MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM JANUARY 26, 2018

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1 MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM JANUARY 26, 2018

2 Introduction LABOR MARKET INTERMEDIATION Recruiting Sector aka Labor Market Intermediaries aka Headhunters aka Middlemen January 26,

3 Introduction LABOR MARKET INTERMEDIATION Recruiting Sector aka Labor Market Intermediaries aka Headhunters aka Middlemen Develop Monopolistically Competitive Recruiting Model Moen (1997 JPE), Shimer (1996) Bilbiie, Ghironi, and Melitz (2012 JPE) Pissarides (1985 AER) Based on components of these frameworks January 26,

4 Introduction LABOR MARKET INTERMEDIATION Recruiting Sector aka Labor Market Intermediaries aka Headhunters aka Middlemen Develop Monopolistically Competitive Recruiting Model Moen (1997 JPE), Shimer (1996) Bilbiie, Ghironi, and Melitz (2012 JPE) Pissarides (1985 AER) Based on components of these frameworks Wage model Implications for aggregate matching Effects between recruiting-market matches and non-recruiting matches Implications for general equilibrium MAIN QUESTIONS January 26,

5 Introduction LABOR MARKET INTERMEDIATION Ordering of events N Mt-1 n t-1 δn Mt-1 recruiters exit market New recruiters enter matching sector Number of active recruiters in period t: N Mt = (1-δ)N Mt-1 + N MEt Participation by s and v in labor submarkets Production occurs and goods markets clear N Mt n t Aggregate state realized ρn t-1 jobs separate Number of jobs in period t: n t = (1-ρ)n t-1 + ρ(n Mt )(N Mt )m(s t,v t ) Period t-1 Period t Period t+1 January 26,

6 Introduction LABOR MARKET INTERMEDIATION Ordering of events N Mt-1 n t-1 δn Mt-1 recruiters exit market New recruiters enter matching sector Number of active recruiters in period t: N Mt = (1-δ)N Mt-1 + N MEt Participation by s and v in labor submarkets Production occurs and goods markets clear N Mt n t Aggregate state realized ρn t-1 jobs separate Number of jobs in period t: n t = (1-ρ)n t-1 + ρ(n Mt )(N Mt )m(s t,v t ) Period t-1 Period t Period t+1 January 26,

7 Related Literature RELATED LITERATURE Related Literature Rubinstein and Wolinsky (1987 QJE) Masters (2007 IER) Wright and Wong (2014 IER) Nosal, Wong, and Wright (2015 JMCB) Farboodi, Jarosch, and Shimer (2017)... January 26,

8 Related Literature RELATED LITERATURE theory Related Literature Rubinstein and Wolinsky (1987 QJE) Masters (2007 IER) Wright and Wong (2014 IER) Nosal, Wong, and Wright (2015 JMCB) Farboodi, Jarosch, and Shimer (2017)... empirics Autor, Katz, and Krueger (1998 QJE) Nakamura et al (2009 Studies of Labor Market Intermediation) Stevenson (2008 NBER WP) The Internet and Job Search Kroft and Pope (2014 J. Labor) Does Online Search Increase Matching Efficiency? Evidence from Craigslist Kuhn and Skuterud (2004 AER) Internet Job Search and Unemployment Duration January 26,

9 January 26,

10 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Recruiter ij profit maximization Cost minimization (directed-search optimization) Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation January 26,

11 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Recruiter ij profit maximization Cost minimization (directed-search optimization) Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

12 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Recruiter ij profit maximization Cost minimization (directed-search optimization) Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

13 Recruiting Sector MONOPOLISTIC RECRUITING MARKET Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry January 26,

14 Recruiting Sector MONOPOLISTIC RECRUITING MARKET Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry FACTOR MARKETS (search and vacancies) (Symmetric equilibrium for all i in j, and for all j) CREATES NEW EMPLOYMENT MATCHES RECRUITING SECTOR j MATCHING Aggregate recruiting firm j 1 1 N Mjt ms (, ) 0 v di Sell differentiated matches to matching bundler j DIFFERENTIATED RECRUITER 1j DIFFERENTIATED RECRUITER 2j DIFFERENTIATED RECRUITER NMj DIFFERENTIATED/ SPECIALIZED RECRUITERS IN LABOR MARKET j Measure N M of monopolistic recruiters, each of which produces a differentiated match January 26,

