The Non-Stochastic Steady State

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1 July 25, 24 Techical Appedix: The Dyamic System I this appedix I preset the full derivatio of the approximated dyamic system, its properties, ad the procedure for computig secod momets. I rely heavily o Burside (1997) ad follow his otatio. 1 A The No-Stochastic Steady State I the steady state two relatios determie the vacacy rate ad the uemploymet rate: (i) All labor force variables grow at the same gross rate G L. Start from the uemploymet dyamics relatio: 4U = δn + 4L M Dividig throughout by N ad usig the steady state relatioship: This becomes: 4L L = 4U U = GL 1 Which ca also be writte as: (G L 1) U N = δ +(GL 1)(1 + U N ) M N (G L 1) + δ = m = Qv Isertig the matchig fuctio: ³ u σ ³ v 1 σ (G L 1) + δ = (1) (ii) Vacacy creatio is give by the combiatio of the relevat F.O.C.: φv +(1 φ)qv Θ (φ +(1 φ)q)( N ) γ G X β = Q [1 (1 δ)g X β] π (2) 1 I would like to express my gratitude to Craig Burside for his kid advice ad for the provisio of his programs ad extesive otes. 1

2 Profits are give by: ½ ¾ φv +(1 φ)qv 1 α π =[1 η] (1 α) Θ( ) γ+1 N 1 ηpλ (3) Notig that: This becomes: π = Thewageshareisgiveby: s = η (1 α)+θ 1 λ = G X β [1 (1 δ)g X β] π φv +(1 φ)qv [1 η] (1 α) Θ( N 1+ηP G X β [1 (1 δ)g X β] ) γ+1 h 1 α 1 io (1 α) φv +(1 φ)qv ( ) γ+1 + P N G X β [1 (1 δ)g X β] π (4) (5) B The Log-Liear Approximatio For each variable Y deote b Y t = Y t Y Y l Y t l Y where Y is the steady state value. B.1 The Itratemporal Coditio This coditio is give by: Defie: Θ (φ +(1 φ)q t.t+1 )( φv t +(1 φ)q t,t+1 V t N t ) γ = Q t,t+1 λ t (6) eq t,t+1 = φ +(1 φ)q t,t+1 The: Θ eq t,t+1 ( e Q t,t+1 V t N t ) γ = Q t,t+1 λ t (7) 2

3 Approximatig the itratemporal coditio, we get: b e Q t,t+1 + γ( b e Qt,t+1 + bv t b t )= b Q t,t+1 + b λ t (8) Usig the defiitio of Q ad u =1 : Approximatig: Defie: The: σ vt Q t,t+1 = (9) 1 t bq t,t+1 = σ bv t + 1 b t b e Q t,t+1 = Ω (1 φ)q Q φ +(1 φ) Q b t,t+1 (1 φ)q φ +(1 φ) Q (1) b e Q t,t+1 = Ω bq t,t+1 Isertig ito the approximated coditio: ΩQ b t,t+1 + γ(ωq b t,t+1 + bv t b t ) = Q b t,t+1 + λ b t bq t,t+1 (Ω + γω 1) + γbv t = γb t + λ b t σ bv t + 1 b t (Ω + γω 1) + γbv t = γb t + λ b t [ σ(ω + γω 1) + γ] bv t = γ + σ (Ω + γω 1) b t + λ 1 b t (11) Whe φ =the Ω =1ad the expressio becomes: (1 σ)γbv t =(γ + σγ 1 )b t + λ b t Whe φ =1the Ω =ad the expressio becomes: 3

4 B.2 The Flow Equatio [σ + γ] bv t = γ σ 1 Approximatig the dyamic equatio for employmet: b t + b λ t t+1 G L t+1 =(1 δ t,t+1 ) t + Q t,t+1 v t (12) We get (assumig G bl t+1 =): Usig (1): We get: G L b t+1 =(1 δ)b t δ b δ t,t+1 + Qv( bq t,t+1 + bv t ) (13) G L b t+1 =(1 δ)b t δ b δ t,t+1 + Qv G L b t+1 = Qv(1 σ)bv t δ b δ t,t+1 B.3 The Itertemporal Coditio This coditio is give by: σ bv t + 1 b t + bv t (14) (1 δ) Qvσ b t (15) 1 λ t G X t+1 " ³ ³ (1 α)+θ = E t β t+1 1 (1 α) γ+1 ( φv t+1+(1 φ)q t+1,t+2 v t+1 t+1 ) (1 η) ηp t+1,t+2 λ t+1 (16) Deote: +E t (1 δ t+1,t+2 )β t+1 λ t+1 π t+1 = "Ã ³ +Θ 1 (1 α) (1 α) ( v t+1 e Q t+1,t+2 t+1 ) γ+1! (1 η) ηp t+1,t+2 λ t+1 G X t+1 F t+1/n t+1 F t /N t 4

