemployment and a larger share of the job ofers are skilled jobs. Conversely, for low-educated workers we obtain an increase in . In the new steady-state equilibrium they match at a higher rate, but ea larger proportion of these jobs are skilled jobs, resulting in a reduction of the overall matching rate

These cohort-size efects for high-educated workers are absent in AV. In their model the overall labour market tightness, is invariant to changes in and . Thus, any increase in is ofset by an equivalent reduction in . The only variable that changes is therefore and this variable afects but leaves invariant . On the contrary, allowing for on-the-job search e eleads to cohort- size efects for both types of workers.22

4.2 “Skill-biased” technological change

Let us now consider a situation in which technological change is biased towards high-educated workers, resulting in an increase of their productivity to , for given . From (24), it follows immediately that the increase in raises the profits of skilled jobs. The locus will shift to the right. Furthermore, since is independent of , the locus remains unchanged.

The overall efects of an increase in are illustrated in Figure 5. The rightward shift of the locus leads to an increase in and . From the increase in we can immediately conclude that “skill-biased” technological change reduces , while the efect on is priori unclear. For a given value of e e, the share of unemployed workers with low skills, can only increase if decreases. But this efect is at least partially ofset by the increase in which tends to raise for given values of .

²²As mentioned earlier, this result is in the same spirit as the one obtained by Shimer (2001). However, his result refers to cyclical/transitional unemployment rates and not to equilibrium unemployment, as in our case.

Proposition 3 Under Assumption A.1, “skill-biased” technological change reduces while the efect on is ambiguous.

The above Proposition is again clearly diferent from the results in AV who obtain that and . A first diference is theree efore that “skill-biased” technological change may improve the employment prospects of high-educated workers.

It should be noted that this result does not depend on our assumptions about the common cost of vacancy creation, and unemployment income, b. In our model technological change would be neutral if , b and c all grew at the same rate. Nonetheless, if skill-biased technological change implies diferent growth rates of and , it would still reduce ewhen the cost of skilled vacancies and the unemployment income of higheducated are indexed to . What does seem to depend on the values of c and b is the efect of changes in on . In our numerical simulations ebelow we find that skill-biased technological change raises for common values of b and eInterestingly enough, however, when we allow for a higher cost of skilled vacancies it is easy to construct examples in which skill-biased technological change reduces

This latter result points at an important diference between “skill-biased” technological change and “skill-upgrading”. Both changes induce the creation of more skilled jobs which exerts a negative congestion externality on the unskilled jobs in the market. On top of that skill-upgrading also results in an increase in the share of high-educated job seekers which further reduces the profits of unskilled jobs. By contrast, in the case of skill-biased technological change, more high-educated workers are drawn into skilled jobs. As a result, the share of low-educated job seekers, , tends to go up which exerts a positive efect on the profits of unskilled jobs. When the cost of vacancy creation are the same for both jobs, the congestion efect dominates over the composition efect and goes up. However, when skilled jobs are esuficiently more expensive to create, so that a large share of high-educated workers end up in unskilled jobs, the positive composition efect generated by a higher productivity diferential may actually dominate so that goes down.

5 Simulations

In this section we perform a few illustrative simulations with the model. Our aim is to gauge quantitatively the comparative-statics efects of the model following the two changes discussed in the previous section, namely, skill upgrading and skill-biased technological change.

The model is calibrated using a standard Cobb-Douglas matching function, , together with the following parameter configuration: (equal productivity in unskilled jobs), , and . Thus, in the baseline version of the model, the proportion of low-educated workers is 75% of the (unit mass) population. Under this choice of parameters, Table 1 reports the equilibrium values of the vector of unknowns plus the unemployment rates of both types of workers, and

