LABOR MATCHING MODELS: EFFICIENCY PROPERTIES FEBRUARY 1, PDF Free Download

LABOR MATCHING MODELS: EFFICIENCY PROPERTIES FEBRUARY, 209

Eiciency Considerations LABOR-MATCHING EFFICIENCY Social Planning problem Social Planner also subject to matcing TECHNOLOGY t max ( t ct, vt, n uc t+ t= 0 c + v = z n ( + ( n b t t t t t t n = ( t x n + t m( + nt, vt Fix = And n = u February, 209 2

Eiciency Considerations LABOR-MATCHING EFFICIENCY Social Planning problem Social Planner also subject to matcing TECHNOLOGY FOCs t max ( t ct, vt, n uc t+ t= 0 c + v = z n ( + ( n b t t t t t t n = ( t x n + t m( + nt, vt u'( c = 0 t (, 0 t + tm2 nt vt = Fix = And n = u + + + + E z b + E ( m ( n, v = 0 t t t t t t x t t t Eliminate multipliers Multipliers λ t μ t February, 209 3

Eiciency Considerations LABOR-MATCHING EFFICIENCY Social Planning problem u '( c t+ m( nt, vt ( x = Et zt+ b + m2 ( nt, vt u '( ct m2( nt, vt m2( nt, vt February, 209 4

Eiciency Considerations LABOR-MATCHING EFFICIENCY Social Planning problem Cobb-Douglas matcing m( u, v = u v u '( c t+ m( nt, vt ( x = Et zt+ b + m2 ( nt, vt u '( ct m2( nt, vt m2( nt, vt m( u, v m ( u, v u v k ( = = m(, = = = AND u m 2 ( u, v ( u v = = ( m( u, v k ( m( = =, = v Combine and rearrange AND (, m ( u v = k (, ( m ( 2 u v = k k u'( ct ( x E t zt ( zt t ( b + = + + + + + + ( t u '( ct k ( t+ KEY IDEAS Taking te pricing kernel as given, te only unknown process ere is θ t Eiciency in job-postings is governed by eicient market tigtness February, 209 6

Eiciency Considerations LABOR-MATCHING EFFICIENCY k Socially-eicient vacancy posting caracterized by u'( ct ( x E t zt ( zt t ( b + = + + + + + + ( t u '( ct k ( t+ Decentralized vacancy posting caracterized by u'( c t+ ( x = E t zt+ wt + + k ( t u '( ct k ( t+ AND ( w = z + + b t t t February, 209 7

Eiciency Considerations MORTENSEN-HOSIOS CONDITION Cobb-Douglas matcing tecnology + Nas bargaining Eicient level o job-creation requires η = α Mortensen (982 AER, Hosios (990 ReStud February, 209 9

Eiciency Considerations MORTENSEN-HOSIOS CONDITION Cobb-Douglas matcing tecnology + Nas bargaining Eicient level o job-creation requires η = α Mortensen (982 AER, Hosios (990 ReStud Intuition: searc activity generates externalities One extra individual (irm searcing or a job (worker LOWERS te probability tat all oter individuals (irms will ind a matc but RAISES te probability tat all oter irms (individuals will ind a matc Congestion externality searc imposes bot positive and negative externalities (on opposite sides o te market Nas bargaining: η governs te private returns to searc Sare o total matc surplus kept by individual Cobb-Douglas matcing: α governs te social returns to searc Elasticity o aggregate number o matces wit respect to u Eiciency requires equating private and social returns: η = α February, 209

Eiciency Considerations HOSIOS CONDITION Also olds under some more general conditions Endogenous searc intensity Endogenous vacancy posting intensity (Pissarides Capter 5 Pissarides (2000, p. 98:..we are not likely to ind intuition or it RSW (2005 JEL p. 982: genuinely surprising result Is te Hosios condition empirically relevant? Wo knows?...it s a nongeneric parameterization but valuable because eliminates wage-determination rictions but retains matcing tecnology Is Nas bargaining empirically relevant? February, 209 2

