Comparing Two Variants of Calvo-Type Wage Stickiness

Transcription

1 Comparing Two Varians of Calvo-Type Wage Sickiness Sephanie Schmi-Grohé Marín Uribe Ocober 10, 2006 Absrac We compare wo ways of modeling Calvo-ype wage sickiness. One in which each household is he monopolisic supplier of a differeniaed ype of labor inpu (as in Erceg, e al., 2000) and one in which households supply a homogenous labor inpu ha is ransformed by monopolisically compeiive labor unions ino a differeniaed labor inpu (as in Schmi-Grohé and Uribe, 2006a,b). We show ha up o a log-linear approximaion he wo varians yield idenical equilibrium dynamics, provided he wage sickiness parameer is in each case calibraed o be consisen wih empirical esimaes of he wage Phillips curve. I follows ha economeric esimaes of New Keynesian models ha rely on log-linearizaions of he equilibrium dynamics are mue abou which ype of wage sickiness fis he daa beer. In he conex of a medium-scale macroeconomic model, we show ha he wo varians of he sicky-wage formulaion give rise o he same Ramsey-opimal dynamics, which call for low volailiy of price inflaion. Furhermore, under boh specificaions he opimized operaional ineresrae feedback rule feaures a large coefficien on price inflaion and a mue response o wage inflaion and oupu. JEL Classificaion: E52, E61, E63. Keywords: Nominal Wage Rigidiy, Wage Phillips Curve, Opimal Moneary Policy. We hank Andy Levin for commens and Sarah Zubairy for research assisance. Duke Universiy, CEPR, and NBER. grohe@duke.edu. Duke Universiy and NBER. uribe@duke.edu.

2 1 Inroducion Nominal wage sickiness is a cenral characerisic of poswar U.S. aggregae daa. Exising esimaes of he wage Phillips curve documen a low sensiiviy of changes in wage inflaion wih respec o deviaions of real wages from he marginal disuiliy of labor. A number of recen sudies model wage sickiness as arising from Calvo-ype saggering. The purpose of his paper is o compare wo varians of he Calvo-ype wage-sickiness model. The firs varian we consider is he one due o Erceg, Henderson, and Levin (2000), hereafer EHL. In his model, each household is he monopolisic supplier of a differeniaed ype of labor inpu and equilibrium effor inensiy varies across households. The oher version of wage sickiness we consider is he one developed in Schmi-Grohé and Uribe (2006a,b), hereafer SGU. In his varian, households supply a homogeneous labor inpu ha is ransformed by monopolisically compeiive labor unions ino a differeniaed labor inpu and every household works equal hours in equilibrium. We embed boh versions of wage sickiness ino he medium-scale macroeconomic model of Alig e al. (2005). This model feaures a large number of nominal and real rigidiies, including price sickiness, money demand by households and firms, habi formaion, variable capaciy uilizaion, invesmen adjusmen coss, and imperfec compeiion in produc and labor markes. The Alig e al. model is of paricular empirical ineres because i has been shown o accoun well for he observed effecs of moneary and supply-side shocks. We firs esablish analyically ha up o a log-linear approximaion he SGU and EHL varians of wage sickiness yield idenical expecaions-augmened wage Phillips curves. I follows ha economeric esimaes of his relaionship are necessarily mue abou which ype of wage sickiness fis he daa beer. We derive he precise mapping from he wage Phillips curve coefficien o he deep srucural parameer governing he degree of wage sickiness in he SGU and EHL models. We find ha in he conex of he EHL framework, he available empirical esimaes of linear wage Phillips curves imply ha nominal wages are reopimized every 3 o 4 quarers. A he same ime, we find ha according o he SGU model, available esimaes of he relaionship beween wage inflaion and wage markups imply ha nominal wages are reopimized much less frequenly, only abou every 10 o 12 quarers. We demonsrae ha up o a log-linear approximaion he SGU and EHL varians of wage sickiness yield idenical equilibrium condiions, provided he wage sickiness parameer is in each case calibraed o be consisen wih empirical esimaes of he wage Phillips curve. This resul implies ha, for a given policy regime, boh varians of wage sickiness give rise o idenical equilibrium dynamics up o firs order. A furher consequence of his equivalence resul is ha he impossibiliy of idenifying wheher wage sickiness sems from he SGU 1

3 or he EHL mechanism is no limied o economeric sudies esimaing he wage Phillips curve in isolaion, bu exends o sudies esimaing he deep srucural parameers of he model from he complee se of log-linearized equilibrium condiions (such as Alig e al., 2005, Levin e al., 2006, and Smes and Wouers, 2004). We characerize Ramsey opimal policy under SGU and EHL wage sickiness. The fac ha he SGU and he EHL models induce he same equilibrium dynamics up o firs order does no imply ha Ramsey opimal policy under boh ypes of wage sickiness is also idenical. This is because he firs-order condiions of he Ramsey problem include no only he complee se of equilibrium condiions, bu also addiional consrains involving he derivaives of he equilibrium condiions wih respec o all endogenous variables. We find, however, ha Ramsey dynamics are numerically he same under he SGU and EHL formulaions. Noably, he Ramsey policy calls for sabilizing price inflaion. The opimal sandard deviaion of inflaion is 0.6 percen a an annualized rae. We characerize opimal, operaional ineres-rae rules for he SGU and EHL formulaions of wage sickiness. We consider ineres-rae rules whereby changes in he ineres rae are se as a linear funcion of lagged price and wage inflaion and lagged oupu growh. We find ha he same operaional ineres-rae rule is opimal in he SGU and EHL models. The opimal rule is a pure inflaion argeing rule, feauring a large coefficien on price inflaion and a mue response o wage inflaion and oupu growh. This resul is in sharp conras wih he findings of Erceg e al. (2000), Canzoneri, Cumby, and Diba (2005), and Levin e al. (2006). These auhors find ha under EHL wage sickiness, he bes rule akes he form of a pure wage-inflaion argeing rule. We show ha his discrepancy in resuls is aribuable o differences in he definiion of wha consiues an operaional ineres-rae feedback rule. Our findings ha Ramsey dynamics and he opimal operaional ineres-rae rule are numerically he same under he SGU and EHL varians of Calvo-ype wage sickiness lead us o conjecure ha he welfare funcion of he Ramsey planner mus be quie similar under boh formulaions. We confirm his conjecure by deriving analyically he welfare crierion of he Ramsey planner under SGU and EHL wage sickiness in he much simpler economic environmen of Erceg e al. (2000). In paricular, we show ha up o second order he uncondiional expecaions of he period uiliy funcion are he same under he SGU and EHL varians as he subjecive discoun facor approaches uniy. The remainder of he paper is organized in six secions. Secion 2 presens he SGU and EHL varians of nominal wage sickiness. Secion 3 esablishes ha available esimaes of linear wage Phillips curves canno discriminae beween he SGU and EHL models. I also derives he mapping from he coefficien of he Phillips curve o he wage sickiness parameers in he wo models. Secion 4 demonsraes ha Ramsey dynamics are he same 2

