Techncal Appendx o The Equvalence of Wage and Prce Saggerng n Moneary Busness Cycle Models Rochelle M. Edge Dvson of Research and Sascs Federal Reserve Board Sepember 24, 2 Absrac Ths appendx deals he dervaon of a number of resuls repored n The Equvalence of Wage and Prce Saggerng n Moneary Busness Cycle Models, whch appears n he Revew of Economc Dynamcs. JEL classfcaon codes: E24, E3, E32 Mal Sop 77, 2h and C Srees NW, Washngon DC 255. Emal: rochelle.m.edge@frb.gov.
Ths appendx provdes full dervaons of he saggered wage and prce models dscussed n he Revew of Economc Dynamcs paper The Equvalence of Wage and Prce Saggerng n Moneary Busness Cycle Models. The appendx s organzed as follows. Secon A presens he saggered wage model. Secons B and C derve he saggered prce model wh homogeneous and frm-specfc facors, respecvely. The fnal secon compares he saggered prce model wh frm-specfc facors o he correspondng frm-specfc facor model derved by Char, Kehoe, and McGraen 2. In he dervaons ha follow, I refer o a number of equaons from he man ex of he paper. These are denoed wh a p o dsngush hem from he appendx equaons. For example, 5p refers o equaon 5 n he paper descrbng he evoluon of household s holdng of he capal sock, whle 5 refers o appendx equaon 5 he demand for household s dfferenaed labor. A Dervaon of he Saggered Wage Model A. The Frm s Problem The frm, akng as gven he real wage on aggregae labor w and he real renal rae of capal r chooses aggregae labor h and capal k o mnmze s cos of producng oupu y subjec o s producon funcon. Specfcally, he frm solves: mn {h,k } w h + r k subjec o h α k α y where α represens he elascy of oupu wh respec o capal. The Lagrangan s wren as: The frs-order condons are: w = λ α ] L = w h + r k λ [h α k α y. k h α, r = λα h k α, and y = h α k α. The frs wo frs-order condons mply ha w r = α k α h. Togeher wh he hrd frs-order condon, hs gves he facor demand schedules: α α α w α α α w h = Y and k = Y. α α r r
Subsung equaons no w h + r k yelds an expresson for real oal cos c = α y w r α α α. Real margnal cos mc s herefore mc = A.2 The Inermedary s Problem α α w r. 2 α α The nermedary, akng as gven he real wages { w} = se by each of he households for her dfferenaed labor npu, chooses { h } = o mnmze s producon coss subjec o he aggregaor funcon. Specfcally, he nermedary solves: mn {h } = w h d subjec o h σ σ σ σ d h. The Lagrangan s wren as: L = The frs-order condons are: w h d λ w = λ [ h σ σ ] σ σ d h. h σ σ σ d h σ 3 h σ σ σ σ d = h. 4 If he lef- and rgh-hand sdes of equaon 3 are boh rased o he power σ : w σ = λ σ and hen negraed over he un nerval: w we are lef wh λ = yeld σ d = λ σ w = h σ h σ σ d σ d h σ σ h σ σ d, w σ d σ. Ths can be subsued for λ n equaon 3 o σ w d σ h σ σ σ d }{{} 2 h σ h σ.
Furher manpulaon yelds he demand curves for each ype of dfferenaed labor: h = w w σ d σ σ h. 5 To calculae he real cos of aggregae labor, noe ha he real oal cos of producng h call c h s equal o w h d. Subsung n for each h usng equaon 5, we oban ha: c h = h σ σ w d σ w σ d = h σ w σ d. The frm ses he real wage on aggregae labor compevely, ha s, equal o margnal cos mc h, so ha: w = mc h = σ w σ d. 6 Ths expresson for he aggregae prce of labor can be subsued no equaon 5 o yeld a smpler form for he frm s demand for household s labor: h = w σ h. 7 w The assumpon of wo-perod wage saggerng mples ha equaon 6 can be wren as: 2 w σ w = w d + P P 2 σ σ w d = 2 P where w s defned as he nomnal wage ha s se n perod. σ + 2 w P σ σ 8 A.3 The Household s Problem A household who s able o rese s nomnal wage n perod akes as gven he nomnal neres rae, he gross nflaon rae, he real renal rae on capal, he real wage rae on aggregae labor, aggregae labor demand, and N-perod wage sckness, and chooses s consumpon c, real money balances M P, nomnal wage W, and capal sock k o maxmze s uly equaon 3p subjec o s budge consran equaon 4p, he evoluon of he capal sock equaon 5p, and he demand for s dfferenaed labor 3