15 Recruiting Sector ENDOGENOUS ENTRY IN RECRUITING MARKET Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Free Entry in Recruiting Markets Representative Recruiter j Cost of creating new differentiated m(.) and entering market { N Mjt, NMEjt } t 0 NMjt max E 0 t 0 ( mc ) m( s, v ) di 0 Mt N t0 N (1 ) N N jt MEjt Mjt Mjt1 MEjt Cost of entry Γ Mt Technological R&D Regulatory... Mt TECH R& D REG Mt Mt Mt January 26,

16 Recruiting Sector ENDOGENOUS ENTRY IN RECRUITING MARKET Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Free Entry in Recruiting Markets Representative Recruiter j Cost of creating new differentiated m(.) and entering market { N Mjt, NMEjt } t 0 NMjt max E 0 t 0 ( mc ) m( s, v ) di 0 Mt N t0 N (1 ) N N jt MEjt Mjt Mjt1 MEjt Free-entry condition determines new recruiting agencies N MEjt Mt ( mc jt ) m( s, v ) (1 ) E t t 1 t Mt 1 w/ i NMjt January 26,

17 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog January 26,

18 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog Incentive for Entry vs. Welfare Benefit of Increasing Returns to Scale Larger number of monopolistic competitors Larger number of monopolistic competitors Smaller profits for potential new entrants Positive (negative) spillovers in production January 26,

19 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog aka, love of variety NOTE: cannot include this in utility function in our model. Incentive for Entry vs. Welfare Benefit of Increasing Returns to Scale Larger number of monopolistic competitors Larger number of monopolistic competitors Smaller profits for potential new entrants Positive (negative) spillovers in production January 26,

20 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog Incentive for Entry vs. Welfare Benefit of Increasing Returns to Scale Dixit-Stiglitz Technology Efficiently Balances Tradeoff Translog and Benassy Technologies Inefficiently Balance Tradeoff January 26,

21 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog Dixit-Stiglitz technology 1 N 1 Mjt jt 0 m m di j labor-market j aggregator dmd_fct January 26,

22 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Measure [0, 1] of recruiting markets Perfectly-competitive index by j Measure [0, N Mj ] of monopolistic submarkets in recruiting market j Index by ij N Mj endogenously determined via free entry Matching Aggregator Dixit-Stiglitz ( Benassy ) Translog Demand function for recruiter ij (Dixit-Stiglitz) 1 1 m m jt i j January 26,

23 Monopolistic Recruiter ij RECRUITER ij PROFIT-MAXIMIZATION January 26,

24 Monopolistic Recruiter ij RECRUITER ij PROFIT-MAXIMIZATION 1 mc jt PERFECT CSE: ε = infinity (recovers Moen 1997) Gross matchingmarket markup marginal cost of creating new job match Generally (symmetric equilibrium) ( N ) ( N ) mc( N ) Mt Mt Mt January 26,

25 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Profit-maximizing (ρ *, m * (s, v )) chosen Monopolistic recruiter ij s recruiting problem Recruiting firm ij must attract firms to post vacancies in submarket ij Recruiting firm ij must attract active job searchers to send résumés to (i.e., search in) submarket ij January 26,

26 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Profit-maximizing (ρ *, m * (s, v )) chosen Monopolistic recruiter ij s recruiting problem Recruiting firm ij must attract firms to post vacancies in submarket ij Recruiting firm ij must attract active job searchers to send résumés to (i.e., search in) submarket ij Definitions J( w ) ( ) W w value to goods-producing firm of successfully hiring worker in submarket ij value to worker of successfully finding a job in submarket ij U outside option of worker if unsuccessful in finding a job in submarket ij January 26,

27 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiting agency ij operates matching technology m( s, v ) Profit function of recruiting firm ij ms (, v ) p s p s v Recruiting firm ij Pays p sjt to s searchers Pays p vjt to v vacancies posted jt jt v Question: In context of bargained wage models, who own/operates matching technology?... Question: In context of bargained wage models, do people get paid for their search effort?... Recruiter ij must incentivize labor suppliers seeking new jobs Recruiter ij must incentivize labor demanders to post new job openings January 26,

