Multidimensional Sorting Under Random Search

Transcription

1 Multidimensional Sorting Under Random Search Ilse Lindenlaub Fabien Postel-Vinay July 217 Abstract We analyze sorting in a standard market environment with search frictions and random search, where both workers and jobs have multi-dimensional characteristics. We offer definitions of multi-dimensional positive and negative assortative matching in this frictional environment. We say that matching is positive assortative in dimension (j, k) if workers with higher endowments in skill dimension k are matched to a distribution of jobs with higher values of job attribute j, in the sense of first-order stochastic dominance, and similarly for negative assortative matching. We then provide conditions on the primitives of the economy (technology and distributions) under which sorting arises in equilibrium. An essential condition for positive sorting is a single-crossing property on the technology, although in general further conditions on type distributions are needed. Guided by our theoretical framework, we develop an empirical test of the standard assumption that heterogeneity in the data is one-dimensional. We conduct simulation exercises (i) to show that the test correctly reveals misspecification and (ii) to quantify the errors in assessing sorting, mismatch and policy caused by the restriction to one-dimensional heterogeneity when our test indicates that said restriction is not valid. Keywords. Multidimensional Heterogeneity, Random Search, Sorting, Assortative Matching We would like to thank Hector Chade and Jan Eeckhout for useful comments, as well as Carlos Carrillo- Tudela and Jim Albrecht for their insightful discussions of this paper at the 216 Cowles Foundation Macro Conference and the 216 Konstanz Search and Matching Conference. We further thank seminar audiences at Essex, Yale, the Chicago Fed, Einaudi, Konstanz, the NBER Summer Institute, MIT, Simon Fraser University, McGill, EUI, and at the SED 217 for their comments. All remaining errors are ours. Yale University, Address: Department of Economics, Yale University, 28 Hillhouse Avenue, New Haven, 652, US. ilse.lindenlaub@yale.edu UCL and IFS. Address: Department of Economics, University College London, Drayton House, 3 Gordon Street, London WC1H AX, UK. f.postel-vinay@ucl.ac.uk

2 1 Introduction Labor market frictions impede the efficient assignment of heterogeneous workers to heterogeneous jobs and cause mismatch. Measuring mismatch in frictional markets its nature, its extent, and its implications for efficiency, inequality and welfare is the focus of a growing body of literature in structural labor economics. With very few exceptions, that literature works under the assumption that job and worker heterogeneity can both be captured by scalar indexes, i.e. heterogeneity is one-dimensional. 1 While this restriction to one-dimensional heterogeneity is a natural starting point and convenient for modeling, it is at odds with the fact that typical data sets describe both workers and jobs in terms of many different productive attributes (e.g. cognitive skills, manual skills, or psychometric test scores for workers, and numerous task-specific skill requirements for jobs). 2 If the productive characteristics of workers and firms are truly multi-dimensional, what are the features of the data, relating to sorting and mismatch, that we miss by modeling them as one-dimensional scalars? This is the question we ask in this paper. We proceed in two steps. First, we develop a theory of the assignment of workers to jobs when both workers and jobs differ along several dimensions in a random search environment. We propose a notion of assortative matching (or sorting) in this setting and analyze how the economy s primitives shape equilibrium sorting. Second, we use our theory to develop an empirical test to assess whether the assumption of one-dimensional (1D) heterogeneity in the random search framework is valid. We show on simulated data that the test accurately reveals misspecification of the one-dimensional model. We then highlight some consequences of approximating job and worker characteristics by 1D summary indexes when it is not justified, i.e. when the test indicates that the data is multi-dimensional. We show that using the misspecified 1D model to measure sorting and mismatch (as opposed to the multi-dimensional one) leads to quantitative and qualitative errors as well as misguided policy recommendations. We now describe the two steps of our analysis theory and application in more detail. We begin with developing a theoretical framework of multi-dimensional sorting under random search. Our environment is a standard random search model, except that workers and firms (also referred to as jobs) are endowed with vectors of productive attributes, x = (x 1,, x X ) 1 A recent exception is Lise and Postel-Vinay (215), who focus on the accumulation of skills in various dimensions within a model that can otherwise be seen as a special case of the theoretical framework we develop here. 2 Beyond search models, a growing applied literature takes explicit account of these multiple dimensions of productive heterogeneity. Recent examples include Yamaguchi (212), Sanders (212), and Guvenen et al. (215). 1

3 for workers and y = (y 1,, y Y ) for jobs. Employed and unemployed workers receive job offers drawn at random from an exogenous sampling distribution of job attributes. Employers face no capacity constraint. Utility is transferable: workers and firms are joint surplus maximizers. The fact that agents base their match acceptance decisions on a scalar value (i.e. the match surplus that summarizes all underlying multi-dimensional heterogeneity) makes our multi-dimensional problem tractable. In order to interpret the assignment patterns that arise in equilibrium, we define a notion of positive assortative matching (PAM) and negative assortative matching (NAM) in this environment. This notion is based on first-order stochastic dominance (FOSD) ordering of the marginal distributions of job attributes across workers with different skills: if workers with a higher endowment of some skill x k are matched to jobs with better (in the first-order stochastic dominance sense) attributes y j, then PAM occurs in dimension (x k, y j ). For instance, if workers with more cognitive skills are matched to a distribution of jobs whose cognitive contents stochastically dominates that of less cognitively skilled workers, then we call this sorting pattern PAM between cognitive skills and cognitive job contents. It is important to note that sorting is thus defined dimension by dimension, meaning that PAM can arise in one particular dimension (here in the cognitive dimension) while NAM occurs in another. Using this definition of sorting, we present three main sets of theoretical results. The first one is about the sign of sorting. We provide conditions on the economy s primitives under which positive or negative sorting arises in equilibrium. For ease of exposition and clarity of interpretation, we focus in the main body of the paper on a baseline model: it features two-dimensional heterogeneity on the job side, bilinear technology, wage setting by sequential auctions, and positive surplus of any possible match all assumptions that we can and do relax. In this baseline case, we find that matching in, say, dimension (x 1, y 1 ) is positive assortative if and only if the technology satisfies a single crossing condition implying that the complementarity between worker skill x 1 and job attribute y 1 dominates the complementarity in the competing dimension (x 1, y 2 ). This condition is distribution-free: it only involves restrictions on the production technology. We generalize this analysis of the sign of sorting to cases where (1) not all possible matches generate positive surplus (implying that there also is sorting on the unemployment-to-employment margin), (2) heterogeneity on the job side is of dimension higher than two, and (3) the technology is monotone in at least one job attribute but not necessarily bilinear or it is non-monotone but separable. We show that in these more general environments the conditions for sorting not 2

