Endogenous ranking in a two-sector urn-ball matching process

This paper contributes to the debate concerning the micro-foundation of matching functions in frictional labor markets. The focus is on a particular matching regime, i.e., the so-called urnball process. It is shown that in a two-sector economy, even in the presence of heterogeneous workers, the assumption of applicants-ranking may be misleading. Instead, the choice concerning the adoption of either ranking or no-ranking behavior is endogenous and it is affected by both the tightness of the two sectors and the composition of the labor force in terms of skills. Moreover it is proved that exogenous shocks may change the form of the matching function. This result casts additional doubts on the assumption of exogenous matching functions often made in empirical works aimed at assessing the effectiveness of policy measures. JEL J63 J64


Introduction
The matching function (from now on MF) represents an important tool that allows labor economists to model employment out- ‡ows and in- ‡ows in the presence of frictional labor markets (Diamond, 1982;Mortensen, 1989;Pissarides, 1987). In early theoretical models the functional form of MF has generally been assumed to satisfy some desirable properties such as concavity in the arguments and constant returns to scale. Recently, the issue of micro-foundation of MF has attracted researchers'attention giving rising to a ‡ourishing of studies highlighting that the assumed MF should be consistent with labor market behavior of …rms and workers (among others see Stevens, 2007). Moreover, according to Lagos (2000), Neugart (2003) and Brown et al. (2011) since agents' behavior can be a¤ected by labor market policies and institutions, the MF could be endogenous implying instability and vulnerability with respect to the Lucas critique.
This work enters the existing literature by exploiting a particular matching regime known as urn-ball process analyzed by Butters (1977) and Hall (1979) among the …rst. The urn-ball process, that nowadays is a popular mechanism among labor economists, is considered the …rst example of micro-founded MF and has proved to be a convenient instrument to describe the labor market when workers are heterogeneous since it makes possible to specify individuals'exit rate from unemployment as a function of their own characteristics (Blanchard and Diamond, 1994). This study considers the case of a urn-ball process in the presence of heterogeneous workers operating in a perfectly segmented two-sector economy. In particular, according to Gavrel (2009) and Moen (1999) the economy is characterized by graduate (high-tech) and undergraduate (low-tech) sector. Agents are heterogeneous and have to decide the sector they want to enter. Once the entry decisions have been taken, the pure matching process starts following the lines set out by standard matching models. However, di¤erently from Gavrel (2009) and Moen (1999), in the present paper it is not assumed ex-ante that …rms rank amongst the applications they receive. Instead, the ranking decision is left to be determined by agents' optimal actions. Using this framework it is shown that, although in the presence of heterogeneous workers the assumption of ranking may seem obvious, there can be standard economic environments where the speci…c form of the hiring process results from a more complex strategic behavior. The rationale behind this result turns out to be straightforward when the issue of sector tightness in terms of labor supply and demand is taken into account along with the composition of the labor force in terms of productivity. Indeed, …rms set the hiring behavior to maximize their expected actual value which depends on both the productivity of employees and the probability of …lling vacancies. Therefore, in the presence of a tight market the adoption of no-ranking may be suitable for …rms as far as it increases the labor supply and, consequently, the …rm's expected value.
The main implication of this model is that the resulting form of the urn-ball MF is endogenous and it is shaped by agents'microeconomic behavior. A corollary of this …nding is that exogenous shocks in ‡uencing agents'decisions may not only determine the number of matches formed in each sector but, most importantly, they may shape the entire form of the MF. This is extremely relevant when assessing the impact of speci…c policy measures on labor market equilibria. In particular, this paper explicitly considers the case of exogenous changes in elements driving access to the graduate sector, namely the selectivity of the higher education system showing that it may in ‡uence the functional form of the matching process. This result is undoubtedly important for empirical works aimed at evaluating policies a¤ecting matching process. Indeed, these works often assume exogenous MF to estimate elasticities with respect to the numbers of vacant jobs and job seekers (for a survey see Petrongolo and Pissarides, 2001) although more recent studies highlight that existing estimates of the matching function elasticities are likely to be exposed to an endogeneity bias arising from the search behavior of agents. In particular, Borowczyk-Martins et al. (2011) argue that random shocks to matching e¢ ciency determine the number of matches formed both directly through the matching technology and indirectly through …rms'vacancy-posting behavior. From an empirical point of view, this means that simple OLS regressions between the number of job matches and that of job seekers and vacancies fail to account for that endogeneity and deliver misleading predictions. The present paper shows that the parameterbias problem arising when estimating MF elasticities may be even more severe since the entire functional form of the MF can be a¤ected by exogenous policy measures.
The outline of the article is as follows. In Section 2 a brief summary of the existing literature on the urn-ball process is presented. Section 3 sets up the theoretical model and Section 4 evaluates the equilibria discussing the endogeneity of the hiring regime. Conclusions are presented in Section 5.