4 under SGU- and EHL-ype wage sickiness. Secion 5 shows ha he same operaional rule is opimal under boh formulaions of wage sickiness. Secion 6 characerizes analyically second-order accurae welfare funcions. Secion 7 concludes. Malab code o replicae he numerical resuls presened in his paper is available on he auhors websies. 2 Modeling Calvo-Type Wage Sickiness In his secion, we derive he SGU and EHL varians of Calvo-syle wage sickiness. We develop in deail he condiions describing equilibrium in he labor marke, as his is he dimension along which he wo approaches differ. The remaining blocks of he macroeconomic model ino which we embed he SGU and EHL wage sickiness mechanisms are shared by hese wo formulaions and are hose presened in Schmi-Grohé and Uribe (2006b). The Schmi-Grohé and Uribe (2006b) model is a medium-scale economy feauring a number of nominal and real rigidiies. In addiion o nominal wage sickiness, he model allows for sicky prices, money demands by households and firms, monopolisic compeiion in produc and labor markes, habi formaion, invesmen adjusmen coss, and variable capaciy uilizaion. Conrary o a common pracice in he relaed lieraure, we do no allow for subsidies in produc and labor markes o undo he disorions semming from imperfec compeiion in hose markes. In his model economy, business cycles are driven by sochasic variaions in he growh rae of oal facor produciviy, invesmen-specific echnological progress, and governmen spending. 2.1 SGU Wage Sickiness The economy is assumed o be populaed by a large represenaive family wih a coninuum of members. Consumpion and hours worked are idenical across family members. The household s preferences are defined over per capia consumpion, c, and per capia labor effor, h, and are described by he uiliy funcion E 0 =0 β U(c bc 1,h ), (1) where E denoes he mahemaical expecaions operaor condiional on informaion available a ime, β (0, 1) represens a subjecive discoun facor, and U is a period uiliy index assumed o be sricly increasing in is firs argumen, sricly decreasing in is second argumen, and sricly concave. The parameer b>0 inroduces habi persisence. To faciliae comparison wih he EHL model, we will a imes appeal o he following spe- 3

5 cific funcional form of he uiliy index, which is separable in consumpion and leisure, logarihmic in habi-adjused consumpion, and iso elasic in labor: U(c bc 1,h )=ln(c bc 1 ) h1+ξ 1+ξ. (2) Firms hire labor from a coninuum of labor markes of measure 1 indexed by j [0, 1]. In each labor marke j, wages are se by a monopolisically compeiive union, which faces a demand for labor given by ( W j ) η /W h d. Here W j denoes he nominal wage charged [ ] 1 by he union in labor marke j a ime, W W j 1 η 1/(1 η) 0 dj is an index of nominal wages prevailing in he economy, and h d is a measure of aggregae labor demand by firms. A formal derivaion of his labor demand funcion is presened in Schmi-Grohé and Uribe (2006b). In each paricular labor marke, he union akes W and h d as exogenous. The case in which he union akes aggregae labor variables as endogenous can be inerpreed as an environmen wih highly cenralized labor unions. Higher-level labor organizaions play an imporan role in some European and Lain American counries, bu are less prominen in he Unied Saes. Given he wage i charges, he union is assumed o supply enough ( ) labor, h j, o saisfy demand. Tha is, h j w j η = w h d, where w j W j /P and w W /P. The oal number of hours allocaed o he differen labor markes/unions mus saisfy he resource consrain h = 1 0 hj dj. Combining hese wo resricions yields ( ) η 1 h = h d w j dj. (3) w 0 Households are assumed o have access o a complee se of nominal sae-coningen asses. Specifically, each period 0, consumers can purchase any desired sae-coningen nominal paymen X +1 in period + 1 a he dollar cos E r,+1 X +1. The variable r,+1 denoes a sochasic nominal discoun facor beween periods and + 1. The household s period-by-period budge consrain is given by: E r,+1 x +1 + c = x ( ) η 1 + h d w j w j dj. (4) π 0 w The variable x X /P 1 denoes he real payoff in period of nominal sae-coningen asses purchased in period 1. The variable π P /P 1 denoes he gross rae of consumer-price inflaion. We inroduce nominal wage sickiness by assuming ha each period in a fracion α [0, 1) of randomly chosen labor markes he nominal wage canno be reopimized. In hese labor 4

6 markes nominal wages are indexed o pas price inflaion, denoed π 1, and long-run real wage growh, denoed µ z. The household chooses processes for c, h, x +1, and w j so as o maximize he uiliy funcion (1) subjec o (3) and (4), he wage sickiness fricion, and a no-ponzi-game consrain, aking as given he processes w, h d, r,+1, and π, and he iniial condiion x 0. The household s opimal plan mus saisfy consrains (3) and (4). In addiion, leing β λ w / µ and β λ denoe Lagrange mulipliers associaed wih consrains (3) and (4), respecively, he Lagrangian associaed wih he household s opimizaion problem is { L = E 0 β U(c bc 1,h )+λ [h d =0 [ + λ 1 ( ) w w h h d i η di]}. µ w 0 1 w i 0 ( w i w ) η di c r,+1x +1 + x π The firs-order condiions wih respec o h and w i, in ha order, are given by ] U h (c bc 1,h )= λ w µ (5) and { w i = w if w i is se opimally in w 1 i µ z π, 1/π oherwise where w denoes he real wage prevailing in he 1 α labor markes in which he union can se wages opimally in period. Because he labor demand curve faced by he union is idenical across all labor markes where he wage rae is opimized in period, and because he cos of supplying labor is he same for all markes, one can assume ha wage raes, w, are idenical across all labor markes updaing wages in a given period. In any labor marke i in which he union could no reopimize he wage rae in period, he real wage is given by w 1µ i z π 1 /π, as nominal wages are fully indexed o pas price inflaion and long-run produciviy growh. I remains o derive he opimaliy condiion wih respec o he wage rae in hose markes where he wage rae is se opimally. To his end, i is of use o rack he evoluion of real wages in a paricular labor marke ha las reopimized in period. In general, s periods afer he las wage reopimizaion in a paricular labor marke, he real wage ( ) prevailing in ha marke is given by w s µz π +k 1 k=1 π +k. And similarly, s periods afer he las wage ( reopimizaion ( in a )) paricular labor marke, he labor demand in ha marke η is given by µz π +k 1 h d +s. The par of he household s Lagrangian ha is w w +s s k=1 π +k 5