4 only involve single-crossing of the technology but also interactions between the technology and sampling distribution of jobs. We also show that these results do not hinge on the assumed sequential auctions wage setting but hold for several other commonly used wage setting protocols like Nash bargaining, sequential auctions with worker bargaining power, and wage posting. Second, our model predicts sorting based on specialization rather than on absolute advantage. This implies that uniformly more skilled workers do not sort into jobs with uniformly higher skill requirements. Rather, they sort into jobs with a higher requirement for the skill in which they are relatively strong, possibly at the cost of a lower requirement for the other skills. Our third set of results highlights that sorting generally cannot be positive between all skill and job dimensions. Instead, there are sorting trade-offs. We provide conditions under which PAM arises in all dimensions except the one that is characterized by the weakest complementarities in production, where forces push towards NAM. An important insight from our analysis is that multi-dimensional heterogeneity per se is a cause of sorting. That is, in the 1D version of this model, where match surplus depends on scalar worker type x and job type y and is increasing in y, workers all rank jobs in the same way (there is a single job ladder), regardless of their skill x: their common strategy is to accept any job with a higher y than their current one, which rules out sorting. By contrast, in the multi-dimensional world where workers have skill bundles, what matters is not just to match with a productive job in any component of y. Rather, a worker wants to match with a job that puts much weight on the skill in which he is strong. Thus, workers with different skill bundles (and hence different strengths) accept and reject different types of jobs, they climb different job ladders, which is why sorting arises. In the second part of our analysis, we show that our theory of multi-dimensional sorting has important implications for applied work. Based on the theoretical insight that multi-dimensional job ladders are heterogenous across workers, we design a test of the assumption that job and worker heterogeneity can be captured by scalar indices (i.e. the single index assumption). If the data is well-approximated by 1D worker and job types, any two workers with the same (scalar) type should face the same job ladder. Conversely, if the single index assumption is not justified, approximating different multi-dimensional worker types by the same 1D worker type will produce job ladder heterogeneity within 1D worker types, revealing misspecification. To assess the accuracy of this test, we simulate data from several two-dimensional models that comply with our theory and fit misspecified 1D models to those data. We show that the test 3

5 correctly reveals misspecification of the 1D model when the data is multi-dimensional. We then use our simulations to assess the potential errors in the analysis of sorting and mismatch caused by imposing the single-index assumption when it is not justified (i.e. when our test indicates that this restriction is violated). We find that the misspecified model is largely uninformative on true sorting patterns (it predicts sorting when the true data features none and vice versa). Perhaps most importantly, a misspecified 1D model suggests a first-best allocation that is very different from the true first-best under multi-dimensional types. Implementing the first-best allocation suggested by the misspecified 1D model can cause sizable welfare losses. While much is known about sorting under one-dimensional heterogeneity without frictions (Becker, 1973; Legros and Newman, 27) and one-dimensional heterogeneity with frictions (Shimer and Smith, 2; Smith, 26; Eeckhout and Kircher, 21), little is known about sorting on multi-dimensional types. One exception is Lindenlaub (217) who studies multidimensional sorting in a frictionless assignment game. 3 To the best of our knowledge, this paper is the first to develop a theory of multi-dimensional sorting under random search an environment of great importance for applied work since it carries a well-defined notion of mismatch and allows for policy analysis. Our work not only shifts the focus to multi-dimensional heterogeneity but also differs in another important way from that theoretical literature on sorting. All of the aforementioned papers analyze assignment problems, meaning that agents on either side of the market can be matched with at most one agent from the other side (matching is one to one). While this assumption is certainly appropriate in partnership models of the marriage market, it is much less common in analyses of the labor market, where a firm usually employs many workers. In this paper we assume, as is almost invariably done in the applied structural search literature, that firms operate technologies with constant returns to scale and face no capacity constraints. 4 The no-capacity-constraint assumption greatly increases the tractability of structural search models. 5 Its cost, though, is that it tends to rule out sorting when job and worker heterogeneity 3 Our restriction on the technology to obtain sorting is most closely related (but not equivalent) to that needed in the multi-dimensional, frictionless assignment problem by Lindenlaub (217), who also has to discipline complementarities across tasks to ensure PAM within tasks. But compared to Lindenlaub, the introduction of random search and no-capacity constraint changes the technical nature of the problem (for instance, we cannot rely on optimal transport theory here) as well as the insights (in our framework with frictions, we can analyze sorting on the UE and EE margin, heterogenous job ladders, mismatch and policy). 4 Examples of widely-used models without firm capacity constraint are Burdett and Mortensen (1998) and Postel-Vinay and Robin (22). 5 For instance, it is instrumental in circumventing a complex existence proof involving a fixed point problem in the distribution of the unmatched agents that is common in one-to-one assignment models under search frictions (see Shimer and Smith, 2). 4

