MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM FEBRUARY 13, 2019

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1 MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM FEBRUARY 3, 209

2 Introduction LABOR MARKET INTERMEDIATION Recruiting Sector aka Labor Market Intermediaries aka Headunters aka Middlemen Develop Monopolistically Competitive Recruiting Model Moen (997 JPE), Simer (996) Bilbiie, Gironi, and Melitz (202 JPE) Pissarides (985 AER) Based on components o tese rameworks Wage model Implications or aggregate matcing Eects between recruiting-market matces and non-recruiting matces Implications or general equilibrium MAIN QUESTIONS February 3, 209 2

3 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Recruiter ij proit maximization Cost minimization (directed-searc optimization) Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3, 209 3

4 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Recruiter ij proit maximization Cost minimization (directed-searc optimization) Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3, 209 4

5 Recruiting Sector MONOPOLISTIC RECRUITING MARKET Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry FACTOR MARKETS (searc and vacancies) (Symmetric equilibrium or all i in j, and or all j) CREATES NEW EMPLOYMENT MATCHES RECRUITING SECTOR j MATCHING Aggregate recruiting irm j N Mjt ms (, ) 0 v di Sell dierentiated matces to matcing bundler j DIFFERENTIATED RECRUITER j DIFFERENTIATED RECRUITER 2j DIFFERENTIATED RECRUITER NMj DIFFERENTIATED/ SPECIALIZED RECRUITERS IN LABOR MARKET j Measure N M o monopolistic recruiters, eac o wic produces a dierentiated matc February 3, 209 5

6 Recruiting Sector ENDOGENOUS ENTRY IN RECRUITING MARKET Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry Free Entry in Recruiting Markets Representative Recruiter j Cost o creating new dierentiated m(.) and entering market { N Mjt, NMEjt } t = 0 ( N ) Mjt max E 0 t 0 ( mc ) m( s, v ) di Mt N 0 t= 0 N = ( ) N + N jt MEjt Mjt Mjt MEjt Cost o entry Γ Mt Tecnological R&D Regulatory = Mt TECH R& D REG Mt Mt Mt February 3, 209 6

7 Recruiting Sector ENDOGENOUS ENTRY IN RECRUITING MARKET Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry Free Entry in Recruiting Markets Representative Recruiter j Cost o creating new dierentiated m(.) and entering market { N Mjt, NMEjt } t = 0 ( N ) Mjt max E 0 t 0 ( mc ) m( s, v ) di Mt N 0 t= 0 N = ( ) N + N jt MEjt Mjt Mjt MEjt Free-entry condition determines new recruiting agencies N MEjt = Mt ( mc jt ) m( s, v ) + ( ) E t t + t Mt + w/ i = NMjt February 3, 209 7

8 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry Matcing Aggregator Dixit-Stiglitz ( Benassy ) Translog Incentive or Entry vs. Welare Beneit o Increasing Returns to Scale Dixit-Stiglitz Tecnology Eiciently Balances Tradeo Translog and Benassy Tecnologies Ineiciently Balance Tradeo February 3, 209 8

9 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry Matcing Aggregator Dixit-Stiglitz ( Benassy ) Translog Dixit-Stiglitz tecnology N Mjt jt = 0 m m di j labor-market j aggregator dmd_ct February 3, 209 9

10 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, ] o recruiting markets Perectly-competitive index by j Measure [0, N Mj ] o monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via ree entry Matcing Aggregator Dixit-Stiglitz ( Benassy ) Translog Demand unction or recruiter ij (Dixit-Stiglitz) = m m jt i j February 3, 209 0

11 Monopolistic Recruiter ij RECRUITER ij PROFIT-MAXIMIZATION = mc jt PERFECT CSE: ε = ininity (recovers Moen 997) Gross matcingmarket markup marginal cost o creating new job matc Generally (symmetric equilibrium) ( N ) = ( N ) mc( N ) Mt Mt Mt February 3, 209

