Technical Appendix for DSGE Models for Monetary Policy Analysis

Technical Appendix for DSGE Models for Moneary Policy Analysis by Larence J Chrisiano, Mahias Traband, Karl Walenin A Daa Appendix A Daa Sources FRED2: Daabase of he Federal Reserve Bank of S Louis available a: hp://researchslouisfedorg/fred2/ BLS: Daabase of he Bureau of Labor Saisics available a: hp://blsgov/ FR: Fujia and Ramey (26) daa on job separaions and job findings available a: hp://philadelphiafedorg/research-and-daa/economiss/fujia/ransiion_raesxls SH: Daa on job separaions and job findings available a Rober Shimer s Homepage: hp://robershimergooglepagescom/ NIPA: Daabase of he Naional Income And Produc Accouns available a: hp://beagov/naional/nipaeb/indexasp BGOV: Daabase of he Board of Governors of he Federal Reserve Sysem available a: hp://federalreservegov/econresdaa/defaulhm CONFB: Daabase of he Conference Board available a: hp://conference-boardorg/economics/helpwanedcfm A2 Ra Daa Nominal GDP (GDP ): nominal gross domesic produc, billions of dollars, seasonally adjused a annual raes, NIPA GDP Deflaor (P ) : price index of nominal gross domesic produc, index numbers, 25=, seasonally adjused, NIPA Nominal nondurable consumpion (C nom nondurables ):nominal personal consumpion expendiures: nondurable goods, billions of dollars, seasonally adjused a annual raes, NIPA Nominal durable consumpion (C nom durables ):nominal personal consumpion expendiures: durable goods, billions of dollars, seasonally adjused a annual raes, NIPA Nominal consumpion services (C nom services) :nominal personal consumpion expendiures: services, billions of dollars, seasonally adjused a annual raes, NIPA Nominal invesmen (I nom ):nominal gross privae domesic invesmen, billions of dollars, seasonally adjused a annual raes, NIPA Price index: nominal durable consumpion (PC nom durables ):price index of durable goods, index numbers, 25=, seasonally adjused a annual raes, NIPA Price index: nominal invesmen (PI nom ):price index of nominal gross privae domesic invesmen, index numbers, 25=, seasonally adjused a annual raes, NIPA Employmen (E): civilian employmen, CE6OV, seasonally adjused, monhly, housands, persons 6 years of age and older, FRED2 Federal Funds Rae (FF): effecive federal funds rae, H5 seleced ineres raes, monhly, percen, averages of daily figures, FRED2

Treasury bill rae (TBill): 3-monh reasury bill: secondary marke rae, H5 seleced ineres raes, monhly, percen, averages of business days, discoun basis, FRED2 Populaion (POP): civilian noninsiuional populaion, no seasonally adjused, monhly, housands, FRED2 Capaciy uilizaion (CAP): capaciy uilizaion, G7 - indusrial producion and capaciy uilizaion, UTL: manufacuring (SIC) G7/CAPUTL/CAPUTLB4SQ, seasonally adjused, percenage, BGOV Job separaion rae (S): separaion rae: E o U, seasonally adjused, monhly, 976M- 28M2, FR Spliced ih corresponding daa from Rober Shimer for he sample before 976, quarerly SH Job finding rae (F ): Job finding rae: U o E, seasonally adjused, monhly, 976M- 28M2, FR Spliced ih corresponding daa from Rober Shimer for he sample before 976, quarerly SH Vacancies (V ): index of help aned adverising in nespapers, HELPWANT, The Conference Board, seasonally adjused, monhly, index 987=, Unemploymen rae (U): unemploymen rae labor force saus: unemploymen rae, LNS4, seasonally adjused, percen, 6 years and over, monhly frequency, BLS Nominal age (W ): nominal hourly compensaion, PRS8563, secor: nonfarm business, seasonally adjused, index, 992 =, BLS Average hours (H avg ): average eekly hours, PRS85623, secor: nonfarm business, seasonally adjused, index, 992 =, BLS Paricipaion rae (LabF orce) :civilian paricipaion rae, CIVPART, he employmen siuaion, seasonally adjused, monhly, percen, BLS A3 Daa Transformaions Ra daa are ransformed as follos POP is seasonally adjused using he X2 (muliplicaive) mehod The indices for W and H avg are normalized such ha 25= E, FF,TBill,POP,V,U and LabF orce are convered o quarerly frequencies by averaging monhly observaions For he job finding rae F, e compue he quarerly measure from monhly daa as follos: F q = F m +( F m )F m2 + ( ( F m )F m2 )F m3, here F q denoes he findingraeofquarerandf m,f m2,f m3 are he corresponding monhly finding raes The case for he separaion rae, S, follos accordingly Due o missing daa e use TBill as a proxy for he FF prior 954Q3 All daa are available from 948Q excep for vacancies for hich he firs observaion is 95Q We calculae he folloing ime series hich, among ohers, is used in he VAR: real GDP = GDP P POP hours = Havg E Pop nominal consumpion = Cnondurables nom + Cservices nom nominal invesmen = I nom + Cdurables nom 2