6 are one-dimensional. As discussed above, if technology is monotone in firm type, workers share a common ranking of firms and all seek to climb a single, economy-wide job ladder. The lack of capacity constraint on the firm side then implies that all workers match with the same distribution of jobs in equilibrium. This result is independent of the complementarities in production. Getting the standard structural search framework without firm capacity constraint to produce a meaningful notion of equilibrium sorting is not straightforward. 6 In this paper, we extend that framework by allowing for multi-dimensional heterogeneity of jobs and workers. This change drastically alters the model s predictions. Multi-dimensional skill bundles cause different workers to rank jobs differently and face different job ladders, which is why sorting arises. Our model and definition of multi-dimensional sorting can guide the measurement of sorting in the data. It can be used to study efficiency losses due to mismatch or the role of sorting and mismatch in wage inequality topics that are of great importance from an applied point of view. The rest of the paper is organized as follows: Section 2 illustrates our main theoretical insights via a simple example. Section 3 introduces our model. Section 4 provides a definition of sorting in multiple dimensions under random search. Section 5 contains our main results on the sign of sorting, established within our baseline model with bilinear technology and twodimensional job attributes. Generalizations are discussed in Section 6 (and presented formally in Appendix A). Section 7 investigates sorting based on absolute advantage vs specialization and the interdependence of sorting patterns across different heterogeneity dimensions. Section 8 contains our application, including our test of the single index restriction. Section 9 concludes. 2 An Illustrative Example We begin by illustrating our main theoretical insights using a simple example, the foundations of which will be explored later as we introduce our full model. Consider a labor market where workers are characterized by a two-dimensional skill vector x = (x 1, x 2 ) R 2 ++ (e.g. cognitive and manual skills). Workers can either be employed or unemployed. Either way, they face search frictions in that they sample job offers randomly and sequentially. Jobs are also characterized by a two-dimensional vector of attributes y = (y 1, y 2 ) R 2 ++ describing their contents 6 Bagger and Lentz (216), who build on Lentz (21), include endogenous search intensity into an otherwise standard 1D sequential auction model. Under complementarities in production, high worker types have more to gain from being with high firm types, hence search more intensively, and therefore end up in better firms in equilibrium. Thus, their model features sorting due to endogenous search intensity. Another way to introduce sorting is to abandon the assumption that technology is monotone in firm type. 5

7 in terms of skill dimensions 1 and 2 (e.g. cognitive and manual skill requirements). A match between a worker with skills x and a job with attributes y generates a surplus σ(x, y). We also assume that jobs and workers are joint surplus maximizers. Hence, a meeting between a type-x unemployed worker and a type-y job will result in a match if and only if σ(x, y). And a meeting between a type-x worker, employed in job y, and an alternative type-y job will result in the worker accepting the type-y job if and only if σ(x, y ) > σ(x, y). For illustration, we assume that the match surplus function is bilinear: σ(x, y) = 2x 1 y 1 + x 2 y 2 (It will become clear in our baseline model below under which conditions it suffices to focus on this flow surplus σ, which is a primitive.) In this example, complementarities in production within task dimension 1 and within dimension 2 are positive while between-task complementarities are zero. This implies that complementarities are larger in dimension (x 1, y 1 ) than in (x 1, y 2 ), ensuring that the single crossing condition [ ] σ/ y1 > x 1 σ/ y 2 holds. Note that, in this example, σ(x, y) for all pairs (x, y), implying that all matches out of unemployment will be accepted. Consider two unemployed workers, one with skills (x 1, x 2 ) = (1, 1) and the other one with (x 1, x 2 ) = (2, 1), so the second worker is relatively specialized in skill x 1. Assume for illustration that both workers receive the same sequence of job offers over time, (y 1, y 2 ) = {(2, 2), (3, 1), (3.4, )}, implying the following acceptance/rejection decisions: Offers: y = (2, 2) y = (3, 1) y = (3.4, ) Worker x = (1, 1): accept accept reject Worker x = (2, 1): accept accept accept The last job offer with high y 1 but very low y 2 is only accepted by the worker with higher x 1 since for him the surplus comparison yields σ ((2, 1), (3.4, )) = 13.6 > 13 = σ ((2, 1), (3, 1)) while for the worker with lower x 1, σ ((1, 1), (3.4, )) = 6.8 < 7 = σ ((1, 1), (3, 1)). This simple example illustrates our main theoretical insights in an intuitive way. Even though both workers are inclined to accept jobs with higher y 1 at the expense of lower y 2 6

8 because complementarities in dimension (x 1, y 1 ) are stronger for workers with higher x 1 this sorting trade-off is more pronounced: they are willing to accept jobs with particularly low y 2 in exchange for a higher y 1. This is why in equilibrium, for given x 2, higher-x 1 workers will be matched to a distribution of jobs that is better (in the FOSD sense) in terms of y 1, compared to workers with lower x 1. But, in turn, higher-x 1 workers will be matched to jobs with worse y 2 (again in the FOSD-sense) relative to workers with lower x 1. According to our definition of PAM, there is PAM in the dimension (x 1, y 1 ) (which is the dimension of relatively strong complementarities in production), while there is NAM in dimension (x 1, y 2 ) (which is the dimension of relatively weak complementarities). These sorting patterns arise because workers with different skill bundles have different specializations and thus rank jobs in different ways. As a consequence, the two workers in this example climb different job ladders over time. It is important to note that if there were no differences in the relative complementarities across dimensions, then both workers in our example would share the same ranking of jobs. To illustrate this, change the technology to σ(x, y) = 2x 1 y 1 + 2x 1 y 2 + x 2 y 1 + x 2 y 2, which implies that [ σ/ y1 σ/ y 2 ] / x 1 =. One can easily check that both workers would reject the last job offer y = (3.4, ) in this case. Independent of their different degrees of specialization, they would both climb the same job ladder, and no sorting would arise. The labor market would be similar to one with 1D heterogeneity, where workers have a single skill x and jobs a single attribute y. In such a 1D world, if the technology is monotone in y, workers share the same ranking of jobs and climb the same job ladder over time (as is the case, e.g., in Postel-Vinay and Robin, 22). 3 The Model 3.1 The Environment. Time t R + is continuous. The economy is populated by infinitely lived workers and firms. There is a fixed unit mass of workers, each characterized by a time-invariant skill bundle x = (x 1,, x X ) X = X k=1 [x k, x k ], where X denotes the number of different skill dimensions. 7 We normalize the lower support of worker skills to x k = and allow x k R + {+ }. Skills are distributed with cdf L and strictly positive density l. 8 Firms can be thought of as single 7 For early influential work on the importance of workers characteristics being bundled in the labor market context, see Heckman and Scheinkman (1987). 8 We adopt the following notational conventions throughout the paper: we denote all CDFs using uppercase letters (e.g. L), densities by the associated lowercase letter (e.g. l), and survivor functions using bars over CDFs (e.g. L = 1 L). Also, we will state a function is increasing/decreasing or positive/negative if this is the case in the weak sense. Strict properties will be mentioned explicitly. 7

