WAGE RIGIDITIES IN AN ESTIMATED DYNAMIC, STOCHASTIC, GENERAL EQUILIBRIUM MODEL OF THE UK LABOUR MARKET*

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1 The Mancheser School Supplemen 203 doi: 0./manc.2023 WAGE RIGIDITIES IN AN ESTIMATED DYNAMIC, STOCHASTIC, GENERAL EQUILIBRIUM MODEL OF THE UK LABOUR MARKET* by RENATO FACCINI Queen Mary Universiy STEPHEN MILLARD Bank of England and FRANCESCO ZANETTI Universiy of Oxford We esimae a New Keynesian model wih maching fricions and nominal wage rigidiies on UK daa. We show ha in a model wih maching fricions, wheher nominal wage rigidiies are relevan or irrelevan for inflaion dynamics depends on he paramerizaion. A he esimaed equilibrium, we find ha wage rigidiies are irrelevan, despie improving he empirical performance of he model. The reason is ha wih maching fricions, marginal coss depend on uni labour coss and on an addiional componen relaed o search coss. Wage rigidiies affec boh componens in opposie ways leaving marginal coss and inflaion virually unaffeced. INTRODUCTION Dynamic, sochasic, general equilibrium (DSGE) models based on he New Keynesian paradigm have become a powerful ool o invesigae he propagaion of shocks and inflaion dynamics. In his framework price rigidiies esablish a link beween nominal and real aciviy: if nominal prices are saggered, flucuaions of nominal aggregaes rigger flucuaions of real aggregaes. Using his framework, seminal work by Gali and Gerler (999) * Manuscrip received 3.9.2; final version received The auhors wish o hank George Brasiois and hree anonymous referees for exremely useful commens and seminar paricipans a he Bank of England, Durham Universiy, he Universiy of Noingham, he Sociey for Nonlinear Dynamics and Economerics 8h Annual Symposium, he Royal Economic Sociey meeings, he Third World Conference EALE-SOLE 200, he 6h inernaional conference of he Sociey for Compuaional Economics, he 6h Dynare Conference, 23rd annual EALE Conference he 25h congress of he European Economic Associaion and he 202 MMF Annual Conference in Dublin for very helpful suggesions. This paper represens he views and analysis of he auhors and should no be hough o represen hose of he Bank of England or he Moneary Policy Commiee members. See Smes and Wouers (2003, 2007) for an exensive applicaion of his framework. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser 66

2 Wage Rigidiies in a DSGE Model of he UK Labour Marke 67 has documened ha he dynamic behaviour of inflaion is ighly linked o firms marginal cos (represened by uni labour cos), whose dynamics crucially depend on he funcioning of he labour marke. Gali and Gerler (999) assume fricionless labour markes, whereas we develop he analysis in a model based on labour marke fricions. We believe ha including labour marke fricions is imporan for wo reasons. Firs, hey provide a comprehensive descripion of he labour marke and enable us o inroduce unemploymen ino he model. Imporanly for he sudy in his paper, as shown in Krause and Lubik (2007), hese fricions, once incorporaed in a New Keynesian model, enrich he noion of marginal cos, by incorporaing he coss of esablishing a work relaionship over and above he uni labour cos, hereby, in principle, alering he dynamics of inflaion. Second, a growing number of empirical sudies documen ha embedding labour marke fricions ino a sandard New Keynesian model improves he model s empirical performance and enables a more accurae descripion of inflaion dynamics. 2 The conribuion of our paper is wofold. Firs, we build on previous sudies documened below o esimae a New Keynesian model characerized by labour marke fricions using UK daa. The esimaion allows us o esimae he srucural parameers of he UK economy, he unobservable shocks and sudy heir ransmission mechanism. Second, we invesigae how saggered nominal wage negoiaions affec he propagaion of shocks and he abiliy of he model o fi he daa. To his end, he heoreical framework allows, bu does no require, nominal wage rigidiies o affec he model s dynamics, herefore leaving he daa o esablish he imporance of wage rigidiies. In addiion, his esimaion sraegy allows us o invesigae he effec of nominal wage rigidiies on inflaion dynamics. Our findings are he following. Firs, we esimae imporan srucural parameers ha characerize he Briish economy. In paricular, we idenify a relaively low Frisch elasiciy of labour supply, reflecing he fac ha employmen is more volaile along he exensive margin han he inensive margin. The bargaining power of he workers is esimaed o equal abou 0.9. A he esimaed equilibrium, firm s surplus is approximaely 0 per cen relaive o produciviy, which is higher han he value of 2.5 per cen calibraed by Hagedorn and Manovskii (2008) o mach he US daa. The habi persisence parameer is close o zero, differenly from he high value of approximaely 0.6 in US daa. We find ha on average prices adjus every wo quarers, in line wih UK micro-esimaes in Bunn and Ellis (2009), whereas he average frequency of wage negoiaions is abou hree quarers. We also provide esimaes for he moneary auhoriy s reacion funcion. We find ha he moneary auhoriy s response o inflaion and oupu is srong 2 Noiceable examples, documened below, are Gerler e al. (2008), Chrisoffel e al. (2009b), Krause e al. (2008b), Zanei (20) and Ravenna and Walsh (2008). The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