The price of invesmen is calculaed as a Torn price index using PI nom,pcdurables nom,inom and Cdurables nom The resuling price index PIT and quaniy index QIT are used o calculae he relaive price of invesmen as follos: PIT Inom relaive price of invesmen= P QIT B Scaling of Variables in Medium-sized Model We adop he folloing scaling of variables The neural shock o echnology is z and is groh rae is μ z, : z = μ z z, The variable, Ψ, is an invesmen specific echnology shock and i is convenien o define he folloing combinaion of our o echnology shocks: z + Ψ α α z, μ z +, μ α α Ψ, μ z, (B) Capial, K, and invesmen, I, are scaled by z + Ψ Consumpion goods C, governmen consumpion G and he real age, W /P are scaled by z + Also, υ is he muliplier on he nominal household budge consrain in he Lagrangian version of he household problem Tha is, υ is he marginal uiliy of one uni of currency The marginal uiliy of a uni of consumpion is υ P The laer mus be muliplied by z + o induce saionariy Oupu, Y, is scaled by z + Opimal prices, P, chosen by inermediae good firms hich are subjec o Calvo price seing fricions are scaled by he price, P,of he homogeneous oupu good Similarly, opimal ages, W, chosen by monopoly unions hich are subjec o Calvo age seing fricions are scaled by he age, W, of he homogenous labour inpu Thus our scaled variables are: k + = K + z + Ψ, k + = K + z + Ψ,i = I z +,c = C, (B2) Ψ g = G z +,ψ z +, = υ P z +, = W z +, ỹ = Y P z +, p = P, = W P W We define he scaled dae price of ne insalled physical capial for he sar of period + as p k, and e define he scaled real renal rae of capial as r k : p k, = Ψ P k,, r k = Ψ r k here P k, is in unis of he homogeneous good The inflaion rae is defined as: π = P P 3 z +

C Equilibrium Condiions for he Medium-sized Model C Firms We le s denoe he firm s marginal cos, divided by he price of he homogeneous good The sandard formula, expressing his as a funcion of he facor inpus, is as follos: s = ³ r k P α α W R α α P z α When expressed in erms of scaled variables, his reduces o: s = r k α α R α α (C) Producive efficiency dicaes ha s is also equal o he raio of he real cos of labor o he marginal produc of labor: α μψ, R s = ³ (C2) ki, ( α) /H μ z + i, α, The only real decision aken by inermediae good firmsisoopimizepriceheniis seleced o do so under he Calvo fricions The firs order necessary condiions associaed ih price opimizaion are, afer scaling: # πf,+ E "ψ z,y + + βξ π p F f + F f =, (C3) + πf,+ E ψ z,y + s + βξ p K f + K f =, (C4) π + p = ξp ³ πf, λ ξ f p π ξ p πf, + ξ p p π, (C5) ³ πf, λ ξ f p π ξ p ( ) = Kf, (C6) F f π f, π (C7) When e log-linearize abou he seady sae, e obain, βξp ξp ˆπ = βe ˆπ + + bs, ξ p here a ha indicaes log-deviaion from seady sae 4

