Dynamic firm profit-maximization problem. max ( ( )) Total output sold in perfectlycompetitive

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1 LABOR SEARCH MODELS: BASIC DSGE IMPLEMENTATION NOVEMBER 2, 200 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem f f Ξ 0 y wn g v v, n + = 0 ma ( ( Discoun facor beween ime 0 and because dynamic firm problem; in equilibrium, = ouseold socasic discoun facor Number of vacancies o pos (ow many job adverisemens Desired arge fuure firm employmen Toal oupu sold in perfeclycompeiive goods marke Toal wage bill depends on bo eensive and inensive employmen Toal cos of posing v vacancies Subjec o (perceived law of moion for firm s employmen sock Baseline model Su down inensive margin: = Linear posing coss: g(v = γv Firm producion funcion: y = z * n Wage-seing (process aken as given wen posing vacancies November 2, 200 2

2 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem f f ma 0 (, f Ξ zn wn γ v v n + = 0 s.. n f = ( f f ρ n + + vk ( θ Perceived law of moion for evoluion of employmen sock Number of eising jobs a do no end: ρ eogenous separaion rae, bu can also endogenize Eac vacancy as probabiliy k f (θ of aracing a prospecive employee: depends on a marke variable, θ, so aken as given FOCs wi respec o v, n + f γ + μ k ( θ = 0 { + ( } μ + E Ξ z w + ( ρ μ = 0 Combine November 2, FIRM VACANCY-POSTING PROBLEM Vacancy posing condiion (aka job creaion condiion f ( ρ γ γ = k ( θ E Ξ+ z+ w+ + f k ( θ+ γ/k f is capial value of an eising employee because one less worker firm as o find in e 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 Macing probabiliy (can ofen inerpre probabiliies as prices wic depends on e marke variable θ November 2,

3 HOUSEHOLD PROBLEM Dynamic ouseold uiliy-maimizaion problem A coninuum [0, ] of ouseolds (a sandard assumpion A coninuum [0, ] of aomisic individuals live in eac ouseold Tus represenaive ouseold as a coninuum of family members ma E0 β u( c c, a = 0 s.. c + a = nw + ( n b+ Ra An (arbirary asse o make pricing ineres raes eplici Wage (-seing process aken as given by ouseold Measure n of family members earn labor income (because ey work (and recall we ve normalized = Measure -n of family members receive unemploymen benefis and/or engaged in ome producion November 2, HOUSEHOLD PROBLEM Dynamic ouseold uiliy-maimizaion problem A coninuum [0, ] of ouseolds (a sandard assumpion A coninuum [0, ] of aomisic individuals live in eac ouseold Tus represenaive ouseold as a coninuum of family members KEY: Assuming infinie family srucure delivers full consumpion insurance i.e., all employed and unemployed individuals ave equal consumpion! Tus individual family members are risk-neural wi respec o eir labor-marke realizaion Analogy wi Hansen-Rogerson srucure (see Andolfao 996 AER ma 0 ( c, a E β u c = 0 s.. c + a = nw + ( n b+ Ra Measure n of family members earn labor income (because ey work (and recall we ve normalized = An (arbirary asse o make pricing ineres raes eplici Wage (-seing process aken as given by ouseold Measure -n of family members receive unemploymen benefis and/or engaged in ome producion β u'( c + Consumpion-savings opimaliy condiion: = RE u'( c Socasic discoun No labor-supply/par. margin in basic model facor Eac family member eier works or is looking for work November 2,