3 68 The Mancheser School and here is a mild degree of ineres rae ineria. The esimaed model allows us o characerize he ransmission of shocks. We invesigae how he model variables reac o supply and demand shocks, and we find ha neural and invesmen-specific echnology shocks are more imporan han oher shocks in explaining he daa. We esablish ha saggered wage seing enables he model o fi he daa more closely. However, we find ha a he esimaed equilibrium wage rigidiies are irrelevan for inflaion dynamics. This resul echoes he findings by Krause and Lubik (2007). In a fricional labour marke inflaion depends on uni labour coss and on an addiional erm which is relaed o labour marke fricions, i.e. o he expeced change in he search coss incurred in finding a mach. Following a shock, wage rigidiies have a direc effec on he uni labour cos. However, he conribuion of uni labour coss o marginal coss is offse by he conribuion of he componen relaed o labour marke fricions. For insance, following a posiive mark-up shock, nominal wage rigidiies aenuae he drop in uni labour coss and induce a fall in he fricional componen of marginal coss compared wih a flexible wage regime. As a resul, marginal coss and inflaion dynamics behave similarly in he wo seings. A similar resul holds for all he shocks in our model economy and sands in sharp conras wih hose obained in New Keynesian models wih compeiive labour markes. Absen search fricions in he labour marke, he dynamics of inflaion are only driven by he uni labour coss. I follows ha wage rigidiies generae inflaion persisence by making uni labour coss more persisen (see Chrisiano e al., 2005). In his paper we also invesigae wheher he irrelevance of wage rigidiies for inflaion dynamics is buil ino he model, which poenially may preven wage rigidiies from affecing he dynamics of inflaion, or i is a consequence of he specific esimaion. To answer his quesion we simulae he heoreical framework by drawing 000 imes from he parameer poseriors and we compare he variance of inflaion from he baseline model wih wage rigidiies and an oherwise idenical model where wages are flexible. I clearly emerges ha he variance of inflaion in he model wih flexible wages is greaer han in he model wih sicky wages for a wide range of parameer values. So we find ha in our model he irrelevance resul obained by Krause and Lubik (2007) only holds for some paricular calibraions. However, for he esimaed parameers he variance of inflaion is idenical in he wo models. This shows ha he heoreical framework allows nominal wage rigidiies o have relevan consequences for inflaion dynamics, bu he daa prefer a version of he model in which wage rigidiies are irrelevan for inflaion dynamics. This resul sands in conras wih hose found by Gerler e al. (2008) in a similar analysis of he US labour marke. The paper is relaed o several sudies. As in Krause and Lubik (2007), Krause e al. (2008a, 2008b), Ravenna and Walsh (2008), Zanei (20) and Chrisiano e al. (20), we inernalize he imporance of labour marke The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

4 Wage Rigidiies in a DSGE Model of he UK Labour Marke 69 fricions o describe inflaion dynamics, bu we also exend he framework o incorporae and es he empirical relevance of saggered wage seing. In his respec, our approach is similar o Gerler e al. (2008). However, our work differs from heirs as we allow firms o change he labour inpu along boh he exensive and he inensive margin, and we simplify he modelling of wage rigidiies following Thomas (2008). Moreover, by assuming ha newly hired workers become immediaely producive we inroduce an insananeous channel from wages o inflaion wihou deparing from efficien bargaining on hours. As shown by Trigari (2006), under efficien bargaining on hours and a delay in he iming of he maching funcion, here is no link beween curren period wages and marginal coss. The inuiion is sraighforward: if i akes ime for workers o conribue o producion, firms can change oupu only by changing hours. As a resul, curren period marginal coss will only depend on hours. Bu when hours are efficienly bargained he number of hours will depend only on he raio beween he marginal rae of subsiuion and he marginal produc of labour, which in urn are independen of wages. In order o inroduce a link beween curren period wages and marginal coss, a number of auhors have abandoned he assumpion of efficien bargaining o invesigae he implicaions of righ o manage (Chrisoffel and Linzer, 2006; Chrisoffel and Kueser, 2008; Maesini and Rossi, 2008; Chrisoffel e al., 2009a; Zanei, 20). We build on his lieraure by showing ha a conemporaneous iming of he maching funcion resores a wage channel in he presence of efficien bargaining on hours. However, we find ha a he esimaed equilibrium he wage channel is unable o affec inflaion dynamics. This paper is also relaed o he sudies of Erceg e al. (2000) and Chrisiano e al. (2007) who invesigae he effec of wage rigidiies on he dynamics of New Keynesian models and inflaion. We add o his line of research by exending he analysis o labour marke fricions. Finally, differenly from all he aforemenioned sudies, we are he firs o esimae a model wih labour marke fricions and nominal wage rigidiies on he UK economy. The remainder of he paper is organized as follows. Secion 2 ses up he model and deails he specificaion of marginal coss. Secion 3 presens he resuls of he esimaion. Secion 4 uses impulse response funcions o lay ou he ransmission mechanism of he model. I hen evaluaes he imporance of each shock in explaining he dynamics of he endogenous variables, and finally uses he reduced form of he model o recover he dynamics of he unobserved shocks. Secion 5 invesigaes wheher he irrelevance of wage rigidiies for inflaion dynamics is buil ino he heoreical framework. Finally, Secion 6 concludes. 2 THE MODEL The model combines he search and maching framework in Krause e al. (2008a) wih he saggered wage-seing mechanism in Thomas (2008). The The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

5 70 The Mancheser School economy consiss of: households; firms, comprised of a coninuum of producers indexed by j [0, ] and reailers; a moneary auhoriy and a fiscal auhoriy. In wha follows we explain he srucure of he labour marke and he problems faced by households and firms. We conclude by deailing he specificaion of marginal coss. 2. The Labour Marke The maching of workers and firms is esablished by he sandard maching ξ ξ funcion MU (, V)= muv, which represens he aggregae flow of hires in a uni period. 3 The variable U denoes aggregae unemploymen and V aggregae vacancies, m > 0 capures maching efficiency and 0 < ξ < denoes he elasiciy of he maching funcion wih respec o unemploymen. During each period, vacancies are filled wih probabiliy q(θ ) = M /V, where θ = V /U denoes labour marke ighness. Consan reurns o scale in he maching funcion imply ha workers find a job wih probabiliy θ q(θ ). We assume ha new hires sar working a he beginning of each period, and a he end of each period a consan fracion of workers loses he job wih probabiliy ρ. 4 Consequenly, he evoluion of aggregae employmen N is 5 N N M = ( ρ) + () Workers who lose he job a ime can look for a job a he beginning of ime. The sock of workers searching for a job a ime is herefore given by he number of workers who did no work in, N, plus hose who los heir job a he end of he period, ρn. The evoluion of aggregae unemploymen can be wrien as U = ( ρ)n. 2.2 Households The economy is populaed by a uni measure of households whose members can be eiher employed or unemployed. We follow Merz (995) and Andolfao (996) in assuming ha members of he represenaive household perfecly insure each oher agains flucuaions in income. The problem of he represenaive household is o maximize an expeced uiliy funcion of he form s βζ+ s s= 0 E σ + μ ( c+ s ςc+ s ) hj+ s χ+ s nj+ s j σ d + μ 0 (2) 3 Noe ha U ujdj and V vjd j. 4 The assumpion ha new hires conribue o employmen during period is suppored by microeconomic evidence as in Davis e al. (203) and references herein. = 0 5 Noe ha N n d j. = 0 j = 0 The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