12 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Proit-maximizing (ρ *, m * (s, v )) cosen Monopolistic recruiter ij s recruiting problem Recruiting irm ij must attract irms to post vacancies in submarket ij Recruiting irm ij must attract active job searcers to send résumés to (i.e., searc in) submarket ij Deinitions J( w ) ( ) W w value to goods-producing irm o successully iring worker in submarket ij value to worker o successully inding a job in submarket ij U outside option o worker i unsuccessul in inding a job in submarket ij February 3, 209 2

13 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij total proit unction V ( mc ) ( s, v jt ) m( s, vij ) M i jt t Marginal proit conditions V s M = ( mc jt ) ms V v M = ( mc jt ) mv s V M v V M February 3, 209 3

14 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction max w, ( mc ) m jt v February 3, 209 4

15 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction subject to max w, ( mc ) m jt v F p k ( ) J( w ) X = 0 v jt H p + k ( ) W( w ) + ( k ( )) U X = 0 s t jt February 3, 209 5

16 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction subject to, ( mc ) max ( ) k ( ) w jt Suppose m( s, v) = s v F p k ( ) J( w ) X = 0 v jt H p + k ( ) W( w ) + ( k ( )) U X = 0 s t jt February 3, 209 6

17 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction subject to, ( mc ) max ( ) k ( ) w jt Suppose m( s, v) = s v F p k ( ) J( w ) X = 0 v jt multipliers H p + k ( ) W( w ) + ( k ( )) U X = 0 s t jt κ (given CRS m(.), only one multiplier needed) FOCs wrt w and θ February 3, 209 7

18 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) k ( ) k ( ) k ( ) ( ) ( ) + ( ) = 0 H ( mc jt ) J w ( W w Ut ) February 3, 209 8

19 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) k ( ) k ( ) k ( ) + ) = 0 H ( mc jt ) ( ) J( w ) ( W( w Ut ) 0 MONOPOLISTICALLY competitive recruiting sector February 3, 209 9

20 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) k ( ) k ( ) k ( ) + ) = 0 H ( mc jt ) ( ) J( w ) ( W( w Ut ) 0 MONOPOLISTICALLY competitive recruiting sector Crucial or EFFICIENT matcing in decentralized economy elast February 3,

21 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) Cobb-Douglas matcing m( s, v) k ( ) k ( ) k ( ) + ) = 0 H ( mc jt ) ( ) J( w ) ( W( w Ut ) = s v Combine and rearrange k k m( s, v) k ( ) ( ) = = m(, ) = = ( ) s AND m( s, v) ( ) = = m(,) = k ( ) = v February 3, 209 2

22 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistically Competitive Surplus Saring Condition ( ) ( w ) ) mc + ( ) W( ) U = J( w ) ( jt payo accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer February 3,

23 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistically Competitive Surplus Saring Condition ( ) ( w ) ) mc + ( ) W( ) U = J( w ) ( jt payo accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer substitute mc = ρ/μ symmetric equilibrium unctional dependence o ρ(.) and μ(.) on N M (see Bilbiie, Gironi, Melitz 2008 NBER WP, 206 NBER WP) February 3,

24 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistically Competitive Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) payo accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer Observations ( N ) Mt As ( NMt ) 0 surplus saring condition PERFECTLY competitive ( NMt ) February 3,

25 Analytical Result SURPLUS SHARING Proposition. Perectly Competitive Surplus Saring Condition ( w ) ( ) W( ) U = J( w ) t t surplus accruing to new employee and new employer are EFFICIENT contributions to m(.) in decentralized CSE Observations ( N ) Mt As ( NMt ) 0 surplus saring condition PERFECTLY competitive ( NMt ) February 3,

