(Dis)Assortative Matching and (Un)Directed Search
1Luca Paolo Merlino
Universitat Autònoma de Barcelona October 2007
1I would like to thank Melvyn Coles and Antoni Calvó Armengol for help and guidance; Caterina Calsamiglia, Emiliano Carluccio and participants in the Microeconomic Workshop at the Universitat Autònoma de Barcelona for their helpful com- ments. Usual disclaimers apply. Financial support from the Universitat Autònoma de Barcelona is gratefully acknowledged.
Abstract
When there is heterogeneity in a labour market, a natural question is whether the good jobs go to the good workers and viceversa. I propose a model of a frictional labour market where assortativity depends on the ability of the workers to direct their search strategies to a particular type of vacancy. This allows to study the welfare e¤ects of assortativity in a calibrated version of the model. Surprisingly, while total unemployment is reduced because of the reduction of the sorting externality, assortativity does not make everybody better o¤. Factotum workers face a decrease in the arrival rate of o¤ers and hence are more likely to be unemployed. The reduced competition among types of vacancies results in a higher proportion of low tech vacancies and lower production. Hence, only unskilled workers receive higher wages.
JEL Codes: J21, J31, J64.
Keywords: unemployment, search, assortative matching.
1 Introduction
When there is heterogeneity in a labour market, a natural question is whether the good jobs go to the good workers and viceversa. In other words, if the matching between posted vacancies and workers with di¤erent characteristics is assortative or not. In models of the labour market with frictions, usually multiple outcomes can be sustained in equilibrium encompassing the two polar cases, and the issue is in which region of the parameters’values there is one or the other result.
Here, I propose a model delivering a unique prediction in terms of the assortativity of the matches which can be either assortative or disassortative depending on the ability of the workers to direct their search or not. Hence, the existence of di¤erent channels where di¤erent types of vacancies are posted favours the assortativity of the matches by forcing the workers to choose where to look for a job.
Furthermore, when equilibria are multiple, di¤erent regions of the parameters space typically sustain di¤erent equilibrium outcomes and a comparison between assortative and disassortative matching is not possible. Without the hurdle of multiplicity, a welfare analysis of disassortativity is possible in a calibrated version of the model. Such an exercise is interesting because, while assortativity reduces the overall level of unemployment, total output is reduced and the welfare e¤ect on the workers will depend on their skill level. Quite surprisingly, the less skilled workers are the ones that bene…t the most by the possibility to direct search strategies to only one particular type of vacancy.
The environment is the following. There is a continuum of workers types, which we will refer as skills. Ex ante homogenous …rms open either one of two types of vacancies: a traditional one where all the workers have the same productivity, and a high-tech one whose productivity depends on the skills of the particular worker …lling it. The proportion of vacancy opened of each type is determined in equilibrium. Hence, …rms are ex ante homogenous, but they will end up being di¤erent depending on the type of vacancy they decide to post. We label low skilled the workers which apply only for traditional vacancies, high skilled the ones that apply only for high-tech vacancies. There can also be factotum workers which apply for both types of vacancies. These classes are endogenously determined in the model, depending on the equilibrium acceptance sets of …rms and workers.
The matching between type of vacancy and skills of the workers is assortative when it is never the case that a less skilled worker gets a job with higher skill requirements than a more skilled one. In the model, this happens in equilibrium precisely when such a middle class of workers does not exist. I show that in the proposed framework complete assortativity will never occur if the matching is totally random, à la Pissarides (2000). On the other hand, if the workers are able to direct their search to one type of vacancy, the equilibrium will be always assortative.
Although search is directed in a way close to Moen (1997), I am not relaxing the assumption of Nash bargaining of wages in order to allow a welfare comparison with the undirected case, at the cost that the equilibrium does not deliver competitive wages.
Even if the possibility of directed search reduces the overall level of unemployment, it does not make everyone better o¤, neither in terms of unemployment rate nor in terms of wage. Indeed, assortativity has three contrasting e¤ects on the labour market. First, it eliminates the sorting externality present in the random matching scenario. As a result, total unemployment is lower.
Second, it reduces the arrival rate of o¤ers for some of the workers that before were candidates for
di¤erent types of vacancies, the factotum, which now are unemployed with a higher probability.
Lastly, directed search softens the competition among di¤erent types of vacancies, modifying the composition of vacancies which are posted in favour of the low-tech ones. As a result, total production is lower and the skilled workers end up receiving a slightly lower wage.
In both scenarios, assortative and disassortative, skill-biased technological change reduces over- all unemployment rate and inter-group inequality while increasing intra-group inequality for the most skilled workers.
The most related work is Albrecht and Vroman (2002), which studies the e¤ect of skill biased technological changes in a frictional labour market. In their model, they consider a labour market with random matching with only two types of workers, skilled and unskilled, and a production technology very severe with the unskilled workers: their productivity is zero when employed in the high-tech jobs. Both an assortative and a disassortative equilibrium are possible depending on the values of the parameters. In the present model, by introducing a continuum of workers’
characteristics, I obtain that a complete assortative matching is possible only in the limit case when the labour market tightness goes to in…nity. On the other hand, we obtain similar e¤ects of a skill biased technological change on inequality, even though here technological change does not increase overall unemployment as in their model.