6 Wage Rigidiies in a DSGE Model of he UK Labour Marke 7 where β is he discoun facor, ζ is a preference shock and χ is a labour supply shock. The variable c denoes consumpion of he represenaive household a ime, while C denoes aggregae consumpion in period, and ς is an index of exernal consumpion habis. The variable n j denoes he number of household members employed in firm j, and h j denoes he corresponding number of hours. The parameer σ governs he degree of risk aversion and μ is he inverse of he Frisch elasiciy of labour supply. Consumpion c is a Dixi Sigliz aggregaor of a bundle of differeniaed goods: c = [ 0 c( j) ( ε ) ε ε ( ε ) d j], where ε is he sochasic elasiciy of subsiuion among differeniaed goods. Denoing by p j he price of a variey produced by a monopolisic compeior j, he expendiure minimizing price index associaed wih he represenaive consumpion bundle c is: p = p( j) ε 0 j ( ε ) d. The household faces he following budge consrain: I B c R B w j + + = + p p p n j h jd j + ( n ) b + r kk + d + T (3) 0 which dicaes ha expendiure, on he lef-hand side (LHS), mus equal income, on he righ-hand side (RHS). The households expendiure is invesmen, I, consumpion, c, and he acquisiion of bonds, B /p. Households income is he sock of bonds B /p from previous period which pay a gross nominal ineres rae R, he proceedings from working in he firms indexed by j, 0 ( w j p ) n j h jd j, 6 and he unemployed benefis, b, earned by each unemployed member of he household. In addiion, he household earns proceedings from rening capial, k, o he firms a he rae r k, he real dividends from owning he firms, d, and he ne governmen ransfer T. The household chooses c, B and k + o maximize he uiliy funcion (2), subjec o he budge consrain in equaion (3) and he law of moion for capial, k+ = φi + ( δk) k (4) where δ k denoes he rae of capial depreciaion and ϕ denoes an invesmen-specific echnology shock. By subsiuing equaion (4) ino (3), and leing λ denoe he Lagrange muliplier on he budge consrain, he firs-order condiions wih respec o c, B and k + are and σ λ = ζ( c ςc ) (5) λ = βe( λ+ R π+ ) (6) 6 Noe ha wih his noaion w j, n j and h j are he wage employmen and hours of work a firm j a ime respecively. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

7 72 The Mancheser School [ ] λ = βeλ+ r k + + ( δk ) (7) where π + = p +/p denoes he gross inflaion rae. Equaion (5) saes ha he Lagrange muliplier equals he marginal uiliy of consumpion. Equaions (6) and (7), once equaion (5) is subsiued in, are he sandard household s Euler equaions ha describe he consumpion and capial decisions respecively. To conclude he descripion of he household we need o define he marginal value of being employed and unemployed. The marginal value of employmen a firm j, W j E, is given by + μ w W p h h E j j U E j = λ j ζ χ E W Wj + + [ + ] μ β λ ρ ( ρ ) (8) which saes ha he marginal value of a job for a worker is given by he real wage bill ne of he disuiliy of work plus he expeced-discouned value from being eiher employed or unemployed in he following period. The marginal value of unemploymen, W U,is { } (9) W = λ b+ βe λ [ θ q( θ )] W + ( ρ) θ q( θ ) Wˆ U U E ˆ + + E where EW = 0Wj E dj is he expeced value of employmen in +. This equaion saes ha he marginal value of unemploymen is he sum of unemploymen benefis plus he expeced-discouned value from being eiher employed or unemployed in +. Using equaions (8) and (9) we deermine E U he household s ne value of employmen a firm j, Wj W, denoed by W j, as + μ w j W p h b hj j = λ j λ ζ χ E Wj q ( ) + + μ β λ ρ θ ( θ + ) Wˆ + (0) where EW ˆ = Wj j + 0 d Firms We assume wo ypes of firms: producers and reailers. Producers hire workers in a fricional labour marke and ren capial in a perfecly compeiive marke. They manufacure a homogeneous inermediae good and sell i o reailers in a perfecly compeiive marke. Reailers ransform inermediae inpus from he producion secor ino differeniaed goods and sell hem o consumers. As i is sandard in he New Keynesian lieraure, we assume saggered price adjusmen àlacalvo (983). In wha follows we describe he problems of he producers and reailers in deail Producers. There is a coninuum of producers of uni measure selling homogeneous goods a he compeiive price φ. During each period, The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