26 Analytical Result SURPLUS SHARING Proposition. Perectly Competitive Surplus Saring Condition ( w ) ( ) W( ) U = J( w ) t t Suppose m( s, v) = s v surplus accruing to new employee and new employer are EFFICIENT contributions to m(.) in decentralized CSE Observations ( N ) Mt As ( NMt ) 0 surplus saring condition PERFECTLY competitive ( NMt ) Depends on matcing process m(s, v) February 3,

27 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Recruiter ij marginal proit unction max w, ( mc ) m jt s Marginal proit conditions V s M = ( mc jt ) ms V M v = ( mc jt ) mv February 3,

28 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Recruiter ij marginal proit unction, ( mc ) max k ( ) w jt Suppose m( s, v) = s v Marginal proit conditions V s M = ( mc jt ) ms V M v = ( mc jt ) mv February 3,

29 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Recruiter ij marginal proit unction, ( mc ) max k ( ) w jt Suppose m( s, v) = s v Marginal proit conditions V s M = ( mc jt ) ms V M v = ( mc jt ) mv m( s, v) m s m v = = s v k ( ) m( s, v) = ( ) k ( ) = = ( ) k k ( ) = = m(, ) = s m( s, v) ( ) m( = =,) = v February 3,

30 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Recruiter ij marginal proit unction, ( mc ) max k ( ) w jt Suppose m( s, v) = s v Marginal proit conditions V s M = ( mc jt ) ms V M v = ( mc jt ) mv m( s, v) m s m v = = s v = ( ) k k ( ) ( ) = = ( ) k k m( s, v) ( ) = = m(, ) = s m( s, v) ( ) m( = =,) = v Ratio caracterizes EFFICIENT matcing (more to say soon..) Ratio crucial or EFFICIENT matcing in decentralized economy February 3,

31 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Marginals and elasticities s ms k ( ) k ( ) / ms, m(, s v) = = k ( ) k ( ) / v m mv, v ms (, v) February 3, 209 3

32 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Marginals and elasticities s ms k ( ) k ( ) / ms, m(, s v) = = k ( ) k ( ) / v m mv, v ms (, v) = θ FOCs February 3,

33 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Marginals and elasticities s ms k ( ) k ( ) / ms, m(, s v) = = k ( ) k ( ) / v m mv, v ms (, v) = θ Suppose k ( ) = m s = m( s, v) = s v k ( ) = ( ) m v = ( ) ms, = mv, February 3,

34 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Explicit-orm wage expression ms, = mv, Suppose m( s, v) = s v w = z + + E + r t t t t + February 3,

35 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Explicit-orm wage expression ms, = mv, Suppose m( s, v) = s v w = z r ( ) ( ) E t t t t February 3,

36 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Explicit-orm wage expression ms, = mv, Suppose m( s, v) = s v w = z r ( ) ( ) E t t t t m m s v = February 3,

37 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Marginals and elasticities s ms k ( ) k ( ) / ms, m(, s v) = = k ( ) k ( ) / v m ms (, v) Suppose dhrw m(.) m( s, v) = sv mv, v ( ) / = θ s + v February 3,

38 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Marginals and elasticities s ms k ( ) k ( ) / ms, m(, s v) = = k ( ) k ( ) / v m ms (, v) Suppose dhrw m(.) m( s, v) = sv mv, v ( ) / s + v k k m( s, v) ( ) = = m(, ) = s ( ) m(,) = θ + m( s, v) = = = v + / / ms, mv, = k ( ) = + ( + ) / k ( ) = + + ( ) February 3,

39 Dependence on Matcing Process MARGINALS VS. ELASTICITIES Explicit-orm wage expression ms, mv, = Suppose dhrw m(.) m( s, v) = sv ( s + v ) / w = z + + E r t t t t + February 3,

40 February 3,

41 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction subject to max w, ( mc ) m jt s Suppose dhrw m(.) F p k ( ) J( w ) X = 0 v jt H p + k ( ) W( w ) + ( k ( )) U X = 0 s t jt February 3, 209 4