e eThe equilibrium value of θ is 1.435 which implies an unemployment duration of 10 months, in line with the average duration of unemployment in some European countries, like Spain, where unemployment hysteresis has been strong (see Bentolila and Jimeno, 2003). Accordingly, the unemployment rates for low and high-educated workers are 11.1% and 7.8%, respectively, giving rise to an overall unemployment rate (u) of 10.2%, in line with the average EU rate during the last fifteen years. The proportion of mismatched workers in the pool of job seekers, is 23.7%, while the share of high-educated workers applying for unskilled jobs, , equals 38.3%.This fraction captures the NSE on firms creating those vacancies Likewise, the share of unskilled vacancies in total vacancies, is 67.4% and the fraction of low-educated workers in the pool of searchers, is 81.0%. Further, turns out to be 1.36 while and are 0.86 and 0.78, respectively. Thus, in agreement with Proposition 1, mismatched workers get paid less than low-educated workers in unskilled jobs.²³ Henceforth, we use the ratio to capture within-group wage inequality (W GI henceforth), and the ratio between a weighted average of and and to represent the average skill premium by education (AW I henceforth).²⁴ Note that WGI is a good proxy for the penalty to over-education. In our baseline estimation WGI is 1.74 while AWI is 1.37.

Table 1 Equilibrium values in Baseline Model

Figure 6 displays four panels illustrating the efects of skill upgrading, captured by a continuous fall in from 0.75 to 0.50, on and e eNSE, WGI and AWI, respectively. Note that, in order to follow the correct direction of changes as decreases, the graphs should be looked from right to left since the horizontal axis displays increasing values of . Although, as argued in section 4, u(h) can go up or down since the efect of a change of on ecannot be unambiguously signed, we find that for our choice of parameters it declines from 7.8% to 7.3% when goes down from 75% to 50%, illustrating in this way the cohort-size efect discussed before. By contrast, increases from 11.1% to 13.8%. Regarding NSE, we find that it increases from 38.3% to almost 50%. Finally, both WGI and AWI increase, the reason being that the reduction in the share of unskilled jobs, decreases the ouside option value of low-educated workers, , whereas the corresponding increase in increases . Thus, for unchanged productivities, typically increases with “skill-upgrading” whilst both and decrease, leading to the observed widening of both WGI and AWI.

y(n, h) > y(n, l)

w(n, l). w(n, l).

^{w(n, h)} ≥

e23Had we allowed for y(n, h) > y(n, l), then we could have obtained that

1 µ u(h) e(n,h) 1−µ−u(h)−e(n,h) 1 µ u(h) 1−µ−u(h)

e(n,h) e(n,h) 1 µ u(h) 1−µ−u(h)

²⁴The corresponding weights are and

Figure 6: The efects of skill upgrading

Figure in turn, shows the efects of skill biased technical progress, captured by a continuous increase in from 1.5 to 2.0. In accord with Proposition 2, falls to 7.2% whereas goes up to 11.5%. Similarly, NSE e edrops by almost six percentage points reflecting the reduction in the proportion of mismatched workers in unskilled jobs. As with skill upgrading, both and AWI increase for similar reasons as above.

Figure 7: The efects of “skill-biased” technological change

So far, the numerical results corroborate our theoretical predictions. In the remainder we shall consider two slight generalisations of the model. In the first case we allow for diferent costs of vacancy creation, , while the second example considers job-specific separation rates, 25

Diferent costs of vacancy creation are a plausible assumption. In particular, it seems reasonable to assume that skilled job openings are more costly to create than unskilled job openings. Furthermore, from our discussion in the previous section we know that this assumption may change the response of to changes in . This feature is illustrated in Figure 8. eIn this example the cost of skilled vacancies, , is assumed to be 1, while the cost of unskilled jobs, , is kept constant at its benchmark value of . As can be seen, in this case follows a U-shaped pattern in It initially falls up to values of around 1.80, beyond which it starts to increase again. The reason for this increase is that for, for high values of , the congestion efect dominates over the weaker . Nonetheless for , the case in which the cost of job creation is perfectly indexed to , the unemployment rate of low-educated workers is still below the value at . Hence, in economies with relatively high costs of skilled job creation, skill-biased technological change may therefore help to reduce the unemployment of low-educated workers over a certain range of productivity diferentials .

²⁵Appendix B contains the derivation of the equations determining equilibrium in this more general setup.