C2 Households We no derive he equilibrium condiions associaed ih he household We firs consider he household s consumpion saving decision We hen urn o is age decision The Lagrangian represenaion of he household s problem is: E j = " β { h j, ln (C bc ) A L υ W,j h,j + X k K + R B + a,j P C + Ψ I B + P P k, +ω +( δ) K + The firs order condiion ih respec o C is: # I S I K + } I bβ E = υ P, C bc C + bc or, afer expressing his in scaled erms and muliplying by z + : ψ z +, = c b c βbe c μ z + + μ z +,+ bc, (C8) The firs order condiion ih respec o is, afer rearranging: P P k, = ω υ (C9) The firs order condiion ih respec o I is: I ω S S I I + E βω + S I+ I I I I I+ I 2 = P υ Ψ Making use of (C9), muliplying by Ψ z +, rearranging and using he scaled variables, μz,μ S + Ψ, i μz,μ S + Ψ, i μz,μ + Ψ, i (C) ψ z +,p k, i +βψ z +,+p k,+s μz +,+μ Ψ,+ i + i i i+ i i 2 μ z+,+μ Ψ,+ = ψ z+,, Opimaliy of he choice of K + implies he folloing firs order condiion: ω = βe υ + X k + + βe ω + ( δ) =βe υ + X k + + P + P k,+ ( δ) Using (C9) again, υ = E βυ + X k + + P + P k,+ ( δ) P P k, 5 = E βυ + R+, k (C)

here R k + denoes he rae of reurn on capial: Muliply (C) by P z + R k + Xk + + P + P k,+ ( δ) P P k, and express he resuls in scaled erms: ψ z +, = βe ψ z +,+ (C2) π + μ z +,+ Expressing he rae of reurn on capial, (??), in erms of scaled variables: R+ k = π + u + r + k a(u + )+( δ)p k,+ (C3) μ Ψ,+ p k, The firs order condiion associaed ih capial uilizaion is: Ψ r k = a (u ), or, in scaled erms, r k = a (u ) The firs order condiion ih respec o B + is: Muliply by z + P : υ = βυ + R ψ z +,+ R k + (C4) ψ z +, = βe R (C5) μ z,+π + + Finally, he la of moion for he capial sock, in erms of scaled variables is as follos: k + = δ μz,μ k + S + Ψ, i i (C6) μ z,μ + Ψ, C3 Resource Consrain i The resource consrain afer scaling by z + is given by: k y = g + c + i + a (u ) (C7) μ ψ, μ z +, In appendix D e derive a relaionship beeen oal oupu of he homogeneous good, Y, and aggregae facors of producion hich in scaled form looks as follos: α λ y =( p ) f k H α ϕ, (C8) μ Ψ, μ z +, here Finally, GDP is given by: k = k u gdp = g + c + i (C9) (C2) 6

C4 Wage Seing by he Monopoly Union We urn no o he equilibrium condiions associaed ih he household age-seing decision Consider he j h household ha has an opporuniy o reopimize is age a ime We denoe his age rae by W This is no indexed by j because he siuaion of each household ha opimizes is age is he same In choosing W, he household considers he discouned uiliy (neglecing currenly irrelevan erms in he household objecive) of fuure hisories hen i canno reopimize: " # E j (βξ ) i (h j,+i ) A L + υ +i W j,+i h j,+i, here υ is he muliplier on he household s period budge consrain The demand for he j h household s labor services, condiional on i having opimized in period and no again since, is: Ã! λ W π,+i π,+ h j,+i = H +i W +i Here, i is undersood ha π,+i π,+ hen i = Subsiuing his ino he objecive funcion and opimizing (see appendix F for deails) yields he folloing equilibrium equaions associaed ih age seing: ẘ = ( ξ ) π,+ = W + W = +z + +P + z + P = +μ z+,+π +, (C2) ³ ξ π, π, ξ h = ẘ λ H, (C22) λ λ π, + ξ ẘ π, λ (C23) In addiion o (C23), e have folloing equilibrium condiions associaed ih sicky ages 2 : 2 Log linearizing hese equaions abou he nonsochasic seady sae e obain, η b + η E b + η 2 b + + η 3ˆπ + η 4ˆπ + + η 5ˆπ =, +η 6 ˆψz+, + η 7 Ĥ + η 8ˆμ z+, + η 9ˆμ z+,+ here b = [λ σ L ( λ )] [( βξ )( ξ )],η = b ξ,η = σ L λ b +βξ 2,η2 = b βξ, η 3 = b ξ ( + βκ ),η 4 = b βξ,η 5 = b ξ κ,η 6 =( λ ), η 7 = ( λ )σ L,η 8 = b ξ,η 9 = b βξ 7