9 jobs (possibly vacant), or as collections of independent, perfectly substitutable jobs. Jobs are characterized by a vector of time-invariant productive attributes, or skill requirements y = (y 1,, y Y ) Y = Y j=1 [y j, y j], where Y denotes the number of different job attributes and where y j R + and y j R + {+ }. 9 Workers search for jobs at random, both on and off the job. They can be matched to a job or be unemployed. If matched, they lose their job at Poisson rate δ, and sample alternate job offers drawn from an exogenous sampling distribution distribution Γ at rate λ 1. We assume Γ to have a strictly positive density γ, which is twice continuously differentiable. Unemployed workers sample job offers from the same sampling distribution at rate λ. The output flow in a match between a worker with skills x and a job with attributes y is p(x, y), where p : R X R Y R. 1 We denote the income flow of an unemployed worker with skill x by b(x). We stress that there is no capacity constraint on the firm side (firms are happy to hire any worker with whom they generate positive surplus) and matched jobs do not search for other workers. As such, this set-up is really a (partial equilibrium) model of the labor market rather than one of symmetric, one-to-one matching of the marriage market. 3.2 Rent Sharing and Value Functions Workers and firms are risk-neutral and have equal time discounting rates ρ >. Under those assumptions, the total present discounted value of a type-(x, y) match is independent of the way in which it is shared, and only depends on match attributes (x, y). We denote this value by P (x, y). We further denote the value of unemployment by U(x), and the worker s value of being employed under his current wage contract by W, where W U(x) (otherwise the worker would quit into unemployment), and W P (x, y) (otherwise the firm would fire the worker). Assuming that the employer s value of a job vacancy is zero, the total surplus generated by a type-(x, y) match is P (x, y) U(x). We assume in the main text that wage contracts are set as in the sequential auction model without worker bargaining power of Postel-Vinay and Robin (22). This is mainly to simplify exposition. We show in Appendix B that most of our results extend to other common wage setting rules, as Nash bargaining (Mortensen and Pissarides, 1994; Moscarini, 21), wage/contract 9 The restriction to x and y having positive elements is not necessary but simplifies the economic interpretations. Also, we can relax that X is a product of intervals but assume it for symmetry with job attributes. 1 We assume that the production function is defined over the entire space R X R Y, not just the set X Y of observed (x, y). This is to streamline some proofs and can be relaxed. Details are available upon request. 8

10 posting (Burdett and Mortensen, 1998; Moscarini and Postel-Vinay, 213), or sequential auctions with worker bargaining power (Cahuc, Postel-Vinay and Robin, 26). In the sequential auction model, firms offer take-it-or-leave-it wage contracts to workers. Wage contracts are long-term contracts specifying a fixed wage that can be renegotiated by mutual agreement only. In particular, when an employed worker receives an outside offer, the current and outside employers Bertrand-compete for the worker. Consider a type-x worker employed at a type-y firm and receiving an outside offer from a firm of type y. Bertrand competition between the type-y and type-y employers results in the worker matching with the employer where the total match value is higher, while extracting the full surplus from the lowersurplus match. This implies that he stays in his initial job if P (x, y) P (x, y ), moves to the type-y job otherwise, and ends up with a new wage contract worth W = min {P (x, y), P (x, y )} (provided that W exceeds the value of the worker s initial contract, W, as otherwise the worker would not have initiated the contract renegotiation in the first place). It follows that the total value of a type-(x, y) match, P (x, y), solves the equation: ρp (x, y) = p(x, y) + δ [U(x) P (x, y)]. The annuity value of the match, ρp (x, y), equals the output flow p(x, y) plus the expected capital loss δ[u(x) P (x, y)] of the firm-worker pair from job destruction. 11 Given that U(x) is independent of firm type, the optimal mobility choices of workers hinge on the comparison of match surplus P (x, y) U(x) across jobs. It solves (ρ+δ) [P (x, y) U(x)] = p(x, y) ρu(x). In what follows, we will mostly reason in terms of the match flow surplus: σ(x, y) = p(x, y) ρu(x). A worker x employed in a job y accepts an offer from a job y if and only if P (x, y ) U(x) > P (x, y) U(x). This is equivalent to σ(x, y ) > σ(x, y), so that the optimal strategy to accept/reject a job is entirely based on the comparison of flow surpluses. Likewise, an unemployed worker x accepts a job of type y if σ(x, y). In turn, the optimal strategy of firm y is to accept any worker x if σ(x, y). It follows that the dynamic optimization problem of the 11 Note that, under the sequential auction model, the realization the other risk faced by the firm-worker pair, namely the receipt of an outside job offer by the worker, generates zero capital gain for the match: either the worker rejects the offer and stays, in which case the continuation value of the match is still P (x, y), or the worker accepts the offer and leaves, in which case he receives P (x, y) while his initial employer is left with a vacant job worth, so that the initial firm-worker pair s continuation value is again P (x, y). 9