servce h = h W /P w perods {Nk} k= solves: max { } c, M,W P,k = E = σ. Specfcally, a household who reses s nomnal wage W n ln b v M c + b P v v + η ln W /P w σ h subjec o WNk =... = W Nk+ k, and B +M R B +M +P W /P σ w σ h +P r k P c P k J k + k + δ. 9 A household who s unable able o rese s nomnal wage n perod solves a smlar problem bu akes s prese wage W as gven. The frs-order condons for real money balances, consumpon, and he capal sock for all households are gven by: β R b c v β c b c v M v = + b M P P β b c v P c b c v M v = E R β + + b P + c + P β c = E b c v b c v + b M β+ c + Equaon smplfes o: P v b c v + J k + k b c v M v + + b + P + b M P v b c v + b M P v, b c v + b c v M v + + b + P + + δ k r + J +2 k+ + δ =, and 2 δ k +2 k k+ J +2 k+ +. M P = c b v. 3 b R Equaons and 2 also smplfy and equaon 3 can be subsued for M P o yeld: [ ] R β U c c, R = E U c c +, R + and Π + 4 4
k+ U c c, R J = E [βu c c +, R + k where U c c, R = c b + δ r + J k+2 k + + δ v b v + b v v R v 5 k ] +2 k+2 J + δ k + k +. The frs-order condon for wages for he households able o se her wages n perod assumng ha N s wo perods s: = β w U c c, R /P σ h σ + ση w /P σ w h w P w /P σ w h w/p 6 +β + w E U c c +, R + /P σ h + σ + ση w /P σ Π + w h+ + Π + w + P + w /P σ Π + w h+ w/p. + Equaon 6 smplfes o: w P = σ σ η h w/p w σ + βe h + h σ w Π + w + η w /P σ h Π + w + + ] σ. 7 h U c c, R + βe [h w + Π + Π + w Uc + c +, R + Noe ha n wrng household s frs-order condons above I have dropped he superscrp from c, M, and k ; he mplcaon s ha he values of hese varables are he same across all households. In general hs would no be he case snce households receve dfferen wages and [ ] ] work dfferen hours dependng on wheher hey are members of, 2 or 2, ; as a resul, her accumulaed wealh and hus her c, M, and k profles are lkely o dffer. To allow a sngle c, M, and k profle o characerze all households requres he assumpon ha asse porfolos can be consruced so as o provde he household wh complee nsurance agans any dosyncrac rsk. Consequenly, a household s wealh s ndependen of he perod n whch ses s wage. Snce c = c d, M = M d, and k = k d, hs assumpon allows he superscrps o be dropped from consumpon, real money balances, and he capal sock n equaons 3 o 5. The superscrps reman on h and W snce wages and hours worked wll vary by household dependng on he perod n whch he frm reses s nomnal wage; he varable h, however, does no appear n equaons 6 and 7 snce has been subsued ou wh equaon 7 and he varable W n equaons 6 and 7 appears only for frms reseng wages n perod and has been replaced wh he varable w. 5
A.4 Solvng he Fully Specfed Model Equlbrum n hs economy consss of an allocaon { { h } =,h,k,c, M P,y } = and sequence {Π, w P,w,r,µ,R,mc } =. The equlbrum allocaon and sequence sasfy he followng condons: he frs-order condons from he frm s cos-mnmzaon problem p equaons and 2; he frs-order condons from he nermedary s cos-mnmzaon problem 2p equaons 7 and 8; he frs-order condons from he households uly-maxmzaon problems 6p and 7p equaons 3 o 5 and 7; v he moneary auhory follows 8p; v he goods marke clears y = c + k J k + k + δ ; and v facor markes clear. Ths s gven he nal condons, k, µ, M P, w P, and he sequence of moneary polcy shocks {ε } =. The model s loglnearzed equlbrum condons are gven n able A.. +δβ β ω = +β Table A. µ = ζ µ + ε Eq. 8p ĥ = ŷ αŵ + α r Eq. k = ŷ + α ŵ α r Eq. mc = α ŵ + α r = Eq. 2 [ ] ĥ = ĥ σ w P + σŵ for, 2 Eq. 7 ĥ = ĥ σ w P + σŵ + σ Π ] for 2, Eq. 7 2ŵ = w P + w P Π Eq. 8, 3p M P = ĉ v R Eq. 3 ρ cc E ĉ + ρ cr E R + = ρ cc ĉ + ρ cr R E Π + Eq. 4 R E Π + = E r + + J δ βe k +2 + β k + + k +δβ β ρ hhĥ + βρ hhĥ + ρ ccĉ βρ cc ĉ + ρ R cr βρ R cr + ŷ = c y ĉ + c y δ E k + δ k δ Here, ρ cc =, ρ cr = b v v v b v + b v v v v v, ρ hh = h h, and c y = δα ρ Eq. 5 Eq. 7 Y-Clearng where ρ, he equlbrum real renal rae, s equal o β +δ. In general, I calbrae he model wh he parameer values used by Huang and Lu 999; hese are summarzed and dscussed n secon.7. of my paper. The log-lnearzed frs-order condons gven n able A. can be reduced o he sysem of dfference equaons descrbed by equaon 9p n secon.7.. 6