Existing background 2.1 The Basic Framework
In its simplest version, the urn-ball MF can be described as follows. The economy is assumed to have homogeneous …rms and workers who search for each other in the labor market. There is a coordination failure arising because workers simultaneously apply for jobs not knowing where other workers send their applications. This implies that some vacancies may remain un…lled, while others may get one or more applications. When …rms receive more than one application they can choose randomly among applicants. As discussed in details in Butters (1977) and Hall (1979) this process can be described as an urn-ball process where …rms are urns and workers are balls. Hence, by indicating with a(:) the probability that a worker receives a job o¤er, it can be shown that this follows a Poisson process with: where = U=V indicates the tightness of the labor market given by the ratio between the number of workers looking for a job (U ) and available vacant jobs (V ). By indicating with M the number of matches in the labor market, the MF is given by:

Extensions
Eqs. (1)- (2) have been enriched in two main directions. On the one hand, Albrecht et al. (2003 and2004) allow for multiple applications of job seekers. The authors prove that, although with multiple applications it is very likely that every vacancy will get at least one application, still a coordination failure may arise because of competition among …rms for single candidates. In fact, an applicant may receive more than one job o¤er and vacancies may remain un…lled because the chosen candidate is hired away by a competing …rm. As a consequence, allowing for more applications per worker may not increase the matching e¢ ciency. On the other hand, Blanchard and Diamond (1994), Gavrel (2009) and Moen (1999) model situations in which workers are heterogeneous in terms of their productivity. This implies that when …rms receive more than one application, they are not indi¤erent among applicants, hence they do not choose randomly. These authors assume that …rms rank applicants according to their productivity. In this case, the probability that a worker receives a job o¤er is a function of his/her characteristics. By indicating with the individual productivity and assuming distributed according to a continuous and strictly increasing cumulative distribution ( ); whose density function is ( ); over a support [ ; ] where 1 < (so ( ) = 0 and ( ) = 1), the probability that a -type worker receives a job o¤er can be written as follows: where the probability of receiving the o¤er increases with individual ability ( @a( ; ) @ > 0) and if = then a( ; ) has a unit value since -types get any job they apply for. By integrating a( ; ) over [ ; ], it is possible to obtain the unconditional probability of being hired, called a( ). Therefore the MF can be written as: Gavrel (2009) and Moen (1999) present analytical derivations of the previous expressions. Blanchard and Diamond (1994) set out the conditions under which eq.
(3) applies in a continuous time setting -as that presented in this paper -giving rising to a steady-state unemployment equilibrium. The present paper shows that in a two-sector model the choice between eqs. (1) and (3) should be solved endogenously. It is proved that both speci…cations can be consistent with a pro…t maximizing behavior conditional upon labor market institutions. Furthermore, it is discussed that some policies may induce a switch from (1) to (3) and vice-versa, implying instability of the MF and vulnerability with respect to the Lucas critique.