11 agents is solved simply by flow surplus comparisons (as hinted at in the example of Section 2). Finally note that, in the sequential auction case, the value of unemployment, U(x), is given by ρu(x) = b(x), implying σ(x, y) = p(x, y) b(x), i.e. σ is pinned down by technology. Thus, optimal mobility decisions are entirely determined by technology. 3.3 Steady-State Distribution of Skills and Skill Requirements In our analysis of sorting below, the key object will be the steady-state equilibrium measure of type-(x, y) matches, denoted by h(x, y), which indicates who matches with whom. It is determined by the following flow-balance equation, which embeds the optimal mobility decisions, 12 { [ { δ + λ1 E Γ 1 σ(x, y ) > σ(x, y) }]} h(x, y) = λ γ(y)1 {σ(x, y) } u(x) + λ 1 γ(y) 1 { σ(x, y) > σ(x, y ) } h(x, y )dy, (1) where u(x) is the measure of type-x unemployed workers in the economy. Note that we use primes () to denote random variables with respect to which expectations are taken. The left-hand side (l.h.s.) of (1) is the outflow from the stock of type-(x, y) matches, comprising matches that are destroyed at rate δ and matches that are dissolved because the worker receives a dominant outside offer. The flow probability of this latter event is λ 1 E Γ [1 {σ(x, y ) > σ(x, y)}], the product of the arrival rate of offers λ 1 and the probability of drawing a job type y that yields a higher flow surplus with the worker than the current type-y job. The right-hand side (r.h.s.) of (1) is the inflow into the stock of type-(x, y) matches and composed of two groups: unemployed type-x workers who draw a type-y job with flow probability λ γ(y) and accept it (which they do if the flow surplus is positive); and type-x workers employed in any type-y job who draw an offer from a type-y job with flow probability λ 1 γ(y) and accept it (which they do if the flow surplus with that job exceeds the one with their initial type-y job). The measure of type-x unemployed workers solves the following flow-balance equation with similar interpretation: [ { λ E Γ 1 σ(x, y ) > ) }] u(x) = δ h(x, y )dy. (2) Finally note that, consistently with (1) and (2), the total measure of workers with skill bundle x in the economy solves l(x) = u(x) + h(x, y )dy. 12 Throughout the paper, we use the notation E Γ to distinguish expectations taken w.r.t. the sampling distribution Γ from expectations w.r.t. the equilibrium distribution of matches, which we simply denote by E. 1

12 Note that the job acceptance rule of an employed worker in a type-(x, y) match hinges on the comparison of two scalar random variables, σ(x, y ) and σ(x, y), despite the underlying multidimensional heterogeneity of workers and firms. It is therefore convenient to introduce the conditional sampling distribution F σ x of flow surplus σ, given x (with density f σ x ). With this notation, the job acceptance probability for an employed worker x is E Γ [1 {σ(x, y ) > σ(x, y)}] = F σ x (σ(x, y)) and that for an unemployed worker is E Γ [1 {σ(x, y ) > )}] = F σ x (). Substituting these elements into (1), we show in Appendix C that the matching density h(x, y) has the following closed-form: h(x, y) l(x)γ(y) = δλ 1 {σ(x, y) } δ + λ 1 F σ x () [ δ + λ F σ x () δ + λ1 F σ x (σ(x, y)) ] 2. This equation also implies that the equilibrium conditional density of job types y given worker types x among employed workers is given by: h (y x) = [ δ1 {σ(x, y) } δ + λ1 F σ x () ] γ(y) [ F σ x () δ + λ1 F σ x (σ(x, y)) ] 2 = g σ x (σ(x, y)) γ(y), (3) f σ x (σ(x, y)) where for any s R, the steady-state cross-section cdf of flow surplus among employed workers of type x is G σ x (s) := 1 δ + λ 1F σ x () F σ x () F σ x (s) δ + λ 1 F σ x (s) and g σ x is its density. 4 Equilibrium Sorting 4.1 Measuring Sorting We first specify a measure of sorting in this multi-dimensional environment under frictions. A criterion that has been proposed for multi-dimensional positive assortative matching in a frictionless context is that the Jacobian matrix of the equilibrium matching function be a P -matrix, meaning that all its principal minors are positive (Lindenlaub, 217). This criterion captures the way in which a worker s job type y improves or deteriorates as one varies the worker s skill bundle x when matching is pure, i.e. when any two workers with the same skill bundle are matched to the exact same type of job. By contrast, in our frictional environment with random search the equilibrium assignment is generally not pure there is mismatch. A 11