7 relevan for opimal wage seing is given by ( s µz π +k 1 L w = E ( αβ) s k=1 λ +s s=0 w +s π +k ) η h d +s [ w 1 η s ( ) µz π +k 1 k=1 π +k w η w +s µ +s Using equaion (5) o eliminae µ +s, he firs-order condiion wih respec o w is given by E s=0 ( αβ) s λ +s w ( s µz π +k 1 k=1 w +s π +k ) η h d +s [ ( η 1) w η s ( ) µz π +k 1 k=1 π +k ]. ] U h( + s) =0. λ +s (6) This expression saes ha in labor markes in which he wage rae is reopimized in period, he real wage is se so as o equae he union s fuure expeced average marginal revenue o he average marginal cos of supplying labor. The union s marginal revenue s periods afer is las wage reopimizaion is given by η 1 ( w s µz π +k 1 η k=1 π +k ). Here, η/( η 1) represens he markup of wages over marginal cos of labor ha would prevail in he absence of wage sickiness. In urn, he marginal cos of supplying labor is given by he marginal rae of subsiuion beween consumpion and leisure, or U h(+s) λ +s = w +s µ +s. The variable µ is a wedge beween he disuiliy of labor and he average real wage prevailing in he economy. Thus, µ can be inerpreed as he average markup ha unions impose on he labor marke. The weighs used o compue he average difference beween marginal revenue and marginal cos are decreasing in ime and increasing in he amoun of labor supplied o he marke. We wish o wrie he wage-seing equaion in recursive form. To his end, define and f 2 ( ) η 1 f 1 = w E η = w η E s=0 (β α) s λ +s ( w+s w ) η h d +s s k=1 (β α) s w η +sh d +s U h(c +s bc +s 1,h +s ) s=0 One can hen express f 1 and f 2 recursively as ( ) η 1 f 1 = η ( ) η ( w w λ h d w + αβe π+1 µ z π ( π+k µ z π +k 1 s k=1 ) η 1 ( w+1 ( π+k w ) η 1 µ z π +k 1 ) η. ) η 1 f 1 +1, (7) ( ) η ( ) η f 2 w = U h (c bc 1,h ) h d w+1 π +1 + αβe f w µ z w π (8) Wih hese definiions a hand, he firs-order condiion of he household s problem wih 6

8 respec o w collapses o f 1 = f 2. (9) Aggregaion in he Labor Marke ( ) Recall ha he demand funcion for labor of ype j is given by h j W j η = W h d. Taking ino accoun ha a any poin in ime he nominal wage rae in all labor markes in which wages are se opimally in period is idenical and equal o W, i follows ha labor demand in period in every labor marke in which wages were reopimized in period, which we denoe ( ) η by h, is given by h = W W h d. This expression, ogeher wih he facs ha each period a fracion 1 α of unions reopimizes nominal wages and ha h = s h ds, where s h denoes he demand for labor in period in every labor marke where he wage rae was las reopimized in period s, implies ha h = (1 α)h d s=0 ( s W s α s k=1 (µ z π ) η +k s 1). W Le s (1 α) ( W s s ) η s=0 αs k=1 (µ z π +k s 1) W be a measure of he degree of wage dispersion across differen ypes of labor. Then he above expression can be wrien as h = s h d. (10) In urn, he sae variable s evolves over ime according o ( ) η w s =(1 α) + α w ( w 1 w ) η ( ) η π s 1. (11) (µ z π 1 ) We noe ha because all job varieies are ex-ane idenical, any wage dispersion (i.e., s > 1) is inefficien. This is refleced in he fac ha s is bounded below by 1 (see Schmi-Grohé and Uribe, 2006b, for a proof). Inefficien wage dispersion inroduces a wedge ha makes he number of hours supplied o he marke, h, larger han he number of producive unis of labor inpu, h d. In an environmen wihou long-run wage dispersion, up o firs order he dead-weigh loss creaed by wage dispersion is nil even in he shor run. Formally, a firs-order approximaion of he law of moion of s yields a univariae auoregressive process of he form ˆ s = αˆ s 1. This siuaion emerges, for example, when nominal wages are fully indexed o long-run produciviy growh and lagged price inflaion, as is he case in he presen model. When wages are fully flexible, α = 0, wage dispersion disappears, and hus s equals 1. 7

9 [ ] 1 I follows from our definiion of he wage index, W W j 1 η 1/(1 η), 0 dj ha he real wage rae w evolves over ime according o he expression w 1 η =(1 α) w 1 η + αw 1 η 1 ( ) 1 η µz π 1. (12) π Equaions (7)-(12) describe equilibrium in he labor marke in he SGU model. The Wage Phillips Curve in he SGU Model Because mos available esimaes of he degree of wage sickiness are based on linearized versions of he wage Phillips curve, i is of use o compare his relaionship in he SGU and EHL seups. Here we derive a log-linear approximaion o he wage Phillips curve in he SGU model. To faciliae comparison wih he linearized wage Phillips curve ha obains in he EHL model (o be derived in he nex secion), we assume ha he period uiliy funcion akes he specific form given in equaion (2). Le π W W /W 1 denoe wage inflaion in period. Then log-linearizing equaions (7)-(12) around a seady sae yields ˆπ W ˆπ 1 = βe (ˆπ W +1 ˆπ )+γ [ξĥd ŵ ˆλ ], (13) where (1 αβ)(1 α) γ =. (14) α A ha on a variable denoes is log-deviaion from he deerminisic seady sae. Our economy displays long-run sochasic growh in wages and he marginal uiliy of consumpion semming from variaions in he growh rae of neural and invesmen specific facor produciviy (see Schmi-Grohé and Uribe, 2006b). For his reason, he variables w and λ display a sochasic rend in equilibrium. For hese wo variables, a ha denoes deviaions of a saionariy-inducing ransformaion of he respecive variable. The inflaion rae π appears in he wage Phillips curve because of our mainained assumpion ha wages are indexed o pas inflaion. Exising empirical sudies (e.g., Alig e al., 2005 and Levin e al. 2006) esimae he linear wage Phillips curve given above. Such esimaes do no provide direcly a measure of he degree of wage sickiness α. Insead, hey deliver an esimae of he coefficien γ on he wage markup. In he SGU model, γ is relaed o α by equaion (14). In he nex secion, we show ha he EHL model implies a wage Phillips curve idenical o he one given in (13). The only difference beween he linear Phillips curve in he SGU and EHL models lies in he mapping beween γ and α. 8