Figure 8: Diferent job creation costs

Next, “skill-biased” technological change in our model reduces the average tenure on unskilled jobs, while the tenure on skilled jobs is fixed. Nonetheless, if we were to consider fully endogeneous job-destruction rates, the increase in the relative profits of skilled jobs would give rise to a lower separation rate on them. To capture this efect in our set-up, we allow for a reduction in the separation rate of skilled jobs, , from a value of 0.1 to 0.075 while keeping constant at its benchmark value of 0.1. Figure 9 illustrates the efect of this change on .²⁶ It turns out that a drop in the rate of eturnover on skilled jobs reduces for any given value of , although ein contrast to what happened with the increase in 2 keeps on being increasing in e. The explanation is somewhat similar to the argument for the cohort-size efects. When skilled jobs are more stable than unskilled jobs, high-educated workers return less frequently to the unemployment pool. Other things equal, the pool of searchers is therefore comprised by a larger share of low-educated workers and this exerts a positive efect on the profits of unskilled jobs. Improving the stability of skilled jobs might therefore be beneficial for both types of workers in the presence of “skill-biased” technical change.

y(s).

u(h)

²⁶The efect of such a change on are not reported since it obviously falls for given values of

Figure 9: Diferent separation rates (a)

This last observation seems to point out to the need for diferential labour market policies. In particular, in order to avoid the from frequent unemployment spells of high-educated workers on low-educated workers, there may be scope for employment protection legislation that is diferentiated across worker and/or job type. To illustrate somewhat the implications of this selective policies, Figure 10 shows the combined efects of both an increase in , from 0.5 to 1, and a reduction in , from 0.1 to 0.075. The change in would capture less jobs creation whereas the drop in would imply less job destruction in the skilled sector, in accord with the standard efects of employment protection legislation which makes both hiring and firing of workers more expensive. As can be observed, when increases, both unemployment rates fall relative to their benchmark values. The explanation is that, for the chosen configuration of parameter values, the net efect of both changes reduces (i.e., the efect of the reduction in is stronger than the rise in whilst, at the same time, it makes fall as well the efect of the lower implied by both the inecrease in and the drop in dominates over the direct negative efect on stemming from the increase in . Of course, our example cannot ebe generalised to any value of the productivity diferential and the explicit introduction of firing cost in this model exceeds the scope of this paper. This issue is left for future research.

Figure 10: The efects of firing costs in skilled jobs

6 Conclusions

In this paper, we have analysed the properties of a matching model with two types of workers (low and high-educated) and two types of jobs (unskilled and skilled). High-educated workers can perform both jobs while the loweducated ones are only productive in unskilled jobs. A skilled job performed by a high-educated worker produces the highest output and both types of workers are equally productive in unskilled jobs. The novel element in the model is to allow for on-the-job search by mismatched high-educated workers while keeping a Nash bargaining rule to share the surpluses within each market. We show that this model can account for some of the stylised facts of the labour market in countries where a large tertiary educational upgrading which has taken place over the last decades.

In particular, we show that, with on-the-job search, skill upgrading generally decreases the unemployment rate of high-educated workers whereas it increases the unemployment rate of low educated workers. Thus we get a cohort-size efect whereby, in a market with search frictions, an increase in the supply of high-education ends up increasing the demand for skilled jobs by so much that the unemployment rate of the high-educated workers falls. Conversely, skill-biased technical change, while decreasing the unemployment rate of the high-educated workers, can have ambiguous efects on the unemployment rate of the low-educated workers. The intuition is that mismatched workers in unskilled jobs create a negative search externality on firms opening unskilled vacancies which hampers the creation of these jobs. In situations where skilled vacancies are more expensive to open than unskilled jobs, and therefore skilled jobs are relatively scarce, skill-biased technical change can lead to a large improvement in the profitability of those jobs, and a reduction in the unemployment rate of the low-educated workers. Finally, under the assumption of equal productivity of both types of workers in unskilled jobs, our result that mismatched workers get a lower wage than correctly matched low-educated workers, ofers a new channel to explain a widening of within-group wage inequality and a higher average skill premium which has been observed in many OECD countries.

The model could be extended in a number of ways. One extention is to endogeneise the skill distribution by allowing workers to invest in education. A second extension would be to consider a model of directed search. In that environment workers can target their search to diferent types of jobs but nonetheless high-educated may consciously decide to apply for unskilled jobs (with some probability). Finally, one could consider the possibility of allowing for multiple meetings so that the model can address issues of ranking of applicants by firms.