4 WAGE BARGAINING (Generalized Nas Bargaining η ( W w U w ( J w V w ma ( ( ( ( w η Bargaining over ow o divide e surplus Ne payoff o an individual/ouseold of agreeing o wage w and beginning producion Ne payoff o a firm of agreeing o wage w and beginning producion Asse values W: value o (represenaive ouseold of aving one addiional member employed U: value o (represenaive ouseold of aving one addiional member unemployed and searcing for work J: value o (represenaive firm of aving one addiional employee V: value o (represenaive firm of aving a job a goes unfilled Free enry in vacancy-posing V = 0 Define W and U in erms of ouseold problem i.e., based on envelope condiions of ouseold value funcion November 2, WAGE BARGAINING (Generalized Nas Bargaining ( ma W( w U( w J( w w η η Bargaining over ow o divide e surplus Ne payoff o an individual/ouseold of agreeing o wage w and beginning producion Ne payoff o a firm of agreeing o wage w and beginning producion Te Nas surplus-saring rule η W '( w U'( w J( w = ( η( J'( w W( w U( w ( ( (FOC wi respec o w Presen in any model wi Nas bargaining (Mos labor searc models (Mos money searc models Poliical bargaining games (Albanesi 2007 JME Mus specify value equaions W(., U(., J(. November 2,

5 VALUE EQUATIONS Eac searcing individual as probabiliy k (θ of finding a job opening: depends on a marke variable, θ, so aken as given Individual/ouseold value equaions (consruced from ouseold problem Conemporaneous reurn is wage { ρ ρ } W( w = w + E Ξ ( W( w + U( w Epeced fuure reurn akes ino accoun ransiion probabiliies { θ θ } U( w = b+ E Ξ k ( W( w + ( k ( U( w Value o ouseold of aving e marginal individual employed Value o ouseold of aving e marginal individual unemployed and searcing Conemporaneous reurn is unemploymen benefi/ome producion Epeced fuure reurn akes ino accoun ransiion probabiliies November 2, VALUE EQUATIONS Eac searcing individual as probabiliy k (θ of finding a job opening: depends on a marke variable, θ, so aken as given Individual/ouseold value equaions (consruced from ouseold problem Conemporaneous reurn is wage { ρ ρ } W( w = w + E Ξ ( W( w + U( w Epeced fuure reurn akes ino accoun ransiion probabiliies { θ θ } U( w = b+ E Ξ k ( W( w + ( k ( U( w Value o ouseold of aving e marginal individual employed Value o ouseold of aving e marginal individual unemployed and searcing Conemporaneous reurn is unemploymen benefi/ome producion Epeced fuure reurn akes ino accoun ransiion probabiliies Firm value equaion { + ρ + } J( w = z w + E Ξ ( J( w Value o firm of e marginal employee Conemporaneous reurn is marginal oupu ne of wage paymen Epeced fuure reurn akes ino accoun ransiion probabiliies November 2,

6 WAGE BARGAINING Te Nas surplus-saring rule η ( W '( w U'( w J( w ( η( J'( w ( W( w U( w = (FOC wi respec o w Inser marginal values ( ηj ( w = ( η W( w U( w Firm s surplus J a consan fracion of ouseold s surplus W U NOTE: NOT a general propery of Nas bargaining; ere due o e lineariy of W, U, and J wi respec o wage November 2, 200 WAGE BARGAINING Te Nas surplus-saring rule η ( W '( w U'( w J( w ( η( J'( w ( W( w U( w = (FOC wi respec o w Inser marginal values ( ηj ( w = ( η W( w U( w Using definiions of W, U, and J, e job-creaion condiion, and some algebra Firm s surplus J a consan fracion of ouseold s surplus W U NOTE: NOT a general propery of Nas bargaining; ere due o e lineariy of W, U, and J wi respec o wage NOTE: Wi CRS macing funcion, θ = k (θ/k f (θ [ ] ( w = η z + γθ + η b Conemporaneous marginal oupu and a erm a capures e social savings on fuure posing coss if mac coninues Bargained wage a conve combinaion of gains from consummaing e mac and e gains from walking away from e mac November 2,