Overview
Consider an economy characterized by a continuum of risk-neutral individuals and …rms matching in the labor market following the lines set out by Diamond-Mortensen-Pissarides. Before entering the job-market, it is assumed that these agents have to make a choice concerning the sector they want to enter. According to Moen (1999), there are two sectors in this economy: Graduate (high-tech) and undergraduate (low-tech) sector. 1 The graduate sector is characterized by workers who invested in human capital and by …rms with (costly) high technology. Conversely, no particular investment is required to …rms and workers in order to enter the undergraduate sector. The mass and the distribution of agents, de…ned in due course, remain constant over time. As discussed in details in the next paragraph, individuals are assumed to be heterogeneous with respect to their pre-university individual skills which determine their productivity on the job and a¤ect the cost of entering the graduate sector. 2 From now on, these individual characteristics are simply de…ned as ability. On the demand side, each …rm can post a limited number of vacancies, normalized to 1, and it decides the sector where posting the vacancy on the basis of a technological choice. In particular, a …rm can choose to operate either within the high-or the low-technological sector. In order to simplify notation, from now on this paper refers to graduate versus undergraduate choice for both …rms and individuals. However the reader should keep in mind that individuals make an educational choice while …rms take a technological decision. Once the educational/technological choices have been made, the pure matching-process starts. As in Moen (1999) and Gavrel (2009) the two sectors are assumed to be perfectly segmented, i.e., graduates and undergraduates can be matched only with high-tech and low-tech …rms respectively.

Individuals
Consider a continuum of individuals of mass 1. According to the notation introduced in Section 2, individuals are characterized by heterogeneous individual ability . ( ) and ( ) are the c.d.f. and p.d.f. respectively and both are assumed to be stationary over time. Indicate with e = fg; ugg the educational choice made by individuals in order to maximize their expected discounted utility (g stands for graduate while ug stands for undergraduate). For the sake of simplicity individuals are assumed to have no income if unemployed (no unemployment bene…ts). As a consequence, once the educational choice has been made, in each instant of time the individual's utility function W (e) is given by: 0 if unemployed w ug if undergraduate and employed w g if graduate and employed (5) where w ug and w g indicate wage for employed undergraduate and graduate workers respectively. The cost of acquiring education ug is normalized to zero while, when individuals decide to acquire education g, on top of monetary costs, they have to sustain a cost c( ) > 0 related to their individual ability with @c @ < 0. Monetary costs are assumed to be the same for all individuals, while the e¤ort required to achieve a degree quali…cation is determined by personal ability. From now on, j @c @ j indicates a measure of the selectivity of the higher education sector. In words, the more the cost of education rises when ability decreases the more selective may be considered the higher education sector. It will be shown that the selectivity of the higher education system shapes the tightness of the two sectors and a¤ects …rms' optimal behavior in terms of ranking.

Firms
Consider a continuum of …rms of mass 1. Indicate with T = fg; ugg …rm's investment in graduate and undergraduate vacancies respectively. The cost of entering the g sector is given by > 0. The cost of entering the ug sector is normalized to zero. 3 Firms are assumed to be heterogeneous with respect to the cost they have to sustain in order to enter the g sector. In fact, in the growth theory literature, the cost of advanced technology has been considered typically related to the actual …rm's technological endowment. The closer is a …rm to the technological frontier the lower is the cost it needs to sustain in order to update its technology. The concept of technological frontier has been introduced by Nelson and Phelps (1966). Acemoglu et al. (2006) study empirically the relation between R&D expenditure and the distance from the technological frontier and build up a model where …rms di¤er in terms of costs to adopt new technologies. In the present model, …rms are assumed to be distributed according to a continuous and strictly increasing cumulative distribution ( ) whose density function is ( ); over a support [ ; ] where 0 < < (so ( ) = 0 and ( ) = 1). (:) and (:) are stationary over time.
Following Acemoglu (1997), the production function is given by: where y > 0 is a constant. Relation (6) indicates that there is homogeneity in the undergraduate sector, i.e., when individuals work in the ug sector they produce an output y independently on their ability. Conversely, graduate technologies are complementary only to graduate workers and the intensity of such complementarity is given by individual's ability . In fact, in eq. (6) skill-ability complementary technology has been assumed. This conjecture regarding the centrality of the positive interaction between technologies and ability is largely consistent with the empirical evidence. 4 Finally, Q indicates the cost of maintaining a vacancy 8T , and it is assumed that in the steady-state vacancies yield zero pro…t (free-entry condition). 5 Once the technological decision has been made, in each instant of time each …rm realizes a pro…t (T ) given by:

Interaction Process and Bellman Equations
The interaction process evaluated in this paper consists in the following stages. At the …rst stage, individuals and …rms conditional on their own type (ability and distance to the frontier) simultaneously decide the sector they want to enter, i.e., they choose between graduate and undergraduate sectors. Also at this stage, …rms set out the ranking behavior they want to adopt. Once the educational/technological choices have been made and the hiring process has been established, individuals and …rms enter the labor market as unemployed and with un…lled vacancies respectively, and then the matching process starts. Finally, when a match is realized, standard individual Nash-bargaining axiomatic solution is applied.
In order to solve the model, a backward procedure is adopted. Firstly, the actual expected value functions for individuals and …rms are evaluated using a standard dynamic programming method; secondly, by using the obtained results the Bayesian Nash Equilibrium (BNE) of the simultaneous game in which agents decide, conditional upon their own type, educational level and technological contents to maximize their expected steady-state payo¤s is established. Then, the hiring regime characterizing the BNE is set out.

The Frictional Labor
where a e ( e ) is the unconditional probability that an individual with education e is employed, expressed as a function of the tightness of the e sector with e = U e =V e : Crucially, it is assumed that the functional form of a e ( e ) is endogenous. In particular, indicate with a e (:) the probability that an individual with ability and education e receives a job o¤er. This probability is given by In the …rst line of eq. (9) the probability of receiving a job o¤er increases along with individual ability (as in eq. 3) while, when no-ranking applies all workers have the same job …nding rate and this is equal to the average arrival rate of jobs to workers (as in eq. 1). Consider the g sector. By integrating a g ( g ; ) over [ ; ], whose lower bound is the threshold-ability of individuals in the g sector (it is determined in the BNE), it is possible to indicate the unconditional probability of being hired in a g position, a g ( g ); as follows: Mutatis mutandis, in the ug sector the unconditional probability of being hired is given by: Now, it is useful to describe the urn-ball process from …rms' perspective. The probability that a T …rm hires a -type individual, indicated with T (:), can be written as follows: T ( e ; ) = 8 > < > : The …rst line of eq. (12) contains the probability that a T …rm does not meet any applicant of ability greater than times the probability of matching a worker with ability . In the no-ranking case, the probability of hiring a -type contains the probability that the …rm receives an application times the probability that this application is from an individual with ability : Consider the case of a g …rm. When integrating g ( g ; ) over [ ; ] the unconditional probability that a g vacancy is …lled is obtained and it can be de…ned as follows: Mutatis mutandis, for a ug …rm the probability of …lling a vacancy is given by: Having …xed this formalism, it is crucial to point out that the pure matching process can be solved as a function of the parameters a e ( e ; ), a e ( e ), T ( e ; ), and T ( e ). Put di¤erently, given the sequential structure of the interaction process, it is possible to solve the matching part of the model by not imposing either ranking or no-ranking behavior. Then, by using the obtained payo¤s in terms of wages and pro…ts, the educational/technological choices are established. Simultaneously, …rms'behavior in terms of hiring process is set out.

The value functions
The notation for actual expected values is set in Box 1. By indicating with r > 0 the intertemporal interest rate, the value functions can be written as follows.

Box 1: Notation for actual expected values
Undergraduate individuals: Graduate individuals: Firms with undergraduate job-positions: Firms with graduate job-positions: Notice that relations above represent pretty standard value functions for twosector matching models.

Equilibrium Wages
In order to set the equilibrium of the model, it is crucial to solve the last stage of the interaction process, i.e., to establish the payo¤s resulting from the matching process in the two sectors. Since individual Nash-bargaining solution is applied, when a match is realized the generated surpluses for …rm and worker must be equal conditional upon agents'characteristics and labor market opportunities. Formally: By combining the relative value functions, the following wage expressions for undergraduate and graduate individuals are obtained: w ug = y[r + b + a ug (:)] a ug (:) + ug (:) + 2b + 2r : w g = y [r + b + a g (:)] g (:) + a g (:) + 2r + 2b : As expected -given the perfect segmentation between the two sectors -wage equations are similar to those of standard matching models. Moreover, since graduates' ability is reveled once the match is realized, in this sector the wage is expressed as a function of . Now it is possible to proceed backward to determine the sectorchoice for …rms and individuals.