Wong (2003) modi…ed the random matching scenario of Albrecht and Vroman (2002) using a production technology that penalises less dramatically the low skilled workers employed in the high-tech …rms. By doing so, the e¤ects of technological change on wages and unemployment di¤erentials are the same as in this paper. But since there are only two types of workers, there are di¤erent types of equilibria in her model, some of them assortative and others disassortative, and a full ‡edged welfare analysis is not possible. This highlights that the great deal of heterogeneity introduced using a continuum of workers’ skills is necessary to always get disassortativity in a random matching model.
In the directed search literature, even when the outcome in terms of assortativity is unique, a comparison between assortative and disassortative matching usually is not possible. With a discrete number of types, the matches are not completely assortative for example in Shi (2002) or Shimer (2005a). Introducing a continuum of skill levels of the workers, the analysis is clear cut in terms of prediction of the type of matches resulting in equilibrium, providing a benchmark to compare the welfare e¤ects of assortativity. The main di¤erence with these papers though is that here we do not consider a wage posting game. The search of the workers is directed in the sense that they are able to choose the type of job they are looking for as in Moen (1997), as if the two vacancies were posted in two di¤erent islands that constitute the economy. But here wages are set using Nash bargaining solution rather than by the competition among market makers. As a result, the equilibrium is not competitive, but has the nice property of being comparable with the equilibrium of the completely random matching scenario, allowing for welfare comparisons.
The paper develops as follows. In Section 2, the model is formally introduced. In Section 3, the completely random matching is considered: there is only one search channel where both type of vacancies are posted and the workers have to decide their application strategies. After analysing the decision making of the agents, the acceptance sets of the two positions are determined as a function of the equilibrium labour market tightness and the proportion of types of …rms. The steady state equilibrium is characterised, using the free entry conditions for both kinds of positions and the equilibrium between in‡ow and out‡ow from unemployment status. In Section 4, directed search is introduced: the two vacancies are posted in two di¤erent channels, or islands, and the
workers have to decide which channel to use. The welfare of di¤erent agents in the two scenarios is analysed and compared in Section 5, using the model calibrated on the US economy. Section 6 concludes.
2 The Model
There is a continuum of workers and …rms, both risk neutral, in…nitely lived and normalised to mass one. Time is continuous.
Each worker di¤ers in the sense of its personal characteristics, which can be thought of as skills, or ability in occupying a skilled position. This is not human capital, in the sense that it is not a decision variable. Hence, we label each worker by its type, x, which we assume is distributed according to a uniform distribution between 0 and 1.
Jobs are even vacant or …lled. Once a job is …lled, production takes place and a wage is paid.
The wage is determined through the generalised Nash bargaining solution, with a proportion of the surplus being given to the workers. The position is in place inde…nitely until it is hit by shock at an exogenous rate . In this case, the position is closed and the agent goes back to the labour market as an unemployed worker. There is no recall of the matches. Renegotiation of the contract is not possible, and there is not on-the-job search.
It is possible to produce the unique good in two possible ways. One is a traditional technology, labelled by L, which gives raise to a production of y units of the good independently of the type of the worker. The other is a high technology, labelled by H, whose productivity depends linearly on the type of the worker. Consider 1and 1 < y < , then:
Y (x; y) = y if y = L x if y = H
Assume that the market for capital is perfect, so that we can consider each position as a …rm.
The productivity of a match between a worker and a …rm can be depicted as in Figure 1 depending of the type of worker a …rm is matched to and the kind of position a worker is matched to.
In the labour market, unemployed workers and vacancies are assumed to meet each other randomly according to a matching function, M (u; v), which is constant returns to scale. Hence, M (u; v) = M (1;uv)u = m( )u, = uv being the labour market tightness of the labour market.
Assume Mu(u; v); Mv(u; v) > 0, limu!0M (u; v) =1 and the usual Inada conditions limv!0M (u; v) = 0, that translates into m0( ) > 0, lim !1m( ) =1 and lim !0m( ) = 0.
Each worker hears about vacancies at the same rate, independently of her type. The vacancies are of two types: low skilled and high skilled. is the proportion of the vacancies that are low skilled. Hence, since the arrival rate of vacancies to the workers is m( ), such rate for low skilled and high skilled vacancies is m( ) and (1 )m( )respectively.
All vacancies have the same arrival rate of workers m( ), but di¤erent types of workers show up each time according to a function u(x) that describes the number of the unemployed workers across types. Note that in general the distribution of unemployment is di¤erent from the distribution of types across the population of the workers and it is determined endogenously.
In the good market, there is free entry of …rms, so that they will enter in the labour market until it is pro…table.
Figure 1: The productivity of the two technologies employing di¤erent kinds of workers.
Before meeting, agents are imperfectly informed about the location of the workers’and …rms’
types, as capture by the matching technology. But once they met, they recognise immediately the others’type. Hence, even if agents cannot select their partner ex ante, they can do so ex post.