10 2.2 EHL Wage Sickiness In his secion, we presen he household problem under he EHL wage-sickiness assumpion. The goal is o derive counerpars o equaions (7)-(14). In he EHL model, he economy is assumed o be populaed by a large number of differeniaed households indexed by j wih j [0, 1]. The preferences of household j are defined over per capia consumpion, c j, and labor effor, h j, and are described by he uiliy funcion E 0 =0 β U(c j bc j 1,hj ). (15) To faciliae aggregaion, we assume ha he period uiliy index is separable in habiadjused consumpion and labor effor and isoelasic in labor as in equaion (2). Household j is assumed o be a monopolisic supplier of labor of ype j. The household faces a demand for labor given by ( W j ) η /W h d. The household akes W and h d as exogenous. Given he wage i charges, household j is assumed o supply enough labor, h j,o saisfy demand. Tha is, ( ) η h j W j = h d W. (16) The household s period-by-period budge consrain is given by: E r,+1 x j +1 + cj = xj π + h d wj ( ) η w j. (17) w Each period, a random fracion α of households canno reopimize heir nominal wage. Household j chooses processes for c j, h j, x j +1, and wj, so as o maximize he uiliy funcion (15) subjec o (16) and (17), he wage sickiness fricion, and a no-ponzi-game consrain, aking as given he processes w, h d, r,+1, and π, and he iniial condiion x j 0. The household s opimal plan mus saisfy consrains (16) and (17). In addiion, leing β λ j w / µ j and β λ j denoe Lagrange mulipliers associaed wih consrains (16) and (17), respecively, he Lagrangian associaed wih household j s opimizaion problem is ( ) L j = E 0 β U(cj bc j 1,h j )+λ j w j w j η h d c j r,+1 x j +1 + xj w =0 π ( ) + λj w h j w j η µ j h d w. 9

11 The firs-order condiions wih respec o h j and w j, in ha order, are given by U h (h j )= λj w µ j (18) and { w j = w j w j 1µ z π 1 /π if w j is se opimally in oherwise, where w j denoes he real wage se by household j if i is free o reopimize his variable in period. The household s firs-order condiion wih respec o w j is given by ( s µz 0=E ( αβ) s λ j wj π +k 1 k=1 +s s=0 w +s π +k ) η h d +s [ ( η 1) w j η s ( ) µz π +k 1 k=1 π +k ] U h(h j +s) λ j. +s (19) Following EHL (2000), we assume ha households can insure agains he risk of no being able o reopimize he nominal wage. This assumpion implies ha he marginal uiliy of income, λ j, is indeed he same across all households j [0, 1]. From his resul and he fac ha every household faces he same labor demand funcion, i follows ha all households reopimizing wages in a given period will choose o se he same nominal wage. Therefore, we drop he superscrip j from he variables λ j and w j. Then, he only difference beween he opimaliy condiion (19) and is counerpar in he SGU model, equaion (6), is ha he argumen of he marginal disuiliy of labor is household-specific in he EHL model (and given by h j +s) whereas in he SGU model he argumen of he marginal disuiliy of labor is aggregae per capia labor effor (h +s ). I follows ha in he special case of preferences ha are linear in hours worked (ξ = 0 in equaion (2)), he SGU and EHL models deliver idenical expressions for he opimaliy condiion wih respec o he wage rae. This resul is inuiive: if agens are risk-neural wih respec o hours worked, i is irrelevan wheher households pool employmen uncerainy (as in he SGU model) or no (as in he EHL model). A his poin exising models ha use wage sickiness à la EHL (2000) proceed o loglinearizing he opimaliy condiion for wages given in equaion (19), yielding he well-known linear wage Phillips curve. Because our evenual goal is o compue Ramsey dynamics and o perform welfare evaluaions, we are ineresed in deriving he rue nonlinear wage Phillips curve. To his end, we derive a recursive represenaion of equilibrium condiion (19). We explain his derivaion in some deail, as i is a novel feaure of he presen paper. Le f 1 be defined as in equaion (7). As before, le he number of hours worked in period + s by 10

12 a household who received a wage change signal for he las ime in period be denoed by h +s. Then define f 2 as f 2 = w η E (β α) s w η +sh d +su h ( h +s ) s=0 s k=1 ( π+k µ z π +k 1 ) η. Wih hese definiions a hand, he firs-order condiion wih respec o w, given in equaion (19), can be wrien as f 1 = f 2. To express f 2 recursively proceed as follows. Noe ha h +s is given by h +s = = ( ) W j η +s h d +s ( W +s w s w +s k=1 µ z π +k 1 π +k ) η h d +s. Clearly, he variable h +1+s is relaed o +1 h +1+s by he expression, h +1+s = ( w µ z π w +1 π +1 ) η +1h +1+s. Now we resor o he assumpion made earlier ha U h is homogenous of degree ξ in h (see equaion (2)). Then we have ha ( ) ξ η w µ z π ( ) U h ( h +1+s )= U h +1h w +1 π +1+s. +1 This expression allows us o rewrie f 2 recursively as follows: where h is given by ( ) η ( ) η(1+ξ) f 2 w = U h ( h ) h d w+1 π +1 + αβe f w µ z w π +1, 2 (20) ( ) η W h = h d. (21) W Labor marke equilibrium in he EHL model is given by equaion (7), (9), (12), (20), and (21). 11