5 So: λ t G X t+1 Approximatig this coditio we get: = E t β t+1 π t+1 + E t (1 δ t+1,t+2 )β t+1 λ t+1 λ G (b λ X t bg X t+1) =E t βπ( β b t+1 + bπ t+1 )+(1 δ)βλ( β b t+1 + λ b t+1 ) δβλ b o δ t+1,t+2 Approximatig π t+1 : (17) πbπ t+1 = ηpλ( P b t+1,t+2 + λ b t+1 ) (18) (1 α) +Θ 1 (1 η)( v eq )γ+1 (1 + γ)(bv t+1 + ΩQ b t+1,t+2 b t+1 ) The first term o the RHS may be derived from: Usig (1) oe period forward: Isertig ito (18): πbπ t+1 = ηpλ +Θ 1 P t+1,t+2 = Q t+1,t+2v t+1 U t+1 = Q t+1,t+2v t+1 1 t+1 bp t+1,t+2 = b Q t+1,t+2 + bv t+1 + bp t+1,t+2 = 1 b t+1 (19) 1 (1 σ)b t+1 +(1 σ)bv t+1 (2) 1 (1 σ)b t+1 +(1 σ)bv t+1 + λ b t+1 (1 α) (1 η)( v eq )γ+1 (1 + γ)(bv t+1 + Ω Thus isertig ito (17): 5 σ bv t b t+1 b t+1 )

6 λ G (b λ X t bg X t+1) = E t [βπ +(1 δ)βλ] β b t+1 (21) +(1 δ)βλλ b t+1 δβλ b δ t+1,t+2 h ηpλ (1 σ)b 1 t+1 +(1 σ)bv t+1 + b i λ t+1 ³ +β +Θ 1 (1 α) (1 η)( v Q e )γ+1 (1 + γ) bvt+1 + Ω σ bv t+1 + bt+1 1 b t+1 Re-arragig: = " βηpλ 1 (1 σ)+βθ 1 (1 α) (1 η)( v eq )γ+1 (1 + γ) 1+Ωσ E t b t+1 1 +[(1 δ)βλ βηpλ] λ b t+1 λ λ G b X t (22) " (1 α) βηpλ(1 σ) βθ(1 η) 1 ( v eq )γ+1 (1 + γ)[1 σω] E t bv t+1 λ G X E t b G X t+1 [βπ +(1 δ)βλ] E t b βt+1 +δβλe t b δt+1,t+2 C The System i Matrix Form The dyamic system is solved by the regular methods of differece equatios. I briefly outlie here the mai steps; for a extesive discussio, albeit i a differet cotext, see Kig, Plosser ad Rebelo (1988) ad Burside (1997). The geeral problem is give by: xt M cc u t = M cs + M λ ce z t (23) t MssE xt+1 t + Mss λ 1 xt = M t+1 λ t sce t u t+1 + Mscu 1 t + MseE t z t+1 + Msez 1 t. (24) 6

7 where x,λ are the edogeous state ad co-state variables, u are the cotrol variables, ad z is a vector of exogeous variables. Combiig we get or (Mss MscM cc 1 xt+1 M cs )E t +(Mss λ 1 M 1 t+1 scmcc 1 M cs ) (Mse + MscM cc 1 M ce )E t z t+1 +(Mse 1 + MscM 1 cc 1 M ce )z t M sse xt+1 t + M ss 1 xt = M λ t+1 λ t see t z t+1 + M sez 1 t xt λ t = Solvig for the edogeous variables: xt+1 E t = ( λ M t+1 ss) 1 1 xt M ss +( λ M t ss) 1 M se E t z t+1 +( M ss) 1 1 M se z t xt = W + RE λ t z t+1 + Qz t. (25) t I this paper the followig defiitios apply (LHS variables use otatio above while RHS variables are the actual variables here): x t = t λ t = λ t u t = v t z t = bg X t bβ t b δt,t+1 The other variables i the model - u, s, Q ad P - shall be cotaied i the vector f t ad are expressible as a liear combiatio of the cotrol, state ad exogeous variables: Here: f t = F c u t + F x x t + F z z t f t = bu t bs t bq t,t+1 bp t,t+1 \ Q t,t+1 V t 7