13 natural extension of this measure of sorting to our environment is to consider changes in the quantiles of the conditional matching distribution of job types y as one varies worker type x. 13 Formally, let H j (y x) denote the marginal cdf of y j (the jth component of job attribute vector y) conditional on the matched workers having skill bundle x. Using (3), we can express this as H j (y x) = 1 { y j y } h ( y x ) dy = δ [ δ + λ 1 F σ x () ] F σ x () } 1 {σ(x, y ) } 1 {y j y [ δ + λ1 F σ x (σ(x, y )) ] 2 γ(y )dy. (4) We are interested in signing, for each job attribute j, the elements of the gradient H j (y x) = ( H j (y x)/ x 1,, H j (y x)/ x X ), i.e. the Jacobian of H(y x). A situation of particular interest is when a component of this Jacobian, H j (y x)/, has a constant sign over the support of y j. If that sign is negative [positive], then H j ( x) is increasing [decreasing] in x k in the sense of FOSD: positive [negative] assortative matching then occurs in dimension (y j, x k ), as a worker with higher type-k skill is matched to jobs with greater (in the FOSD sense) type-j skill requirement compared to a worker with lower type-k skill. 14 We will thus use the following formal definition of sorting: Definition 1 (Positive and Negative Assortative Matching). Matching is positive [negative] assortative in dimension (y j, x k ) if and only if H j (y x)/ is negative [positive] for all y [y j, y j ] and all x X. We will use the acronyms PAM and NAM for positive and negative assortative matching. Alternatively, we will also refer to PAM (or NAM) as positive (or negative) sorting. 15 To avoid duplication of some results, we focus on positive sorting throughout most of the paper. 4.2 A Decomposition Result We begin our analysis by showing how equilibrium sorting can be decomposed into sorting on the unemployment-to-employment (UE) margin and on the employment-to-employment (EE) 13 We choose to analyze the equilibrium matching distribution of y given x and not that of x given y for the following reason. While workers sample job types from an exogenous sampling distribution γ, jobs sample workers from an endogenous distribution (the distribution of workers across employment statuses and job types), which in itself is a complex equilibrium object. The acceptance decisions of firms would impact the distribution of x across employment statuses and job types, and the distribution of x, in turn, impacts the acceptance decisions of the firms. Analyzing the matching distributions of x given y would therefore require us to deal with a complicated fixed point problem, which proved intractable. 14 FOSD has been used to characterize sorting under frictions and 1D-heterogeneity (e.g. Chade, 26). 15 Note that Definition 1 is global in the sense that it imposes a sign restriction on H j (y x)/ for all skill bundles x X. Alternatively, we could have opted for a local definition by only imposing the weaker condition that H j (y x)/ be positive or negative at a given skill bundle x. In what follow we use the global version of this definition as sorting is commonly envisaged as a global property in the literature. 12

14 margin. As we show in Appendix C, a typical element of the gradient of H j (y x), which we use to characterize sorting patterns (Definition 1), has two parts, indicating that a marginal increase in the worker s skill x k affects his equilibrium distribution of job types y j in two ways. 16 H j (y x) = C UE + C }{{}}{{} EE (1): UE margin (2): EE margin (DC) where components C UE and C EE are given by [ σ(x, y ] ) C UE := g σ x () E Γ σ(x, y ) = [ { Pr Γ y x j y σ(x, y ) = } H j (y x) ] k { +Pr Γ y j y σ(x, y ) = } { [ σ(x, y ] [ ) σ(x, y E Γ σ(x, y ) =, y j ]} ) y E Γ σ(x, y ) = and C EE := + 2λ 1 f σ x (s)g σ x (s) { Pr Γ y j y σ(x, y ) = s } δ + λ 1 F σ x (s) { [ σ(x, y ] [ ) σ(x, y E Γ σ(x, y ) = s, y j ] } ) y E Γ σ(x, y ) = s ds. First, a marginal increase in skill x k affects the boundary of the set of profitable matches for that worker, i.e. the set of job types y such that σ(x, y). An increase in skill may render some matches between unemployed workers and jobs profitable that were unprofitable before. This is reflected in the first term of expression (DC) our label for decomposition which only works through selection on the UE margin: term C UE contains (multiplicatively) the density of marginally profitable matches for type-x workers, g σ x (). If the worker s type x is such that σ(x, y) > for all job types y (i.e. if the worker accepts any job type when unemployed), then there are no such marginal matches, g σ x () =, and sorting on the UE margin is shut down. Second, a marginal increase in x k affects the job acceptance probability for employed workers as well. More specifically, an increase in x k changes the comparison between any two potential matches involving the worker: for any two job types (y, y ), the difference σ(x, y ) σ(x, y) generally varies with x k. This, in turn, changes how employed workers reallocate between jobs through on-the-job search. This effect, given by term C EE of expression (DC), operates through 16 Two important technical notes: Pr Γ {A} is used to denote the probability of A occurring following a random draw of a job type y from the sampling distribution γ. Second, it may be that the joint event ( σ(x, y ) = s, y j y ) on which some of the expectations in (DC) are conditioned have zero probability in γ. As explained in the detailed derivation, we set expectations conditional on zero-probability events to zero by convention. 13

15 sorting on the EE margin. If we shut down sorting on the UE margin (for example by assuming σ(x, y) > for all (x, y)-pairs), then H j (y x)/ = C EE. In turn, if we shut down sorting on the EE margin (for example by setting λ 1 =, or by specifying a flow surplus function such that the job acceptance probability of employed job seekers is independent of skills x), then H j (y x)/ = C UE. In what follows, we will say that PAM [NAM] occurs on the EE margin whenever C EE is negative [positive], and that PAM [NAM] occurs on the UE margin whenever C UE is negative [positive]. In other words, we analyze sorting on the EE margin as if sorting on the UE margin were shut down, and vice-versa. This is obviously an expositional convention: in general, both margins of sorting are present, and the contributions to overall sorting of both terms C UE and C EE need to be signed in order to determine the sign of H j (y x)/, as indicated by (DC). To offer some economic intuition for (DC), consider the EE margin first. Term C EE is negative if E Γ [ σ(x, y )/ σ(x, y ) = s, y j y ] E Γ [ σ(x, y )/ σ(x, y ) = s] for all s. In the one-dimensional case (Y = 1), one can show that this will be true if 2 σ/ y, i.e. if x k and y are complements in the production technology, echoing the familiar finding from the literature on one-dimensional, frictionless sorting that supermodularity implies PAM. While things are more complex in the multi-dimensional world, we show below that complementarities remain a key determinant of sorting patterns in that case. Note that a similar expression, [ ] E Γ σ(x, y )/ σ(x, y ) =, y j y E Γ [ σ(x, y )/ σ(x, y ) = ], is present in term C UE, hinting at the importance of productive complementarities for UE sorting as well. Beyond this basic intuition about the driving forces of sorting on the UE and EE margin, those two margins involve complex interactions between the technology σ and the sampling distribution of job types γ. This implies that terms C UE and C EE in (DC) cannot easily be signed without further assumptions on the primitives. In order to make progress towards a characterization of the sign of sorting, we focus in the next section on a class of technologies for which we can derive clean and (with one exception) distribution-free conditions for positive sorting. We investigate generalizations in the following section and in the appendix. 5 The Sign of Sorting: Bilinear Technology in Two Dimensions 5.1 Assumptions The following two assumptions simplify decomposition (DC) considerably. 14