8 Wage Rigidiies in a DSGE Model of he UK Labour Marke 73 firm j manufacures y j unis of goods according o he following producion α α echnology yj = A ( njhj ) kj, where A is a sochasic variable capuring neural echnology shocks. We assume consan reurns o scale in producion implying ha all firms have he same capial labour raio k j/n jh j = k /n h for all j. Consequenly, he marginal produc of labour is also equalized across firms such ha mpl j = mpl. Firms open vacancies a ime o choose employmen in he same period; he cos of opening vacancies is C( vj ) = av ε j c, where a > 0 is a scaling facor and ε c > is he elasiciy of hiring coss wih respec o vacancies. The vacancy cos funcion is assumed o be convex in order o produce an equilibrium where all he firms pos vacancies. If he vacancy cos funcion were linear all firms would face he same marginal vacancy posing cos. Since we assume saggered nominal wage negoiaions, i follows ha only he firm wih he lowes wage would hire a equilibrium. In our model wage dispersion implies ha firms wih high wages face low marginal reurn from search and low marginal vacancy posing coss since hey hire only a relaively small number of workers. The problem of he firm is o choose v j, n j and k j+ o maximize he presen value of fuure discouned profis: w s s j max E β λ + ϕ sy s ( ) λ p n h C v k r s= 0 k + + j+ s j+ s j+ s j+ s + s subjec o he producion funcion and he law of moion for employmen: n = ( ρ) n + v q( θ ) () j j j Since households own he firms, fuure profis are discouned a he rae β s λ +s/λ. Leing J j denoe he Lagrange muliplier on he employmen consrain (), he firs-order condiions wih respec o k j+, v j and n j are α α r = ϕ ( α) A( n h ) k (2) k j j j C ( v j ) q( ) = J j θ (3) α w α j Jj A njhj kj hj p h j E λ+ = ϕα ( ) + β( ρ) J j+ λ (4) Equaion (2) implies ha reurns o capial equalize he marginal revenue produc. Equaion (3) implies ha he per period cos of filling a vacancy C ( v j ) imes he average vacancy duraion /q(θ ) mus equal he shadow value of employmen J j. Equaion (4) shows ha he shadow value of employmen o he firm equals curren period profis, i.e. he marginal revenue produc of employmen ne of wage coss, plus he coninuaion value. Subsiuing equaion (3) ino equaion (4) yields he sandard job creaion condiion: The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

9 74 The Mancheser School C ( v j ) ( ) = ( ) α w α j ϕα A njhj kj hj q p h j + β ( ρ ) E λ θ λ C ( v ) q( θ ) + j+ + (5) which saes ha he cos of hiring an addiional worker (LHS) equals he marginal benefi (RHS) ha he addiional worker brings ino he firm Reailers. There is a uni measure of reailers who ransform homogeneous goods from he producion secor ino differeniaed goods. Monopolisic compeiion implies ha each reailer j faces he following demand for is own produc c j pj = p ε c (6) where c is aggregae demand of he consumpion bundle. Each reailer produces c j unis of oupu using he same amoun of inpus from he producion secor. We assume price sickiness àlacalvo (983), meaning ha during each period a random fracion of firms, δ p, are no allowed o rese heir price. The problem of he reailers is o choose p j o maximize max E s= 0 s s + s p δβ λ λ p p j + s ϕ c + s j+ s subjec o he demand funcion (6). The opimal pricing decision is E s= 0 s s s p δβ λ + * ε p s λ ϕ + p+ s ε = 0 (7) where p * is he opimal price chosen by all firms renegoiaing a ime. This implies ha forward-looking firms choose he opimal price such ha he ime-varying mark-up is equal o ε /(ε ). Since firms are randomly seleced o change price, he law of moion for he aggregae price level is p ε = δ p ε + ( δ )( p*) ε (8) p p 2.4 Wage Bargaining Similarly o he price-seing decision, we assume saggered nominal wage negoiaions, meaning ha each period only a random fracion of firms, δ w, is allowed o renegoiae on nominal wages. Following Thomas (2008) we assume ha he wage se by he renegoiaing firm j saisfies he following sharing rule: Wj * ηj* j = ( η) (9) λ The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

10 Wage Rigidiies in a DSGE Model of he UK Labour Marke 75 where η ηε η is he sochasic bargaining power of he workers and he superscrip * denoes renegoiaing workers and firms. This sharing rule implies ha renegoiaing workers obain a fracion of he oal surplus equal o heir bargaining power. Noice ha his is differen from Nash bargaining. Wih Nash bargaining wages maximize a weighed average of he join surplus. Nash bargaining delivers he sharing rule, equaion (9), only if wages are coninuously renegoiaed. As shown by Gerler and Trigari (2009), Nash bargaining implies ha, in he presence of saggered nominal wage negoiaions, he share parameer η in equaion (9) would flucuae over he cycle even if i were no subjec o shocks. This follows from he fac ha workers and firms face differen ime horizons when hey consider he effecs of differen wages. However, Gerler and Trigari (2009) sugges ha his horizon effec has quaniaively negligible implicaions. We herefore choose o follow Thomas (2008) and adop he sharing rule in equaion (9) as i simplifies he analysis considerably. Wih saggered wage negoiaions, he shadow value of employmen a firm j o he household ha is allowed o renegoiae can be rewrien from equaion (0) as follows: Wj * w* j p h w E λ + Wj+ Wj* + = j j + β ( ρ) δw ( δ w) λ λ λ+ λ+ (20) where he worker s opporuniy cos of holding he job, w j, is equal o w j + μ ζχ hj W = b+ E q ( ) + ( + ) + λ μ β ˆ ρθ θ λλ λ The ne value of employmen o he household condiional on wage renegoiaion a ime (equaion (20)), equals he ne flow income from employmen, ( w j / p ) hj w j, plus he coninuaion value, which is he las erm on he RHS. The laer is equal o he sum of he marginal discouned value of employmen in + condiional on he wage se a ime, if he firm does no renegoiae wih probabiliy δ w, and he value of employmen in + condiional on a renegoiaion, wih probabiliy δ w. Similarly, he shadow value of employmen o he renegoiaing firm j can be wrien w* j Jj wj p h E λ * = j + ( ) + ρ [ J j+ + ( ) J * j+ ] λ δ w δ (2) w where w j = ϕmplhj denoes he marginal revenue produc. The marginal value of employmen for a renegoiaing firm equals he ne flow value of he mach plus he coninuaion value. In urn, his equals he marginal value of The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