A.5 Solvng he Smplfed Model Equlbrum n he core model of secon.7.2 of he paper s an allocaon {y } = and a sequence { w P,R } = ha sasfy equaons p o 2p, wh he equlbrum condons noed n pons a o e of secon.7.2 mposed. Specfcally, y w P = σ σ 2 = R βe R y + ηy + βe y w/p σ and 2 R 2 R + 2 w + /P + w 2 /Pσ w+ /P + σ σ σ + βe 2 w /P σ σ w /P w /P [ 2 w + /P + σ w /P σ 2 R + σ σ, 8 ηy + σ y + 2 + w /P σ σ + ], 9 y = y µ exp [ε ]. 2 Ths s gven y and w P and he sequence of moneary polcy shocks {ε } =. Equaons 8 o 2 can be log-lnearzed o yeld equaons 4p o 6p n secon.7.2 of he paper. Of he hree equaons ha characerze equlbrum n he smplfed saggered-wage model only equaon 9 s somewha arduous o log-lnearze. Ths equaon log-lnearzes as follows: w P = + ρ hh ŷ σρ hh w + β P + + β + β + ρ hh E ŷ + + σ + β + β + β 2 w P + σ + ρ hh R E w + E P + 2 E R + + σ w P + σ E + w E P + where ρ hh, he elascy of labor subsuon, s: V h h V h = h h. Ths rearranges o: + σρ hh w P R = + ρ hh ŷ + βe ŷ + + R 2 + βe R + +β + σρ hh E w + E P +. 7
Dvdng hrough on boh sdes by + σρ hh and seng β, he dscoun facor and by mplcaon,, he gross nomnal neres rae equal o uny yelds equaon 5p. The equlbrum pahs of ŷ, w P, and R can be found from he log-lnearzed sysem equaons 4p o 6p. The equlbrum pah of R can be derved mmedaely. By akng equaon 6p forward one perod and hen akng expecaons for perod one fnds ha he lef-hand sde of equaon 4p s equal o zero. Ths means ha E R + = 2 R, whch mples ha R = 2 E R + = lm k 2 k E R +k =. Ths fndng elmnaes R and E R + from he log-lnearzed labor supply schedule equaon 5p, so yeldng: w P = γ ŷ + E ŷ + + E w + E P +. 2 The log-lnearzed expressons for money demand and he marke-clearng condon M P = ĉ = ŷ can be subsued for ŷ n equaon 2 o yeld: P = γ M P + E M + E P + + E w + E P +. 22 w + 2 w ; subsung hs no equaon 22 yelds a second-order The prce level can be elmnaed from equaon 22 by nong ha equaon 3p loglnearzes o P = 2 w dfference equaon n w wh M as he drvng process: E + γ w + 2 w γ + w = 2γ M + E γ M +. The varables E w +, w, and w can be expressed usng lag operaors and he symmerc lag polynomal can be facorzed o oban: + γ L L 2 2 L + w = L a L a L w = 2γ M + E γ γ M + 23 where a = γ + γ. Noe ha snce γ >, a <. Equaon 23 can be re-wren as: and re-arranged o al w or alernavely a al al w = 2γa γ M al + E M + w = a w + 2γa γ s= = 2γ M + E γ M + = 2γa γ s= a s E M +s + E M ++s, a s E M +s + E M ++s. 24 8
The money supply process gven by 2p whch log-lnearzes o M = M + ε mples ha E M +s = M for all values of s >. Equaon 24 hus becomes: w = a w + 2γa γ a s 2 M = a w + s= 2 γ + γ M = a w + a M. 25 Equaon 25 can be subsued no he log-lnearzed verson of equaon 3p o gve: P = 2 w + 2 w = a 2 w + a M + a 2 2 w + a M 26 = a 2 w + 2 2 w + 2 a M + M = a P + 2 a M + M. }{{} P I subsue for P usng P = M ŷ whch yelds: M ŷ = a M ŷ + 2 a M + M. 27 Re-arrangng equaon 27 yelds he equlbrum pah of ŷ gven ŷ and he sequence of shocks {ε } = : ŷ = aŷ + 2 + a M M = aŷ + + γ ε. 28 The equlbrum pah of { w P } = can be found by re-wrng equaon 25 and he loglnearzed versons of equaon 3p P = 2 w + 2 w as: P = a w P a P P + a M P and 29 w P P = w P + w P. 3 Equaon 3 can be subsued for P P n equaon 29 whle ŷ can be subsued for M P. Ths yelds + a w P = a ŷ, whch mples ha he equlbrum pah of w P gven ŷ and he sequence of shocks {ε } = s w P = γŷ = a γ γŷ + + γ ε. Thus he responses of ŷ, w P, and R, o a moneary shock ε gven ha y = can be wren as equaon 25p. 9