F, = ψ z +, λ K, = ẘ λ h + βξ E + ẘ λ h A L π,+ π,+ + βξ E π,+ ³ ξ π, π, ξ π,+ λ () + λ F,+ λ () K,+ F, = K, (C24) (C25) (C26) π,+ = π κ π ( κ) μ z + (C27) C5 Equilibrium Equaions The equilibrium condiions of he model correspond o he folloing 28 equaions, (C), (C2), (C3), (C4), (C5), (C6), (C7), (C6), (C8), (C), (C4), (C5), (C22), (422), (C7), (C8), (424), (C9), (425), (C3), (C2), (C24), (C25), (C26), (C23), (C27), (C2), (C2), hich can be used o solve for he folloing 28 unknons: r k,,r,s,π,p k,,k +, k +,u,h,h,i,c,ψ z +,,y, K f,f f, π f,, p,k,,f,, π,r k,s,a(u ), ẘ,π,,gdp D Resource Consrain in he Medium-sized Model We begin by deriving a relaionship beeen oal oupu of he homogeneous good, Y, and aggregae facors of producion We firs consider he producion of he homogenous oupu good: Y sum = = = Z Z Z = z α Y i, di (z H i, ) α Ki, α z + ϕ di α Ki, H i z + ϕ di z α K H H i α Z H i di z + ϕ, here K is he economy-ide average sock of capial services and H is he economy-ide average of homogeneous labor The las expression explois he fac ha all inermediae good firms confron he same facor prices, and so hey adop he same capial services 8

o homogeneous labor raio This follos from cos minimizaion, and holds for all firms, regardless heher or no hey have an opporuniy o reopimize Then, The demand for Y j, is so ha say, here Dividing by P, or, Y Z Y i, di = p = ξp The preceding discussion implies: Y sum Z = z α K α H α z + ϕ P P i, P Y P i, " Z P = P p = Z Pi P = Y i, Y, di = Y P i, ³ π λ ξ f p π ξ p di # di ³ P λf, (D), π + ξ p π p λ Y =( p ) f λ Y =( p ) f z α K α H α z + ϕ, (D2) or, afer scaling by z +, y =( p ) α k H α ϕ, μ Ψ, μ z +, here k = k u Finally, e adjus hours orked in he resource consrain so ha i corresponds o he oal number of people orking, as in (F6): " α # λ y =( p ) f k ẘ λ h α ϕ μ Ψ, μ z +, 9

I is convenien o also have an expression ha exhibis he uses of he homogeneous oupu, z + y = G + C + Ĩ, or, afer scaling by z + : k y = g + c + i + a (u ) μ ψ, μ z +, E Opimal Price Seing in he Medium-sized Model The profi funcion of he i h inermediae good firm ih he subsiued demand funcion is given by, λ X f β j Pi,+j Pi,+j υ +j P +j Y +j { s+j }, or, here E E j= X j= P +j P +j β j υ +j P +j Y +j {(X,j p ) s +j (X,j p ) P i,+j P +j = X,j p,x,j ( π+j π + π +j π +,j>, j= }, The i h firm maximizes profis by choice of p Thefachahisvariabledoesnohave an index, i, reflecs ha all firms ha have he opporuniy o reopimize in period solve hesameproblem,andhencehavehesamesoluion Differeniaing is profi funcion, muliplying he resul by p + E X j=, rearranging, and scaling e obain: βξp j A +j [ p X,j s +j ]=, here A +j is exogenous from he poin of vie of he firm: A +j = ψ z +,+jy +j X,j Afer rearranging he opimizing inermediae good firm s firs order condiion for prices, e obain, p = E P j j= βξp A +j s +j P E j = Kf, j= βξp A +j X,j F f say, here X K f j E βξp A +j s +j F f = E j= j βξp A +j X,j j=