7 LABOR MARKET MATCHING NOTE: Wi CRS macing funcion, θ = k (θ/k f (θ Aggregae macing funcion displays CRS mu (, v u = n is measure of individuals searcing for work For any given individual vacancy or individual (parial equilibrium, macing probabiliies depend only on v/u mu (, v u f = m, = m( θ, k ( θ v v mu (, v v = m, = m, k u u v θ u ( θ ( θ Probabiliy a given vacancy/job posing aracs a worker Probabiliy a given individual finds a job opening Marke igness: measures relaive number of raders on opposie sides of marke November 2, LABOR MARKET MATCHING NOTE: Wi CRS macing funcion, θ = k (θ/k f (θ In macing models, θ is e key driving force of efficiency and erefore opimal policy prescripions (Hosios 990 ReSud e key reference Aggregae macing funcion displays CRS mu (, v For any given individual vacancy or individual (parial equilibrium, macing probabiliies depend only on v/u mu (, v u f = m, = m( θ, k ( θ v v mu (, v v = m, = m, k u u v θ u Marke igness an allocaional signal Because macing probabiliies depend on i e.g., e iger (lower is v/u, e easier (arder i is for a given individual o find a job opening November 2, ( θ ( θ u = n is measure of individuals searcing for work Probabiliy a given vacancy/job posing aracs a worker Probabiliy a given individual finds a job opening Marke igness: measures relaive number of raders on opposie sides of marke 7

8 LABOR-MARKET EQUILIBRIUM Aggregae law of moion of employmen N = ( ρ N + + m( u, v Flow equilibrium condiions (an accouning ideniy f mu (, v = uk ( θ = vk ( θ Vacancy-posing (aka job-creaion condiion f ( ρ γ γ = k ( θ E Ξ+ z+ w+ + f k ( θ+ Wage deerminaion [ ] ( w = η z + γθ + η b Basic labor-eory lieraure: impose ss on ese and analyze, do comparaive saics, ec. (eogenous real ineres rae Pissarides Caper, RSW 2005 JEL November 2, GENERAL EQUILIBRIUM Aggregae law of moion for employmen Vacancy-posing (aka job-creaion condiion Wage deerminaion Te labor marke equilibrium (parial equilibrium from e perspecive of e enire environmen Consumpion-savings opimaliy condiion (endogenizes real ineres rae β u'( c + = RE u'( c Aggregae resource consrain Ofen inerpreed as e oupu of a ome producion secor only e unemployed produce in e ome secor c + g + γ v = z N + ( N b Vacancy posing coss and ouside opion are real uses of resources Eogenous LOMs for any driving processes (TFP, ec November 2,

9 Long-Run Analysis STEADY STATE OF LABOR MARKET Imposing deerminisic seady sae on labor-marke equilibrium condiions ( u = ( ρ ( u + m( u, v (using N = u (2 f ( ρ γ γ = βk ( θ z w+ f k ( θ w negaively and nonlinearly relaed o θ (given CRS macing funcion (3 w [ ] ( w= η z+ γθ + η b w posiively and linearly relaed o θ Pissarides Figure. job-creaion curve wage curve Labor supply curve and labor demand curve replaced by wage curve and job-creaion curve Te relevan quaniy variable θ bu can also ink of θ as a price because i governs macing probabiliies θ November 2, Long-Run Analysis STEADY STATE OF LABOR MARKET Imposing deerminisic seady sae on labor-marke equilibrium condiions muv (, + ρ For a given (w,θ, v and u ( u = negaively relaed (given CRS macing funcion ρ f v ( ρ γ For a given (w,θ, v and u (2 γ = βk z w + posiively relaed (given CRS u f v macing funcion k u v Beveridge curve Job-creaion curve Pissarides Figure.2 BEVERIDGE CURVE: Empirical relaionsip in bo long run and sor run (i.e., cyclical November 2, u 9

10 Long-Run Analysis STEADY STATE OF LABOR MARKET Labor-marke equilibrium is (w, u, θ saisfying (, (2, (3 Comparaive saics A rise in b raises w lowers θ...lowers v and raises u Higer value (ue benefi of unemploymen requires a iger wage o induce individuals o work, wic 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 Higer real rae and/or faser job separaions (i.e., faser depreciaion of employmen sock makes posing jobs (FOR FIXED u less aracive for firms (bo erode firm profis See Pissarides Caper and RSW (2005 JEL for more Ne: dynamic socasic parial equilibrium (Simer 2005, Hall 2005, Hagedorn and Manovskii 2008, Pissarides 2009 November 2,