B Dervaon of he Saggered Prce Model B. The Frm s Problem The frs par of frm j s problem s o choose labor h j and capal kj o mnmze producon coss, akng as gven he real wage on homogeneous labor, w, he real renal rae of homogeneous capal, r, and he producon funcon. Specfcally, frm j solves: mn {h j,kj } w h j + r k j subjec o h j α k j α y j. Ths problem s very smlar o ha solved by he frm n he saggered wage model dealed n secon A.. The only dfference s ha n secon A. he varable w denoed he real wage on he aggregae labor sock used n he frms producon process, whle now he varable w denoes he real wage on he homogeneous labor sock ha he frm uses. The seps aken o solve he problem are exacly he same as hose followed n secon A., and he soluons ha emerge are smlar n form, ha s: h j α α α = y j w, k j α α α r = y j w α r α, and mc = α α w r. α α 3 The problem for frms who se new prces n perods {Nk} k= s o choose {P j } = so as o maxmze he presen dscouned value of her profs, akng as gven he real margnal cos of producng y j, he aggregae prce level, aggregae demand, he nomnal neres rae, N-perod prce sckness, and he demand curve faces for y j. Specfcally, he frm solves: max {P j } = { E Q, P j P mc j = subjec o y j = y y j } where Q, = and Q, = s= P j θ P and PNk =... = P Nk+ k. R s In choosng he prce P j ha wll reman n effec for he nex N perods where N here s assumed o equal wo he frm solves: max P j P j P mc j P j θ y + E P j P R P +mc j P j θ + y +. 32 P + The frs-order condon s: P j θ = θ y + θ P P j θ P mc j y P P j
+ P j θ P j θ P+ mc E j + θ y + + θ y + R P + P + P j whch can be rewren as: p θ = θ y + θ where p P mc j P + E θ R p P p θ mc j P y θ y + + θ Π + p P Π + θ mc j + y + denoes he rao of prces se hs perod p o he aggregae prce level P and denoes he real margnal cos of producon for frm j n perod. Dvdng hrough by p P θ yelds: θ p P y + θmc j y + R E [ θ Π + θ p ] θ y + + θ mc j + P Π y + =, + [ ], 2 : whch when rearranged yelds he frs-order condon for prces for frm j where j [ p y θ mc + E Π+ R y + mc + = [ ] P θ θ. 33 y + E R y + Π + Π + θ ] Ths s equaon 3p n secon 2.7.2 of he paper. Noe ha snce he real wage and renal rae are he same across all frms, real margnal cos s also he same, so mc can be wren whou he j superscrp. B.2 The Inermedary s Problem The nermedary akes as gven he prces {P j } = se by each frm for s dfferenaed oupu, and chooses {y j } = o mnmze s producon coss subjec o he aggregaor funcon. Specfcally, he nermedary solves: The Lagrangan s wren as: max P j {y} j yj dj subjec o j= L = P j yj dj λ θ y j θ θ θ dj θ y j θ θ θ dj y. y.
The frs-order condons are: P j = λ y j θ θ θ dj y j θ 34 θ y j θ θ θ dj = y. 35 If he lef- and rgh-hand sdes of equaon 34 are boh rased o he power θ : P j θ = λ θ and hen negraed over he un nerval: we are lef wh λ = yeld: P j θ dj = λ θ y j θ θ dj y j θ θ y j θ θ dj y j θ θ dj P j θ θ dj. Ths can be subsued for λ n equaon 34 o P j = P j θ dj θ y j θ θ θ dj }{{} y θ y j θ. Furher manpulaon yelds he demand curves for each ype of dfferenaed oupu: y j = P j P j θ dj θ θ y. 36 To calculae he prce of aggregae oupu P one noes ha he nomnal oal cos n perod of producng y s equal o P j yj dj. Subsung n for each yj usng equaon 36 mples ha: Nomnal Toal Cos = Y θ P j θ dj θ P j θ dj = Y P j θ θ dj. Snce he nermedary produces aggregae oupu compevely s prce, whch s equal o nomnal margnal cos, s gven by: P = P j θ θ dj. 37 2
Ths expresson for he prce level can be subsued no equaon 36 o yeld a smpler expresson for he nermedary s demand for good j: y j P j θ = y. 38 P The assumpon of wo-perod prce saggerng mples ha equaon 37 can be wren as: 2 P = p θ d + 2 θ p θ d = 2 p θ + p θ θ 2 where p s defned as he prce rese n perod. Dvdng hrough by P yelds: = 2 p θ + p P 2 P θ Π θ. 39 B.3 The Household s Problem Household chooses {c, M P, h, k} = o maxmze s uly equaon 3p subjec o s budge consran and he evoluon of he capal sock equaons 29p and 5p, akng as gven he nomnal neres rae, he gross nflaon rae, he real renal rae on capal, and he real wage rae on labor. Specfcally, household solves: max { } c, M,h P,k = E = ln b v M c + b P v v [ ] η ln h subjec o: B + M R B + M + P w h + P r k P c P k J k + k + δ. The frs-order condons for real money balances, consumpon, and capal supply are dencal o hose gven by equaons o 2, and can be rearranged n he same way as hey were n he saggered-wage model n order o yeld equaons 3 o 5. frs-order condon for h s now: = β c b c v β η b c v M v w + b h, P The 3