A match will be formed only if it is pro…table for both parties and …rms will pay di¤erent wages to di¤erent types of workers. Given the Nash bargaining determination of wages, the matches are consummated only if the joint surplus that would be generated by the match if formed is nonnegative. De…ne E(x; y) and J (x; y) as the value of the job for workers of type x and for …rms using technology y respectively, U (x) as the value of being unemployed for the worker of type x and V (y) as the value of a vacancy of type y. Thus, I can write that when a match (x; y) is formed, a job contract is signed if:
E(x; y) + J (x; y) U (x) + V (y) (1)
Call MLand MH the acceptance set of a …rm of type L and H respectively, i.e., the set of types of workers that she will be willing to employ. Hence, when a worker of type x and a …rm of type y meet, the match is consummated if and only if x 2 My. Formally,
My =fx 2 [0; 1] : E(x; y) + J(x; y) U (x) + V (y)g; y = H; L
When the contract is signed, wages are set such that the surplus of the workers will be a fraction of the total surplus. Hence:
E(x; y) U (x) = [E(x; y) + J (x; y) U (x) V (y)] (2) The value of a job for a …rm of type y employing a worker of type x at a wage w(x; y) is given by:
J (x; y) = Y (x; y) w(x; y) + V (y)
r + (3)
where r is the discount rate used by both the …rms and the workers. This follows by the usual
formula for the asset function of a job for a …rm of type y employing a worker of type x at a wage w(x; y), namely
rJ (x; y) = Y (x; y) w(x; y) + [V (y) J (x; y)]
Similarly, the value for the worker of the same match is given by:
E(x; y) = w(x; y) + U (x)
r + (4)
Let u(x) be the distribution function of the unemployed workers across the workers’types and a the cost of a vacancy. Then, the instant value for a …rm of a posted vacancy of type y is given by
rV (y) = a + m( )Z
My
[J (x; y) V (y)]u(x)dx (5)
The workers show up to vacancies according to the distribution of unemployment, u(x). On the other hand, the …rm is potentially interested in a worker only if she belongs to her acceptance set My.
Hence, for a vacancy to be posted, it is necessary m( )Z
My
[J (x; y) V (y)]u(x)dx a
In equilibrium, since there is free entry of …rms, V (H) = V (L) = 0, and the condition (1) for both type of jobs when …rms meet a worker of type x becomes:
y rU (x) x rU (x) Hence, the acceptance sets can be written as:
ML=fx 2 [0; 1] : y rU (x)g (6)
MH =fx 2 [0; 1] : x rU (x)g (7)
The value of being unemployed for a worker of type x is given by:
rU (x) = b + m( ) maxfE(x; L) U (x); 0g (8)
+(1 )m( ) maxfE(x; H) U (x); 0g
being the proportion of the vacancies that are low skilled and 1 < b < y the instantaneous value of leisure, or the unemployment bene…t.
Using (2) the wage paid when a job contract is formed can be derived. The wage is given by the sum between the productivity of a worker and her value of being unemployed, weighted by her bargaining power, that is:
w(x; y) = Y (x; y) + (1 )rU (x) (9) There are three possible cases as regarding the sorting of consummated matches across types:
(i) unskilled workers: types that accept only a job in the traditional sector;
(ii) skilled workers: types that accept only a job in the high technology sector;
(iii) factotum workers: types that accept both kinds of jobs.
Following Burdett and Coles (1999), one can think at these three cases as di¤erent classes of workers. The measure of each class is endogenously determined, since it depends on the conditions of the labour market. Obviously the di¤erence in productivity of the two technologies across the workers’types is an important determinant, but it is not the only factor. Indeed, the value function of being unemployed is di¤erent for each class, re‡ecting the di¤erent opportunities each type of worker is facing.
Depending on the composition of these categories of workers in equilibrium, it is possible to de…ne the assortativity of the consummated matches, that is, whether the better workers go to the better technology.
De…nition 1 An equilibrium is assortative when the following conditions are satis…ed:
(i) if x 2 ML, then 8x0 < x, x0 2 M= H. (ii) if x 2 MH, then 8x0 > x, x0 2 M= L.
In other words, the matches are assortative whenever, considering two workers of di¤erent types, it is not possible that the better worker is employed in a …rm employing the traditional technology and the other is employed in a high-tech …rm, and viceversa. Hence, assortativity is a property which depends on the shape of the acceptance sets. Following (6) and (7), these depend on the shape of U (x), which is di¤erent whether the search strategy of the workers is directed or not.
This is why the two scenarios will deliver di¤erent outcomes in terms of assortativity.
3 The Random Matching Case
When the workers are not able to direct their search to one type of vacancy only, all the workers are in the same market and the matching rates between workers and …rms depend on the total labour market tightness and the proportion of …rms o¤ering each type of vacancy. As a result, the value to a worker of being unemployed can take three di¤erent values according to the acceptable matches she has in the labour market.
Case 1 - When a worker accepts only low type positions, from (8) the value they get from unemployment is
rU (x) = b + m( ) [E(x; L) U (x)]
Figure 2: The function U (x) as the upper envelope of the U (x) of the workers accepting di¤erent types of vacancies.