11 76 The Mancheser School employmen in + condiional on he previous period wage, wih probabiliy δ w, and he marginal value condiional on a wage renegoiaion, wih probabiliy δ w. Ieraing equaions (20) and (2) forward and using he sharing rule in equaion (9), i yields w s s s s j E β λ + * ar ( ρ) δw hj+ s w + s = 0 s= 0 λ p+ s (22) ar where wj+ s= η+ swj+ s+ ( η+ s) w j+ sis he oal real wage paymen o he worker on which boh paries would agree if wages were fully flexible. Subsiuing for w j+ s and w j+ s he arge real wage bill can be wrien + μ h ar ζχ j + wj+ s = ηϕ mplhj + ( η ) b+ E λ + μ + β λλ ( Wˆ + ρ) θ+ q ( θ+ ) λ (23) Equaion (23) is sandard in he search and maching lieraure. The arge real wage bill is expressed as a weighed average beween he marginal revenue produc of he worker and he opporuniy cos of holding a job a he level of hours worked h j. Given ha renegoiaing firms are randomly chosen, he law of moion for he aggregae nominal wage is given by w = δ w + ( δ ) w* (24) w w = 0 jd. where w w j 2.5 Hours Bargaining We assume ha hours and wages are bargained simulaneously and ha bargaining on hours is efficien. Hence, hours saisfy he Nash bargaining crierion: h j Wj * = arg max λ η ( J*) j η Using he sharing rule (9), he firs-order condiion becomes χζ h ϕ Aα n h k λ μ 2 α j = j j α j α This equaion saes ha he marginal rae of subsiuion, on he LHS, equals he marginal produc of hours, on he RHS. Since he marginal reurn o he labour inpu is equalized across firms a equilibrium, i follows ha The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

12 Wage Rigidiies in a DSGE Model of he UK Labour Marke 77 members of he household employed in differen firms work he same amoun of hours, i.e. h j = h. Solving he firs-order condiion for hours yields h j A n k = η ϕ α α χζ 2 α j j λ + μ α (25) 2.6 Price and Wage Inflaion Following Calvo (983), using equaions (7) and (8) we derive he sandard New Keynesian Phillips Curve: π = kp ( ϕˆ + ε)+ βeπ (26) + where a ha superscrip denoes he variable s deviaion from is seady sae, and he coefficien k p is equal o: k p [( βδ p)( δ p)]/δ p. Similarly, following Thomas (2008), using equaions (22) and (24) we obain he following equaion for wage inflaion: ( ) ar πw = kw wˆ wˆ + hˆ + β( ρ) Eπ (27) w+ where ŵ denoes he real wage and he coefficien k w is equal o: k w {[( β( ρ)δ w]( δ w)}/δ w. Equaion (27) saes ha he real wage inflaion depends on he gap beween he acual and arge real wage bill, wˆ hˆ + and ŵ ar respecively. Inflaion maerializes whenever he real wage bill is below arge, i.e. whenever he wage bill is below he level ha would prevail if wages were perfecly flexible. 2.7 Closing he Model The moneary auhoriy ses he nominal ineres rae following he ρ r r ρ r π y r Taylor rule: ( R R* )= ( R R* ) ( π π* ) ( y y* ) ε R, where an aserisk superscrip denoes he seady-sae values of he associaed variables. The parameer ρ r represens ineres rae smoohing, and r y and r π govern he response of he moneary auhoriy o deviaions of oupu and inflaion from heir seady-sae value. The error erm ε R denoes an independen and idenically disribued (i.i.d.) moneary policy shock. The fiscal auhoriy is assumed o run a balanced budge: B /p = R (B /p )+T + b( n ). 2.8 Marginal Coss In his secion we compare he specificaion of marginal coss in our model agains alernaive formulaions in he lieraure. This is imporan o unveil some key properies of he model and undersand he findings deailed in he nex secion. In a maching model wih efficien bargaining on hours, Trigari The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

13 78 The Mancheser School (2006) shows ha whenever firms pos vacancies a ime o conrol employmen in he following period, he real marginal cos a ime is independen of wages a ime. The independence of curren period marginal coss from curren period real wages is ypically referred o, in he lieraure, as he lack of a wage channel. The inuiion is sraighforward. Since curren hires conribue o nex period employmen, in he curren period firms can change producion only by adjusing hours. This implies ha he marginal cos of producion a ime depends solely on hours. Wih efficien bargaining he number of hours worked is deermined by he marginal rae of subsiuion beween consumpion and leisure and he marginal produc of labour, and herefore i is independen of wages in he curren period. I follows ha curren wages are irrelevan for curren period marginal coss. Following Trigari (2006), a number of auhors such as Chrisoffel and Kueser (2008), Chrisoffel and Linzer (2006) and Zanei (2007) have resored he ransmission channel from wages o prices by resoring o alernaive bargaining schemes such as he righ o manage. In our model we are able o resore a wage channel a ime while preserving efficien Nash bargaining. We do so by changing he iming assumpion of he maching funcion. Tha is, we allow firms o conrol employmen a ime by choosing vacancies in he same period, as described by equaion (). Under his iming assumpion, he cos of increasing producion a he margin depends on he cos of hiring an addiional worker, which is represened by he wage paid o he new hire. This can be seen by solving he job creaion condiion in equaion (4) for marginal coss φ : λ+ J βe ( ρ) J w h λ ϕ = + p mpe mpe + (28) α where mpe = A ( njhj ) α α k j hj denoes he marginal produc of employmen. From equaion (28), as shown by Krause and Lubik (2007), real marginal coss are equal o he sum of he uni labour cos and an addiional erm relaed o maching fricions. Given ha he shadow value of employmen J equals he expeced hiring cos, he second erm on he RHS of equaion (28) can be inerpreed as he expeced change in search coss. By equaion (3), his erm depends on he expeced value of labour marke ighness in he nex period relaive o he curren period. Had we assumed ha newly hired workers were unable o conribue o producion immediaely, he decision on vacancies would only affec nex period marginal coss, leaving curren period marginal coss solely dependen on he number of hours, which, due o efficien wage bargaining, are independen of wages. 7 7 Noe ha in his insance fuure period marginal coss include wages. However, since fuure periods are discouned, he dynamics of fuure wages have a more limied effec on inflaion. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