whch smplfes o: U c c, R w = η h. 4 Snce all households receve he same real wage and renal rae, and hence supply he same amouns of labor and capal, her real wealh and hus {c, M P,h,k} = wll be dencal. As a resul, he households frs-order condons equaons 3 o 5, and 4 can be wren whou he superscrps. B.4 Solvng he Fully Specfed Model Equlbrum s an allocaon {{h j } j=,h,{k j } j=,k,c, M P,{y j } j=,y } = {Π, p P,w,r,µ,R,mc } = and a sequence. The equlbrum allocaon and sequence sasfy he followng condons: he frs-order condons from he frms cos-mnmzaon problem 26p and prof-maxmzaon problem 27p equaons 3 and 33; he frs-order condons from he nermedary s cos-mnmzaon problem 28p equaons 38 and 39; he frs-order condons from he households uly-maxmzaon problems 3p equaons 3 o 5 and 4; v he moneary auhory follows 8p; v he goods marke clears y = c +k J k + k + δ ; and v facor markes clear h = hj dj and k = kj dj. Ths s gven he nal condons, k, µ, M P, p P, and he sequence of moneary polcy shocks {ε } =. The model s log-lnearzed frs-order condons are gven n able B.. The model s calbraed wh he parameer values gven n able of he paper. The log-lnearzed frs-order condons gven n able B. can be reduced o he sysem of dfference equaons descrbed n secon 2.7. of he paper. B.5 Solvng he Smplfed Model Equlbrum n he core model of secon 2.7.2 of he paper s an allocaon {y } = and a sequence { p P,R } = ha sasfy equaons p, 2p, and 3p, wh he equlbrum condons noed n pons a o f of secon 2.7.2 mposed. Specfcally, y 2 = R βe R y + 2 R + 2 p + /P + p /P θ θ, 4 4
+δβ β p θ P = θ Table B. µ = ζ µ + ε Eq. 8p M P = ĉ v R Eq. 3 ρ cc E ĉ + ρ cr E R + = ρ cc ĉ + ρ cr R E Π + Eq. 4 R E Π + = E r + + J δ +δβ β βe k +2 + β k + + k Eq. 5 y 2 [ P θ + ĥ j = ŷj αŵ + α r Eq. 3 k j = ŷj + α ŵ α r Eq. 3 mc = α ŵ + α r = Eq. 3 p P = +β mc + β +β E mc + + β +β E Π + Eq. 33, 3p ŷ j = ŷ θ p [ ] P for j, 2 Eq. 38 ŷ j = ŷ θ p P + θ Π ] for j 2, Eq. 38 Π = p P + p P Eq. 39, 32p ŵ = ρ hhĥ ρ cc ĉ ρ R cr Eq. 4 [ ] ] 2, ĥ = 2ĥk + 2ĥl where k k = k 2 k + k 2 l where k, 2] [, 2 and l ] and l 2, ŷ = cĉ + c δ E k + δ δ k 2 R ηy 2 2 θ θ θ P ] +E y + 2 y + E R y + ηy R 2 + 2 R + [ + P + θ + θ P θ θ θ + 2 P + θ θ θ + 2 P + ] H-Clearng K-Clearng Y-Clearng P +θ +θ θ θ + 2 P + 42, and 2 p /P θ θ p /P y = y µ exp [ε ]. 43 Ths s gven y and p P and he sequence of moneary polcy shocks {ε } =. 5
Equaons 4 o 43 can be log-lnearzed o yeld equaons 33p o 35p n secon 2.7.2 of he paper. Of he hree equaons ha characerze equlbrum n he smplfed saggered-prce model, only equaon 9 s somewha arduous o log-lnearze. Ths equaon log-lnearzes as follows: p P = + R 2 + ρ hh ŷ + R R 2 R + ŷ R R + + R 2 + ρ hh E ŷ + + R 2 E R + + + θ p P +E p + E P + R R + E ŷ + + θ p P + E p + E P + R where ρ hh, he elascy of labor subsuon, s: V h h V h = h h. Ths rearranges o: p P = + ρ hh ŷ + E ŷ + + 2 R + E R + + E p+ E P +. Seng β, he dscoun facor and, by mplcaon,, he gross nomnal neres rae equal o uny yelds equaon 35p wh γ equal o + ρ hh and φ equal o. The equlbrum pahs of ŷ, p P, and R whch are gven by equaon 36p can be found from he log-lnearzed equaons 33p o 35p by followng exacly he same seps oulned n secon.7.2 of he paper and presened n more deal n secon A.6 of he appendx. C Dervaon of he Saggered Prce Model wh Frm-Specfc Facors C. The Frm s Problem As noed n secon 3.2 of he paper he problem for frms n he saggered prce model wh frm-specfc labor npus s very smlar o he problem faced by frms n he saggered prce model wh homogeneous labor; he dfferences are ha h j and kj now have he nerpreaon of beng frm j s demand for s specfc labor and capal npus and ha frms now face real wage and real renal raes w j and rj assocaed wh her specfc facors. The problem ha frms frs solve s herefore: mn {h j,kj } wj hj + rj kj subjec o h j α k j α y j. 6
