LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012

Transcription

1 LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012

2 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem ma 0 ( ) f Ξ v, n + 1 = 0 ( f y wn h g v ) Discoun facor beween ime 0 and because dynamic firm problem; in equilibrium, = household sochasic discoun facor Number of vacancies o pos (how many job adverisemens ) Desired arge fuure firm employmen Toal oupu sold in perfeclycompeiive goods marke Toal wage bill depends on boh eensive and inensive employmen Toal cos of posing v vacancies Subjec o (perceived) law of moion for firm s employmen sock Baseline model Shu down inensive margin: h = 1 Linear posing coss: g(v) = γv Firm producion funcion: y = z * n Wage-seing (process) aken as given when posing vacancies April 12,

3 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem ma, f v n + 1 = 0 f f ( zn wn γ v) Ξ 0 s.. n f = 1 (1 ) f f ρ n + + vk ( θ) Perceived law of moion for evoluion of employmen sock Number of eising jobs ha do no end: ρ eogenous separaion rae, bu can also endogenize Each vacancy has probabiliy k f (θ) of aracing a prospecive employee: depends on a marke variable, θ, so aken as given FOCs wih respec o v, n +1 f γ + µ k ( θ ) = 0 ( ) { } µ + E Ξ z w + (1 ρ ) µ = 0 Combine April 12,

4 FIRM VACANCY-POSTING PROBLEM Vacancy posing condiion (aka job creaion condiion) f γ = k ( θ ) E Ξ z w f (1 ρ ) γ k ( θ+ 1) γ/k f is capial value of an eising employee because one less worker firm has o find in he fuure EMPLOYEES ARE ASSETS Cos of posing a vacancy Epeced benefi of posing a vacancy = (probabiliy of aracing a worker) (epeced fuure benefi of an addiional worker) = marginal oupu wage paymen + epeced asse value of an addiional worker Vacancy-posing is a ype of invesmen decision Ineremporal dimension makes discoun facor poenially imporan i.e., makes general equilibrium effecs poenially imporan Two prices affec posing decision (aside from ineremporal price) (Fuure) wage Maching probabiliy (loosely inerpre probabiliies as prices) which depends on he marke variable θ April 12,

5 HOUSEHOLD PROBLEM Dynamic household uiliy-maimizaion problem A coninuum [0, 1] of households (a sandard assumpion) A coninuum [0, 1] of aomisic individuals live in each household Thus represenaive household has a coninuum of family members ma E0 β u( c ) c, a = 0 s.. c + a = nwh + (1 n) b+ Ra 1 An (arbirary) asse o make pricing ineres raes eplici Wage (-seing process) aken as given by household Measure n of family members earn labor income (because hey work) (and recall we ve normalized h = 1) Measure 1-n of family members receive unemploymen benefis and/or engaged in home producion April 12,

6 HOUSEHOLD PROBLEM Dynamic household uiliy-maimizaion problem A coninuum [0, 1] of households (a sandard assumpion) A coninuum [0, 1] of aomisic individuals live in each household Thus represenaive household has a coninuum of family members KEY: Assuming infinie family srucure delivers full consumpion insurance i.e., all employed and unemployed individuals have equal consumpion! Thus individual family members are risk-neural wih respec o heir labor-marke realizaion Analogy wih Hansen- Rogerson srucure (see Andolfao 1996 AER) Consumpion-savings opimaliy condiion: No labor-supply/par. margin in basic model ma E0 β u( c ) c, a = 0 s.. c + a = nwh + (1 n) b+ Ra Measure n of family members earn labor income (because hey work) (and recall we ve normalized h = 1) Each family member eiher works or is looking for work An (arbirary) asse o make pricing ineres raes eplici April 12, Wage (-seing process) aken as given by household Measure 1-n of family members receive unemploymen benefis and/or engaged in home producion β u'( c 1) + 1 = RE u'( c ) Sochasic discoun facor

7 WAGE BARGAINING (Generalized) Nash Bargaining η ( Ww Uw) ( Jw Vw) 1 ma ( ) ( ) ( ) ( ) w η Bargaining over how o divide he surplus Asse values W: value o (represenaive) household of having one addiional member employed U: value o (represenaive) household of having one addiional member unemployed and searching for work J: value o (represenaive) firm of having one addiional employee V: value o (represenaive) firm of having a job ha goes unfilled Free enry in vacancy-posing V = 0 Define W and U in erms of household problem Ne payoff o an individual/household of agreeing o wage w and beginning producion Ne payoff o a firm of agreeing o wage w and beginning producion i.e., based on envelope condiions of household value funcion April 12,