28 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij total profit function ms (, v ) p s p s v jt jt v Zero fixed costs of creating new job match Operates a constant-returns-to-scale (CRS) matching technology Marginal cost of creating a match = average cost of creating a match is invariant to the quantity of matches created mc is NOT a function mc(quantity of matches) Re-express recruiter ij total profit function ms (, v ) mc m( s, v ) jt January 26,

29 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij total profit function mc jt ms (, v ) January 26,

30 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal profit function max w, mc m jt v January 26,

31 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal profit function subject to max w, mc m jt v f F p k ( ) J( w ) X 0 v jt h h H p k ( ) W( w ) (1 k ( )) U X 0 s t jt m_s January 26,

32 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal profit function subject to, f max mc (1 ) k ( ) w jt Suppose m( s, v) s v 1 f F p k ( ) J( w ) X 0 v jt h h H p k ( ) W( w ) (1 k ( )) U X 0 s t jt January 26,

33 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) Recruiter ij marginal profit function subject to f max mc (1 ) k ( ) w, jt Suppose m( s, v) s v 1 f F p k ( ) J( w ) X 0 v jt multipliers 1 h h H p k ( ) W( w ) (1 k ( )) U X 0 s t jt κ (given CRS m(.), only one multiplier needed) FOCs wrt w and θ January 26,

34 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs with respect to w and θ 1) J( w ) W( w ) f H h k ( ) k ( ) 0 w w H k k f h ( ) ( ) 1 b/c zero proportional taxation on wage =-1 =1 2) f f h k ( ) k ( ) k ( ) (1 ) ( ) ( ) 0 H mc jt J w W w Ut January 26,

35 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs with respect to w and θ 1) J( w ) W( w ) f H h k ( ) k ( ) 0 w w H k k f h ( ) ( ) 1 b/c zero proportional taxation on wage =-1 =1 2) f f h k ( ) k ( ) k ( ) ) 0 H mc jt (1 ) J( w ) W( w Ut 0 MONOPOLISTICALLY competitive recruiting sector January 26,

36 Monopolistic Recruiter ij RECRUITER ij COST-MINIMIZATION (DUAL) FOCs with respect to w and θ 1) J( w ) W( w ) f H h k ( ) k ( ) 0 w w H k k f h ( ) ( ) 1 b/c zero proportional taxation on wage =-1 =1 2) Cobb-Douglas matching m( s, v) f f h k ( ) k ( ) k ( ) ) 0 H mc jt (1 ) J( w ) W( w Ut s v Combine and rearrange 1 k k h f h m( s, v) k ( ) 1 ( ) m(1, ) (1 ) s AND f m( s, v) 1 1 ( ) m(,1) k ( ) v January 26,

37 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Profit maximization Cost minimization (directed-search optimization) CRUCIAL Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

38 Analytical Result SURPLUS SHARING ξ is elasticity of m ij (.) wrt s ij Wage Model Proposition 1. Monopolistic Surplus Sharing Condition w ) mc (1 ) W( ) U J( w ) (1 jt payoff accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer January 26,

39 Analytical Result SURPLUS SHARING ξ is elasticity of m ij (.) wrt s ij Wage Model Proposition 1. Monopolistic Surplus Sharing Condition w ) mc (1 ) W( ) U J( w ) (1 jt payoff accruing to monopolistic recruiter ij surplus accruing to new employee surplus accruing to new employer substitute mc = ρ/μ symmetric equilibrium functional dependence of ρ(.) and μ(.) on N M (see Bilbiie, Ghironi, Melitz 2008 NBER WP, 2016 NBER WP) January 26,

40 Analytical Result SURPLUS SHARING ξ is elasticity of m ij (.) wrt s ij Wage Model Proposition 1. Monopolistic Surplus Sharing Condition ( N ) Mt (1 ) ( NMt ) (1 ) W( wt) U J( wt) ( NMt ) payoff accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer Observations ( N ) Mt As ( NMt ) 0 surplus sharing condition PERFECTLY competitive ( NMt ) January 26,

41 Analytical Result SURPLUS SHARING ξ is elasticity of m ij (.) wrt s ij Wage Model Proposition 1. Monopolistic Surplus Sharing Condition ( N ) Mt (1 ) ( NMt ) (1 ) 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 sharing condition PERFECTLY competitive ( NMt ) Matching elasticity ξ in (0,1). From where do extra resources arise? January 26,