These objecs have he folloing convenien recursive represenaions: # π,+ E "ψ z,y + + βξ π p F f + F f = + π,+ E ψ z,y + s + βξ p K f + K f = Turning o he aggregae price index: π + P = = Z P i di ( λf ) λ ξp P f + ξ p ( π P ) ( λf ) (E) Afer dividing by P and rearranging: ³ ξ π, p π =( p ) (E2) ξ p This complees he derivaions of opimal decisions ih respec o firms price seing F Opimal Wage Seing in he Medium-sized Model The objecive funcion ih he subsiued labor demand funcion looks as follos: ³ λ W π,+i π,+ E j (βξ ) i W +i H +i [ A L Ã! λ W π,+i π,+ +υ +i W π,+i π,+ H +i ] W +i Recalling he scaling of variables, (B2), e have here W π,+i π,+ = W π,+i π,+ W +i +i z +i + P +i ³ W W /W = +i z + P X,i = W +i z + X,i P ³ W /W = +i X,i = π,+i π,+,i> π +i π +i π + μ z +,+i μ z +,+ =, X,i = +i X,i, (F) (F2)

I is ineresing o invesigae he value of X,i in seady sae, as i Thus, In seady sae, X,i = (π π +i ) κ (π i ) ( κ) μ i z + π +i π +i π + μ z +,+i μ z +,+ X,i = (πi ) κ (π i ) ( κ) μ i z + = π i μ i z + Simplifying using he scaling noaion, ³ λ E j (βξ ) i +i X,i H +i [ A L or, or, E j +υ +i W +i +i X,i λ X,i H+i ], +i ³ λ E j (βξ ) i +i X,i H +i [ A L λ +ψ z,+i + X,i X,i H+i ], +i (βξ ) i [ A L +ψ z,+i + λ + ³ λ +i X,i H +i λ X,i X,i H+i ] +i λ () Differeniaing ih respec o, ³ λ E j (βξ ) i +i X,i H +i [ A L λ ( + φ) λ () +ψ z,+i λ λ + X,i X,i H+i ]= +i Dividing and rearranging, Ã! λ E j (βξ ) i [ A L X,i H+i +i + ψ z +,+i λ λ () X,i X,i H+i ]= +i (F3) 2

Solving for he age rae: () = E j E j P ³ λ (βξ ) i A L +i X,i H +i P ³ λ (βξ ) i ψ z +,+i λ X,i +i X,i H +i here K, = E j F, = E j = A LK, F, Ã! λ (βξ ) i X,i H+i +i (βξ ) i ψ λ z +,+i X,i X,i H+i λ +i Thus, he age se by reopimizing households is: AL K, = F, We no express K, and F, in recursive form: or, K, = E j () Ã (βξ ) i X,i +i + βξ π κ π ( κ) μ z + + π + μ z +,+ = H Ã +(βξ ) 2 + K, = H λ H+i! λ +2 (π π + ) κ (π 2 ) ( κ) μ 2 z + π +2 π + μ z +,+2μ z +,+ π κ + E βξ π ( κ) μ z + + π + μ z +,+ +βξ + π κ + π ( κ) μ z + +2 π +2 μ z +,+2 λ H+! λ H +2 λ () {H + H+2 + } = H = H π κ + βξ E π ( κ) μ z + + π + μ z +,+ + βξ E π,+ π,+ λ () K,+, λ () K,+ (F4) 3

using, Also, or, F, = E j π,+ = W + W = +z + +P + z + P = +μ z+,+π + = ψ z +, H λ +βξ ψ z +,+ λ (βξ ) i ψ λ z +,+i X,i X,i H+i λ +i + +(βξ ) 2 ψ z +,+2 + Ã λ λ π κ +2 λ (π π + ) κ (π 2 ) ( κ) μ 2 z + π +2 π + μ z +,+2μ z +,+ π κ μ z + π + μ z +,+! + λ H +2 + λ H+ F, = ψ z +, H λ +βξ + λ π κ λ + π κ +βξ +2 +} = ψ z +, H + βξ λ π κ μ z + π + μ z +,+ + π κ μ z + π +2 μ z +,+2 + π,+ π,+ + λ { ψ z +,+ + λ + λ F,+, λ H + ψz +,+2 λ H +2 so ha F, = ψ + λ z +, + π,+ H + βξ λ E F,+ (F5) π,+ We obain a second resricion on using he relaion beeen he aggregae age rae and he age raes of individual households: ³ W = ( ξ ) W + ξ ( π, W ) Dividing boh sides by W and rearranging, = ³ ξ π, π, ξ λ 4