The soluons ha emerge from he problem are: h j α α = y j α w j α r j, k j α = α α Y j w j α r j, and mc j = w j α α r j α. α 44 Frm j s prce-seng problem s dencal o ha solved n 32 of secon B. and so he soluon s very smlar o ha gven by equaon 33. Noe, however, ha snce he real wage and renal rae dffer across frms, real margnal cos wll also dffer; consequenly mc j s wren wh s j superscrp. The frs-order condon for prces, herefore, for a frm j whch reses s prce n perods {2k} k= s: [ p y θ mc j + E Π+ = P θ y + Ths s equaon 4p n secon 3.7.2 of he paper. Π + θ ] R y + mc j + [ ] θ. 45 E R y + Π + C.2 The Inermedary s Problem The nermedary s problem s dencal o ha solved n secon B.2. C.3 The Household s Problem The household s problem, ha of choosng {c, M P,h,k} =, changes only o reflec he fac ha wh frm-specfc facors real wages w and real rens r as well as hours worked h and capal suppled k vary across households. The household s problem becomes: max { } c, M,h P,k = E = ln b v M c + b P v v [ ] η ln h 46 subjec o: B + M R B + M + P w h + P r k P c P k J k + k + δ, akng as gven he nomnal neres rae, he gross nflaon rae, he real renal rae on s capal, and he real wage rae on s labor. The frs-order condons for real money balances and consumpon are dencal o hose gven by equaon and and can be 7
rearranged n he same way ha hey were n he wo prevous models o yeld equaons 3 and 4. The frs-order condons for k and h are now: β c = E b c v b c v + b M β+ c + whch smplfy o: U c c, R P v b c v + J k + k b c v M v + + b + P + and = β c = E [βu c c +, R + k J + k + δ + δ k r+ J +2 k+ + δ b c v b c v + b M P k r+ J +2 k+ + δ v w β η h, δ k +2 k k+ J +2 k+ + k +2 k k+ J +2 k+ + δ and U c c, R w = η h. 48 I make he same assumpon as n he saggered-wage model ha asse porfolos can be consruced so as o provde he household wh complee nsurance agans any dosyncrac rsk. Consequenly, a household s wealh s ndependen of he wage and renal rae ha faces and he amoun of labor and capal ha supples. ] 47 Ths allows me o wre he households frs-order condons equaons 3, 4, 47, and 48 whou he subscrps on consumpon or real money balances. C.4 Solvng he Fully Specfed Model Equlbrum s an allocaon {{h j } j=,{h } =,{kj } j=,{k } =,c, M P,{y j } j=,y } = and a sequence {Π, p P,{w j } j=,{w } =,{rj } j=,{r } =,µ,r,{mc j } j= } =. The equlbrum allocaon and sequence sasfy he followng condons: he frs-order condons from he frms cos-mnmzaon problem 37p and prof-maxmzaon problem 27p equaons 44 and 45 ; he frs-order condons from he nermedary s cos mnmzaon problem 28p equaons 38 and 39; he frs-order condons from he households uly-maxmzaon problem 39p equaons 3, 4, 47, and 48; v he moneary 8
auhory follows 8p; v he goods marke clears y = c + k J k + + δ d; k and v facor markes clear h j = h, k j = k, w j = w, and r j = r. Ths s gven he nal condons, k, µ, M P, p P, and he sequence of moneary polcy shocks {ε } =. The model s log-lnearzed frs-order condons are gven n able C.. +δβ β Table C. µ = ζ µ + ε Eq. 8p M P = ĉ v R Eq. 3 ρ cc E ĉ + ρ cr E R + = ρ cc ĉ + ρ cr R E Π + Eq. 4 ŷ j = y θ p [ ] P for j, 2 Eq. 38 ŷ j = ŷ θ p P + θ Π ] for j 2, Eq. 38 p P = +β Π = p P + p P Eq. 39, 32p ĥ j = ŷj αŵj + α rj Eq. 44 k j = ŷj + α ŵj α rj Eq. 44 mc j = α ŵj + α rj = Eq. 44 mc j + β +β E mc j + + β +β E Π [ ] + for j, 2 Eq. 45, 4p R E Π + = E r + + J δ +δβ β βe k +2 + β k + + k Eq. 47 ŵ = ρ hhĥ ρ cc ĉ ρ cr R Eq. 48 ĥ j = ĥ and ŵ j = ŵ where = j [, ] k j = k and r j = r where = j [, ] ŷ = cĉ + c δ E k + δ d δ k d H-Clearng K-Clearng Y-Clearng The model s calbraed wh he parameer values gven n able of he paper. The loglnearzed frs-order condons gven n able C. can be reduced o he sysem of dfference equaons descrbed n secon 3.7. of he paper. C.5 Solvng he Smplfed Model Equlbrum n he core model of secon 3.7.2 of he paper s an allocaon {y } = and a sequence of prces { p P,R } = ha sasfy equaons p, 2p, and 4p, wh he equlbrum condons noed n pons a o f of secon 3.7.2 mposed. Specfcally, equlbrum 9