8 So: f t = F v bv t + F b t + F z z t The relevat equatios are: bu t = 1 b t (26) bq t,t+1 = σ bv t + 1 b t (27) Note too that hirig is give by Q t,t+1 V t ad that i log deviatios from steady state this is: Q\ t,t+1 V t = bq t,t+1 + bv t = σ bv t + 1 b t [1 σ] bv t σ 1 b t + bv t Ad give: bp t,t+1 = 1 (1 σ)b t +(1 σ)bv t (28) s t = η (1 α)+θ 1 1 α ( φv t +(1 φ)q t,t+1 v t ) γ+1 + P t,t+1 λ t Ã t = η (1 α)+θ 1 1 α! γ+1 vt Qt,t+1 e + P t,t+1λ t t We have: Isertig: bs t = η s Θ 1 (1 α) ( v eq )γ+1 (1 + γ)(bv t + ΩQ b t,t+1 b t ) + η s Pλ ³ bpt,t+1 + b λ t 8

9 bp t,t+1 = 1 (1 σ)b t +(1 σ)bv t bλ t = γ + σ (Ω + γω 1) b t +[ σ(ω + γω 1) + γ] bv t 1 bq t,t+1 = σ bv t + 1 b t We get: bs t = η s Θ (1 α) 1 ( v eq )γ+1 (1 + γ)(bv t + Ω σ bv t + 1 b t b t ) + η s Pλ 1 (1 σ)b t +(1 σ)bv t γ + σ (Ω + γω 1) b t +[ σ(ω + γω 1) + γ] bv t 1 " η = s Θ (1 α) 1 ( v eq )γ+1 (1 + γ)(1 σω)+ η s Pλ(1 σ)+η Pλ[ σ(ω + γω 1) + γ] bv t s " ³ η + Θ 1 (1 α) ( v eq s )γ+1 (1 + γ)(1 + σω ) 1 + η Pλ (1 σ) η Pλ b γ + σ t (Ω + γω 1) s 1 s 1 Thematricesaregiveby: 9

10 F v = F = F e = " ³ η Θ 1 (1 α) ( v Q e s )γ+1 (1 + γ)(1 σω) + η s Pλ(1 σ)+ η Pλ[ σ(ω + γω 1) + γ] s σ (1 σ) [1 σ] " ³ 1 η Θ 1 (1 α) ( v eq s )γ+1 (1 + γ)(1 + σω ) 1 + η Pλ (1 σ) η Pλ γ + σ (Ω + γω 1) s 1 s 1 σ 1 (1 σ) 1 σ 1 1

11 M cc = [ σ(ω + γω 1) + γ] M cs = γ + σ (Ω + γω 1) 1 1 M ce = [] G L M ss = M 1 ss = M sc = M 1 sc = M se = ³ +βθ βηpλ (1 σ) 1 1 (1 α) (1 η)( v Q e (1 + γ) 1+Ωσ 1 (1 δ) Qvσ 1 M 1 se = [] " ³ βθ(1 η) λ G X βηpλ(1 σ) 1 (1 α) Qv(1 σ) δ λ [βπ +(1 δ)βλ] δβλ G X )γ+1 ( v eq )γ+1 (1 + γ)[1 σω] [(1 δ)βλ βηpλ] The implied matrices are: M ss = (Mss MscM cc 1 M cs ) M ss 1 = (Mss 1 MscM 1 cc 1 M cs ) M se = (Mse + MscM cc 1 M ce ) M se 1 = (Mse 1 + MscM 1 cc 1 M ce ) W = ( M ss) 1 M ss 1 R = ( M ss) 1 M se Q = ( M ss) 1 M se 1 11

12 D The Dyamic System D.1 Derivatio of the Solutio Thesystem(25)isgiveby: bt+1 bλ t+1 bt = W bλ t bg X t+1 +R bβ t+1 +Q b δt+1,t+2 bg X t bβ t b δt,t+1 (29) Defie W = P ΛP 1,whereΛ is a diagoal matrix with the eigevalues of W o its diagoal, ad P is a matrix whose colums are liearly idepedet eigevectors of W. The Λ is costructed with the eigevalues i icreasig order of modulus ad is decomposed ito Λ1 Λ = Λ 2 with all elemets of Λ 1 less tha 1, ad all values of Λ 2 greater tha 1. As a result the equatio for b t+1 should be solved backward, while the equatio for λ b t+1 should be solved forward. Partitio the matrices W, R, Q, P ad P 1 as follows: W11 W W = 12 Rb Qb R = Q = W 21 W 22 R bλ Q bλ ad P = P11 P 12 P 21 P 22 P P 1 11 P = 12 P 21 P 22. Sice W = P ΛP 1 we have W11 W 12 P11 Λ = 1 P 11 + P 12 Λ 2 P 21 P 11 Λ 1 P 12 + P 12 Λ 2 P 22 W 21 W 22 P 21 Λ 1 P 11 + P 22 Λ 2 P 21 P 21 Λ 1 P 12 + P 22 Λ 2 P