42 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 0 N Mjt 1 1 m di j Dixit-Stiglitz Aggregation ( N t ) 1 1 IRTS effect (elasticity) Dixit-Stiglitz N Mt t t integrate over i (integrate over j) 1 m( s, v ) Requires BOTH monopolistic competition AND costs of entry ala Romer endogenous growth model January 26,

43 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 0 N Mjt 1 1 m di j Dixit-Stiglitz Aggregation ( N t ) 1 1 IRTS effect (elasticity) Dixit-Stiglitz 1 integrate over i (integrate over j) 1 N N m( s, v ) Mt Mt t t Requires BOTH monopolistic competition AND costs of entry ala Romer endogenous growth model January 26,

44 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 0 N Mjt 1 1 m di j Dixit-Stiglitz Aggregation ( N t ) 1 1 IRTS effect (elasticity) Dixit-Stiglitz 1 integrate over i (integrate over j) 1 N N m( s, v ) Mt Mt t t more generally Requires BOTH monopolistic competition AND costs of entry ala Romer endogenous growth model ( N ) N m( s, v ) Mt Mt t t Holds for any aggregator January 26,

45 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz (1 ) NMt (1 ) W( wt) U J( wt) markup effect IRTS effect January 26,

46 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 1 N 1 Mjt N Mt m di j Benassy Aggregation integrate over i (integrate over j) Requires BOTH monopolistic competition AND costs of entry ala Romer endogenous growth model ( N ) N m( s, v ) Mt Mt t t Holds for any aggregator January 26,

47 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 1 N 1 Mjt 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 of entry ala Romer endogenous growth model ( N ) N m( s, v ) Mt Mt t t Holds for any aggregator January 26,

48 Analytical Result AGGREGATE INCREASING RETURNS AGGREGATE increasing returns in matching 1 N 1 Mjt N Mt m di j Benassy Aggregation ( N t ) IRTS effect INDEPENDENT of markup effect integrate over i (integrate over j) N N ms (, v ) Mt Mt t t Requires BOTH monopolistic competition AND costs of entry ala Romer endogenous growth model ( N ) N m( s, v ) Mt Mt t t Holds for any aggregator January 26,

49 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy (1 ) NMt (1 ) W( wt) U J( wt) markup effect IRTS effect φ measures increasing returns to scale (independent of ε) 1 (1 ) N Mt (1 ) W( wt) U J( wt) markup effect IRTS effect Incentive for Entry Welfare Benefit of Aggregate IRTS January 26,

50 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy (1 ) NMt (1 ) W( wt) U J( wt) markup effect IRTS effect lim φ 0 1 (1 ) 1 (1 ) ( wt) ( wt ) W U J markup effect Incentive for Entry Welfare Benefit of Aggregate IRTS Declines under Benassy aggregation as φ 0 January 26,

51 Analytical Result SURPLUS SHARING AGGREGATOR-DEPENDENCE Dixit-Stiglitz Benassy (1 ) NMt (1 ) W( wt) U J( wt) markup effect IRTS effect φ measures increasing returns to scale (independent of ε) 1 (1 ) N Mt (1 ) W( wt) U J( wt) Translog markup effect IRTS effect 1 N Mt 1 NM N Mt exp 1 1 N 2 N Mt M NMt (1 ) (1 ) W( wt ) U J( wt ) January 26, 2018 markup effect IRTS effect 51

52 January 26,

53 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity of m ij (.) wrt s ij Proposition 1. Monopolistic Surplus Sharing Condition ( N ) Mt (1 ) ( NMt ) (1 ) W( wt) U J( wt) ( NMt ) payoff accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer January 26,

54 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity of m ij (.) wrt s ij Proposition 1. Monopolistic Surplus Sharing Condition ( N ) Mt (1 ) ( NMt ) (1 ) W( wt) U J( wt) ( NMt ) payoff accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer substitute W(.), U, and J(.) Monopolistic Wage (explicitform) w z f ( k, n ) (1 ) (1 ) E t t n t t t t1 t t1 ( N ) Mt ( N 1) Mt (1 ) ( NMt ) (1 ) Et t1 t ( NM t1) ( NM t) ( NM t1) January 26,