s characerzed by equaons 4 and 43 from secon B.6, as well as p θ = P θ ηy 2 2 R + E y p /P θ ηy R 2 + 2 R + θ y + 2 p + /P θ θ + y + E R y + p /Pθ 2 p+ /P + θ θ θ p /P+θ +θ 2 p+ /P θ θ + Ths s gven y and p P and he sequence of moneary polcy shocks {ε } =. Equaons 4, 43, and 49 can be log-lnearzed o equaons 33p o 35p, ha are hen used n secon 3.7.2 of he paper o fnd he equlbrum pahs of ŷ, p P, and R. Of he hree equaons ha characerze equlbrum only equaon 49 s somewha dffcul o log-lnearze. Ths equaon log-lnearzes as follows: p P = + R 2 + ρ hh ŷ + R R 2 θρ hh R + + + + Ths rearranges o θρ hh + + θρ hh p P p P + ŷ. 49 R 2+ρ hh E ŷ + + 2 E R + ++θ p P +E p + E P + R E p + E P R + + E ŷ + +θ p P +E p + E P + R. = + ρ hh ŷ + E ŷ + + + + θρ hh E p + E P +. 2 R + E R + Dvdng hrough on boh sdes by + θρ hh and seng β, he dscoun facor and by mplcaon,, he gross nomnal neres rae equal o uny yelds equaon 35p wh γ equal o +ρ hh +θρ hh and φ equal o +θρ hh. The equlbrum pahs of ŷ, p P, and R whch are gven by equaon 4p can be found from he log-lnearzed sysem defned by equaons 33p o 35p; he seps nvolved are exacly he same as hose oulned n secon.7.2 of he paper and presened n more deal n secon A.6. 2
D Comparson wh Char, Kehoe, and McGraen s Frm- Specfc Facor Resuls Char, Kehoe, and McGraen 2 ncorporae frm-specfc facors no her model by assumng ha frms produce her dfferenaed oupus usng homogeneous labor and some fxed facor specfc o her producon process. Specfcally, Char e al. assume ha frms face a producon funcon of: y j = h j ψ j ψ where h j s frm j s use of he homogeneous labor npu, j s frm j s fxed, undeprecable, frm-specfc npu, and ψ represens he elascy of oupu wh respec o he frm-specfc npu. Clearly, assumng ψ = reurns he model o ha of secon 2 of he paper. I s assumed ha all frms are endowed wh dencal quanes of her frm-specfc npu and ha each frm s specfc npu s useful only o self. Consequenly, frms face no prce for her frm-specfc npu. The frm s demand for homogeneous labor can be found by smply rearrangng he producon funcon: h j = y j ψ j ψ ψ. 5 Snce he wage bll w h j s he only cos of producon, oal and margnal cos are gven by: c j = w y j mc j = ψ j ψ ψ and w y j ψ ψ j ψ ψ. 5 ψ The frm s prce-seng problem s dencal o ha solved n secon 2.2 secon B.2 and secon 3.2 secon C.2. The frs-order condons for prces for frms who rese her prces n perods {2k} k= assumng wo-perod prce sckness s gven by equaon 3p n secon 2.2 of he paper wh he superscrp js on margnal cos reaned. The loglnearzed verson of equaon 3p s smlar o ha gven n ables B. and C.; snce I wll be obanng analycal soluons, however, I employ he smplfyng approxmaon ha β = and mplcly =. Ths yelds: p P = 2 mcj + 2 E mc j + + 2 E Π + for j [, ] 2 52 2
where mc j and E mc j + are gven by he log-lnear approxmaons of equaon 5: mc j = ŵ + ψ ψ ŷj and E mc j + = E ŵ + + ψ ψ E ŷ j +. 53 The nermedary s problem s dencal o ha oulned n secon 2.3 of he paper; s demand for he jh good s gven by equaon 38 whle he prce ndex for he aggregae good s gven by equaon 32 n secon 2.2 of he paper or equaon 39 n secon B.2. The household s problem s smlar o ha oulned n secon 2.4 of he paper, alhough whou capal. The smplfyng assumpon ha v mples ha he household s money demand curve, Euler equaon, and labor supply schedule are hose gven by M P = y, equaon p, and w = V h c 2 R. The smplfyng assumpon ha ζ = allows he money growh process o be wren as equaon 2p. Equaon 53 can be smplfed by subsung ou for ŷ j and E ŷ j + from he nermedary s demand for he dfferenaed goods and for ŵ and E ŵ + from he household s labor supply curve. The log-lnearzed approxmaon of he nermedary s demand for he dfferenaed goods, gven n ables B. and C., mples ha for frms who rese her prces n perod, ŷ j and E ŷ j + are: ŷ j = ŷ p θ P and E ŷ j + = E p ŷ + θ + θe P Π +. 