A Baseline DSGE Moel Jesús Fernánez-Villavere Duke Universiy, NBER, an CEPR Juan F. Rubio-Ramírez Duke Universiy an Feeral Reserve Bank of Alana Ocober 10, 2006 1
1. Inroucion In hese noes, we presen a baseline sicky prices-sicky wages moel. The basic srucure of he economy is as follows. A represenaive househol consumes, saves, hols money, supplies labor, an ses is own wages subjec o a eman curve an Calvo s pricing. The final oupu is manufacure by a final goo proucer, which uses as inpus a coninuum of inermeiae goos manufacure by monopolisic compeiors. The inermeiae goo proucers ren capial an labor o manufacure heir goo. Also, hese inermeiae goo proucers face he consrain ha hey can only change prices following a Calvo s rule. Finally, here is a moneary auhoriy ha fixes he one-perio nominal ineres rae hrough open marke operaions wih public eb. 1.1. Househols There is a coninuum of househols in he economy inexe by j. The househols maximizes he following lifeime uiliy funcion, which is separable in consumpion, c j,realmoney balances, m j /p (where p is he price level), an hours worke, l j : E 0 X =0 β (log (c j hc j 1 )+υ log mj p ) ϕ ψ l1+γ j 1+γ where β is he iscoun facor, h is he parameer ha conrols habi persisence, γ is he inverse of Frisch labor supply elasiciy, is an ineremporal preference shock wih law of moion: log = ρ log 1 + σ ε, where ε, N (0, 1), an ϕ is a labor supply shock wih law of moion: log ϕ = ρ ϕ log ϕ 1 + σ ϕ ε ϕ, where ε ϕ, N (0, 1). Noe ha he preference shifers are common for all househols. Also, we have selece a uiliy funcion (log uiliy in consumpion) whose marginal relaion of subsiuion beween consumpion an leisure is linear in consumpion o ensure he presence of a balance growh pah wih consan hours. Househols can rae on he whole se of possible Arrow-Debreu commoiies, inexe boh by he househol j (since he househol faces iiosyncracic wage-ajusmen risk ha we will escribe below) an by ime (o capure aggregae risk). Our noaion a j+1 inicaes he amoun of hose securiies ha pay one uni of consumpion in even ω j,+1, purchase 2
by househol j a ime a (real) price q j+1,. To save on noaion, we rop he explici epenence of q j+1, an a j+1 on he even when no ambiguiy arises. Summing over ifferen iniviual asses we can price securiies coningen only on aggregae saes. Househols also hol an amoun b j of governmen bons ha pay a nominal gross ineres rae of R. Then, he j h househol s buge consrain is given by: c j + x j + m j p + b Z j+1 + p q j+1, a j+1 ω j,+1, = w j l j + r u j μ 1 a [u j ] k j 1 + m j 1 b j + R 1 + a j + T + z p p where w j is he real wage, r he real renal price of capial, u j > 0 he inensiy of use of capial, μ 1 a [u j ] is he physical cos of use of capial in resource erms, μ is an invesmenspecific echnological shock o be escribe momenarily,, T is a lump-sum ransfer, an z are he profis of he firms in he economy. We assume ha a [1] = 0, a 0 an a 00 > 0. Invesmen x j inuces a law of moion for capial k j =(1 δ) k j 1 + μ 1 S xj x j 1 x j where δ isheepreciaionraeans [ ] is an ajusmen cos funcion such ha S [Λ x ]=0, S 0 [Λ x ]=0, an S 00 [ ] > 0 where Λ x ishegrowhraeofinvesmenalonghebalance growh pah. We will eermine ha growh rae below. Noe our capial iming: we inex capial by he ime is level is ecie. The invesmen-specific echnological shock follows an auoregressive process: μ = μ 1 exp (Λ μ + z μ, ) where z μ, = σ μ ε μ, an ε μ, N (0, 1) The value of μ is also he inverse of he relaive price of new capial in consumpion erms. Given our escripion of he househol s problem, he lagrangian funcion associae wih i is: ¾ mj ½log (c j hc j 1 )+υlog ϕ ψ l1+γ j p 1+γ E 0 X β =0 ( λ j c j + x j + m j p w j l j r u j μ n 1 Q j + b j p + R q j+1, a j+1 ω j,+1, a [u j ] k j 1 m j 1 b p R j 1 1 p a j T z h i o xj k j (1 δ) k j 1 μ 1 S x j 1 x j where hey maximize over c j, b j, u j, k j, x j, w j, l j an a j+1 (maximizaion wih respec o ) 3
money holings comes from he buge consrain), λ j is he lagrangian muliplier associae wih he buge consrain an Q j he lagrangian muliplier associae wih insalle capial. The firs orer coniions wih respec o c j, b j, u j, k j,anx j are: (c j hc j 1 ) 1 hβe +1 (c j+1 hc j ) 1 = λ j λ j = βe {λ j+1 } Π +1 r = μ 1 a 0 [u j ] Q j = βe (1 δ) Qj+1 + λ j+1 r+1 u j+1 μ 1 +1a [u j+1 ] ª xj λ j + Q j μ 1 S S 0 xj xj + βe Q j+1 μ x j 1 x j 1 x +1 S 0 xj+1 j 1 x j R xj+1 x j 2 =0. We o no ake firs orer coniions wih respec o Arrow-Debreau securiies since, in our environmen wih complee markes an separable uiliy in labor, heir equilibrium price will be such ha heir eman ensures ha consumpion oes no epen on iiosyncracic shocks (see Erceg e. al., 2000). If we efinehe(marginal)tobin sqasq j = Q j λ j, (he raio of he wo lagrangian mulipliers, or more loosely he value of insalle capial in erms of is replacemen cos) we ge: (c j hc j 1 ) 1 hβe +1 (c j+1 hc j ) 1 = λ j λ j = βe {λ j+1 } Π +1 r = μ 1 a 0 [u j ] ½ λj+1 q j = βe (1 δ) qj+1 + r +1 u j+1 μ 1 λ +1a [u j+1 ] ¾ j xj 1=q j μ 1 S S 0 xj xj λ j+1 + βe q j+1 μ x j 1 x j 1 x +1 S 0 j 1 λ j x j R xj+1 xj+1 The las equaion is imporan. If S [ ] =0(i.e., here are no ajusmen coss), we ge: x j 2. q j = 1 μ i.e., he marginal Tobin s Q is equal o he replacemen cos of capial (he relaive price of capial). Furhermore, if μ =1, as in he sanar neoclassical growh moel, q j =1. The firs orer coniion wih respec o labor an wages is more involve. The labor use by inermeiae goo proucers o be escribe below is supplie by a represenaive, 4
compeiive firm ha hires he labor supplie by each househol j. The labor supplier aggregaes he iffereniae labor of househols wih he following proucion funcion: Z 1 l = 0 1 l j 1 j (1) where 0 < is he elasiciy of subsiuion among ifferen ypes of labor an l is he aggregae labor eman. 1 The labor packer maximizes profis subjec o he proucion funcion (1), aking as given all iffereniae labor wages w j an he wage w. Consequenly, is maximizaion problem is: Z 1 max w l w j l j j l j 0 whose firs orer coniions are: w Z 1 1 0 1 l j j 1 1 1 1 1 lj w j =0 j Diviing he firs orer coniions for wo ypes