Hence, using (3), (4), (5) and (9), we get
rU (x) = b(r + ) + m( )y r + + m( )
Case 2 - When a worker accepts only a high technology positions, from (8) the value of being unemployed is given by:
rU (x) = b(r + ) + (1 )m( ) x r + + (1 )m( )
Case 3 - When a worker accepts both types of positions, from (8) the value of being unemployed is given by:
rU (x) = b(r + ) + m( )y + (1 )m( ) x r + + m( )
Notice that U (x) is the upper convex envelope of the functions U (x) in the three intervals, as depicted in Figure 2. Furthermore, from its derivation, it is clear that U (x) is continuous on [0; 1]
in the workers’types.
Since U (x) is increasing in x, we can write the acceptance set of the …rms as reservation rules.
According to (9), the wages paid to the workers depend both on their productivity and on the value of being unemployed. Since the productivity of the low-tech position is constant, the …rms o¤ering such a kind of position will accept a worker as long as her type is low enough that the wage that needs to be paid is not higher than the surplus generated by the position. On the other hand, the productivity of the high-tech positions is increasing in the workers’types. Hence, the
…rms o¤ering this kind of positions will be willing to employ workers whose productivity is high enough to generate enough surplus to allow for the wage to be paid. Thus, the acceptance sets are:
ML =fx 2 [0; 1] : x xg MH =fx 2 [0; 1] : x xg
Using U (x), we can derive these thresholds. The fact that y > b assures that an unskilled worker will always accept a low-tech job. But as the workers’type increases, so does U (x), and she will be more willing to accept a job in the advanced sector as well. Using (6), we know this will not happen as long as:
b(r + ) + m( )y r + + m( ) > x
U (x) actually is a constant for the workers accepting only the low-tech positions and does not depend on x. This means that there is a unique threshold below which all the types will look only for a low skilled job. This threshold x is given by the workers’type such that the worker is indi¤erent whether to accept a high-tech position or not. Hence,
x = min b(r + ) + m( )y
(r + + m( )); 1 (10)
The other threshold we are interested in is the one beyond which a worker will accept only a high-tech position. This, using (7), is re‡ected by the condition:
y < b(r + ) + (1 )m( ) x r + + (1 )m( )
Notice that the right-hand side of the expression is a constant, while U (x) is increasing (strictly) in x, which implies that U (x) crosses the line y at most once. Then, the fact that r + > 0 and b 0 ensures that the lines actually do cross. The threshold x then is given by:
x = min (y b)(r + ) (1 )m( ) + y
; 1 (11)
Note that if x = 1, it means that nobody will look for highly skilled positions only, while if x= 1 nobody will look for low skilled positions only.
Hence, in this framework an equilibrium is assortative when x and x coincide, because if not it is possible that skills and requirements are not aligned after the matches randomly took place.
Then:
Proposition 1 If an equilibrium exists with < 1 and both type of jobs are o¤ered, it is disas- sortative.
Proof. The fact that ML\ MH 6= ; follows immediately by the fact that y > b and r + > 0 implies x < y= and x > y= .
The heterogeneity of the workers and the friction on the labour market create a sorting exter- nality that translates into a crowding out between workers of di¤erent types.
The proposition stresses that when we move from a scenario where there are only two types of workers as in Albrecht and Vroman (2002) to one with a continuum of types, an assortative equilibrium can emerge only as a limiting case when the labour market tightness goes to in…nity.
Hence, if workers are allowed to look for both kinds of positions, a positive measure of them will always do it, and the equilibrium will be disassortative. The result is not due to the fact that Albrecht and Vroman (2002) uses a more discriminatory technology, i.e. one were the unskilled workers are not productive if employed in skilled positions. Indeed, Wong (2003) …nds that one of the equilibria is assortative allowing for such a possibility.
If we call u(x) the function that describes the proportion of type x workers that are unemployed, we can de…ne an equilibrium as the values of , and the function u( ) which are consistent with the optimization problems of the workers, described by U (x), the optimization problem of the …rms with free entry, described by the zero pro…t conditions, and with the equilibrium on the labour market, meaning that in a steady state the measure of workers that lose their job in each period because hit by a shock has to be equal to the measure of workers that …nd a job in that period.
This steady state condition has to take into consideration the fact that the workers are divided into three di¤erent classes, each of which has a di¤erent probability to …nd a job because of the di¤erent arrival rates of o¤ers. Because of the uniform distribution of the workers’types, one gets:
(1 u1(x)) = u1(x)m( ) for 0 x < x (1 u2(x)) = u2(x)m( ) for x x x (1 u3(x)) = (1 )u3(x)m( ) for x < x 1
From these steady state conditions, we obtain a distribution function of unemployed workers which is a step function (see Figure 3) de…ned as follows:
u(x) = 8>
<
>:
+ m( ) for 0 x < x
+m( ) for x x x
+(1 )m( ) for x < x 1
(12)
The rate of unemployment then di¤ers among classes of workers depending on and , since both a¤ect the job arrival rate faced by each worker type. Indeed, in the undirected search scenario, the factotum workers, the ones that look for both kinds of vacancies, are rewarded in terms of unemployment because they will consummate each random match proposed to them. The total unemployment rate u is derived by the integral of u(x) over the whole interval [0; 1].