The Entry Game
Individuals and …rms have to decide, conditional on their ability and distance to the frontier, the level of education and the technology they want to acquire respectively. In order to solve the game, it is assumed that agents ground their decisions considering the parameters a ug (:), a g (:), ug (:), and g (:) as if they were at their steady-state values. Put di¤erently, agents choose their strategy in order to maximize the payo¤s they obtain in the steady-state. 6 Once they make their choice, they enter labor market(s) as unemployed individuals and as …rms with un…lled vacancies and then the matching process starts. The interaction process is Bayesian since each agent knows his own type (ability/distance to the frontier) and just the distribution of types of player to whom he may be matched. Since individual's ability is revealed only when a match is realized, i.e., the expected payo¤ of a g …rm that matches a g worker, need to be evaluated. Notice that this interaction process considers pure strategies of …rms and individuals that are best responses to each other, conditional on the type of player. As a consequence, the evaluation of the BNE gives the shares of individuals and …rms that acquire higher education and invest in graduate positions respectively and it provides a measure of the relative tightness of the two sectors in steady-state.

Proposition 1 It exists a unique BNE in which only individuals with ability
set e = g and only …rms with set T = g.
Proof. Consider the …rm's choice. Indicate with the probability (it is a density) that the individual sets e = g: In this case, a …rm invests in g position only if: Given the assumption on the monotonicity of (:), it is possible to indicate with the cuto¤ level of distance to the frontier for which relation (27) is satis…ed. Now, indicate with the probability that a …rm set T = g and consider the individual's educational choice. Setting e = g is optimal for an individual only if: Given the assumption on the monotonicity of (:) and given that @c @ < 0, it is possible to indicate with the cuto¤ ability level for which relation (28) is satis…ed.
Hence, the following pair characterizes the BNE: Intuitively, a …rm invests in a g position only if the associated expected payo¤ is greater than that associated with a ug position. Crucially, this depends on the distribution of within individuals that decide to acquire education g, on the relative markets'tightness, and on …rm's distance to the technological frontier (eq. 27). At the same time, worker's decision of investing in education g is a function of the number of …rms that decide to create g positions and of his own ability (eq. 28). Relation (29) contains the shares that are best response to each other and these can be considered as the shares of agents that represent the only steady-state of the interaction process. 7

Analysis of the BNE
In order to simplify the discussion concerning the hiring process adopted by …rms, it is worthwhile to undertake an in-depth analysis of the BNE established in the previous paragraph. This investigation is particularly useful since it allows for the identi…cation of two di¤erent types of BNE each of them consistent only with a speci…c hiring regime. Moreover, this analysis is important since it considerably eases the assessment of the e¤ect that exogenous shocks may have on the form of the MF discussed in paragraph 4.4.
As already pointed out, the BNE gives a measure of the tightness of the two sectors. By focusing on the cuto¤ level , i.e., the one that satis…es relation (27) as an equality it is possible to graphically describe the BNE. In fact, since the greater the larger the share of g …rms in the considered economy, approximates the share ( ) of …rms creating graduate-complementary positions. To evaluate relation (27) has to be spelled out. By combining eqs. (20) and (22) it is possible to write the cuto¤ level in relation (27) as follows: where A, B, C, D, and F summarize strictly positive constants. 8 Relation (30) gives the best response function in terms of share of …rms investing in graduate positions. Since the best response is evaluated when the share of graduates is ( ), eq. (30) represents the intersection of the best responses and, as a consequence, it describes the BNE of the game. Notice that in eq. (30) and Before turning to the discussion of ranking behavior, it is useful to evaluate how the share changes in equilibrium as changes. By di¤erentiating eq. (30) with respect to using the Leibniz rule for di¤erentiation of de…nite integrals it results that: Relation (32) indicates how a variation in the best response in terms of share of graduates ( ) a¤ects in equilibrium the share of …rms investing in graduate positions. The …rst two lines indicate that …rms'expectation positively depends on : The higher the cuto¤ ability level, the higher is the expected productivity of graduates and this induces a composition e¤ect which fosters …rms'investment in graduate jobs. Conversely, the bottom line of eq. (32) shows the negative e¤ect that a rise in has on …rms'expectation: In this case, as the cuto¤ point rises, the probability of …lling a vacancy reduces, inducing a tightness e¤ect that limits the creation of graduate-complementary positions. Assuming satis…ed second order conditions, it is possible to indicate with the share of graduates that ceteris paribus maximizes …rms'investments in graduate positions, i.e., the share of graduates balancing tightness and composition e¤ects: It is important to note that only the appropriate selectivity level j @c @ j can ensure that is actually achieved in equilibrium. If this is the case, the resulting steadystate allows for a perfect balance between tightness and composition e¤ects ( = ).