42 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal proit unction subject to max w, ( mc ) m jt s Suppose dhrw m(.) multipliers m = s ( + ) / F p k ( ) J( w ) X = 0 v jt H p + k ( ) W( w ) + ( k ( )) U X = 0 s t jt κ ( ) m = + s m s = ( + ) ( + ) = = + / / ( + ) + ( + ) FOCs wrt w and θ (given CRS m(.), only one multiplier needed) February 3,

43 ) b/c zero proportional taxation on wage 2) FOCs wit respect to w and θ J( w ) W( w ) H k ( ) + k ( ) = 0 w w =- = s H ( mc jt ) J w + ( W w Ut ) Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) m k ( ) k ( ) ( ) ( ) = 0 V V M v M s = v V M = s V M ( mc jt ) mv ( mc jt ) ms s VM = = m s ( mc jt ) February 3,

44 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = m k ( ) k ( ) ( ) ( ) = 0 s H ( mc jt ) J w + ( W w Ut ) s VM = = m s ( mc jt ) Divide by dk /dθ February 3,

45 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = / m k ( ) ( ) / + ( ) = 0 k ( ) / k ( ) / s H ( mc jt ) J w ( W w Ut ) ms / ( ) / k = ( mc jt ) = k ( ) / k ( ) / = V k s M / ( ) / February 3,

46 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = m / k ( ) / ( ) ( ) = 0 k ( ) / k ( ) / s ( mc jt ) J w ( W w Ut ) ms / ( ) / k = θ ( mc jt ) -є = = k ( ) / k ( ) / = V k s M / ( ) / February 3,

47 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = m / ( ) + ( ) = 0 k ( ) / s ( mc jt ) J w ( W w Ut ) ms / ( ) / k = ( mc jt ) = k ( ) / k ( ) / = V k s M / ( ) / February 3,

48 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = m / ( ) + ( ) = 0 k ( ) / s ( mc jt ) J w ( W w Ut ) dhrw m(.) ms / ( ) / k = ( mc jt ) = ( mc jt ) k ( ) / k ( ) / it j = V k s M / ( ) / February 3,

49 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage 2) =- = m / ( ) + ( ) = 0 k ( ) / s ( mc jt ) J w ( W w Ut ) dhrw m(.) ms / ( ) / k = ( mc jt ) = ( mc jt ) k ( ) / k ( ) / it j = V k s M / ( ) / = -θ -є February 3,

50 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) ( mc jt ) J w ( W w Ut ) ( ) + ( ) = 0 February 3,

51 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) ( mc jt ) J w ( W w Ut ) ( ) + ( ) = 0 ( W( w ) U ) J( w ) ( mc ) = + t jt February 3, 209 5

52 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs wit respect to w and θ ) J( w ) W( w ) H k ( ) + k ( ) = 0 w w H k = k ( ) ( ) = b/c zero proportional taxation on wage =- = 2) ( mc jt ) J w ( W w Ut ) ( ) + ( ) = 0 ( W( w ) U ) J( w ) ( mc ) = + t jt ( ) W( w ) U = J( w ) + mc t jt February 3,

53 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Consider symmetric equilibrium Compare wit C-D m(.) monopolistically-competitive surplus saring direct relationsip between timevarying θ t and time-varying N Mt ( N ) Mt W( wt ) Ut = t J( wt ) + t ( NMt ) ( NMt ) ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) February 3,

54 February 3,

55 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Proit maximization Cost minimization (directed-searc optimization) CRUCIAL Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3,

56 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistic Surplus Saring Condition ( ) ( w ) ) mc + ( ) W( ) U = J( w ) ( jt payo accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer February 3,

57 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistic Surplus Saring Condition ( ) ( w ) ) mc + ( ) W( ) U = J( w ) ( jt payo accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer substitute mc = ρ/μ symmetric equilibrium unctional dependence o ρ(.) and μ(.) on N M (see Bilbiie, Gironi, Melitz 2008 NBER WP, 206 NBER WP) February 3,