As for the free entry conditions, we obtain from (5) seL( ; ) = 1
r +
m( )Z
ML
y rU (x) u(x)dx a = 0 (13)
seH( ; ) = 1 r +
m( )Z
MH
( x rU (x)) u(x)dx a = 0 (14)
which de…ne the equilibrium market tightness and the proportion of low skilled vacancies because
1
0 x x
u1(x)
u2(x) u3(x)
Figure 3: The distribution of unemployment across workers’types when > :5.
using (10), (11) and (12) we are left with two equations in two unknowns. The …rst term of the two conditions is the part of the surplus created by the consummated matches that goes to the
…rm, discounted by the interest rate and the job destruction rate, and pondered by the rate at which the vacancy is …lled. The second term is the cost of opening a vacancy. Hence, the two expressions are the expected net pro…ts in terms of surplus of opening a vacancy.
Three di¤erent kinds of equilibria can arise depending on the types of positions which are o¤ered.
Lemma 1 The labour market tightness and the proportion of low skilled positions are an equilibrium if and only if:
(i) ( ; ) are such that seL( ; ) = seH( ; ) = 0 and both kinds of vacancies are opened.
(ii) ( ; 1) are such that seL( ; 1) = 0 and seH( ; 1) < 0 and only low skilled vacancies are opened.
(iii) ( ; 0) are such that seL( ; 0) < 0 and seH( ; 0) = 0 and only high skilled positions are opened.
It is obvious why (i) is an equilibrium: both free entry conditions are satis…ed and we have an interior solution. An example of this kind of equilibrium is depicted in Figure 4, namely in point E.
As for (ii) and (iii), there is only one type of …rm. Hence the labour market tightness is determined considering only the type of vacancy present. In order for this to be an equilibrium, it is necessary that the other type of …rm does not want to enter because it is not pro…table to do so, i.e., if the value of opening a vacancy as described by (13) and (14) is negative. An example of low-type equilibrium is depicted in Figure 5, provided that when = 1 the pro…ts of a high tech
…rm are negative when they are zero for a low tech …rm.
So far nothing has been said regarding the conditions under which an equilibrium of the model does exist. This is the aim of the next proposition.
Proposition 2 Giving the parameters a 2 R++, 0 b 1 < y, 2 (0; 1) and a matching function satisfying the Inada conditions, there exist and that de…ne an equilibrium of the labour market with density of unemployment described by u( ; ).
Figure 4: An example of mixed type equilibrium when both kinds of …rms open vacancies.
Figure 5: An example where there exists a low type equilibrium, given that when = 1 the pro…ts of a high tech
…rm are negative when they are zero for a low tech …rm.
Proof. In order to simplify the argument, the proof will be given for the case b = 0, but it can be easily generalised to the case 0 < b < y.
Given 2 [0; 1], the functions seL( ; ) and seH( ; ) have the following properties:
(a) lim
!0seL( ; ) = lim
!0seH( ; ) = K > 0. When ! 0, x ! 0 and x = 1, because the value of being unemployed for the workers tends to zero and the …rms get the entire surplus from production.
Furthermore, the fact that m( ) satisfy the Inada conditions imply that lim
!0
m( ) =1, ensuring that the limit is positive for every parameters value.
(b) lim
!1seL( ; ) = lim
!1seH( ; ) = a. When ! 1,the value of being unemployed for the workers tends to the value of the surplus from production, and the …rms still have to pay the cost of the vacancy.
(c) seL( ; )and seH( ; ) are continuous functions, since they are products and sums of continuous functions.
These facts imply that for both functions there exists a value of , e, which equalises the functions to zero. Hence the loci of that equalise the value of opening a vacancy to zero for each value of , describe two correspondences in , that we will call zL( ) and zH( ) for the low skilled and the high skilled positions respectively. For the same argument used above, seL( ; ) and seH( ; ) are continuous also with respect to ( ; ), and hence the two functions zL( ) and zH( ) are continuous. We can hence have two cases:
Case 1 There exists (e; e) such that e = zL(e) = zH(e). Then, this is an equilibrium under (i) of Lemma 2.
Case 2 There does not exists e such that e = zL(e) = zH(e). Then, one of the two things must happen:
(i) when = 0, zL(0) < zH(0) = e;but this means that seL(e; 0) < 0 and seH(e; 0) = 0, and this is a equilibrium where only high-tech vacancies are posted, under (ii) of Lemma 2;
(ii) when = 1, e = zL(1) > zH(1);but this means that seL(e; 1) = 0 and seH(e; 1) < 0, and this is an equilibrium where only low-tech vacancies are posted, under (iii) of Lemma 2;.
Suppose neither (i) nor (ii) were satis…ed. Then, when zL(0) < zH(0) and zL(1) > zH(1). But because of the continuity of zL( ) and zH( ), this would imply that there exists (e; e) such that e = zL(e) = zH(e), a contradiction.
Note that in general, we cannot say that @seL@( ; ) 0 and @seH@( ; ) 0. Even if an increase in the labour market tightness means that the number of vacancies increases with respect to the workers and that the value of opening a vacancy for a …rm is lower, the change in the labour market tightness induces a change in the relative proportion of …rms. Such a composition e¤ect is very strong for values of close to 0 or to 1.