13 3 Can Exising Economeric Evidence Disinguish Beween he SGU and he EHL Models? To derive a log-linear approximaion o he wage Phillips curve in he EHL model, we coninue o assume ha he period uiliy funcion akes he specific form given in equaion (2). Log-linearizing equaions (7), (9), (12), (20), and (21) yields equaion (13) wih γ = (1 αβ)(1 α) α 1 1+ ηξ. (22) Exising economeric sudies ha esimae he degree of wage sickiness using aggregae daa are based on linear models. Sudies ha are limied o esimaing he wage Phillips curve deliver an esimae of he coefficien γ in equaion (13). Because boh he EHL and SGU models give rise o a linearized Phillips curve of he form given in equaion (13), esimaes of γ are no sufficien o ell he SGU and EHL models apar. Furhermore, even if one were o esimae he complee se of linearized equilibrium condiions using aggregae daa (as, for insance, in Alig e al. Levin e al., and ohers), he SGU and EHL models would coninue o be observaionally equivalen. This is because he parameer α appears only in he coefficien γ and because one can show ha up o firs order, all equaions of he general equilibrium model are idenical in he SGU and EHL models (excep, of course, for he mapping beween he coefficien γ and α). Given an esimae of γ, one will draw differen conclusions abou he size of α depending on wheher one assumes ha he model displays EHL or SGU-ype wage sickiness. Le α SGU denoe he degree of wage sickiness ha one would infer from given values for γ, ξ, β, and η in he SGU model of wage sickiness and similarly le α EHL be he inferred degree of wage sickiness in he EHL model. I is clear from equaions (14) and (22) ha α SGU and α EHL are linked by he following implici funcion: (1 α SGU β)(1 α SGU ) = (1 αehl β)(1 α EHL ) 1 α SGU α EHL 1+ ηξ. (23) Clearly, for a given value of γ he implied degree of wage sickiness is always higher in he SGU han in he EHL model, or α SGU > α EHL. Figure 1 displays he graph of he implici funcion linking α SGU o α EHL given in equaion (23). In consrucing he graph, we draw from he calibraion used in Alig e al. (2005) and se β =1.03 1/4, ξ = 1, and η = 21. Consider, for example, a value of 0.69 for α EHL, 12

14 Figure 1: The Relaion Beween α SGU and α EHL α SGU α EHL The figure displays he graph of he implici funcion given in equaion (23) for β =1.03 1/4, ξ = 1, and η =

15 which is he degree of wage sickiness esimaed by Alig e al. in heir high-markup case under he assumpion of EHL-ype wage sickiness. The corresponding value for α SGU is Thus, under he Alig e al. esimaion of γ he SGU model implies ha nominal wages are reopimized on average every 13 quarers, whereas he EHL model implies ha hey are reopimized every 3 quarers. 4 Ramsey Policy in he SGU and EHL Models The Ramsey opimizaion problem consiss in maximizing a weighed average of lifeime uiliy of all households in he economy subjec o he se of equaions defining a compeiive equilibrium. Thus far, we have esablished ha for a given moneary regime and up o firs order he SGU and EHL models of wage sickiness resul in idenical equilibrium dynamics provided ha in boh models α is chosen appropriaely. This is because he linearized equilibrium condiions of boh models are he same. Furhermore, as we will show shorly, he objecive funcion of he Ramsey planner is he same in he SGU and EHL models. However, i does no follow direcly from hese resuls ha Ramsey dynamics mus be he same in he SGU and EHL models up o firs order. The reason is ha linear approximaions o he equilibrium condiions of he Ramsey problem involve higher han firs-order derivaives of he compeiive equilibrium condiions. We herefore resor o a numerical analysis o compare Ramsey dynamics under he SGU and EHL models. 4.1 The Ramsey Planner s Objecive Funcion The SGU economy is populaed by a represenaive household. As a consequence, he Ramsey planner s objecive funcion coincides wih ha of he represenaive household. Recalling ha h = s h d (see equaion (10)) and imposing he specific funcional form for he period uiliy funcion given in equaion (2), we can wrie he Ramsey planner s objecive funcion as [ V E β j ln(c +j bc +j 1 ) ( s ] +jh d +j) 1+ξ, (24) 1+ξ j=0 wih ( ) η ( ) η w w 1 µ z π 1 s =(1 α) + α s 1. w w π In he EHL economy, households are heerogeneous. In equilibrium consumpion is idenical across households bu labor supply varies cross secionally. We assume ha he Ramsey planner cares abou all households equally. Therefore, he planner s objecive funcion is 14

16 given by V E j=0 β j [ln(c +j bc +j 1 ) 1 (h i +j )1+ξ 0 1+ξ ] di. Recall ha because households mus saisfy labor demand a he posed wage, we have ha h i =(w/w i ) η h d (see equaion (16)). Le s be defined by s 1+ξ = 1 0 (wi /w ) η(1+ξ) di. I follows ha in he EHL model he Ramsey planner s objecive funcion is given by equaion (24), which is idenical o is counerpar in he SGU model. However, he evoluion of s differs in he wo models. In effec, aking ino accoun ha in he EHL model only a fracion 1 α of households are allowed o reopimize wages in any given period and ha every reopimizing household charges he same wage, we can express he above expression for s recursively as follows: s 1+ξ ( ) η(1+ξ) ( w µz π 1 w 1 =(1 α) + α w π w ) η(1+ξ) s (1+ξ) 1. The Ramsey problem in he SGU model consiss in maximizing equaion (24) subjec o he equilibrium condiions given in secions A.1 and A.2 of he appendix. The Ramsey problem in he EHL model consiss in maximizing equaion (24) subjec o he equilibrium condiions given in secions A.1 and A.3 of he appendix. 4.2 The Opimal Degree of Inflaion Sabilizaion We compue Ramsey dynamics by approximaing he Ramsey equilibrium condiions up o firs order. We calibrae he SGU and EHL models o he U.S. economy following Schmi- Grohé and Uribe (2006b). Table 1 presens he values of he deep srucural parameers implied by our calibraion sraegy. The only srucural parameer ha akes a differen value in he SGU and EHL models is he degree of wage sickiness α. As discussed earlier, here is a one-o-one relaionship beween α SGU and α EHL ha makes boh models consisen wih he empirical esimaes of he wage Phillips curve. Alig e al. esimae α EHL o be The corresponding value for α SGU is We adop hese values in our calibraion of α in he EHL and SGU models, respecively. Table 2 displays he sandard deviaion, firs-order auocorrelaion, and correlaion wih oupu growh of price inflaion, wage inflaion, he nominal ineres rae and he growh raes of oupu, consumpion, and invesmen. The able shows ha he SGU and EHL models imply idenical second momens under he Ramsey policy. This resul suggess ha differences in he characerisics of opimal moneary policy repored in sudies using SGUor EHL-ype wage sickiness mus be aribued no o he way wage sickiness is modeled, 15