58 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistic Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) payo accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer Observations ( N ) Mt As ( NMt ) 0 surplus saring condition PERFECTLY competitive ( NMt ) February 3,

59 Analytical Result SURPLUS SHARING ξ is elasticity o m ij (.) wrt s ij Wage Model Proposition. Monopolistic Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) extra resources?... surplus accruing to new employee surplus accruing to new employer Observations ( N ) Mt As ( NMt ) 0 surplus saring condition PERFECTLY competitive ( NMt ) Matcing elasticity ξ in (0,). From were do extra resources arise? February 3,

60 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing 0 N Mjt m di j Dixit-Stiglitz Aggregation ( N t ) = IRTS eect (elasticity) Dixit-Stiglitz N Mt t t integrate over i (integrate over j) m( s, v ) Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model February 3,

61 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing 0 N Mjt m di j Dixit-Stiglitz Aggregation ( N t ) = IRTS eect (elasticity) Dixit-Stiglitz integrate over i (integrate over j) N N m( s, v ) Mt Mt t t Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model February 3, 209 6

62 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing 0 N Mjt m di j Dixit-Stiglitz Aggregation ( N t ) = IRTS eect (elasticity) Dixit-Stiglitz integrate over i (integrate over j) N N m( s, v ) Mt Mt t t more generally Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model ( N ) N m( s, v ) Mt Mt t t Holds or any aggregator February 3,

63 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz ( ) N Mt + ( ) ( W( wt) U) = J( wt) markup eect IRTS eect February 3,

64 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing N Mjt 0 + N Mt m di j Benassy Aggregation integrate over i (integrate over j) Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model ( N ) N m( s, v ) Mt Mt t t Holds or any aggregator February 3,

65 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing N Mjt 0 + N Mt m di j Benassy Aggregation integrate over i (integrate over j) N N ms (, v ) Mt Mt t t Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model ( N ) N m( s, v ) Mt Mt t t Holds or any aggregator February 3,

66 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matcing N Mjt 0 + N Mt m di j Benassy Aggregation ( N t ) = IRTS eect INDEPENDENT o markup eect integrate over i (integrate over j) N N ms (, v ) Mt Mt t t Requires BOTH monopolistic competition AND costs o entry ala Romer endogenous growt model ( N ) N m( s, v ) Mt Mt t t Holds or any aggregator February 3,

67 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy ( ) N Mt + ( ) ( W( wt) U) = J( wt) markup eect IRTS eect φ measures increasing returns to scale (independent o ε) ( ) N Mt + ( ) ( W( wt) U) = J( wt) markup eect IRTS eect Incentive or Entry Welare Beneit o Aggregate IRTS February 3,

68 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy ( ) N Mt + ( ) ( W( wt) U) = J( wt) markup eect IRTS eect lim φ 0 ( ) ( ) ( ( wt) ) ( wt ) + W U = J markup eect Incentive or Entry Welare Beneit o Aggregate IRTS Declines under Benassy aggregation as φ 0 February 3,

69 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy ( ) N Mt + ( ) ( W( wt) U) = J( wt) markup eect IRTS eect φ measures increasing returns to scale (independent o ε) ( ) N Mt + ( ) ( W( wt) U) = J( wt) Translog ( ) ( N ) markup eect IRTS eect N Mt NM N Mt ( ) exp ( ) ( ( wt ) ) ( wt ) + W U = J + 2 N Mt M NMt February 3, 209 markup eect IRTS eect 69

70 February 3,

71 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity o m ij (.) wrt s ij Proposition. Monopolistic Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) payo accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer February 3, 209 7

72 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity o m ij (.) wrt s ij Proposition. Monopolistic Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) payo accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer substitute W(.), U, and J(.) Monopolistic Wage (explicitorm) w = z ( k, n ) + ( ) + ( ) E t t n t t t t+ t t+ ( N ) Mt ( N ) Mt+ ( ) ( NMt ) ( ) Et t+ t ( NMt+ ) ( NMt) ( NMt+ ) February 3,