14 Wage Rigidiies in a DSGE Model of he UK Labour Marke 79 3 ESTIMATION The model is esimaed wih Bayesian mehods. I is firs loglinearized around he deerminisic seady sae. We hen solve he model and apply he Kalman filer o evaluae he likelihood funcion of he observable variables. The likelihood funcion and he prior disribuion of he parameers are combined o obain he poserior disribuions. The poserior kernel is simulaed numerically using he Meropolis-Hasing algorihm. 8 We firs discuss he daa and he priors used in he esimaion and hen repor he parameer esimaes. 3. Priors and Daa The model is esimaed over he period 97:Q 2009:Q4 using seven shocks and seven quarerly daa series: consumpion, invesmen, inflaion, average hours, employmen, he real wage and he nominal ineres rae. 9 The daa series are from he Office for Naional Saisics daa se; he acronyms are indicaed in ialics. For consumpion, we use daa on household final consumpion expendiure (ABJR) and for invesmen we use daa on business invesmen (NPEL). We define oupu o be he sum of hese wo series and he price level o be he implici deflaor associaed wih his measure of oupu ((NPEK + ABJQ)/(NPEL + ABJR)). Our employmen series comes from he Labour Force Survey (MGRZ) and our series for average hours is calculaed as oal acual weekly hours worked (YBUS) divided by employmen. We define he nominal wage as wages and salaries (ROYJ) divided by oal acual weekly hours worked. The real wage is hen his series divided by our series for he price level. The labour marke variables, employmen, hours and real wages, refer o he whole economy and are inerpreed as proxies for he behaviour of he respecive privae secor variables since our model absracs from he public secor. Finally, our nominal ineres rae series is he London clearing banks base rae (AMIH). The series for consumpion, invesmen, average hours, employmen and real wages are logged and hen all series are passed hrough a Hodrick Presco filer wih smoohing parameer 600. The seven shocks in he model are a preference shock, a mark-up shock, a labour supply shock, a neural echnology shock, a bargaining power shock, an invesmen-specific echnology shock and a moneary policy shock. 8 We use wo blocks of 250,000 draws. The sequence of draws is sable, providing evidence on convergence. An appendix ha deails evidence on convergence is available upon reques from he auhors. 9 This sample period was characerized by muliple moneary policy regimes and changes in labour marke insiuion. To enable he esimaion o accoun for he influence of hese breaks on he parameer esimaes, an alernaive esimaion sraegy would be o exend he esimaion of he DSGE model wih one regime shif proposed by Curdia and Finocchiaro (203) o muliple regime changes. We leave his exension open o fuure research. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

15 80 The Mancheser School TABLE FIXED PARAMETERS Parameers Descripion Values β Discoun facor 0.99 α Labour share 0.69 δ k Capial depreciaion rae ε c Elasiciy of he vacancy cos funcion. b Unemploymen benefis 0.58 ξ Maching funcion elasiciy 0.7 m Consan maching funcion 0.6 ρ Job desrucion rae 0.03 ε Elasiciy of demand All shocks, wih he excepion of moneary policy shock, are assumed o follow a firs-order auoregressive process wih i.i.d. normal error erms such ha ln κ + = ρ κ ln κ + v, where he shock κ {ζ, χ, ε, A, ϕ, ε η }, 0 < ρ κ < and v N(0, σ κ). Moneary policy shocks ε R are i.i.d. The model conains 9 srucural parameers, excluding he shock parameers. Before esimaing he model we calibrae some parameers o mach imporan long-run properies of he daa. This is paricularly imporan for he parameers relaed o he labour marke, since he informaion on labour marke dynamics is limied in he daa se and herefore he esimaion is unable o deliver esimaes ha capure he long-run properies in he daa. We sar by discussing he fixed parameers, whose values are summarized in Table. The discoun facor β is se a 0.99 implying a real ineres rae of 4 per cen. The labour share parameer α is se equal o 0.69 in order o mach he observed labour share over he period of he esimaion and he capial depreciaion parameer δ k is se a o mach an average annual rae of capial desrucion of 0 per cen. The elasiciy of he vacancy cos funcion, ε c, is also fixed. This parameer is se a., a value which is relaively close o he sandard assumpion of linear adjusmen coss, and saisfies he assumpion of convexiy. The unemploymen benefis coefficien, b, is calibraed o mach a replacemen raio of 0.58, as in OECD (2007). This parameer is imporan o generae amplificaion of labour marke variables. As shown by Hagedorn and Manovskii (2008), values of b close o uniy generae responses of unemploymen and vacancies o produciviy shocks ha are close o he daa. When b is high, he value of a job o he worker is very close o he value of unemploymen. In such insance he surplus of a job is very small and iny changes in he produciviy of he labour inpu produce a high percenage change in he oal surplus of a mach, boosing he response of employmen. The elasiciy of he maching funcion, ξ, is se o 0.7, as esimaed by Perongolo and Pissarides (200) for he UK economy. The consan of he maching funcion, m, is se equal o 0.5 o mach he job-finding rae of 35 per cen, in line wih evidence from he Labour Force The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