17 Table 1: Srucural Parameers Parameer Value Descripion β /4 Subjecive discoun facor (quarerly) θ 0.36 Share of capial in value added ψ Fixed cos parameer δ Depreciaion rae (quarerly) ν Fracion of wage bill subjec o a CIA consrain η 6 Price-elasiciy of demand for a specific good variey η 21 Wage-elasiciy of demand for a specific labor variey α 0.8 Fracion of firms no seing prices opimally each quarer α SGU Fracion of labor markes no seing wages opimally in SGU model α EHL 0.69 Fracion of labor markes no seing wages opimally in EHL model b 0.69 Degree of habi persisence φ Transacion cos parameer φ Transacion cos parameer ξ 1 Preference parameer κ 2.79 Parameer governing invesmen adjusmen coss γ Parameer of capaciy-uilizaion cos funcion γ Parameer of capaciy-uilizaion cos funcion χ 0 Degree of price indexaion µ Υ Quarerly growh rae of invesmen-specific echnological change σ µυ Sd. dev. of he innovaion o he invesmen-specific echnology shock ρ µυ 0.20 Serial correlaion of he log of he invesmen-specific echnology shock µ z Quarerly growh rae of neural echnology shock σ µz Sd. dev. of he innovaion o he neural echnology shock ρ µz 0.89 Serial correlaion of he log of he neural echnology shock ḡ Seady-sae value of governmen consumpion (quarerly) σ ɛ g Sd. dev. of he innovaion o log of gov. consumpion ρ g 0.9 Serial correlaion of he log of governmen spending Noe. All parameer values are as in Schmi-Grohé and Uribe (2006b) excep for hose assigned o ψ and ḡ, which change because he number of hours worked in he seady sae in boh models is differen due o differen specificaions for he period uiliy funcion. 16

18 Table 2: Second Momens under Ramsey Opimal Sabilizaion Policy Variable SGU Model EHL Model Sandard Deviaion Nominal Ineres Rae Price Inflaion Wage Inflaion Oupu Growh Consumpion Growh Invesmen Growh Serial Correlaion Nominal Ineres Rae Price Inflaion Wage Inflaion Oupu Growh Consumpion Growh Invesmen Growh Correlaion wih Oupu Growh Nominal Ineres Rae Price Inflaion Wage Inflaion Oupu Growh 1 1 Consumpion Growh Invesmen Growh Welfare Uncondiional (EV 0 ) Condiional (V 0 ) Sandard deviaions are measured in percenage poins per year. Condiional welfare is compued under he assumpion ha he iniial sae is he deerminisic Ramsey seady sae. In compuing (condiional and uncondiional) welfare levels he welfare crierion was appropriaely ransformed o induce saionariy. 17

19 bu raher eiher o oher differences in he heoreical environmens employed across sudies, or o calibraions of he parameer α ha do no saisfy he one-o-one mapping linking hese wo parameers given by equaion (23). Table 2 shows ha under he Ramsey-opimal policy he volailiy of inflaion is low a 0.6 percenage poins a an annual rae. We ake his number o sugges ha inflaion sabiliy should be a cenral goal of opimal moneary policy. Furhermore, he resuls shown in he able sugges ha in he Ramsey-opimal compeiive equilibrium price inflaion is somewha smooher han wage inflaion. Finally, a remarkable feaure of moneary policy is ha a significan degree of inflaion sabiliy is brough abou wih lile volailiy in he policy insrumen. In effec, he Ramsey-opimal sandard deviaion of he nominal ineres rae is only 0.5 percenage poins a an annualized rae. 5 Opimal Operaional Ineres-Rae Rules Thus far, we have esablished ha he Ramsey-opimal policies in he SGU and EHL models induce he same dynamics a business-cycle frequency. In his secion, we ask wha simple ineres-rae feedback rule comes closes o implemening he Ramsey policy in he SGU and EHL models. We are paricularly ineresed in comparing he values of he policy coefficiens under each of he wo sicky-wage formulaions. Our focus is on simple operaional ineres-rae feedback rules. The condiions we impose for a rule o be simple and operaional are hree: (a) The ineres rae mus be se as a funcion of a small number of easily observable macroeconomic variables; (b) he rule mus resul in a locally deerminae compeiive equilibrium; and (c) he rule mus imply dynamics for he nominal ineres rae ha respec he zero lower bound. We consider ineres-rae feedback rules whereby he nominal ineres rae is se as a linear funcion of lagged price and wage inflaion and lagged oupu growh. Formally, we sudy rules peraining o he following wo families: ln ( R R ) ( π 1 ) ( ) ( ) π W = α π ln + α π W ln 1 y 1 + α π y ln µ z y 2 µ z (25) and ( ) R ( π 1 ) ( ) ( ) π W ln = α π ln + α R 1 π W ln 1 y 1 + α π y ln, (26) µ z y 2 µ z where R denoes he arge value for he gross nominal ineres rae, π denoes he arge value for he gross rae of price inflaion, and µ z denoes he long-run gross growh rae of he economy. We se R and π equal o heir respecive values in he deerminisic 18

20 seady sae of he Ramsey equilibrium, which for our calibraion ake he values 4.4 and -0.4 percen per year. We se he parameer µ z o 1.8 percen per year, which coincides wih he seady-sae growh rae of he economy. We limi aenion o ineres-rae rules ha feaure only lagged values of inflaion and oupu because we believe ha i would be unrealisic o assume ha he cenral bank has knowledge of hese indicaors conemporaneously. To he exen ha his is he case, conemporaneous rules would fail o be operaional. A second significan characerisic of he rules we propose is ha he policymaker is assumed o respond o deviaions of oupu growh from is long-run average. This characerisic represens a deparure from much of he relaed lieraure which focuses on an oupu-gap measure defined as deviaions of acual oupu from he level ha would obain in a flexible-price, flexible-wage economy. We say away from his laer measure of he oupu gap because we regard such concep as less operaional han he one we propose. This is because compuing he flexible-price, flexible-wage level of oupu requires knowledge of he values aken by all sae variables on a period-by-period basis. The oupu measure we propose requires knowledge only of he long-run growh rae of he economy. The ineres-rae rule given in equaion (25) is cas in erms of deviaions of he nominal ineres rae from a consan arge value R. This specificaion, hence, presumes knowledge on he par of he cenral bank of he parameer R. In urn, given a arge value of he inflaion rae (π ), knowledge of R requires informaion abou he long-run level of he real ineres rae. When esimaes of he long-run real ineres rae are imprecise, such informaional requiremen may render he policy rule given in (25) nonoperaional. A way o avoid his problem is o formulae he ineres-rae rule in erms of changes in he nominal ineres rae as in equaion (26). The opimal operaional ineres-rae rule is he operaional ineres-rae rule ha maximizes he uncondiional expecaion of he welfare crierion given in equaion (24). To numerically compue he opimal rule, we discreize he policy parameer space (α π,α W,α y ) wih a grid of poins from 0 o 3 and sep size 0.1 for each of he hree policy coefficiens. Tha is, we evaluae uncondiional welfare for 31 3 or 29,791 policy rules. For each of hese specificaions, we compue a second-order accurae approximaion o he uncondiional expecaion of he welfare crierion, EV 0, using he compuer code developed by Schmi-Grohé and Uribe (2004a). (Malab code o compue he opimal operaional rule is available on he auhors websies.) To ensure compliance wih crierion (b) for operaionaliy of he policy rule, which requires he associaed compeiive equilibrium o be locally unique, we eliminae all poins in he policy parameer space for which he equilibrium is eiher indeerminae or locally 19