7 Appendix A : Proofs

Proof of

From (18) and (20), the wage diferential can be expressed as

\[w (n, h) - w (n, l) = (1 - \beta) [ r U (h) - r U (l) - \theta q (\theta) (1 - \eta) S (s, h) ],\]

where and

Then, replacing and into the wage diferential yields

\[r U (h) - r U (l) = \theta q (\theta) \beta [ \eta (S (n, h) - S (n, l)) + (1 - \eta) S (s, h) ],\]

implying that

\[w (n, h) - w (n, l) = (1 - \beta) \theta q (\theta) \beta \eta [ S (n, h) - S (n, l) ].\]

Hence

\[\operatorname{sign} \left[ w (n, h) - w (n, l) \right] = \operatorname{sign} \left[ S (n, h) - S (n, l) \right].\]

This, together with , yields the required inequality

\[w (n, h) - w (n, l) < 0. \blacksquare\]

Proof of Uniqueness

Substituting and from equations (16) and (17) into the free entry conditions yields two equations in two unknowns, θ and φ,denoted in implicit form by and

Skilled jobs: To show that the locus has a negative slope, it is suficient to show that

\[\frac {d \phi}{d \theta} = - \frac {\partial F _ {S} / \partial \theta}{\partial F _ {S} / \partial \phi} < 0.\]

Taking the derivative of with respect to dividing all terms by and denoting the ratio by R, yields

\[\begin{array}{r c l} \frac {1}{\triangle} \frac {\partial F _ {S}}{\partial \theta} & = & \frac {q ^ {\prime} (\theta)}{\lambda_ {1}} (1 - \psi \phi) \left[ R - \frac {\beta \eta \theta q (\theta)}{\lambda_ {2}} \right] \\ & & - \frac {q (\theta)}{[ \lambda_ {1} ] ^ {2}} (1 - \psi \phi) \left[ R - \frac {\beta \eta \theta q (\theta)}{\lambda_ {2}} \right] \frac {\partial \lambda_ {1}}{\partial \theta} \\ & & - \frac {q (\theta)}{\lambda_ {1}} (1 - \phi) \frac {\beta \eta}{\lambda_ {2}} \cdot \frac {\partial \theta q (\theta)}{\partial \theta} + \frac {q (\theta)}{\lambda_ {1}} (1 - \phi) \frac {\beta \eta \theta q (\theta)}{[ \lambda_ {2} ] ^ {2}} \cdot \frac {\partial \lambda_ {2}}{\partial \theta} \\ & & + \frac {q (\theta)}{\lambda_ {1}} \left[ R - \frac {\beta \eta \theta q (\theta)}{\lambda_ {2}} \right] \frac {\partial (1 - \psi \phi)}{\partial \theta}. \end{array}\]

Since the first three terms of the above expression are negative whereas the last two tems, given that 2 , are positive. Nonetheless, combining the the third and fourth terms yield a negative term whilst our assumption that , ensures that a combination of the first and fifth terms is also negative. Thus,

The corresponding expression for is:

\[\begin{array}{r c l} \frac {1}{\triangle} \frac {\partial F _ {S}}{\partial \phi} & = & - \frac {q (\theta)}{[ \lambda_ {1} ] ^ {2}} (1 - \psi \phi) \left[ R - \frac {\beta \eta \theta q (\theta)}{\lambda_ {2}} \right] \frac {\partial \lambda_ {1}}{\partial \phi} \\ & & + \frac {q (\theta)}{[ \lambda_ {1} ] ^ {2}} \left[ R - \frac {\beta \eta \theta q (\theta)}{\lambda_ {2}} \right] \cdot \frac {\partial [ (1 - \psi \phi) ]}{\partial \phi} \\ & & + \frac {q (\theta) (1 - \psi \phi)}{[ \lambda_ {2} ] ^ {2}} \cdot \beta \eta \theta q (\theta) \cdot \frac {\partial \lambda_ {2}}{\partial \phi} \\ & & - \frac {\beta q (\theta)}{\lambda_ {1} \lambda_ {2}} \frac {\partial \eta \theta q (\theta)}{\partial \phi} \end{array}\]

The first two terms are negative since and whereas the last two terms are positive since and . However, if R is suficiently large, then the negative sign of the first two terms will dominate. Thus, when the productivity diferential namely, a threshold value such that, in absolute value, the sum of the first two terms exceeds the remaining two terms (which do not depend on R ) in the above derivative.