73 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity o m ij (.) wrt s ij Proposition. Monopolistic Surplus Saring Condition ( N ) Mt ( ) ( NMt ) + ( ) ( W( wt) U) = J( wt) ( NMt ) payo accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer substitute W(.), U, and J(.) Monopolistic Wage (explicitorm) w = mpn + ( ) + ( ) ( N ) M ( )( ) ( NM ) ( NM ) steady state February 3,

74 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Proit maximization Cost minimization (directed-searc optimization) CRUCIAL Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3,

75 GE Model GENERAL EQUILIBRIUM Introduce random-searc matcing and Nas-bargained wages February 3,

76 GE Model GENERAL EQUILIBRIUM Introduce random-searc matcing and Nas-bargained wages Submarket ij Labor Supply (directed searc) '( lp ) k '( lp ) p t jt+ t+ s = ps + k jt w + Et t+ t k + t k jt+ u c t+ jt+ ( ) ( ) u '( c ) '( ) U ij ( ) W w HH_OPT February 3,

77 GE Model GENERAL EQUILIBRIUM Introduce random-searc matcing and Nas-bargained wages Submarket ij Labor Supply (directed searc) '( lp ) k '( lp ) p t jt+ t+ s = ps + k jt w + Et t+ t k + t k jt+ u c t+ jt+ ( ) ( ) u '( c ) '( ) U ij ( ) W w Submarket ij Labor Demand (directed vacancies) k = pv + k z ( k, n ) w + ( ) E jt t n t t t t + t k p v jt+ jt+ ij J( w ) FIRM_OPT February 3,

78 GE Model GENERAL EQUILIBRIUM Symmetric equilibrium across ij Aggregate law o motion or labor n = ( ) n + ( N ) N m( s, v ) + m( s, v ) t t Mt Mt t t Nt Nt new job matces via monopolistic recruiting new job matces via random searc February 3,

79 GE Model GENERAL EQUILIBRIUM Symmetric equilibrium across ij Aggregate law o motion or labor n = ( ) n + ( N ) N m( s, v ) + m( s, v ) t t Mt Mt t t Nt Nt new job matces via monopolistic recruiting new job matces via random searc Aggregate resource rontier [ Absorption ] = z ( k, n t t t ) (Std. procedure or aggregation: sum BCs, substitute equil. expressions) February 3,

80 Decentralized Economy DEFINITION GENERAL EQUILIBRIUM State-contingent stocastic unctions { c, n, lp, k, N M, N ME, s, v, θ, s N, v N, θ N, w, w N, p v, p s } t=0 tat satisy Searc directed towards monopolistic submarkets Vacancies directed towards monopolistic submarkets Monopolistic wage surplus saring Free-entry condition or recruiters Aggregate law o motion or recruiters Aggregate law o motion or employment s N and v N in random-searc matcing cannel Aggregate LFP (determined by (lp t )/u (c t )) Capital Euler equation Nas wage surplus saring Aggregate goods resource rontier Input prices p vt and p st (markdown o respective marginal products) Deinitions o tigtness θ t and θ Nt Given stocastic process zt t= 0 and initial conditions k 0, n -, N M- (State vector: x t = [ k t, n t-, N Mt-, z t ] ) February 3,

81 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Proit maximization Cost minimization (directed-searc optimization) CRUCIAL Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3, 209 8

82 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matcing aggregation. * * N M = 0 i (eicient bargaining power) = = η is worker s bargaining power ξ is elasticity o m ij (.) wrt s ij February 3,

83 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matcing aggregation. * * N M = 0 i (eicient bargaining power) = = η is worker s bargaining power ξ is elasticity o m ij (.) wrt s ij Lemma (Static Model). * * N M 0 and 0 i * * N M 0 and 0 i (low worker bargaining power) (ig worker bargaining power) February 3,