21 Table 3: Opimal Operaional Ineres-Rae Rules Coefficien SGU Model EHL Model Level Rule α π 3 3 α W 0 0 α y 0 0 Difference Rule α π 3 3 α W 0 0 α y 0 0 The level rule is given by equaion (25), and he difference rule is given by equaion (26). inexisen. Crierion (c) for operaionaliy of he policy rule demands ha he zero bound on nominal ineres raes be observed. This resricion adds an occasionally binding consrain o he se of equilibrium condiions. Because our numerical soluion algorihm is based on a perurbaion argumen, i is ill suied o accommodae occasionally binding consrains. Consequenly, we approximae he zero-bound consrain by requiring ha wo sandard deviaions of he nominal ineres rae implied by he opimal rule no exceed he deerminisic seady-sae value of he nominal ineres rae. Formally, we impose 2 sd(ln R ) < ln(r ). Table 3 displays he opimal operaional ineres-rae-rule coefficiens in he SGU and EHL models of wage-sickiness. We find ha he opimal rule is idenical for boh models. I akes he form of a pure inflaion argeing rule in he sense ha i responds aggressively o deviaions of price inflaion from arge and is compleely insensiive o variaions in eiher wage inflaion or oupu growh. Furhermore, he finding ha he opimal ineres-rae rule responds only o price inflaion holds regardless of wheher he rule is specified in he level of he ineres rae or in is difference. Our opimal operaional ineres-rae rule differs sharply from he ones obained by Erceg e al. (2000), Canzoneri e al. (2005), and Levin e al. (2006) in relaed papers comparing price and wage inflaion argeing in economic environmens wih Calvo-syle nominal ineria in produc and labor markes. These auhors find ha he bes rule akes he form of a pure wage-inflaion argeing rule. Of hese sudies, he one by Levin e al. uses an economic environmen closes o he one sudied in he presen paper. Levin e al. limi aenion o difference rules ha respond only o conemporaneous values of wage and price inflaion and 20

22 find he opimal coefficiens o be α W =3.2 and α π = 0. We noe ha his rule does no qualify as operaional under our definiion, because i does no saisfy our requiremen ha operaional rules respond o lagged measures of inflaion. Moreover, in he conex of our model he Levin e al. pure wage-inflaion argeing rule would coninue o be nonoperaional even if we relaxed he definiion of operaionaliy o allow for curren-looking rules. For he Levin e al. rule violaes our crierion (c) for operaionaliy, which requires he nominal ineres rae no hi is zero bound oo frequenly. Our baseline calibraion assigns a larger degree of sickiness in produc markes han in labor markes (α =0.8 versus α EHL =0.69). This asymmery, however, is no responsible for our resul concerning he opimaliy of a pure price-argeing rule. To subsaniae his claim, we rese he parameers α and α EHL a he common value of This value is abou he average of he degrees of price and wage ineria assumed in our baseline calibraion and coincides wih he one adoped by Erceg e al. (2000) and Canzoneri e al. (2005). We find ha under his parameerizaion he opimal operaional rule coninues o call for argeing price inflaion exclusively (α π = 3 and α W = α y = 0). The same operaional rule emerges as opimal in he SGU model, which requires seing α SGU a o be in line wih a value of 0.75 for α EHL. Furhermore, we find ha a pure wage-inflaion-argeing rule (α W =3 and α π = α y = 0) fails o be operaional in he SGU and EHL models, as i violaes he requiremen of infrequen violaions of he zero bound on nominal ineres raes. This difficuly wih he Levin e al. rule emerges no only in he conex of backward-looking rules, bu also when he rule is conemporaneous. See Primiceri (2006). 6 Second-Order Accurae Welfare Funcions The facs ha he SGU and EHL wage-sickiness formulaions deliver idenical Ramseyopimal dynamics up o firs order, idenical opimal operaional rules, and idenical equilibrium condiions up o firs order sugges ha he welfare funcion associaed wih boh models mus be similar up o second order. This impression is suppored by he welfare levels repored a he boom of able 2. There, we show ha up o second order he Ramsey policy induces an uncondiional level of welfare of under he SGU specificaion and of under he EHL specificaion. In his secion, we esablish analyically he similariy beween he welfare crieria in he SGU and EHL models. To his end, we consider a much simpler economic environmen han he one sudied hus far. In paricular, we consider he sicky-price sicky-wage model of Erceg e al. (2000). This model feaures no habi formaion, no capial accumulaion, no money, no growh, and no variable capaciy uilizaion. Furhermore, he Erceg e al. model 21