4 The Directed Search Case
In the undirected scenario we assumed that all vacancies are posted in the same market, so that in each period there are matches which are produced by the matching technology but that are not consummated by the agents because they are not pro…table enough. But if the two vacancies where posted on two di¤erent markets that constitute the economy, if U (x) is well behaved, this sorting externality might be reduced since the workers have to decide in which market to look for a job. Or, in other words, they have to decide where to direct their search. The framework is similar to Moen (1997): each market can be interpreted as a separated island where only one type of vacancy is posted. In each island the matching technology is the same as in previous section.
The framework di¤ers from Moen (1997) in one important aspect: the competitive mechanism of market creation operating in that model cannot work here because there are only two markets.
Wages are not competitive, but determined through Nash bargaining as before. This allows for an easy welfare comparison between the two scenarios by keeping them as close as possible.
Another interpretation is that the workers can look for a job in di¤erent ways and they have to decide which channel to use given that they can make only one application. This approach is followed in the literature on social networks in the labour market pioneered by Montgomery (1991). For example, in Bentolila et al. (2004), Kugler (2003) and Fontaine (2007), workers have to decide whether to look for a job in a formal channel or through social network, two channels which di¤ers in the e¢ ciency of the matching process, and these are modelled as di¤erent labour market coexisting in the economy.
Here the aim in nonetheless di¤erent: since the workers have to direct their search, the matches will be assortative and hence the model will deliver an assortative equilibrium that we can use as a benchmark to study the welfare e¤ects of disassortativity for the total economy and across the di¤erent classes of workers.
In the economy made of two islands, the labour market tightness of the two channels can di¤er. L and H are the labour market tightness of the unskilled and the skilled labour market respectively. Since in each channel only one type of vacancy is posted, I can now use the framework introduced in Section 2 imposing = 1 in the channel for traditional vacancies and = 0 in the other.
As before, each worker decides which type of job to look for depending on her value of being unemployed, given by:
rU (x) =
( b(r+ )+ m( L)y
r+ + m( L) for x 2 ML
b(r+ )+ m( H) x
r+ + m( H) for x 2 MH
Given Land H, the value of being unemployed in both channels is continuous in x, constant in the market where low-tech vacancies are posted, and increasing in the workers’type in the other.
Hence, there is worker type, ex, which is indi¤erent between the two labour markets. This de…nes the threshold beyond which all workers will apply for a high position, while lesser skilled worker will go to the market for the unskilled positions.
This cut.-o¤ rule derived by the properties of U (x) assures that assortativity of the matching, and the following Proposition follows immediately.
Proposition 3 When the workers are allowed to direct their application, the labour market equi- librium, when it exists and both type of vacancies are posted, is assortative.
In this scenario, every match that takes place will be consummated, and the sorting externality among di¤erent worker types is reduced. The acceptable sets are given by:
ML0 =fx 2 [0; ex]g MH0 =fx 2 [ex; 1]g
As before, the steady state conditions assure that the measure of workers losing a job is the same as the measure of workers that …nd one. But now there are only two conditions since there are only two search strategies:
(1 uL(x)) = uL(x)m( L) for 0 x <ex (1 uH(x)) = uH(x)m( H) for ex x x
These conditions determine uk(x), k = H; L, i.e., the distribution of unemployed workers in each market.
The free entry conditions for low and high positions respectively are:
(1 )
r +
m( L)
L
Z
ML0
y rU (x) uL(x)dx = a (15)
(1 )
r +
m( H)
H
Z
MH0
( x rU (x)) uH(x)dx = a (16)
Note that while (15) is increasing in ex, (16) is decreasing, because changes in ex re‡ect changes in the measure of workers of each market, and hence the probability for a vacancy to reach an unemployed worker. Hence the value of ex in‡uences the labour market tightness, positively for the low market and negatively for the high market. Hence, the equilibrium in the directed search scenario can be de…ned as follows:
De…nition 2 The labour market tightness L and H and the cut-o¤ value ex are an equilibrium of the directed scenario if:
a) given L and H, ex is such that for x ex, b(r+ )+ m(r+ + m( LL))y b(r+ )+ m(r+ + m( HH) x) and for x > ex,
b(r+ )+ m( L)y
r+ + m( L) < b(r+ )+ m(r+ + m( H) x
H) ; b) given ex, L is such that (15) holds;
c) given ex, H is such that (16) holds.
The fact that (15) is continuous, increasing in ex and increasing in L and (16) is continuous, decreasing in ex and increasing in H, ensures that such an equilibrium exists.
Proposition 4 When workers are able to direct their search, there always exist an equilibrium.
An important feature of the directed search scenario is that the solution in each sector depends on the cut-o¤ levelex, the type which is indi¤erent between entering in the either of the two markets.
The value of ex is in general di¤erent from y= because of the friction in the labour market. If ex
were given, we would have two independent standard labour markets, facing free entry conditions downward sloping in the labour market tightness. But it depends on the gains for the workers from looking for a job in each sector, and hence on the labour market tightness. As a result, a change in the labour market tightness has ambiguous e¤ects on the free entry condition.
This translates into an impossibility to obtain analytical results on the welfare of the workers when both type of vacancies are posted in equilibrium. So, I calibrate the model to allow a comparison among the random matching and the directed search scenario.