Subsiuing, ou for from he household s firs order condiion for age opimizaion: A L ³ ξ π, π, ξ λ () F, = K, We no derive he relaionship beeen aggregae homogeneous hours orked, H, and aggregae household hours, h Z h j, dj Subsiuing he demand for h j, ino he laer expression, e obain, h = Z Wj, H W = (W ) λ = ẘ λ H, λ H dj Z (W j, ) λ dj (F6) here Also, W = ẘ W W, W = Z ³ λ ( ξ ) W (W j, ) λ dj λ ³ λ + ξ π, W λ λ This complees he derivaions of he opimal age seing G The Wage-Phillips Curve in he Medium-Sized Model The household faces he folloing demand for he j h ype of labor: By (G), Here,ehaveused, h +i = h,j = W W,j λ H Ã! λ λ W +i H +i = X,i H+i W +i +i W W, W z + P, W +i = W π,+ π,+i, (G) (G2) (G3) 5

and Reriing (??), or, using, e have Bu, E X E ( X,i = (βξ ) i υ +i P +i z + +i X (βξ ) i ψ z +,+i π,+i π,+ π +i π +i π + μ z +,+i μ z +,+ i> i = " W +ih +i h # P +i z +i + A +i L ( + φ) υ +i P +i z +i + ψ z +, = υ P z +, " W +i h +i h # P +i z +i + A +i L ( + φ) ψ z +,+i Then, E X W +ih +i P +i z + +i ³ W W /W h +i = P z + X,i = h +ix,i λ = X,i H+i X,i +i ³ λ λ (βξ ) i ψ z +,+i +i X,i X,i H+i X,i A L +i ( + φ) ψ z +,+i Differeniaing ih respec o, or () H +i X λ E (βξ ) i λ ψ z,+i[ + + X,i H+i X,i λ +i ³ λ λ +i X,i () H +i ( + φ) A L ] λ ( + φ) ψ z +,+i E X (βξ ) i ψ z +,+ih +i X,i λ MRS +i, (G4) Here, MRS+i in (G4) denoes he (scaled) cos of orking for he marginal orker in period + i hoseageasreopimizedinperiod and no again reopimized in periods +,+2,, + i : h φ MRS+i A +i L 6 ψ z +,+i

According o (G4), he union seeks o se he age o a markup, λ, overhecosoforking of he marginal orker, on average We no expand (G4) abou a seady sae in hich =,χ,i =,X,i =, for all i, = λ MRS, π = π, π, = πμ z +, π,+ = πμ z + I is convenien o obain some preliminary resuls Noe, ½ ( κ ˆπ ˆX,i = + + + κ ˆπ +i ) ˆμ z +,+i + +ˆμ z +,+ i> i = ½ ( κ ˆπ ˆχ,i =,+ + + κ ˆπ,+i ) i> i = \MRS +i = ˆψ z +,+i + φĥ +i + φ ³ĥ +i Ĥ +i here Also, from (G2), ½ ĥ +i Ĥ +i = We have = λ κ ˆπ + ˆπ + κ ˆπ, κ ˆπ,+i ˆπ,+ κ ˆπ λ λ (ŵ κ ˆπ,+ κ ˆπ,+i ) i> λ λ ŵ i = φ ŵ +(βξ λ ) φ λ (ŵ κ ˆπ,+ ) λ +(βξ ) 2 φ λ (ŵ κ ˆπ,+ κ ˆπ,+2 ) λ +(βξ ) 3 φ λ (ŵ κ ˆπ,+ κ ˆπ,+2 κ ˆπ,+3 ) λ + φ λ ŵ φ λ (βξ βξ λ λ βξ ) κ ˆπ,+ φ λ (βξ λ βξ ) 2 κ ˆπ,+2 φ λ (βξ λ βξ ) 3 κ ˆπ,+3 Using he las expression, e can rie he discouned sum of he marginal cos of orking as follos: S MRS, (βξ ) i \MRS +i = S o, + φ λ [ŵ S, ] (G5) λ βξ Here, S o, S, h i (βξ ) i ˆψ z +,+i + φĥ +i = ˆψ z +, + φĥ + βξ S o,+ (G6) (βξ ) i κ ˆπ,+i = βξ κ ˆπ,+ + βξ S,+ i= 7 (G7)