23 assumes he exisence of subsidies o producion and labor supply aimed a neuralizing he disorions arising from he presence of monopolisic compeiion in produc and facor markes. We find ha up o second order, he difference in he uncondiional expecaion of he period uiliy funcion under SGU and EHL wage sickiness, which we denoe by E[U SGU U EHL ], is given by: 1 E[U SGU [ ] 1 β α U EHL SGU 1 β αehl huh (h) η ]= Var(ln(π W )). 1 α SGU 1 α EHL 2γ To obain his expression, we use he fac ha up o second order boh he SGU and EHL models give rise o he same volailiy of wage inflaion, Var(ln(π W )). Also, he parameer γ is he wage-markup coefficien in he linearized wage Phillips curve. Given an esimae of he wage Phillips curve, γ akes he same value under SGU and EHL wage sickiness. I is clear from he above expression ha he uncondiional expecaion of he difference in he period uiliy funcions under he SGU and EHL wage-sickiness specificaions vanishes as he discoun facor approaches uniy. Formally, 2 SGU lim E[U U EHL ]=0. β 1 Clearly, because, as shown earlier, α SGU >α EHL, and recalling ha U h < 0, i follows ha welfare under he SGU formulaion is always slighly smaller han under he EHL specificaion, or EU SGU <EU EHL. This resul appears o hold for he more complex model sudied earlier in he paper, as suggesed by able 2. 7 Conclusion The cenral goal of moneary auhoriies around he developed world is he sabilizaion of price inflaion. This policy arge is explici in some counries, like he Unied Kingdom, and implici in ohers, like he Unied Saes. This pracice is in accordance wih he prescripions semming from mos of he vas lieraure on he heory of moneary policy based on he new Keynesian model. A cener sage in his lieraure is he emphasis on nominal rigidiies aking he form of sicky prices. A number of auhors wihin he new Keynesian lieraure, mos noably Erceg e al. (2000), Canzoneri e al. (2005), and Levin e al. (2006), depar from he price-inflaion 1 The derivaion of his expression is available from he auhors upon reques. 2 Of course, his saemen is rue only insofar as U h hvar(ln(π W )) remains bounded as he discoun facor approaches uniy. 22

24 argeing recommendaion and argue ha cenral banks ough o concenrae heir effors on argeing he rae of wage inflaion. These auhors sress he role of nominal wage sickiness in shaping policy advise. In a number of recen papers (Schmi-Grohé and Uribe, 2004b and 2006a,b), we have argued ha price-inflaion coninues o emerge as he opimal arge variable in he conex of an esimaed medium-scale model feauring boh nominal wage and price ineria in conjuncion wih a number of oher real and nominal fricions (including money demands by households and firms, habi formaion, invesmen adjusmen coss, and variable capaciy uilizaion, among ohers). We argue ha his framework is of paricular empirical ineres because i has been shown o accoun well for observed U.S. poswar dynamics in response o demand and supply shocks. Levin e al. (2006) argue ha our emphasis on price sabiliy is a consequence of he paricular varian of he Calvo-syle wage saggering mechanism ha we used. Namely, he SGU wage sickiness model discussed a lengh in he preceeding secions. In heir view, wage-inflaion smoohing would emerge as he opimal moneary policy prescripion if wage sickiness was assumed o be of he EHL ype. Specifically, Levin e al. (2006, p. 261) sae ha our analysis has followed Erceg, Henderson, and Levin (2000) in assuming ha each individual household provides a disinc labor service, whereas Schmi-Grohé and Uribe (2004[b]) assume ha each household has a coninuum of members providing all ypes of labor services and ha he household s uiliy depends only on is oal hours of work. While such assumpions migh seem o be merely echnical deails, in fac hese differences have fairly dramaic consequences for he firs-order dynamics of wage inflaion, he secondorder effecs of cross-secional wage dispersion, and he design of welfare-maximizing policy rules. The resuls of he presen paper show ha firs-order equilibrium dynamics, he Ramsey-opimal dynamics, and he form of he opimal policy rule are he same under he SGU and he EHL varians of Calvo-ype wage sickiness. Furhermore, we esablish ha in he conex of our medium-scale model price inflaion argeing is opimal under boh formulaions of wage sickiness. The fac remains ha a number of exising sudies using essenially he same mediumscale heoreical framework arrive a fairly differen recommendaions for he design of moneary policy rules. The main conribuion of he presen sudy is o esablish ha hese differences are no aribuable o he alernaive varians of Calvo-ype wage sickiness employed. In addiion, our resuls sugges ha par of he dispariy in policy recommendaions originaes from differences in he definiion of wha consiues an operaional ineres-rae rule. In paricular, here appears o be no consensus in he lieraure on wheher i is operaional o include conemporaneous values of oupu and inflaion as arge variables in 23

25 he feedback rule. Also, here is no uniform reamen of he consrains ha he zero lower bound on nominal ineres raes imposes on moneary policy. Mos exising relaed sudies ignore hese consrains alogeher. Our sance in his paper is o admi as operaional only rules ha respond o pas indicaors of inflaion and aggregae aciviy and ha induce low volailiy of he nominal ineres rae o avoid frequen violaions of he zero bound. A full undersanding of wha lies behind he dispariies in opimal policy across sudies warrans furher invesigaion. 24

26 Appendix This appendix liss he complee se of equilibrium condiions in he SGU and EHL models. Suiable saionariy-inducing ransformaions were applied o variables conaining a rend in equilibrium. A derivaion of he equilibrium condiions common o boh models can be found in Schmi-Grohé and Uribe (2006b). In he equaions below, he parameer χ measures he degree of nominal wage indexaion o lagged price inflaion and long-run produciviy growh. To obain he case of full indexaion analyzed in he main body of he paper, se χ =1. A.1. Equilibrium Condiions Common o he SGU and EHL Model [ K +1 =(1 δ) K + I 1 κ µ I, 2 ( I I 1 µ I, µ I ) 2 ] 1 (C µ 1 z,bc 1 ) bβe 1 (µ z,+1c +1 bc ) =Λ [1 + l(v )+v l (v )] Λ Q = E βµ Λ,+1 µ Υ,+1 Λ +1 [ R k +1 u +1 a(u +1 )+Q +1 (1 δ) ] [ Λ =Λ Q 1 κ ( ) 2 ( µi, I µi, I µ I 2 I 1 I 1 ( µ Λ,+1 I +1 +βe Λ +1 Q +1 µ I,+1 µ Υ,+1 I ) ( µi, I κ v 2 l (v )=1 βe µ Λ,+1 Λ +1 Λ 1 π +1 I 1 µ I ) ] ) 2 ( I +1 κ µ I,+1 µ I I ) ( ) η 1 f 1 = η R k = a (u ) ( ) η ( ( ) η 1 W W Λ h W d + αβe π+1 (µ z π ) χ µ z,+1 W +1 W ) η 1 µ Λ,+1µ z,+1f 1 +1, f 1 = f 2 Λ = βr E µ Λ,+1 Λ +1 π +1 Y = C [1 + l(v )] + G +[I + a(u )µ 1 I, K ] 25