5 Calibration and Welfare Comparison
In order to make a welfare comparison of the assortative and disassortative equilibrium, it is necessary to simulate the model assigning the values of its parameters. I calibrate them following the criteria that the parameters of the model and the outcome they deliver should be reasonable, meaning that the model should be able to reproduce the characteristics of an observed labour market. Then I do a sensitivity analysis changing these values, as is usual in these kinds of models (Albrecht and Vroman, 2002; Wong, 2003).
I calibrate the model to US quarterly data following Shimer (2005b) . As for the interest rate, r = 0:012 equivalent to an annual discount factor of 0:953%. The separation rate is …xed to 0:1 consistent with jobs lasting on average two years and a half. The unemployment bene…t b is normalised to zero. As it is most common in the literature, I will use a Cobb-Douglas matching function, i.e., m( ) = M m. I …x m = :72 close to the upper estimates of Petrongolo and Pissarides (2001). In order for the Hosios rule to be satis…ed, the Nash bargaining power of the workers has to be the same to the m; hence I …x = :72.
As for the productivity of the two technologies, I use y = 1:5 and = 3. Hence the most skilled worker produces the double with the high technology compared to what he can do with the traditional technology.
In order to get an unemployment rate around 6%, a, the cost of opening a vacancy need to be 0:015 and M , the constant of the matching function, 1:35.
The equilibrium under these conditions is depicted in Figure 4. It is an equilibrium where both type of vacancies are posted and the proportion of the low-tech …rms is = :34, smaller than the proportion of …rms of the high type. As expected, the unemployment rate is not evenly distributed:
it is 7:5% for workers looking only for a traditional job, 3:44 for factotum workers, and 5:99% for skilled workers. Hence, the distribution of unemployment is such that workers looking only for a low-tech job face a higher unemployment rate than the others. The acceptance set are determined by x = :444 and x = :549, and the total unemployment rate turns out to be 6:39%. The labour market tightness is 2:76, meaning that there are more than two vacancies for each unemployed factotum worker. Given that the other workers accept only one type of vacancy, there are only :93 vacancies for each unemployed unskilled worker and 1:82 for a skilled one.
Let us now turn the attention to the directed search case. The segmentation cut-o¤ value, ex, is :491, between x and x. The labour market tightness for each channel turns out to be di¤erent in equilibrium: L is equal to 1:11, while H is 1:45, meaning that now there are more vacancies for each unemployed unskilled worker and less for a skilled one. Nonetheless, because of the reduction of the crowding out, unemployment rate is reduced for both, from 7:5% to 6:43% and from 5:98% to
Figure 6: Plot of the value of being unemployed: the dotted line represents the value of being unemployed when serach is directed, while the solid line the value of being unemployed in the market when search is undirected.
5:36% respectively. The overall rate of unemployment now is 5:88%, lower than in the undirected search scenario. But for the factotum workers unemployment increased independently on the labour market they ended up by directing their search, since it was 3:44% in the random matching case. Total output is reduced from 2:143 to 2:133.
The interpretation of these results is the following. Assortativity has three e¤ects on the labour market. First, it alleviates the sorting externality so that the total unemployment rate is reduced.
Nonetheless such gains are not spread uniformly across workers’ types. This is because of the second e¤ect: the factotum workers face a reduction in the arrival rate of job o¤ers because they have to direct their search to one of the type of vacancies. Thirdly, since in the directed search scenario the two di¤erent vacancies are posted in di¤erent channels, the competition between the two technologies is reduced. As a result, the proportion of …rms o¤ering low tech vacancies increases. Since there are more …rms that are less productive, total output decreases and the most skilled workers tend to be harmed.
Since wage are set through Nash bargaining in both the scenarios and for each type the pro- ductivity is the same in both scenarios, looking at the value of being unemployed in the cases is enough to compare wages. These are depicted in Figure 6.
Note that not all the factotum workers are hurt: some still enjoy the reduction of the sorting externality. On the other hand, the workers applying for high-tech positions are actually worse o¤ when the matching is assortative, because even if they face a lower level of unemployment, the proportion of …rms posting high-tech vacancies decreased too much. In the …gure, the unemploy- ment value of the workers applying for the high tech-job almost coincide, but it turns out that the line in the case of the random search model is slightly higher.
The results presented insofar do not qualitatively change when the cost of opening a vacancy changes, for example from 0:0015 to 0:0025, an interesting range since these costs give raise to a plausible unemployment rate, which increases both in general and for each workers’ type as a increases. Furthermore Table 1 and Table 2 show that as the cost of a opening a vacancy increases, the proportion of …rms which are low tech decreases, since the increase in the cost harms more the less productive …rms.
a J y S d x x u uL uM uH Y
. 0015 3 1. 5 2. 76 . 34 . 444 . 541 6. 39% 7. 5% 3. 44% 5. 98% 2. 143 . 0018 3 1. 5 2, 46 . 3395 . 439 . 553 6. 88% 8. 13% 3. 73% 6. 45% 2. 132 . 0020 3 1. 5 2. 29 . 339 . 436 . 556 7. 18% 8. 5% 3. 91% 6. 76% 2. 127 . 0025 3 1. 5 1. 99 . 338 . 43 . 563 7. 85% 9. 36% 4. 32% 7. 42% 2. 115
Figure 7: The equilibrium in the model with random search as the cost of opening a vacancy increases.
a J y S
LS
Hæ x u u
Lu
HY
. 0015 3 1. 5 1. 11 1. 45 . 4916 5. 88% 6. 43% 5. 36% 2. 133 . 0018 3 1. 5 . 99 1. 3 . 4907 6. 34% 6. 94% 5. 78% 2. 124 . 0020 3 1. 5 . 925 1. 21 . 4903 6. 64% 7. 27% 6. 05% 2. 118 . 0025 3 1. 5 . 805 1. 06 . 4893 7. 28% 7. 97% 6. 63% 2. 105
Figure 8: The equilibrium in the model with directed search as the cost of opening a vacancy increases.