8 WAGE BARGAINING (Generalized) Nash Bargaining η 1 ( Ww Uw) Jw ma ( ) ( ) ( ) w η Bargaining over how o divide he surplus The Nash surplus-sharing rule η Ne payoff o an individual/household of agreeing o wage w and beginning producion Ne payoff o a firm of agreeing o wage w and beginning producion ( W'( w) U'( w) ) J( w) = (1 η)( J'( w)) ( W( w) U( w) ) (FOC wih respec o w ) Presen in any model wih Nash bargaining (Mos) labor maching models (Mos) moneary search models Poliical bargaining games (Albanesi 2007 JME) Mus specify value equaions W(.), U(.), J(.) April 12,

9 VALUE EQUATIONS Individual/household value equaions (consruced from household problem) { } + 1 ρ + 1 ρ + 1 W( w) = w + E Ξ (1 ) W( w ) + U( w ) Value o household of having he marginal individual employed Conemporaneous reurn is wage Epeced fuure reurn akes ino accoun ransiion probabiliies { h h } + 1 θ + 1 θ + 1 U( w) = b+ E Ξ k ( ) W( w ) + (1 k ( )) U( w ) Value o household of having he marginal individual unemployed and searching Conemporaneous reurn is unemploymen benefi/ home producion Epeced fuure reurn akes ino accoun ransiion probabiliies April 12,

10 VALUE EQUATIONS Each searching individual has probabiliy k h (θ) of finding a job opening: depends on a marke variable, θ, so aken as given Individual/household value equaions (consruced from household problem) Conemporaneous reurn is wage { } + 1 ρ + 1 ρ + 1 W( w) = w + E Ξ (1 ) W( w ) + U( w ) Epeced fuure reurn akes ino accoun ransiion probabiliies { h h } + 1 θ + 1 θ + 1 U( w) = b+ E Ξ k ( ) W( w ) + (1 k ( )) U( w ) Value o household of having he marginal individual employed Value o household of having he marginal individual unemployed and searching Conemporaneous reurn is unemploymen benefi/ home producion Epeced fuure reurn akes ino accoun ransiion probabiliies Firm value equaion { + 1 ρ + 1 } Jw ( ) = z w+ E Ξ (1 ) Jw ( ) Value o firm of he marginal employee Conemporaneous reurn is marginal oupu ne of wage paymen Epeced fuure reurn akes ino accoun ransiion probabiliies April 12,

11 WAGE BARGAINING The Nash surplus-sharing rule η ( W'( w) U'( w) ) J( w) (1 η)( J'( w)) ( W( w) U( w) ) = (FOC wih respec o w ) Inser marginal values ( ) ηjw ( ) = (1 η) Ww ( ) Uw ( ) Firm s surplus J a consan fracion of household s surplus W U NOTE: NOT a general propery of Nash bargaining; here due Using definiions of W, U, o he lineariy of W, U, and J and J, he job-creaion wih respec o wage condiion, and some algebra NOTE: Wih CRS maching funcion, θ = k h (θ)/k f (θ) [ ] (1 ) w = η z + γθ + η b Conemporaneous marginal oupu and a erm ha capures he social savings on fuure posing coss if mach coninues Bargained wage a conve combinaion of gains from consummaing he mach and he gains from walking away from he mach April 12,

12 LABOR MARKET MATCHING Aggregae maching funcion displays CRS mu (, v) u = 1 n is measure of individuals searching for work For any given individual vacancy or individual (parial equilibrium), maching probabiliies depend only on v/u θ v u Marke ighness: measures relaive number of raders on opposie sides of marke April 12,