Wages HOW ARE WAGES DETERMINED? Nas bargaining Underlying alternating oers bargaining game Te relevant outside option as bargaining is occurring? Value o outside market opportunities? Value o continuing negotiations? (Hall and Milgrom 2008 AER Proportional bargaining Rigid real wages (completely rigid or partially rigid Bargaining teoretic wages are ex-post o matc ormation Seems very dierent rom taking wages as given (ex-ante o matc ormation Competitive searc equilibrium Moen (997 JPE: basic static partial labor searc model February, 209 4

COMPETITIVE SEARCH EQUILIBRIUM (CSE FEBRUARY, 209

Eiciency Considerations COMPETITIVE SEARCH EQUILIBRIUM (CSE Question: can a competitive notion o wage-setting be entertained in a searc and matcing model? Wages playing allocational role in determining meeting process In contrast to wage bargaining, wic plays small/no allocational role May be apriori an appealing way o describing labor markets Locating a irm or a worker is costly and time-consuming but once matced, wages are more or less determined by market orces, peraps wit little/no room or bargaining Moen (997 JPE and Simer (996 te pioneers o CSE Static partial equilibrium labor matcing models Small irms (one irm = one job Re-explore CSE ramework Recent burst o work extending CSE model in various dimensions February, 209 3

Eiciency Considerations CSE BASICS OF ENVIRONMENT Need many markets and many irms To rationalize competition Index continuum o labor submarkets by ij Same geograpic region Same career, regardless o geograpy Many ways to interpret.. submarket denotes notion o closeness Several equivalent ways to implement perectly CSE Firms post wages beore individuals searc or job opportunities Perectly-competitive recruiting sector Individuals announce wages beore irms direct teir vacancies Regardless o implementation, market tigtness is eicient February, 209 5

Firm Posting CSE IMPLEMENTATION I Need many markets and many irms To rationalize competition Index continuum o labor submarkets by ij Same geograpic region Same career, regardless o geograpy Many ways to interpret.. submarket denotes notion o closeness Several equivalent ways to implement perectly CSE Firms post wages beore individuals searc or job opportunities Perectly-competitive recruiting sector Individuals announce wages beore irms direct teir vacancies Regardless o implementation, market tigtness is eicient February, 209 6

Firm Posting CSE IMPLEMENTATION I Job seekers direct teir active searc ( send an application to a particular submarket Based on wages announced by goods-producing irms in tat submarket And on probability o contacting an open vacancy in tat submarket Directed searc a key component o CSE but matc ormation still subject to probabilities Ordering o events Wages determined beore searc job seekers actively direct searc according to posted wages ten probability o landing a matc resolved February, 209 7

Firm Posting CSE IMPLEMENTATION I Firm ij payo unction described by vacancy-posting decision = k ( zt w + ( x Et t+ t k ( jt+ Cost o posting a vacancy February, 209 8

Firm Posting CSE IMPLEMENTATION I Cost o posting a vacancy Firm ij payo unction described by vacancy-posting decision = k ( zt w + ( x Et t+ t k ( jt+ Expected beneit o posting a vacancy Note ij subscripts: Matcing probability depends on tigtness o applications at irm ij but uture asset value o employee depends on market j conditions (i.e., replacement value depends on (sub-market conditions = (probability o matcing wit a worker x (contemporaneous payo + continuation payo Value equations or an individual searcing or a matc at irm ij W( w = w + E ( W( w + U t t+ t x jt+ x t+ Ut = b + Et t+ t k ( t+ W( wt + + ( k ( t+ U t+ Wit probability k (θ, individual gets tis payo Wit probability -k (θ, individual gets tis payo February, 209 0

Firm Posting CSE IMPLEMENTATION I Cost o posting a vacancy Firm ij payo unction described by vacancy-posting decision = k ( zt w + ( x Et t+ t k ( jt+ Expected beneit o posting a vacancy Note ij subscripts: Matcing probability depends on tigtness o applications at irm ij but uture asset value o employee depends on market j conditions (i.e., replacement value depends on (sub-market conditions = (probability o matcing wit a worker x (contemporaneous payo + continuation payo Value equations or an individual searcing or a matc at irm ij Wit probability k (θ, individual gets tis payo Wit probability -k (θ, individual gets tis payo Individuals seeking a job optimally direct teir searc so tat expected payo o successul contact wit a job opening at irm ij is k ( W( w + ( k ( U = X W( w = w + E ( W( w + U t t+ t x jt+ x t+ Ut = b + Et t+ t k ( t+ W( wt + + ( k ( t+ U t+ t H Payo o searcing at anoter irm or anoter submarket independent o ij February, 209