16 Wage Rigidiies in a DSGE Model of he UK Labour Marke 8 TABLE 2 PRIOR AND POSTERIOR DISTRIBUTION OF STRUCTURAL PARAMETERS Prior disribuion Poserior disribuion Descripion 5% 95% Mean 5% 95% σ Relaive risk aversion Gamma (0.66, 0.2) ς Habi persisence Bea (0.5, 0.2) μ Inverse Frisch elasiciy Gamma (, 0.) a Cons. vacancy cos funcion Gamma (2.5, ) η Workers bargaining power Bea (0.5, 0.2) δ w Calvo wage parameer Bea (0.5, 0.25) δ p Calvo price parameer Bea (0.5, 0.25) r π Taylor rule response o inflaion Gamma (.5, 0.05) r y Taylor rule response o oupu Gamma (0.25, 0.05) ρ r Taylor rule ineria Bea (0.5, 0.2) Noes: The able repors he prior and poserior disribuion of he esimaed srucural parameers ogeher wih 5 per cen and 95 per cen confidence inervals. The numbers in parenhesis are he mean and sandard deviaion of he disribuion. Survey. The job desrucion rae, ρ, is se o 0.03, as esimaed by Bell and Smih (2002) using Labour Force Survey daa. Finally, he elasiciy of demand, ε, is se o, a value suggesed in Brion e al. (2000), which implies a seady-sae mark-up of 0 per cen. Finally, he seady-sae gross inflaion rae, π, is se equal o. The remaining parameers are esimaed. We use he bea disribuion for parameers ha ake sensible values beween zero and one, he gamma disribuion for coefficiens resriced o be posiive and he inverse gamma disribuion for he shock variances. Tables 2 and 3 repor priors, poserior esimaes and 90 per cen confidence inervals for he srucural and shock parameers respecively. The prior mean of he relaive risk aversion, σ, is se equal o The prior mean of he index of exernal habi, ς, is se a is middle value of 0.5, as in Gerler e al. (2008). The prior mean of he inverse of he Frisch elasiciy of labour supply, μ, is se equal o, as in Krause e al. (2008a). The prior mean of he scaling facor of he cos of posing a vacancy, a, is se equal o 3 such ha he cos of posing a vacancy is approximaely per cen of oal oupu a he seady sae, as in Blanchard and Gali (200). The prior mean of he worker bargaining power, η, is se o 0.5, such ha he firm and he worker hey equally spli he surplus from working. The prior mean of he Calvo parameer on prices, δ p, is se o 0.5 in order o mach an average duraion of prices of abou six monhs, in line wih he evidence in Bunn and Ellis (2009) for he UK economy. We ake an agnosic view on wheher wages are more flexible han prices, and herefore se he prior mean of he Calvo parameer on wages, δ w, o 0.5. We choose he prior means of he Taylor rule response o inflaion, r π, oupu, r y, and he ineres rae smoohing parameer, ρ r, equal o.5, 0.25 and 0.5 respecively. These values are commonly used in he lieraure. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

17 82 The Mancheser School TABLE 3 PRIOR AND POSTERIOR DISTRIBUTION OF SHOCK PARAMETERS Prior disribuion Poserior disribuion Descripion 5% 95% Mean 5% 95% Auoregressive parameers ρχ Labour supply Bea (0.8, 0.2) ρζ Preferences Bea (0.8, 0.2) ρε Mark-up Bea (0.8, 0.2) ρa Technology Bea (0.8, 0.2) ρi Invesmen Bea (0.8, 0.2) ρη Bargaining Bea (0.8, 0.2) Sandard errors σχ Labour supply IGamma (2, 0.3) σζ Preferences IGamma (2, 0.3) σε Mark-up IGamma (2, 0.3) σa Technology IGamma (2, 0.3) σε Moneary policy IGamma (2, 0.3) σi Invesmen IGamma (2, 0.3) ση Bargaining IGamma (2, 0.3) Noes: The able repors he prior and poserior disribuion of he esimaed srucural parameers ogeher wih 5 per cen and 95 per cen confidence inervals. The numbers in parenhesis are he mean and sandard deviaion of he disribuion. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

18 Wage Rigidiies in a DSGE Model of he UK Labour Marke 83 Finally, Table 3 repors he prior disribuions of he shock parameers. The prior mean of he auoregressive parameers is se equal o 0.8 and he prior mean of he sandard errors is se equal o 2 for all he shocks. For he prior mean of he auoregressive parameers we choose values ha are in beween hose seleced by Gerler e al. (2008) and Krause e al. (2008a). 3.2 Parameer Esimaes The hird, fourh and fifh columns of Table 2 show he poserior means of he srucural parameers ogeher wih heir 90 per cen confidence inervals. The poserior mean of he relaive risk aversion σ is equal o 0.73, in he range of esimaes in Brown and Gibbons (985) for he USA. The poserior mean of he index of exernal habis ς is equal o 0.04, which is subsanially lower han he esimae of 0.57 in Smes and Wouers (2003), herefore ruling ou habi in consumpion as an imporan source o generae persisence in he model. 0 The poserior mean of he inverse of he Frisch elasiciy of labour supply μ is equal o.6, which is subsanially higher han is prior, and in line wih microeconomic esimaes as surveyed by Card (994). This high esimae reflecs he fac ha employmen volailiy is higher a he exensive margin han a he inensive margin. Krause e al. (2008a) obain similar resuls for he USA, alhough heir esimae is higher han ours. The poserior mean of he consan of he vacancy cos funcion a is equal o.88, lower han is prior, which indicaes ha he model prefers a low cos of posing a vacancy, which is equal o 0.3 per cen of oal oupu a he esimaed equilibrium, similarly o he esimaes in Silva and Toledo (2009) of approximaely 0.2 per cen on US daa. The poserior mean of he bargaining power of he workers η is equal o 0.87, hereby indicaing ha wages are closer o he marginal produc of labour. This esimae is remarkably close o he value of 0.90 in Gerler e al. (2008) based on US daa. Noe ha he value of η and b deermine he equilibrium firm s surplus ha is a key variable for he response of labour marke variables o shocks, as shown in Hagedorn and Manovskii (2008). In paricular, a low firm s surplus increases he response of labour marke variables. The poserior means of he model implies a firm s surplus of approximaely 0 per cen relaive o produciviy, which is higher han he value of 2.5 per cen in Hagedorn and Manovskii 0 The resul ha he esimaed habi parameer akes a value close o zero is he same as found by Krause e al. (2008a) on US daa. Their model does no have o rely on his source of inrinsic persisence o explain he behaviour of inflaion and consumpion because he esimaed shocks are srongly auocorrelaed. The UK inflaion exhibis very low auocorrelaion in he full sample, and zero auocorrelaion from he mid-980s onward. So our model does no have o rely on any source of persisence o explain inflaion. The persisence of he UK consumpion series is insead explained in our model by he persisence of he preference shocks, which explains 85 per cen of he variance of consumpion on impac. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