13 LABOR MARKET MATCHING NOTE: Wih CRS maching funcion, θ = k h (θ)/k f (θ) In maching models, θ is he key driving force of efficiency and herefore opimal policy prescripions (Hosios 1990 ReSud he key reference) Aggregae maching funcion displays CRS For any given individual vacancy or individual (parial equilibrium), maching probabiliies depend only on v/u Marke ighness an allocaional signal mu (, v) v mu (, v) Because maching probabiliies depend on i e.g., he higher (lower) is v/u, he easier (harder) i is for a given individual o find a job opening April 12, u 1 f = m, 1 = m( θ,1) k ( θ ) v mu (, v) u v = m 1, = m 1, k u v θ u h ( θ ) ( θ ) u = 1 n is measure of individuals searching for work Probabiliy a given vacancy/job posing aracs a worker Probabiliy a given individual finds a job opening Marke ighness: measures relaive number of raders on opposie sides of marke

14 LABOR-MARKET EQUILIBRIUM Aggregae law of moion of employmen Flow equilibrium condiions (an accouning ideniy ) Vacancy-posing (aka job-creaion) condiion Wage deerminaion Basic labor-heory lieraure: impose ss on hese and analyze, do comparaive saics, ec. (eogenous real ineres rae) N = 1 (1 ρ ) N + + m( u, v) h f mu (, v) = uk ( θ ) = vk ( θ ) f γ = k ( θ ) E Ξ z w f [ ] (1 ) w = η z + γθ + η b Pissarides Chaper 1, RSW 2005 JEL (1 ρ ) γ k ( θ+ 1) April 12,

15 GENERAL EQUILIBRIUM Aggregae law of moion for employmen Vacancy-posing (aka job-creaion) condiion Wage deerminaion The labor marke equilibrium (parial equilibrium from he perspecive of he enire environmen) Consumpion-savings opimaliy condiion (endogenizes real ineres rae) β u'( c 1) + 1 = RE u'( c ) Aggregae resource consrain Ofen inerpreed as he oupu of a home producion secor only he unemployed produce in he home secor c + g + γ v = z Nh + (1 N ) b Vacancy posing coss and ouside opion are real uses of resources Eogenous LOMs for any driving processes (TFP, ec) April 12,

16 Long-Run Analysis STEADY STATE OF LABOR MARKET Imposing deerminisic seady sae on labor-marke equilibrium condiions (1) 1 u= (1 ρ )(1 u) + m( u, v) (using N = 1 u) (2) f γ = βk ( θ) z w+ (1 ρ ) γ f k ( θ) w negaively and nonlinearly relaed o θ (given CRS maching funcion) (3) w [ ] (1 ) w= η z+ γθ + η b w posiively and linearly relaed o θ job-creaion curve wage curve Labor supply curve and labor demand curve replaced by wage curve and job-creaion curve Pissarides Figure 1.1 The relevan quaniy variable θ bu can also loosely hink of θ as a price because i governs maching probabiliies θ April 12,

17 Long-Run Analysis STEADY STATE OF LABOR MARKET Imposing deerminisic seady sae on labor-marke equilibrium condiions (1) (2) muv (,) + ρ u = ρ f v (1 ρ ) γ γ = βk z w+ u f v k u For a given (w,θ), v and u negaively relaed (given CRS maching funcion) For a given (w,θ), v and u posiively relaed (given CRS maching funcion) v Beveridge curve Job-creaion curve Pissarides Figure 1.2 BEVERIDGE CURVE: Empirical relaionship in boh long run and shor run (i.e., cyclical) April 12, u

18 Long-Run Analysis STEADY STATE OF LABOR MARKET Labor-marke equilibrium is (w, u, θ) saisfying (1), (2), (3) Comparaive saics A rise in b raises w lowers θ...lowers v and raises u Higher value (ue benefi) of unemploymen requires a higher wage o induce individuals o work, which reduces firm incenives o creae jobs April 12,

19 Long-Run Analysis STEADY STATE OF LABOR MARKET Labor-marke equilibrium is (w, u, θ) saisfying (1), (2), (3) Comparaive saics A rise in b raises w lowers θ...lowers v and raises u Higher value (ue benefi) of unemploymen requires a higher wage o induce individuals o work, which reduces firm incenives o creae jobs A fall in β (or a rise in ρ ) lowers w lowers θ...raises u ambiguous effec on v Higher real rae and/or faser job separaions (i.e., faser depreciaion of employmen sock ) makes posing jobs (FOR FIXED u) less aracive for firms (boh erode firm profis) See Pissarides Chaper 1 and RSW (2005 JEL) for more Ne: dynamic sochasic parial equilibrium (Shimer 2005, Hall 2005, and Hagedorn and Manovskii 2008) April 12,