The Hiring Process
In this paragraph, it is shown that the particular case where = de…ned in eq. (33) separates two di¤erent types of BNE. Then, it is proved that these two types of equilibria are characterized by di¤erent (optimal-)ranking behavior. Consider Figure 1 where the best response function ( ) (which represents the set of all possible BNE) and the ability cumulate distribution ( ) have been drawn. In the particular case depicted in Figure 1, a scenario with tightness problem in the g sector has been represented since > . In words, few individuals have access to Figure 1: A Bayesian equilibrium with tightness dominance Figure 2: A Bayesian equilibrium with composition dominance the g sector and this constrains the creation of graduate complementary jobs. As illustrated in this …gure, a reduction in the selectivity level of the higher education sector (j @c @ j #) induces a rise in the share of graduates ( #) that in turn induces an increase in the share of …rms investing in graduate positions. Now, consider Figure 2. Here, di¤erently from before, the case where < is considered. This equilibrium hides a composition problem within the g sector: A large number of individuals acquire education g implying a low expected productivity of the graduate labor force. This brakes the creation of graduate jobs. In this case an increase in the selectivity level of the higher education sector (j @c @ j ") induces a reduction in the share of graduates ( ") and this generates an increase in the share of …rms investing in graduate positions.
Having established the existence of two di¤erent types of equilibria, it is possible to prove that the hiring process adopted in the graduate sector depends on the particular scenario faced by …rms, i.e., it depends on whether …rms are in the presence of tightness-or composition-related situations.
Proposition 2 a) In tightness-related equilibria g …rms maximize their actual expected value by adopting a no-ranking behavior. b) In composition-related scenarios g …rms' maximize their actual expected value by applying ranking amongst applicants. c) In all scenarios ug …rms rank applicants.
Proof. Part a). Consider g …rms. Consider a BNE characterized by a tightnessrelated scenario and, by contradiction, assume that the application of ranking among applicants represents an optimal choice for g …rms, i.e., it maximizes the expected value for a g …rm in the steady-state. In the presence of ranking, an individual who is at-the-margin, i.e., he has ability just below ( u ) decides not to acquire education g. In the Appendix it is shown that the value of V U g under no-ranking is greater than the value it takes under ranking when u . This implies that the individual at-the-margin would choose to graduate under the no-ranking case, hence would decrease if …rms decide to switch from the ranking to the no-ranking case. By de…nition of tightness-related equilibria, the reduction of raises ex-ante the expected value of all …rms investing in g positions, therefore all …rms …nd convenient to adopt the no-ranking behavior and this leads to a contradiction. Notwithstanding, …rms may still apply a dynamic inconsistent behavior deciding to apply ranking ex-post, i.e., once matches are realized. Since an in…nitely repeated setting is considered and agents care about their future payo¤s (r > 0), by applying the standard folk-theorem it would be possible to set a threshold level of the intertemporal discount rate r under which …rms do not deviate from a no-ranking strategy in order not to lose those graduates at-themargin in the future.
Part b). Consider g …rms. Consider a BNE characterized by a compositionrelated scenario and, by contradiction, assume that the application of no-ranking among applicants represents an optimal choice for g …rms. By replicating mutatis mutandis the argument made above and using the result presented in the Appendix, it is easy to show that increases if …rms decide to switch from the no-ranking to the ranking case. By de…nition of composition-related equilibria, an increase of raises ex-ante the expected value of all …rms investing in g positions, therefore all …rms …nd convenient to adopt the ranking behavior and this leads to a contradiction.
Part c). Consider ug …rms. In this sector, since there is no composition e¤ect, …rms only care about the share of ug workers in the labor market in order to rise the probability of …lling their vacancies. As a consequence ug …rms decide their matching regime to attract as many ug workers as they can. This implies thatindependently on the speci…c scenario generated by the institutional setting -the adoption of ranking represents an optimal action for ug …rms to retain in their sector those individuals near to .