19 84 The Mancheser School TABLE 4 MARGINAL LOG-LIKELIHOOD FUNCTION Sicky wages 3466 Flexible wages 2944 (2008). The unemploymen rae is abou 0 per cen a he esimaed equilibrium, slighly higher han he average value of 7 per cen observed in he daa, possibly accouning for he exisence of workers who are only marginally aached o he labour force, and as such do no qualify as unemployed according o he Inernaional Labour Organizaion (ILO) definiion. This higher unemploymen rae implies ha he job finding rae, around 28 per cen, is somewha lower han our calibraion arge based on Labour Force Survey daa. The poserior means of he Calvo parameers on he frequency of wage and price negoiaions, δ w and δ p, are equal o 0.63 and 0.52 respecively, showing ha prices adjus more frequenly han wages. These values imply an average frequency of wage negoiaions of hree quarers, in line wih Dickens e al. (2007), and an average frequency of price negoiaions of wo quarers, in line wih Bunn and Ellis (2009) for he UK economy. However, alhough he model prefers higher wage rigidiies han price rigidiies, he esimaion shows a sizable uncerainy around is poserior mean. Finally, he esimaes of he Taylor rule parameers are as follows. The poserior means of he ineres rae response o inflaion and oupu, r π and r y, equal o.48 and 0.3, respecively, indicae a srong response o inflaion and oupu. The poserior mean of he degree of ineres rae smoohing, ρ r, equal o 0.53 suggess a mild degree of ineres rae ineria. The hird, fourh and fifh columns of Table 3 show he poserior means of he shock parameers ogeher wih heir 90 per cen confidence inervals. The poserior means of he persisence parameers ρ χ and ρ ζ, equal o 0.84 and 0.88 respecively, indicae ha shocks o he labour supply and preferences are subsanially more persisen han he oher shocks, which is in line wih he esimaes in Caselnuovo (2007) based on Euro Area daa. The poserior means of he shocks variance is close o per cen for all he shocks, wih he excepion of invesmen-specific echnology shocks, σ i, and bargaining shocks, σ η, which are more volaile. In order o esablish wheher saggered wages are imporan o mach he daa, Table 4 repors he value of he marginal log-likelihood funcion for he esimaed models wih sicky and flexible wages respecively. Since he value of he marginal log-likelihood funcion associaed wih he model wih Noe ha when we esimaed he model including ime series for vacancies, he firm s surplus was similar o Hagedorn and Manovskii (2008), alhough he overall fi of he model was worse. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

20 Wage Rigidiies in a DSGE Model of he UK Labour Marke 85 sicky wages is equal o 3466, and subsanially higher han he value of 2944 associaed wih he flexible wage model, saggered wage seing enables he model o fi he daa more closely, hereby suggesing ha wages rigidiies are imporan o replicae UK daa. 4 IMPULSE RESPONSE FUNCTIONS AND VARIANCE DECOMPOSITION In his secion we invesigae, by use of impulse responses, how he shocks are ransmied o he endogenous variables. In order o disenangle he effec of nominal wage rigidiies we use our baseline model and an oherwise idenical model where he Calvo parameer on wages is se o zero (δ w = 0). Figures 5 plo he impulse responses of seleced variables o a one sandard deviaion shock. Each enry compares he responses of he model wih sicky wages (solid line) agains hose wih flexible wages (doed line). 2 Figure shows ha a one sandard deviaion mark-up shock leads o an increase in inflaion. In urn, he resuling increase in he ineres rae decreases consumpion and invesmen. In reacion o he shock, he firm reduces he labour inpu along boh he inensive and he exensive margin o decrease producion. The qualiaive responses of he variables in he saggered wage model are similar o hose in he model wih flexible wage seing, since mark-up shocks do no induce he firm o adjus labour marke variables differenly. However, i is worh noing ha wage rigidiies affec he behaviour of nominal and real wages considerably, bu he reacion of marginal coss and inflaion remains remarkably similar in he wo seings. Why are he inflaion dynamics so similar? As deailed in Secion 2.8, search fricions inroduce an addiional erm ino marginal coss, over and above uni labour coss, which reflecs he expeced change in search coss. Following a posiive mark-up shock, nominal wage rigidiies aenuae he drop in uni labour coss and induce a fall in he fricional componen of marginal coss compared wih a flexible wage regime. As a resul, marginal coss and inflaion dynamics behave similarly in he wo seings. Wage rigidiies aenuae he reacion of uni labour coss as firms are no allowed o renegoiae lower wages. A he same ime, wage rigidiies induce labour marke ighness o fall on impac and hen seadily increase. As a resul, he firm s cos of searching for a worker falls on impac and i hen rises over ime. 3 The rising profile in expeced search coss implies ha he firm can save on fuure hiring coss by increasing curren period hiring. From equaion (28), higher expeced search coss nex period, ranslae in lower marginal coss in he curren period. As 2 We show impulse responses for shocks o he mark-up, neural echnology, moneary policy, preferences and invesmen-specific echnology. An appendix ha deails he impulse responses of he variables o all he shocks in he model is available upon reques o he auhors. 3 Noe ha he average duraion of a vacancy, /q(θ ), depends only on labour marke ighness. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

21 86 The Mancheser School Consumpion Employmen Ineres Rae Oupu Uni Labour Coss Hours Price Inflaion Vacancies Unemploymen Fricional Componen of Marginal Coss Invesmen Wage Inflaion Real Wage Tighness Marginal Coss FIG.. Impulse Responses o a Mark-up Shock Noes: Solid lines denoe he baseline economy wih sicky wages. Dashed lines refer o he economy wih flexible wages. Impulse responses are expressed in percenage deviaions from he seady sae. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser

22 Wage Rigidiies in a DSGE Model of he UK Labour Marke Consumpion 0 Employmen Ineres Rae Oupu Uni Labour Cos Hours Price Inflaion Vacancies Unemploymen Fricional Componen of Marginal Coss Invesmen Wage -6 8 Real Wage Tighness Marginal Coss FIG. 2. Impulse Responses o a Neural Technology Shock Noes: Solid lines denoe he baseline economy wih sicky wages. Dashed lines refer o he economy wih flexible wages. Impulse responses are expressed in percenage deviaions from he seady sae. The Mancheser School 203 John Wiley & Sons and The Universiy of Mancheser