84 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matcing aggregation. * * N M = 0 i (eicient bargaining power) = = η is worker s bargaining power ξ is elasticity o m ij (.) wrt s ij Lemma (Static Model). * * N M 0 and 0 i * * N M 0 and 0 i (low worker bargaining power) (ig worker bargaining power) Distortion (wage) in random searc causes distortion in recruiting sector Despite eicient Dixit-Stiglitz aggregation Quant. Veriication February 3,

85 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matcing aggregation. * * N M = 0 i (eicient bargaining power) = = η is worker s bargaining power ξ is elasticity o m ij (.) wrt s ij Lemma (Static Model). * * N M 0 and 0 i * * N M 0 and 0 i (low worker bargaining power) (ig worker bargaining power) Distortion (wage) in random searc causes distortion in recruiting sector Despite eicient Dixit-Stiglitz aggregation Causality o distortionary spillover does NOT run in opposite direction Intuition: insuicient margins o adjustment Quant. Veriication February 3,

86 Plan OUTLINE Structure o Recruiting Markets Free entry in recruiting markets Proit maximization Cost minimization (directed-searc optimization) CRUCIAL Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k accumulation Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3,

87 Parameters CALIBRATION Utility Aggregate LFP + u( ct ) ( lpt ) = ln ct lp t + / lp ( ) n + s N + s t t t Mt Nt Cobb-Douglas matcing unction ms (, v ) = m s v EFF t t t t (or bot matcing unctions) m EFF larger in recruiting market February 3,

88 Parameters CALIBRATION Utility Aggregate LFP + u( ct ) ( lpt ) = ln ct lp t + / lp ( ) n + s N + s t t t Mt Nt Cobb-Douglas matcing unction ms (, v ) = m s v EFF t t t t (or bot matcing unctions) m EFF larger in recruiting market β = 0.99 Matcing elasticity ξ =0.4 Exogenous job-separation rate ρ = 0.0 Exogenous recruiter exit rate ω = 0.05 Stocastic TFP process ln z = + ln z + z t z t t (Table 4 contains oter baseline parameters) February 3,

89 Quantitative Results SPILLOVER EFFECTS Proposition 2 * * N M = 0 and = 0 i = Lemma * * N M 0 and 0 i * * N M 0 and 0 i February 3, Back to Prop. 2 Outline

90 Plan OUTLINE Structure o Labor Markets Free entry in recruiting markets Proit maximization Cost minimization (directed-searc optimization problem) CRUCIAL Analytical Results Part I Surplus saring condition Aggregate IRS in new job creation General Equilibrium Model Pysical k accumulation Directed searc labor supply and labor demand conditions Aggregate resource rontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion February 3,

91 Conclusion SUMMARY Monopolistically Competitive Recruiting Model Moen (997 JPE), Simer (996), Pissarides (985 AER) Bilbiie, Gironi, and Melitz (202 JPE) Tractable Model Easy to Extend Provides New Competitive Wage Model Aggregate Increasing Returns in Intermediated Matcing Expansion o Aggregate Resource Frontier Eects Between Non-Intermediated and Intermediated Matcing February 3, 209 9

92 February 3,

93 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j A continuum o aggregate recruiting agencies Eac aggregate recruiting agency is perectly competitive Easier to deal wit matematically tan discrete ininity (tools o calculus can be applied) Representative recruiting agency j s proit unction Relative price ρ o submarket recruiter ij N Mjt m( s, v ) m di jt jt 0 Substitute aggregate Dixit-Stiglitz matcing tecnology N Mjt NMjt 0 0 m m di February 3,

94 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Representative recruiter s proit-maximization problem max mi= 0... N Mjt N Mjt N 0 0 FOC wit respect to m (or all ij) Mjt m m di Cooses proit-maximizing quantity o input o eac submarket matc.... ater several rearrangements m = m jt jt = m m i j DEMAND FUNCTION FOR RECRUITER ij February 3,