Firm Posting Optimization CSE IMPLEMENTATION I Firm ij maximizes = k ( zt w + ( x Et t+ t k ( jt+ taking as constraint k ( W( w + ( k ( U = X t Coice variables: w and θ (isomorpic to coosing v or a given number o searcers u H February, 209 2

Firm Posting Optimization CSE IMPLEMENTATION I 2 Firm ij maximizes = k ( zt w + ( x Et t+ t k ( jt+ taking as constraint Coice variables: w and θ (isomorpic to coosing v or a given number o searcers u First-order conditions k ( W( w + ( k ( U = X t k ( k ( W '( w = 0 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U H Taking into account ow matcing probabilities are aected by tigtness is te central idea February, 209 3

Firm Posting Result CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( k ( k ( + ( ( U t = 0 zt w x Et t+ t 2 W w k ( jt+ = J February, 209 4

Firm Posting Result CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U Cobb-Douglas matcing m( s, v = s v Combine and rearrange k k = J m( s, v ( = = m(, = s m( s, v ( m( = =, = v ( w J( w = ( W( U t t t AND k k ( ( = ( = Exactly te Nasbargained saring rule wit knie-edge Hosios condition (η = ξ February, 209 5

Matcing Tecnology? CSE IMPLEMENTATION I m( s, v EFF = m s v Exactly te Nasbargained saring rule wit knie-edge Hosios condition (η = ξ February, 209 6

Matcing Tecnology? CSE IMPLEMENTATION I m( s, v EFF = m s v Cange m EFF to ensure probabalitites lie witin [0, ] boundaries Exactly te Nasbargained saring rule wit knie-edge Hosios condition (η = ξ February, 209 7

Matcing Tecnology? CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U = J Wat i dierent matcing unction? m( s, v = sv ( s + v / denhaan, Ramey, Watson (2000 AER February, 209 8

Matcing Tecnology? CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U = J Wat i dierent matcing unction? m(. = s ( + / denhaan, Ramey, Watson (2000 AER February, 209 9

Matcing Tecnology? CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U = J dhrw matcing m(. = s ( + / k k m( s, v ( = = m(, = / s + m( s, v ( = = m(, = v + / February, 209 20

Matcing Tecnology? CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 k ( k ( zt w + ( x Et t+ t ( w t 0 = k ( jt+ W U dhrw matcing m(. = s ( + / k ( = + k k m( s, v ( = = m(, = / s + m( s, v ( = = m(, = v + = J k ( = + + ( + / ( / February, 209 2

Matcing Tecnology? CSE IMPLEMENTATION I First-order conditions k ( k ( W '( w = 0 W (. = k ( = = k ( 2 ( / x k zt w + ( Et t+ t ( w t 0 = k ( jt+ k ( / W U dhrw matcing m(. = s ( + / k ( = + k k m( s, v ( = = m(, = / s + m( s, v ( = = m(, = v + = J k ( = + + ( + / ( / ( w W( U = J( w t = θ -є (Competition witin submarket j and symmetry across submarkets: drop ij indices Combine and rearrange CSE surplus-saring rule or dhrw m(. t t t February, 209 24

February, 209 25

Recruiting Sector CSE IMPLEMENTATION II Need many markets and many irms To rationalize competition Index continuum o labor submarkets by ij Same geograpic region Same career, regardless o geograpy Many ways to interpret.. submarket denotes notion o closeness Several equivalent ways to implement perectly CSE Firms post wages beore individuals searc or job opportunities Perectly-competitive recruiting sector Individuals announce wages beore irms searc or workers Regardless o implementation, market tigtness is eicient February, 209 26

Recruiting Sector CSE IMPLEMENTATION II Recruiting agency ij total proit unction ( mc jt ms (, v recruiter ij operates matcing tecnology m ij February, 209 27

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction max w, ( mc m jt v February, 209 28