54 The log-lnearzed approxmaon of he households labor supply curve wh = mples ha ŵ and E ŵ + are gven by: ŵ = ρ hhĥ + ĉ + R and E ŵ + = ρ hh E ĥ+ + E ĉ + + E R +. 55 Combnng equaons 53, 54, and 55 mples ha: mc j = ρ hhĥ + ĉ + R + ψ ŷ θ and ψ P 56 E mc j + = ρ hh E ĥ++e ĉ + +E R + + ψ E ŷ + θ +θe ψ P Π +. 57 Equaons 56 and 57 can be smplfed furher. Frs, I can subsue for E Π + p P Equaons 56 and 57 hghlgh he pon emphaszed n secon 4 of he paper ha ncludng frmspecfc facors n he model creaes a feedback effec beween prce adjusmen and margnal cos. I can be seen from he las erm n each equaon ha an ncrease n he prce se by he frm for s dfferenaed oupu reduces he demand for s oupu and n urn reduces s margnal cos. Wha s also clear from equaons 56 and 57 s ha removng frm-specfc facors from he model by seng ψ equal o zero elmnaes he las erm from each equaon and hus removes hs crucal feedback effec from he model. 22
wh E [ p + P + ] by nong ha he prce ndex for he aggregae good equaon 32p or 39 can be wren as Π = p /P θ 2 p /P θ θ. Ths can hen be log-lnearzed, brough forward one perod, and expressed n erms of perod expecaons o yeld E Π + = E [ p + P + ] + p P. Second, goods-marke clearng n he smplfed model wh no nvesmen s y = c, whch mples ha ĉ = ŷ and E ĉ + = E ŷ +. Thrd, labor-marke clearng saes ha h = hj dj. Subsung n he labor demand curve 5 for hj mples ha labor-marke clearng can be wren as: h = y j ψ j ψ ψ dj. Snce all frms are endowed wh he same amoun of he specfc facor, j s a consan and can be aken ousde he negral so ha: h = j ψ ψ y j ψ dj. Subsung n he ndvdual dfferenaed-good demand curves gves: h = j ψ ψ 2 y = j ψ ψ ψ y 2 ψ p θ + p P 2 Y P p P The log-lnearzed approxmaon o hs expresson s: ĥ = ψ ŷ θ p ψ θ Π θ + p P p P Π ψ θ Π. P + } {{ } = I can hus subsue ĉ = ŷ, E ĉ + = E ŷ +, ĥ = ψ ŷ, and E ĥ+ = ψ E ŷ + no equaons 56 and 57 o yeld: mc j = ρ hh ψ ŷ + ŷ + R + ψ p ŷ θ 58 ψ P E mc j + = ρ hh ψ E ŷ + +E ŷ + +E R + + ψ E p ŷ + +θe +. 59 ψ P + ψ. 23
Equaons 58 and 59, along wh E Π + = E [ p + P + ] + p P, can hen be subsued no equaon 52 so ha: p P = 2 + ρ hh + ψ ψ + + ρ hh+ψ 2 ψ whch when rearranged yelds: ŷ + R θψ ψ p P E ŷ + +E R + + θψ ψ E p + + E p + + p, P + 2 P + P + θψ p = ψ P + ρ hh + ψ ŷ + ψ R + + ρ hh + ψ E ŷ + + E ψ R + + + θψ ψ p E +, P + or alernavely: p =γŷ +E ŷ + + φ R +E P R p + + E + where γ = + ρ hh+ψ ψ P + + θψ and φ = ψ + θψ. ψ 6 Equaon 6 s nearly dencal o equaon 34p n secon 2.7.2 of he paper, wh he only dfference beng he values of he parameers γ and φ. Snce he Euler equaon and he money growh rule are all sll gven by equaons p and 2p, and snce condons a o e of secon 2.7.2 sll hold n hs model, her log-lnearzed approxmaons are unchanged from hose gven by equaons 33p and 35p. As n secons 2.7.2 and 3.7.2 of he paper, herefore, he soluon o hs model s gven by: { ŷ, where γ = p P, R + ρ hh+ψ ψ + θψ ψ } = = { γ + γ + γ ε, γ γ + γ + γ ε, } = 6. As n he prevous models, monoone damped responses requre ha he parameer γ s less han one, whch occurs when θ ρ hh ψ sasfed for reasonable parameer values. 2 >, a condon ha s easly Thus, Char, Kehoe, and McGraen s approach o modelng frm-specfc facors yelds monoone-damped real responses o moneary shocks. 2 For example, for 5 θ 2 and ρ hh = H =, he elascy of oupu wh respec o capal, ψ, H 2 need only exceed.3 for θ ρ hh o be greaer han. ψ 24
References [] Char, V. V., Parck J. Kehoe, and Ellen R. McGraen, Scky Prce Models of he Busness Cycle: Can he Conrac Mulpler Solve he Perssence Problem? Economerca, 2, 68, 5-79. [2] Huang, Kevn. D., and Zheng Lu, Saggered Conracs and Busness Cycle Perssence, Federal Reserve Bank of Mnneapols Dscusson Paper 27, 999. 25