95 Recruiting Sector MONOPOLISTIC RECRUITING SUBMARKET ij Focus on proit-maximization o an arbitrary monopolistic recruiter ij Assume zero ixed costs o creating a matc Operates a constant-returns-to-scale (CRS) matcing tecnology in order to create its specialized, dierentiated matc CRS: i all inputs are scaled up by te actor x, total output is scaled up by te actor x Implementation o teory requires speciying neiter te actors o production (i.e., active searc s, vacancies v, etc) nor a matcing unction (m(.)) m( s, v ) = s v February 3,

96 Recruiting Sector MONOPOLISTIC RECRUITING SUBMARKET ij Focus on proit-maximization o an arbitrary monopolistic recruiter ij Assume zero ixed costs o creating a matc Togeter, tese imply a simple description o production Operates a constant-returns-to-scale (CRS) matcing tecnology in order to create its specialized, dierentiated matc CRS: i all inputs are scaled up by te actor x, total output is scaled up by te actor x Implementation o teory requires speciying neiter te actors o production (i.e., active searc s, vacancies v, etc) nor a matcing unction (m(.)) Marginal cost o creating a matc = average cost o creating a matc is invariant to te quantity o matces created i.e., mc is NOT a unction mc(quantity o matces) m( s, v ) = s v February 3,

97 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem Total revenue depends on matc creation and its own submarket ij price. max m( s, v ) mc jt m( sit j, v ) mc is NOT a unction o matces created (due to CRS m(.)) FC = 0 mc = ac February 3,

98 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem Total revenue depends on matc creation and its own submarket ij price. max m( s, v ) mc jt m( sit j, v ) mc is NOT a unction o matces created (due to CRS m(.)) FC = 0 mc = ac Substitute in demand unction or recruiter ij m = m jt Critical point or analysis o monopoly: te recruiter understands and internalizes te eect o its price on te quantity tat it creates. max m( s jt, v jt ) mc jt it j m( sjt, vjt ) Proit-maximization ( stage one ) Compute FOC wit respect to relative price ρ ij February 3,

99 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem max m( s jt, v jt ) mct m( s jt, v jt ) FOC wit respect to ρ ( ) m( s jt, v jt ) + mc jt m( s jt, v jt ) = 0 February 3,

100 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem max m( s jt, v jt ) mct m( s jt, v jt ) FOC wit respect to ρ ( ) m( s jt, v jt ) + mc jt m( s jt, v jt ) = 0 Algebraic rearrangement Optimal relative price o recruiter j is a markup ε/(ε ) over marginal cost o creating specialized/dierent matc. KEY PRICING RESULT OF DIXIT- STIGLITZ THEORY. = mc Gross matcingmarket markup jt Linked only to degree o substitutability across monopolistic recruiters i PERFECT CSE: ε = ininity Monopolistic matcing: ε > and ε < ininity February 3,

101 February 3, 209 0

102 Appendix HOUSEHOLD OPTIMIZATION Houseold utility N t Mjt E0 u( ct ) nt ( knt ) s Nt ( k ) s di dj t= 0 N = uet = ue low budget constraint Mjt = ( + r ) k + w ( ) n + w k s NMjt NMjt ps s ( ) ( ) jt i di dj k s di dj c k T t t t t t t t Nt Nt Nt M F jt knt snt + jt dj +t N w k s di dj perceived LOM or labor FOCs wrt c t, n t, k t+, s Nt, s N Mjt t = ( ) t + Nt Nt n n k s k s di dj GE February 3,

103 Appendix FIRM OPTIMIZATION Firm lietime proit unction E z ( k, n ) r k v 0 t 0 t t t t t N Nt t= 0 N Mjt NMjt v di dj + p v di dj 0 v jt N Mjt 0 t 0 t ( ) t Nt Nt Nt 0 0 t= 0 E w n + w k v + w k v di dj perceived LOM or labor N Mjt t = ( ) t + Nt Nt n n k v k v di dj FOCs wrt k t, n t, v Nt, v GE February 3,

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