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction max w, ( mc m jt v Moen (997 ONE irm = ONE job ( small irms Marginal proit term vacuous in ONE-worker irm Micro origins o searc and matcing ramework: ONE irm = ONE worker Pissarides (985, Mortensen and Pissarides (994 Macro: Goods-producing irms tat need / ire many workers CRTS (k,n: lack o IO structure means number o workers can be ANYTHING February, 209 29

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction max w, ( mc m jt v subject to F k ( J( w X = 0 H k ( W( w + ( k ( U X = 0 t m sij February, 209 30

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction ( mc max ( k ( w, jt Suppose m( s, v = s v subject to F k ( J( w X = 0 H k ( W( w + ( k ( U X = 0 t February, 209 3

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction ( mc max ( k ( w, jt Suppose m( s, v = s v subject to multipliers F k ( J( w X = 0 H k ( W( w + ( k ( U X = 0 t κ H (given CRS m(., only one multiplier needed FOCs wrt w and θ February, 209 32

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w H k ( + k ( = 0 w w 2 k ( k ( k ( ( ( + ( = 0 H ( mc jt J w ( W w Ut = 0 February, 209 33

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w H k ( + k ( = 0 w w 2 k ( k ( H J( w + ( W( w Ut = 0 February, 209 34

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ b/c zero proportional taxation on wage 2 J( w W( w H k ( + k ( = 0 w w k ( k ( H J( w + ( W( w Ut = 0 =- = H k ( = = k ( February, 209 35

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ b/c zero proportional taxation on wage 2 Cobb-Douglas matcing m( s, v J( w W( w H k ( + k ( = 0 w w = s v Combine and rearrange Symmetric equilibrium =- k ( k ( H J( w + ( W( w Ut = 0 k k = m( s, v ( = = m(, = s m( s, v ( m( = =, = v ( w ( W( U = J( w t t t AND H k k k ( = = k ( ( ( = ( = Exactly te Nasbargained saring rule wit knie-edge Hosios condition (η = ξ February, 209 36

Searcer Posting CSE IMPLEMENTATION III Need many markets and many irms To rationalize competition Index continuum o labor submarkets by ij Same geograpic region Same career, regardless o geograpy Many ways to interpret.. submarket denotes notion o closeness Several equivalent ways to implement perectly CSE Firms post wages beore individuals searc or job opportunities Perectly-competitive recruiting sector Individuals announce wages beore irms direct teir vacancies Regardless o implementation, market tigtness is eicient February, 209 37

February, 209 38

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction max w, ( mc m jt s subject to F k ( J( w X = 0 H k ( W( w + ( k ( U X = 0 t m vij February, 209 39

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction ( mc max k ( w, jt Suppose m( s, v = s v subject to F k ( J( w X = 0 H k ( W( w + ( k ( U X = 0 t February, 209 40

Recruiter ij CSE IMPLEMENTATION II Recruiting agency ij marginal proit unction ( mc max k ( w, jt Suppose m( s, v = s v subject to multipliers F k ( J( w X = 0 κ F H k ( W( w + ( k ( U X = 0 t (given CRS m(., only one multiplier needed FOCs wrt w and θ February, 209 4

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w F k ( + k ( = 0 w w 2 k ( k ( k ( ( + ( = 0 F ( mc jt J w ( W w Ut = 0 February, 209 42

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w F k ( + k ( = 0 w w 2 k ( k ( F J( w + ( W( w Ut = 0 February, 209 43

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w F k ( + k ( = 0 w w H k ( = = k ( b/c zero proportional taxation on wage =- = 2 k ( k ( F J( w + ( W( w Ut = 0 February, 209 44

Recruiter ij CSE IMPLEMENTATION II FOCs wit respect to w and θ J( w W( w F k ( + k ( = 0 w w H k = k ( ( = b/c zero proportional taxation on wage =- = 2 Cobb-Douglas matcing m( s, v = s v Combine and rearrange k ( k ( F J( w + ( W( w Ut = 0 k k m( s, v ( = = m(, = s m( s, v ( m( = =, = v ( w ( W( U = J( w t t t AND k k ( ( = ( = Exactly te Nasbargained saring rule wit knie-edge Hosios condition (η = ξ February, 209 45

LABOR MATCHING MODELS: EFFICIENCY PROPERTIES FEBRUARY 1, 2019