16 Assumption 1. (a) The production function p(x, y) is bilinear in (x, y): X Y p(x, y) = (x + a) Qy = q kj (x k + a k )y j k=1 j=1 where Q = (q kj ) 1 k X 1 j Y is a X Y matrix and a = (a 1,, a X ) R X + is a fixed vector; (b) the nonemployment income function b(x) is linear in x: X Y b(x) = (x + a) Qb = q kj (x k + a k )b j k=1 j=1 where b = (b 1,, b Y ) R Y is a fixed vector; (c) there exists j {1,, Y } such that q j (x) := p(x, y)/ y j = X k=1 q kj(x k + a k ) > for all x X ; to fix the notation, we will assume w.l.o.g. that q Y (x) >. Assumptions 1 (a)-(b) restrict the production technology in such a way that the flow surplus function σ(x, y) is bilinear in (x, y). Indeed they imply that: X Y σ(x, y) = p(x, y) b(x) = (x + a) Q(y b) = q kj (x k + a k )(y j b j ) k=1 j=1 The technology matrix Q captures the complementarity structure between all job and worker characteristics and will be crucial to our analysis of sorting. We interpret the vector b as the production technology of the unemployed. In turn, we interpret vector a, which is a technological parameter, as the baseline productivity of workers, noting that a Qy is the output of a type-y job filled with the least skilled worker, x = 1 X. We assume a to be positive (Assumption 1(a)). This ensures that the worker s total input into production, x + a, is positive in all dimensions (recall that x R X + ). While not strictly necessary for our analysis, this restriction ensures that our sorting results do not change with the sign of x + a. Finally, Assumption 1(c) ensures that, for any level of worker skills, there is at least one job attribute, here denoted by y Y, that impacts output positively.note that we do not impose monotonicity of the production function in all job attributes. Nor do we restrict the monotonicity of the production or flow surplus function in worker skills x. Next, we consider: Assumption 2. Each job has Y = 2 attributes, i.e. y Y R 2 +. The sorting results established in the next subsections will rely on Assumptions 1 and 2 (our 15

17 baseline model). In the following section, we provide generalizations of our results on the sign of sorting to other surplus functions and to higher dimensions of job heterogeneity. We now investigate the sign of sorting along both the EE and the UE margin, based on (DC). 5.2 The EE Margin We begin with the following result on the sign of EE sorting. Theorem 1 (EE Sorting, Y = 2, Bilinear Technology). Under Assumptions 1 and 2, PAM occurs in dimension (y 1, x k ) along the EE margin if and only if, for all y Y: ( ) p(x, y)/ y1 > or, equivalently: p(x, y)/ y 2 ( ) q1 (x) >. (SC-2D) q 2 (x) Condition (SC-2D) is a single-crossing condition of the production function (also known as Spence-Mirrlees condition, in its differential form). Note that, in the two-dimensional, bilinear case at hand, (SC-2D) simply amounts to det Q >, which does not depend on any skill bundle x and either holds or fails to hold for all x X. Single-crossing properties have been shown to guarantee positive sorting in several one-dimensional matching problems. 17 The analysis of our multi-dimensional matching model with search frictions and transferable utility further highlights the importance of single-crossing as a driving force toward positive sorting. Condition (SC-2D) states that the marginal rate of substitution between (y 1, y 2 ) is increasing in worker skill x k. This implies that skill x k is more complementary to job attribute y 1 than to y 2, which is why positive sorting occurs between between x k and y 1 (but negative sorting between between x k and y 2 in the dimension of relatively weak complementarity). 18 To illustrate the single crossing condition in our setting and its implication graphically, we consider two workers with skill bundles x = (x 1, x 2 ) and x = (x 1, x 2 ) such that x 1 > x 1 and x 2 = x 2 (the second worker has more of x 1 but both have equal amounts of x 2 ). In Figure 1, we plot for each worker the locus of jobs with which that worker produces the same output as with the job that has attributes A. Single-crossing condition (SC-2D) implies that these isoquants cross only once (at point A). Moreover, because the marginal rate of substitution is increasing in x 1, the curve of the more skilled worker is steeper. Consider point A as a benchmark with 17 In an important paper, Legros and Newman (27) show that a single crossing property is sufficient to guarantee PAM in frictionless one-dimensional problems with non-transferable utility (NTU). Chade, Eeckhout and Smith (217) then demonstrate that several one-dimensional matching problems with transferable utility both in environments with and without frictions can be recast as NTU, frictionless matching problems. After finding the associated NTU problem, the Legros-Newman-condition can be applied and guarantees PAM. 18 This statement about NAM in (y 2, x k ) holds true if p(x, y)/ y 1 > in addition to p(x, y)/ y 2 >. 16