Discussion
The intuition behind Proposition 2 is straightforward. Whether the selectivity level of the higher education sector limits the availability of graduates, …rms …nd optimal not to add additional screening since this practice would lower the expected value of education for individuals at-the-margin, leading to a reduction in the number of graduates and to a worsening of the tightness problem. Conversely, the ranking process represents an optimal choice whenever …rms face composition-related problems since it discourages individuals at-the-margin to enter the graduate sector. Notwithstanding, some arguments are required at this stage since the result of no-ranking among graduates may seem, at a …rst sight, counterfactual.
In this respect, it should be remarked that in this paper a unique level of selectivity of the higher education sector has been modeled and it has been shown that …rms' ranking decisions are conditioned on it. In the presence of heterogeneous selectivity levels, i.e., in the presence of heterogeneous universities, ceteris paribus …rms would ground their ranking decision by conditioning on the institution-speci…c selectivity. Hence, in this case we would observe both ranking and no-ranking behavior. This result is perfectly in line with the existing empirical evidence reporting that employment probability of graduate workers seems to be a¤ected by the characteristics of the attended university in terms of admission's requirements (among others see Hendel et al., 2005; and Ordine and Rose, 2011).
At this stage, in order to have a complete picture of the model's results, …rms' behavior in the ug sector needs to be discussed. The characteristics of the ug sector in terms of ranking are perfectly in line with the main message of this work: When tightness issues are taken into account, the presence of ranking could not be easily determined ex-ante by relying only on the presence of heterogeneous workers'productivity. Ranking may be applied even if …rms operate in a sector characterized by homogeneous workers simply because this hiring regime maximizes the availability of workers in this sector and, consequently, the probability of …lling a vacancy.
A …nal point that needs to be remarked concerns the relevance that the presented results may have for empirical works. Indeed, the analysis presented in paragraphs 4.3.1 and 4.3.2 allows for an immediate assessment of this issue. In particular, from Figure 1 it is easy to check that a change of the selectivity of the university system may induce a switch from the tightness to the composition scenario whenever moves from the RHS to the LHS of . This implies that an exogenous variation of the selectivity level of higher education system may induce a switch in the matching regime going from the no-ranking to the ranking case. Analytically, the functional form of the MF changes too, by relying on the top line instead of the bottom line of eq. (9). This consideration implies that the matching technology changes with exogenous policies and rises concerns about the validity of policy evaluations employing exogenous matching functions. Results of models with exogenous matching regimes could be biased if modelers do not take into account that the matching technology itself may also change with the policy.

Conclusions
This study enters the debate concerning the endogeneity of matching functions by focusing on a particular matching regime known as urn-ball process. In this case, either ranking or no-ranking behavior may be adopted by …rms when choosing among multiple applications. It is argued that the choice of the correct modelling strategy is not an obvious one and it does not only depend on workers'heterogeneity in terms of productivity. Using a simple continuous time two-sector matching model with endogenous technological and educational choice, it has been shown that the speci…c form of the matching process depends on the characteristics of the labor market. In particular, when the two sectors compete to attract workers, …rms evaluate their optimal actions in the light of the tightness of the sector in which they operate. Overall, the study highlights the relevance that endogenous matching process may have in order to correctly capture labor market dynamics and agents' behavior. This has important implications also for empirical works aimed at evaluating policy measures and their e¤ect on workers' employability since the properties usually imposed on exogenous matching functions are justi-…ed on the basis of agents' micro-behavior. Indeed it has been shown that the speci…c form of the matching process can be a¤ected by …rms'behavior resulting from the speci…c institutional setting. As policies are targeted to change agents' choices, these may very well also a¤ect the properties of the matching technology. These aspects should be taken into account by policy evaluators in order to avoid misleading predictions on the e¤ect of policy measures.
By taking logs of both sides in the relation above and by applying a …rst-order Taylor series approximation I have that: It can be easily checked that the RHS of relation (37) is less than -1 8 g > 0.