The e¤ect of skill biased technological change is shown in Table 3 and Table 4, where , the productivity of the high tech jobs, increases from 2 to 5. When the high-tech …rm is not productive enough, namely = 2, in a random search framework, the equilibrium is low-type as the one represented in Figure 5: because of the labour market frictions, only low tech vacancies are posted despite the fact that the most skilled workers are more productive in the high tech job. When
= 2, in the directed search framework we have two possible equilibria: one that is essentially a low type equilibrium (the proportion of high skilled …rms is less than :011%) and the other where around 20% of the posted vacancies are high tech. Note that in a frictionless labour market, there would be exactly one quarter of the …rms of such type.
a J y S d x x u uL uM uH Y
. 0015 2 1. 5 3. 27 1 1 1 5. 29% 5. 29% ? ? 1. 420 . 0015 3 1. 5 2. 76 . 34 . 444 . 541 6. 39% 7. 5% 3. 44% 5. 98% 2. 143 . 0015 4 1. 5 3. 01 . 241 . 325 . 404 6. 04% 8. 96% 3. 24% 4. 83% 2. 972 . 0015 5 1. 5 3. 25 . 178 . 254 . 32 5. 69% 10. 3% 3. 08% 4. 21% 3. 878
Figure 9: The equilibrium in the model with random search with skill biased technological change.
a J y SL SH
æ
x u uL uH Y
. 0015 2 1. 5 1. 69 . 695 . 988 4. 79% 4. 82% 2. 13% 1. 434 . 0015 2 1. 5 1. 49 . 725 . 79 5. 95% 5, 27% 5. 95% 1. 517 . 0015 3 1. 5 1. 11 1. 45 . 4916 5. 88% 6. 43% 5. 36% 2. 133 . 0015 4 1. 5 . 92 1. 88 . 352 5. 5% 7. 35% 4. 48% 2. 965 . 0015 5 1. 5 . 78 2. 23 . 28 5. 15% 8. 14% 3. 99% 3. 842
Figure 10: The equilibrium in the model with directed search with skill biased technological change.
If the high technology is productive enough with respect to the other one, both type of vacancies are present. In such cases, the qualitative results on welfare are the same as before. In contrast with Albrecht and Vroman (2002), the overall level of unemployment decreases with the technological change, while, as in Shi (2002), the gains from such a change are not evenly distributes across all the types of workers. In particular, the unskilled workers are actually damaged in both scenarios.
Analogously, if the productivity of the high tech jobs would increase further, the equilibrium would eventually become a high-type equilibrium in the random search scenario, while for this to happen also in the directed search scenario, a sharper increase would be necessary, again from the fact that the competition among vacancies is reduced in this case.
Concluding, the qualitative e¤ect of a skill biased technological change is the same in both the random and the directed search scenario. Hence, on one side they con…rm the analysis of Shi (2002) while they contrast Albrecht and Vroman (2002). But as already noted by Wong (2003), their results were obtained using a production technology particularly punitive with respect to the unskilled workers when employed in the high tech jobs.
6 Conclusions
I studied an economy with heterogeneous workers and ex-ante homogenous …rms which can post two types of vacancies to compare welfare of the workers under two di¤erent matching processes.
The …rst is a classical random matching framework. Here, given the high degree of heterogeneity of the workers’types, the equilibrium is always disassortative, meaning that there is an imperfect positive assortative matching between the skills of the workers and the type of vacancies posted by the …rms. In the second framework, workers are able to direct their application to one of the two types of vacancies: in this case, the equilibrium always displays perfect positive assortative matching.
In contrast with the previous literature, these two di¤erent institutional frameworks imply di¤erent and unique predictions in terms of assortativity, which does not depend on value of the parameters of the model, as long both type of vacancies are posted in both scenarios.
This allows a welfare analysis of disassortativity in a calibrated version of the model. When search is undirected, some workers do look for both type of positions and, by doing so, they exert
a negative sorting externality on the other workers. In the second scenario this does not happen.
As a result, while these workers face a higher unemployment rate, the other workers face a lower probability to be unemployed and the overall level of unemployment in the assortative labour market is lower.
The elimination of the externality tends to increases the wages that the …rms pay to their workers. But it also favours the existence of the low tech vacancies by reducing the competition among the two types of technologies. As a result, only the unskilled workers receive a higher wage and total output is reduced.
Hence, while assortativity implies a lower total unemployment rate, it a¤ects negatively both total production and some of the workers types, both in terms of wages and unemployment rate.
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