55 Analytical Result WAGE IN MONOPOLISTIC MARKETS ξ is elasticity of m ij (.) wrt s ij Proposition 1. Monopolistic Surplus Sharing Condition ( N ) Mt (1 ) ( NMt ) (1 ) W( wt) U J( wt) ( NMt ) payoff accruing to monopolistic recruiter surplus accruing to new employee surplus accruing to new employer substitute W(.), U, and J(.) Monopolistic Wage (explicitform) w mpn (1 ) (1 ) ( N ) M (1 )(1 ) ( NM ) ( NM ) steady state January 26,

56 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Profit maximization Cost minimization (directed-search optimization) CRUCIAL Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

57 GE Model GENERAL EQUILIBRIUM Introduce random-search matching and Nash-bargained wages January 26,

58 GE Model GENERAL EQUILIBRIUM Introduce random-search matching and Nash-bargained wages Submarket ij Labor Supply (directed search) h'( lfp ) 1 k h'( lfp ) p u '( c ) '( ) h t h t1 s ps k jt w Et t1 t k h t k jt1 u c t1 jt1 jt1 h (1 ) (1 ) U ij ( ) W w HH_OPT January 26,

59 GE Model GENERAL EQUILIBRIUM Introduce random-search matching and Nash-bargained wages Submarket ij Labor Supply (directed search) h'( lfp ) 1 k h'( lfp ) p u '( c ) '( ) h t h t1 s ps k jt w Et t1 t k h t k jt1 u c t1 jt1 jt1 h (1 ) (1 ) U ij ( ) W w Submarket ij Labor Demand (directed vacancies) k f pv k z f ( k, n ) w (1 ) E jt f t n t t t t1 t k p v f jt1 jt1 ij J( w ) FIRM_OPT January 26,

60 GE Model GENERAL EQUILIBRIUM Symmetric equilibrium across ij Aggregate law of motion for labor n (1 ) n ( N ) N m( s, v ) m( s, v ) t t1 Mt Mt t t Nt Nt new job matches via monopolistic recruiting new job matches via random search January 26,

61 GE Model GENERAL EQUILIBRIUM Symmetric equilibrium across ij Aggregate law of motion for labor n (1 ) n ( N ) N m( s, v ) m( s, v ) t t1 Mt Mt t t Nt Nt new job matches via monopolistic recruiting new job matches via random search EXPANSION of aggregate resource frontier [ Absorption ] z f( k, n ) ( N ) N m( s, v ) t t t Mt Mt t t Novel Result Increasing returns in intermediary sector expands aggregate PPF (Std. procedure for aggregation: sum hh BCs, substitute equil. expressions) January 26,

62 Decentralized Economy DEFINITION GENERAL EQUILIBRIUM State-contingent stochastic processes { c t, n t, lfp t, k t+1, N Mt, N MEt, s t, v t, θ t, s Nt, v Nt, θ Nt, w t, w Nt, p vt, p st } t=0 that satisfy Search directed towards monopolistic submarkets Vacancies directed towards monopolistic submarkets Monopolistic wage surplus sharing Free-entry condition for recruiters Aggregate law of motion for recruiters Aggregate law of motion for employment s N and v N in random-search matching channel Aggregate LFP (determined by h (lfp t )/u (c t )) Capital Euler equation Nash wage surplus sharing Aggregate goods resource frontier Input prices p vt and p st (markdown of respective marginal products) Definitions of tightness θ t and θ Nt Given stochastic process zt t 0 and initial conditions k 0, n -1, N M-1 (State vector: x t = [ k t, n t-1, N Mt-1, z t ] ) January 26,

63 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Profit maximization Cost minimization (directed-search optimization) CRUCIAL Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

64 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matching aggregation. N * * M 0 if (efficient bargaining power) η is worker s bargaining power ξ is elasticity of m ij (.) wrt s ij January 26,

65 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matching aggregation. N * * M 0 if (efficient bargaining power) η is worker s bargaining power ξ is elasticity of m ij (.) wrt s ij Lemma (Static Model). N N 0 and 0 iff * * M 0 and 0 iff * * M (low worker bargaining power) (high worker bargaining power) January 26,