13 Solvig the first equatio backward: Eb t+1 = Λ 1 b t +(P 11 R b + P 12 R bλ )E t z t+1 +(P 11 Q b + P 12 Q bλ )z t Solvig the other equatio forward: E tλt+1 b = Λ 2λt b +(P 21 R b + P 22 R bλ )E t z t+1 +(P 21 Q b + P 22 Q bλ )z t bλ t = Λ 1 2 E tλt+1 b Λ 1 2 (P 21 R b + P 22 R bλ )E t z t+1 Λ 1 2 (P 21 Q b + P 22 Q bλ )z t X = (P 21 R b + P 22 R bλ )E t z t+1+j +(P 21 Q b + P 22 Q bλ )E t z t+j. j= Λ (j+1) 2 Recall the AR(1) represetatio for z t : z t+1 = Πz t + Σ t+1 (3) The E t z t+j = Π j z t. Goig back to the solutio for b λ t give above bλ t = = = X j= X j= = Ψz t. " X j= Λ (j+1) 2 (P 21 R b + P 22 R bλ )E t z t+1+j +(P 21 Q b + P 22 Q bλ )E t z t+j Λ (j+1) 2 (Φ E t z t+1+j + Φ 1 E t z t+j ) Λ (j+1) 2 (Φ Π + Φ 1 )Π j z t A explicit formula for the rows of Ψ is obtaied by exploitig the diagoality of Λ 2. Defiig Λ 2i as the ith diagoal elemet of Λ 2,adΦ ji as the ith row of Φ j, j =, 1, it follows that the ith row of Ψ, deotedψ i,isgiveby " X Ψ i = Λ (j+1) 2i (Φ i Π + Φ 1i )Π j j= = Λ 1 2i (Φ iπ + Φ 1i ) X j= Λ j 2i Πj = Λ 1 2i (Φ iπ + Φ 1i )(I e Λ 1 2i Π) 1 13

14 The, the solutio for b t+1 is just b t+1 = (P 11 Λ 1 P11 1 )b t +(P 11 Λ 1 P 12 + P 12 Λ 2 P 22 )(P 22 ) 1 Ψz t + R b Πz t + Q b z t = (P 11 Λ 1 P11 1 )b t + (P 11 Λ 1 P 12 + P 12 Λ 2 P 22 )(P 22 ) 1 Ψ + R b Π + Q b zt = Υ bb b t + Υ bz z t (31) The solutio for b λ t is, ad the solutio for the cotrol is: bv t = Mcc 1 M cs = Υ bvb b t + Υ bvz z t. bλ t = (P 22 ) 1 P 21 b t +(P 22 ) 1 Ψz t = Υ bλb b t + Υ bλz z t (32) I (P 22 ) 1 P 21 b t + D.2 No Stochastic Dyamics M 1 cc M cs (P 22 ) 1 Ψ + M 1 cc M ce z t I order to characterize the properties of the dyamic path of b i the o-stochastic case, set all exogeous variables equal to their steady state values i.e.: o bg X t+1 o t= bβt o t= = for all t (34) bδt ad so the dyamics are give by: t= b t+1 = Υ bb b t (35) All other variables of iterest ca be represeted as liear fuctios of b t alog the uique path: bu t = 1 b t (36) bv t = Υ bvb b t (37) The liear coefficiets i these equatios are the elasticities of the origial variables with respect to deviatios of the employmet rate from its statioary value (as the hatted variables represet log deviatios). Thus they quatify the co-movemet of the differet variables alog the uique dyamic path. 14 (33)

15 D.3 Secod Momets The secod momets were computed usig the followig relatios: Defie vectors of state variables h t ad their iovatios t : bt h t = z t = t Σt Thus: (38) where h t+1 = Gh t + t+1 (39) G = 1 Υ bb Π. (4) The co-state ad cotrol variables are fuctios of this state vector: bλ t = Υ bλb Υ bλz ht (41) bv t = Υ bvb Υ bvz ht Therefore it is possible to represet all the variables, except the state ad exogeous variables, i the form: bλ t bv t = Hh t (42) f t The secod momets are therefore: E(h t h t) = Γ (43) E(h t h t i) = E (G i h t i + t + + G i 1 t i+1 )h t i = G i Γ E h t bλ t bv t f t = E(h t h th ) (44) = Γ H. We use the coefficiet matrix Π ad the variace-covariace matrix of Σ, estimated by the reduced-form VAR, ad the solutio of the model ( 1 ad the Υs)giveitermsof the model s parameters to compute Γ,G ad H. 15