Unskilled jobs: To show that the locus has a positive slope, it is suficient to show that

\[\frac {d \phi}{d \theta} = - \frac {\partial F _ {N} / \partial \theta}{\partial F _ {N} / \partial \phi} > 0.\]

As before, taking the derivative of with respect to and dividing all terms by yields

\[\begin{array}{r c l} \frac {1}{\triangle} \frac {\partial F _ {N}}{\partial \theta} & = & q ^ {\prime} (\theta) \psi \left[ \frac {\phi}{r + s + \theta q (\theta) \eta \beta} + (1 - \phi) \frac {1 - \phi}{\lambda_ {2}} \right] \\ & & + \left[ \frac {q (\theta) \phi}{r + s + \theta q (\theta) \eta \beta} + \frac {q (\theta) (1 - \phi)}{\lambda_ {2}} \right] \cdot \frac {\partial \psi}{\partial \theta} \\ & & - \left[ \frac {\beta q (\theta) \psi \phi}{[ r + s + \theta q (\theta) \eta \beta ] ^ {2}} \right] \cdot \frac {\partial \eta \theta q (\theta)}{\partial \theta} + \frac {\psi \theta q (\theta) (1 - \phi)}{[ \lambda_ {2} ] ^ {2}} \cdot \frac {\partial \lambda_ {2}}{\partial \theta}, \end{array}\]

which is unambiguosly negative since and

Similarly, the derivative is given by

\[\begin{array}{l l} \frac {1}{\triangle} \frac {\partial F _ {N}}{\partial \theta} & = \left[ \frac {q (\theta)}{r + s + \theta q (\theta) \eta \beta} - \frac {q (\theta)}{\lambda_ {2}} \right] \cdot \frac {\partial \psi \phi}{\partial \phi} \\ & - \frac {\beta \psi \phi q (\theta)}{\left[ r + s + \theta q (\theta) \eta \beta \right] ^ {2}} \cdot \frac {\partial \eta \theta q (\theta)}{\partial \phi} \\ & + \frac {q (\theta)}{\lambda_ {2}} \cdot \frac {\partial \psi}{\partial \phi} - \frac {q (\theta) \psi (1 - \phi)}{\left[ \lambda_ {2} \right] ^ {2}} \cdot \frac {\partial \lambda_ {2}}{\partial \phi}, \end{array}\]

where the first three terms are positive since and whilst the last term is negative since However, if and , so that , the second term dominates the fourth term in absolute value. Therefore

8 Appendix B: Equilibrium with unequal separation rates

In Section 5 we simulate the model with diferent job-separation rates for skilled and unskilled jobs, denoted, respectively, by and , with . In this case the steady state conditions for and can be written as

\[\eta \theta q (\theta) \phi u = s (n) (\mu - \phi u)\]

\[\eta \theta q (\theta) (1 - \phi) u = [ s (n) + \theta q (\theta) (1 - \eta) ] e (n, h)\]

\[(1 - \eta) \theta q (\theta) [ (1 - \phi) u + e (n, h) ] = s (s) [ 1 - \mu - (1 - \phi) u - e (n, h) ].\]

To determine the equilibrium value of ,the equations above are need to be complemented by the two free-entry conditions which, following the same arguments as in the derivation of equations (23) and (24), become

\[\frac {c}{q (\theta)} = \psi (1 - \beta) \left[ \phi \frac {y (n) - b}{r + s (n) + \theta q (\theta) \eta \beta} + (1 - \phi) \frac {y (n) - b}{\varsigma_ {2}} \right]\]

\[\frac {c}{q (\theta)} = (1 - \beta) (1 - \psi \phi) \left[ \frac {y (s) - b}{\varsigma_ {1}} - \theta q (\theta) \eta \beta \frac {y (n) - b}{\varsigma_ {1} \varsigma_ {2}} \right],\]