66 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matching aggregation. N * * M 0 if (efficient bargaining power) η is worker s bargaining power ξ is elasticity of m ij (.) wrt s ij Lemma (Static Model). N N 0 and 0 iff * * M 0 and 0 iff * * M (low worker bargaining power) (high worker bargaining power) Distortion (wage) in random search causes distortion in recruiting sector Despite efficient Dixit-Stiglitz aggregation Quant. Verification January 26,

67 Analytical Results SPILLOVER EFFECTS Proposition 2 (Static Model). Assume Dixit-Stiglitz matching aggregation. N * * M 0 if (efficient bargaining power) η is worker s bargaining power ξ is elasticity of m ij (.) wrt s ij Lemma (Static Model). N N 0 and 0 iff * * M 0 and 0 iff * * M (low worker bargaining power) (high worker bargaining power) Distortion (wage) in random search causes distortion in recruiting sector Despite efficient Dixit-Stiglitz aggregation Causality of distortionary spillover does NOT run in opposite direction Intuition: insufficient margins of adjustment Quant. Verification January 26,

68 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Profit maximization Cost minimization (directed-search optimization) CRUCIAL Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k accumulation Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

69 Parameters CALIBRATION Utility Aggregate LFP 1 1 u( ct ) h( lfpt ) ln ct lfpt 1 1/ lfp (1 ) n s N s t t 1 t Mt Nt Cobb-Douglas matching function ms (, v ) m s v EFF 1 t t t t (for both matching functions) m EFF larger in recruiting market January 26,

70 Parameters CALIBRATION Utility Aggregate LFP 1 1 u( ct ) h( lfpt ) ln ct lfpt 1 1/ lfp (1 ) n s N s t t 1 t Mt Nt Cobb-Douglas matching function ms (, v ) m s v EFF 1 t t t t (for both matching functions) m EFF larger in recruiting market β = 0.99 Matching elasticity ξ =0.4 Exogenous job-separation rate ρ = 0.10 Exogenous recruiter exit rate ω = 0.05 Stochastic TFP process ln z ln z z t 1 z t t (Table 4 contains other baseline parameters) January 26,

71 Quantitative Results SPILLOVER EFFECTS Proposition 2 * * NM 0 and 0 if Lemma * * NM 0 and 0 iff * * NM 0 and 0 iff January 26, Back to Prop. 2 Outline

72 Plan OUTLINE Structure of Labor Markets Free entry in recruiting markets Profit maximization Cost minimization (directed-search optimization problem) CRUCIAL Analytical Results Part I Surplus sharing condition Aggregate IRS in new job creation General Equilibrium Model Physical k accumulation Directed search labor supply and labor demand conditions Aggregate resource frontier Analytical Results Part II Quantitative Results Calibration Steady state and IRFs Conclusion January 26,

73 Conclusion SUMMARY Monopolistically Competitive Recruiting Model Moen (1997 JPE), Shimer (1996), Pissarides (1985 AER) Bilbiie, Ghironi, and Melitz (2012 JPE) Tractable Model Easy to Extend Provides New Competive Wage Model Aggregate Increasing Returns in Intermediated Matching Expansion of Aggregate Resource Frontier Effects Between Non-Intermediated and Intermediated Matching January 26,

74 January 26,

75 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j A continuum of aggregate recruiting agencies Each aggregate recruiting agency is perfectly competitive Easier to deal with mathematically than discrete infinity (tools of calculus can be applied) Representative recruiting agency j s profit function Relative price ρ of submarket recruiter ij N Mjt m( s, v ) m di jt jt 0 Substitute aggregate Dixit-Stiglitz matching technology 1 N 1 Mjt NMjt 0 0 m m di January 26,

76 Recruiting Sector MONOPOLISTIC RECRUITING MARKET j Representative recruiter s profit-maximization problem max mi 0... N Mjt 1 N 1 Mjt N 0 0 FOC with respect to m (for all ij) Mjt m m di Chooses profit-maximizing quantity of input of each submarket match.... after several rearrangements m m jt 1 1 jt m m i j DEMAND FUNCTION FOR RECRUITER ij January 26,

77 Recruiting Sector MONOPOLISTIC RECRUITING SUBMARKET ij Focus on profit-maximization of an arbitrary monopolistic recruiter ij Assume zero fixed costs of creating a match Operates a constant-returns-to-scale (CRS) matching technology in order to create its specialized, differentiated match CRS: if all inputs are scaled up by the factor x, total output is scaled up by the factor x Implementation of theory requires specifying neither the factors of production (i.e., active search s, vacancies v, etc) nor a matching function (m(.)) m( s, v ) s v 1 January 26,