18 y 2 Low x 1 High x 1 A B p(x, y) p(x, y A ) p(x, y) p(x, y A ) y 1 Figure 1: Single Crossing Property no sorting (both workers are matched to the same job). Condition (SC-2D) says the following: if the lower-skilled worker weakly prefers job B over job A where B has lower y 2 but higher y 1, then the worker with higher x 1 strictly prefers job B, as is the case in the graph. 5.3 The UE Margin The next result establishes conditions for positive sorting along the UE margin. Theorem 2 (UE Sorting, Y = 2, Bilinear Technology). Under Assumptions 1 and 2 and under single-crossing condition (SC-2D) (from Theorem 1), if for all x X : 1. q 1 (x) > (i.e. p(x, y) is not only increasing in y 2 but also in y 1 ) 2. along all level curves of p(x, ) (i.e. at all y such that p(x, y) = C for some fixed C ): q 2 (x) 2 ln γ y 2 y 1 q 1 (x) 2 ln γ y 2 2 (UE-2D) 3. at the lower support of γ, denoted by y = (y 1, y 2 ): y 2 b 2 and y 1 < b 1 then PAM occurs in dimension (y 1, x k ) along the UE margin. Moreover, (SC-2D) is also necessary for PAM to occur generically under any sampling distribution γ. Theorem 2 highlights the importance of single-crossing also for sorting on the UE margin. It becomes a necessary condition for sorting if all possible sampling distributions are considered. However, contrary to the EE margin, single crossing alone is not sufficient for PAM on the UE margin: additional restrictions on the sampling distribution are needed, given by (UE-2D). 17

19 Recall that sorting on the UE margin occurs when a marginal increase in skill x k affects the boundary of the set of profitable matches, i.e. the locus of y s such that σ(x, y) =. Figure 2 shows how this boundary shifts with x k under the assumptions of Theorem 2, and helps visualize the role of both single crossing and distributional restrictions in that theorem. σ(x, y) = (low x k ) C y 2 Y σ(x, y) = (high x D A B k ) y 2 (b 1, b 2 ) y 1 y 1 Figure 2: Sorting along the UE margin Figure 2 represents the (y 1, y 2 ) plane, where the origin is placed at b = (b 1, b 2 ). The shaded area materializes Y, the support of γ: the (lower) boundaries of Y are the horizontal line at y 2 = y 2 and the vertical line at y 1 = y 1, which are placed in compliance with Condition 3 in Theorem 2. The oblique lines are zero level curves of σ(x, ), which under the assumed linear technology are given by y 2 = b 2 q 1(x) q 2 (x) (y 1 b 1 ). By Condition 1 in Theorem 2, such lines are downward sloping and go through point y = b. The boundary of feasible matches for a given skill bundle x is at the intersection between the zero level curve of σ(x, ) and Y (the emphasized line segment). Note that this boundary lies entirely in the region of Y where y 1 < b 1 : because it is assumed that y 2 b 2, it has to be the case that y 1 < b 1 for surplus to equal zero. Those zero level curves are drawn for two workers x and x with x k > x k (and with the same amount of all other skills). The higher-x k (blue) curve is steeper than the lower-x k (black) one, meaning that for a given y 2, the more skilled worker needs a higher y 1 to generate non-negative surplus. The reason is as follows: by the single crossing property (SC-2D), complementarities in production are stronger between x k and y 1 than between x k and y 2. Thus, the jobs under consideration (with y 1 < b 1 ) are prone to generate surplus losses especially for those workers with high x k. Therefore, for a given y 2, workers with higher x k need jobs with higher y 1 to generate non-negative surplus, which is clearly a force towards PAM. In the figure, this means 18

20 that all job types between the black and the blue line can be profitably matched with the low skilled (x k ) worker, but produce negative surplus with the high-skilled (x k ) worker. This is why all those jobs with relatively low attribute y 1 drop out of his equilibrium matching set. However, complementarities in production alone are not enough to ensure PAM on the UE margin. To see this, consider points A, B, C and D on the figure. After increasing skill k from x k to x k, a worker no longer breaks even with a job at A. Moreover, jobs around B (with higher y 1 but lower y 2 ) are also made unprofitable while jobs around C with lower y 1 but higher y 2 compared to A remain profitable. Therefore, if the sampling distribution γ has most of its mass concentrated around points A, B and C (i.e. there is a negative correlation between y 1 and y 2 ) then workers with higher x k will be matched to jobs with lower y 1 (since jobs around B with higher y 1 have too little of y 2, leading to negative surplus) a force towards NAM. To prevent this, we assume a sufficient degree of positive association between y 1 and y 2 in γ to ensure that more mass is concentrated around points A and D. Notice that the distributional barrier to PAM arising from a negative association of (y 1, y 2 ) becomes more severe the larger is the positive impact of y 2 on the surplus (i.e. the larger is q 2 (x), which makes the zero-surplus lines flatter). How likely is condition (UE-2D) in Theorem 2, guaranteeing a sufficient positive association of job attributes, to hold? Sufficient conditions on the sampling distribution are that the density γ be log-supermodular and its marginals log-concave. This class of distributions is quite broad. For instance, any bivariate distribution of independent random variables that has log-concave marginals (e.g. the uniform distribution with independent random variables) satisfies (UE-2D). Other examples of log-supermodular densities with log-concave marginals are the (truncated) bivariate Gaussian with positive covariance and the multivariate Gamma distribution Taking Stock Both theorems on the sign of sorting show that sorting under multi-dimensional job heterogeneity is fundamentally different from a comparable model with one-dimensional heterogeneity. In such a model, there is no sorting on the EE margin (Postel-Vinay and Robin, 22): the strategy of firms is to accept any worker that yields positive surplus while the strategy of workers is to accept all jobs that yield a higher (flow) surplus than the current one. The assumption that the flow surplus is increasing in y (the one-dimensional version of Assumption 1) implies that 19 The multivariate Gamma distribution is defined by a linear combination of independent random variables that have standard gamma distribution. Log-supermodularity of the multivariate Gamma distribution is implied by Karlin and Rinott (198), Prop. 3.8, and its log-concavity by Shapiro, Dentcheva, Ruszczynski (29), Th