where

References

1. [1] ACEMOGLU, D., “ Changes in Unemployment and Wage Inequality: An Alternative Theory and Some Evidence ”, American Economic Review 89 (1999), 1259-78.
2. [2] ACEMOGLU, D., “ Patterns in Skill Premia ”, Review of Economic Studies 70 (2003), 199-230 .
3. [3] ALBA-RAMIREZ, A., “ Mismatch in the Spanish Labour Market, Journal of Human Resources 28 (1993), 259-278.
4. [4] ALBA-RAMIREZ, A. AND M.T. BLAZQUEZ, “ Mismatch, Overeducation and Labour Mobility in Spain”, (2003) Dept. of Ecomomics, Universidad Carlos III, mimeo.
5. [5] ALBRECHT, J., AND S. VROMAN, “ A Matching Model with Endogenous Skill Requirements ”, International Economic Review 43 (2002), 283-305.
6. [6] AUTOR, D., KATZ, L. AND A. KRUEGER, “Computing Inequality: Have Computers Changed the Labour Market ?”, Quarterly Journal of Economics 113 (1998), 1164-1214.
7. [7] BENTOLILA; S. AND J.F. JIMENO, “ Spanish Unemployment: The End of the Wild Ride ?”, (2003) FEDEA Working Paper 2003-12.
8. [8] BORJAS, G. AND V. RAMEY, “Foreign Competition, Market Power and wage Inequality”, Quarterly Journal ofEconomics 110 (1995), 1075- 1120.
9. [9] BRUNELLO, G.“ Equilibrium Unemployment with Internal Labour Markets”, Economica 63 (1996), 19-35.
10. [10] BURGESS, S. M., “A Model of Competition between Employed and Unemployed Job Searchers: An Application to the Unemployment Outflow Rate in Britain”, Economic Journal 103 (1993), 1190-1204.
11. [11] DOLADO, J. J., FELGUEROSO, F., AND J.F. JIMENO, “Youth Labour Markets in Spain: Education, Training, and Crowding-Out ”, European Economic Review 44 (2000), 943-56.
12. [12] EUROSTAT, “ School Leavers in Europe and the Labour Market efects of Job Mismatches” Population and Social Conditions (2003), Part II of Theme 3-5/2003.
13. [13] GAUTIER , P. A., “ Search Search Externalities in a Model with Heterogeneous Jobs and Workers ”, Economica 273 (2002), 21-40.
14. [14] GAUTIER , P. A., VAN DEN BERG, G. A., VAN OURS, J.C. , AND G. RIDDER, “ Worker Turnover at the Firm Level and Crowding Out of Lower Educated Workers ”, European Economic Review 46 (2002) 523-38.
15. [15] GROOT, W., “ Overeducation and the Returns to Enterprise-related Schooling” Economics of Education Review 12 (1993) 299-309.
16. [16] GROOT, W., “ The Incidence of and Returns to Education in the UK” Applied Economics 28 (1996) 1345-1350.
17. [17] HARTOG, J, AND H. OOSTERBEEK, “ Education, Allocation and Earnings in the Netherlands” Economics of Education Review 7 (1988) 185-194.
18. [18] MCKENNA, C.J., “ Education and the Distribution of of Unemployment, ” European Journal of Political Economy 12 (1996), 113-132.
19. [19] MORTENSEN, D. T., AND C. PISSARIDES, “ Unemployment Responses to ‘Skill- Biased Technology Shocks: The Role of Labour Market Policy, ” Economic Journal 109 (1999), 242-65.
20. [20] OECD, “ Education at the Glance” (2001), Paris.
21. [21] OLIVER, J. AND J.L RAYMOND, “ Educación Formal y Demanda de Cualificación de la Mano de Obra en España: Una Visión A Largo Plazo” Document d Économia Industrial no. 14 (2003) Centre d Économia Industrial, Universitat Autónoma de Barcelona.
22. [22] PISSARIDES, C, “ Search Unemployment with On-the-Job Search”, Review of Economic Studies 61, 457-475.
23. [23] SHIMER, R., “ The Impact of Young Workers on the Aggregate Labor Market, ”, Quarterly Journal of Economics 116 (2001), 986-1007.
24. [24] SHIMER, R., “ A Note on Nash Bargaining with On-the-Job Search” , University of Chicago (mimeo).
25. [25] SICHERMAN, N., “ Overeducation in the Labour Market” Journal of Labor Economics 9 (1991) 101-112.
26. [26] THUROW, L.C., Generating Inequality, Basic Books; New York.
27. [27] VERDUGO, R. AND. N. VERDUGO, “ The Impact of Surplus Schooling on Earnings: Some Additional Findings” Journal of Human Resources 24 (1989) 626-643.

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