78 Recruiting Sector MONOPOLISTIC RECRUITING SUBMARKET ij Focus on profit-maximization of an arbitrary monopolistic recruiter ij Assume zero fixed costs of creating a match Together, these imply a simple description of production Operates a constant-returns-to-scale (CRS) matching technology in order to create its specialized, differentiated match CRS: if all inputs are scaled up by the factor x, total output is scaled up by the factor x Implementation of theory requires specifying neither the factors of production (i.e., active search s, vacancies v, etc) nor a matching function (m(.)) Marginal cost of creating a match = average cost of creating a match is invariant to the quantity of matches created i.e., mc is NOT a function mc(quantity of matches) m( s, v ) s v 1 January 26,

79 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem Total revenue depends on match creation and its own submarket ij price. max m( s, v ) mc jt m( sit j, v ) mc is NOT a function of matches created (due to CRS m(.)) FC = 0 mc = ac January 26,

80 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem Total revenue depends on match creation and its own submarket ij price. max m( s, v ) mc jt m( sit j, v ) mc is NOT a function of matches created (due to CRS m(.)) FC = 0 mc = ac Substitute in demand function for recruiter ij m m jt Critical point for analysis of monopoly: the recruiter understands and internalizes the effect of its price on the quantity that it creates. max 1 m( s jt, v jt ) mc jt it j m( sjt, vjt ) Profit-maximization ( stage one ) Compute FOC with respect to relative price ρ ij January 26,

81 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem max 1 m( s jt, v jt ) mct i jt m( s jt, v jt ) FOC with respect to ρ 1 (1 ) m( s jt, v jt ) mc jt m( s jt, v jt ) 0 January 26,

82 Monopolistic Recruiter ij SUBMARKET ij STAGE ONE OPTIMIZATION Monopolistic recruiter ij s price-maximization problem max 1 m( s jt, v jt ) mct i jt m( s jt, v jt ) FOC with respect to ρ 1 (1 ) m( s jt, v jt ) mc jt m( s jt, v jt ) 0 Algebraic rearrangement Optimal relative price of recruiter j is a markup ε/(ε 1) over marginal cost of creating specialized/different match. KEY PRICING RESULT OF DIXIT- STIGLITZ THEORY. mc 1 Gross matchingmarket markup jt Linked only to degree of substitutability across monopolistic recruiters i PERFECT CSE: ε = infinity Monopolistic matching: ε > 1 and ε < infinity January 26,

83 January 26,

84 Appendix HOUSEHOLD OPTIMIZATION Household utility 1 N t h Mjt h E0 u( ct ) h nt (1 knt ) s Nt ( 1 k ) 0 0 s di dj t0 N uet ue flow budget constraint h Mjt 1 (1 r ) k w (1 ) n 1 w k s NMjt 1 NMjt h ps s (1 ) 1 jt i di dj k s di dj c k T t t t t t t t Nt Nt Nt 1 h M F 0 0 jt knt snt jt dj t 1 N w k s di dj h perceived LOM for labor FOCs wrt c t, n t, k t+1, s Nt, s 1 N h Mjt h t (1 ) t1 Nt Nt 0 0 n n k s k s di dj GE January 26,

85 Appendix FIRM OPTIMIZATION Firm lifetime profit function E z f ( k, n ) r k v 0 t 0 t t t t t N Nt t0 1 N 1 0 Mjt N Mjt v di dj p v di dj v jt 1 N f Mjt f 0 t 0 t (1 ) t1 Nt Nt Nt 0 0 t0 E w n w k v w k v di dj perceived LOM for labor 1 N f Mjt f t (1 ) t1 Nt Nt 0 0 n n k v k v di dj FOCs wrt k t, n t, v Nt, v GE January 26,

MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM FEBRUARY 13, 2019

MONOPOLISTICALLY COMPETITIVE SEARCH EQUILIBRIUM FEBRUARY 3, 209 Introduction LABOR MARKET INTERMEDIATION Recruiting Sector aka Labor Market Intermediaries aka Headunters aka Middlemen Develop Monopolistically