The folloing expression is also useful: S X, = (βξ ) i ˆX,i = βξ i= (βξ ) i ˆμ z +,+i κ ˆπ +i i= (G8) βξ = ˆμz βξ +,+ κ ˆπ + + βξ S χ,+ Because he objec in square brackes in (G4) is zero in seady sae, he expansion of (G4)doesnorequireexpandingheexpressionousidehesquarebracke Takinghisand = λ MRS ino accoun, he expansion of (G4) is: = b +ŵ + SX, S MRS, (G9) βξ We no deduce he resricion across ages implied by he aggregae age index Using heageupdaingequaionandhefachanon-opimizingunionsareselecedarandom, heaggregaeageindexreduceso: ³ W = ( ξ ) W + ξ ( π, W ) Divide by W and use (G3): =( ξ )( ) π, + ξ π, Log-linearize his expression abou seady sae, o obain: ŵ = ξ κ ˆπ, ξ Replace S MRS, in (G9) using (G5) and hen subsiue ou for ŵ using he previous expression: ξ b + κ ˆπ, + S X, (G) βξ βξ ξ = S o, + φ λ ξ κ ˆπ, φ λ S, λ βξ ξ λ βξ Muliply (G) evaluaed a +by βξ and subrac he resul from (G) evaluaed a o obain: ξ b βξ βξ b + + ( κ ˆπ, βξ βξ ξ κ ˆπ,+ ) +(S X, βξ S X,+ ) = (S o, βξ S o,+ )+φ λ ξ ( κ ˆπ, βξ λ βξ ξ κ ˆπ,+ ) φ λ (S, βξ λ βξ S,+ ) 8

Simplify his expression using (G6), (G7) and (G8): ξ b βξ βξ b + + ( κ ˆπ, βξ βξ ξ κ ˆπ,+ ) (G) βξ ˆμz βξ +,+ + κ ˆπ + = ˆψ z +, + φĥ + φ λ ξ ( κ ˆπ, βξ λ βξ ξ κ ˆπ,+ ) φ λ βξ λ βξ κ ˆπ,+ The relaionship beeen age and price inflaion, he change in he real age and echnology groh is given by: b = b +ˆπ, ˆπ ˆμ z, (G2) + Then, ˆπ = b b +ˆπ, ˆμ z +, κ ˆπ ˆπ κ ˆπ = b b +ˆπ, ˆμ z +, κ ˆπ = b b + κ ˆπ, ˆμ z +, Use his o subsiue ou for κ ˆπ + in (G): ξ b βξ βξ b + + ( κ ˆπ, βξ βξ ξ κ ˆπ,+ ) βξ b b + + κ ˆπ,+ βξ = ˆψ z +, + φĥ + φ λ ξ ( κ ˆπ, βξ λ βξ ξ κ ˆπ,+ ) φ λ βξ λ βξ κ ˆπ,+ Collecing erms in b, b +, ˆπ,, ˆπ,+ : βξ βξ b βξ b + βξ βξ βξ βξ ξ + φ λ ξ κ ˆπ, βξ ξ λ βξ ξ βξ [ ξ + βξ φ λ βξ ξ βξ ξ βξ λ βξ ξ φ λ βξ λ βξ ] κ ˆπ,+ = ˆψ z +, + φĥ 9

or, ξ φ λ βξ ξ λ κ ˆπ, = ˆψ z +, + φĥ b + βξ βξ ξ φ λ λ κ ˆπ,+ Le κ = ( ξ )( βξ ) ξ Muliply he previous expression by κ and rearrange: φ λ κ ˆπ, = κ ³ ˆψ + φĥ λ z+, b + φ λ β κ ˆπ,+ λ 2