of labor i an j, wege: w i w j = li l j 1 or: Hence: w j = li l j w j l j = w i l 1 1 wi 1 i l j an inegraing ou: Z 1 0 w j l j j = w i l 1 i Z 1 0 1 l j j = w i l 1 1 i l Now, by he zero profis coniion implie by perfec compeiion w l = R 1 0 w jl j j,wege: w l = w i l 1 1 i l w = w i l 1 i l 1 1 Ofen, papers wrie θ = 1 an 1 θ = 1. For ha reparamerizaion, < θ 1, whereasθ (i.e., as 0), we go o a Leonieff proucion funcion, θ =0(i.e., =1), we have a Cobb-Douglas, an θ =1(i.e., ), a linear proucion funcion. 5
an, consequenly, he inpu eman funcions associae wih his problem are: l j = wj w l j (2) This funcional form shows he effec of elasiciy, on he eman for j h ype of labor. To fin he aggregae wage, we use again he zero profi coniionw l = R 1 0 w jl j j an plug-in he inpu eman funcions: o eliver: w l = Z 1 0 wj w j l j w 1 = w Z 1 w = 0 1 w 1 1 j j. Z 1 0 w 1 j j Iiosyncraic risks come abou because househols se heir wages following a Calvo s seing. In each perio, a fracion 1 θ w of househols can change heir wages. All oher househols can only parially inex heir wages by pas inflaion. Inexaion is conrolle by he parameer χ w [0, 1]. This implies ha if he househol canno change her wage for τ τy Π perios her normalize wage afer τ perios is χ w +s 1 Π +s w j. Therefore, he relevan par of he lagrangian for he househol is hen: max E w j ( X (βθ w ) τ ϕ ψ l1+γ j+τ 1+γ + λ j+τ τ=0 τy Π χ w +s 1 Π +s w j l j+τ ) subjec o à τy l j+τ = Π χ w +s 1 Π +s w j w +τ or, subsiuing he eman funcion (2), we ge: max E w j X (βθ w ) τ τ=0 λ j+τ τy +τ ϕ +τ ψ Π χ w +s 1 Π +s w j τy! l +τ à τy Π χ w +s 1 Π +s Π χ w +s 1 Π +s w j w +τ 1+γ j w j w +τ (1+γ)! l +τ l +τ 1+γ 6
which simplifies o max E w j X (βθ w ) τ τ=0 à τy λ j+τ τy +τ ϕ +τ ψ Π χ w +s 1 Π +s Π χ w +s 1 Π +s w j w +τ w j w +τ 1+γ! 1 w +τl +τ (1+γ) l +τ 1+γ All househols se he same wage because complee markes allow hem o hege he risk of he iming of wage change. Hence, we can rop he jh from he choice of wages an λ j. The firs orer coniion of his problem is: or an E X τ=0 E (βθ w ) τ 1 w E X τ=0 Now, if we efine: à τy (1 ) λ +τ à τy + w +τ ϕ +τ ψ X τ=0 à τy (βθ w ) τ λ +τ à τy (βθ w ) τ +τ ϕ +τ ψ f 1 = 1 w E τ=0 X τ=0 Π χ w +s 1 Π +s Π χ w +s 1 Π +s! 1 w w +τ l +τ w w +τ! Π χ 1 w +s 1 w Π +s Π χ w +s 1 Π +s (βθ w ) τ λ +τ à τy w w +τ! (1+γ) l 1+γ +τ w +τ =0 l +τ =! (1+γ) l 1+γ +τ! Π χ 1 w +s 1 w+τ l Π +s w +τ à X τy! f 2 = E (βθ w ) τ Π χ (1+γ) w (1+γ) +τ ϕ +τ ψ +s w+τ l 1+γ Π +s 1 w +τ we have ha he equaliy f 1 = f 2 is jus he previous firs orer coniion. Noe ha for hosesumsobewellefine (an, more generally for he maximizaion problem o have a à τy! 1 soluion), we nee o assume ha (βθ w ) τ λ +τ goes o zero faser han Π χ w +s /Π +s 1 goes o infiniy in expecaion. 7
an: One can express f 1 an f 2 recursively as: f 2 f 1 = 1 = ψ ϕ w w χ Π w (w ) 1 λ w l + βθ w E Π +1 (1+γ) χ l 1+γ Π w + βθw E Π +1 Now, since f 1 = f 2,wecanefine f = f 1 = f 2, such ha: an: f = 1 f = ψ ϕ w w 1 w +1 w χ Π w 1 (w ) 1 λ w l + βθ w E w +1 Π +1 (1+γ) χ l 1+γ Π w + βθw E Π +1 1 f 1 +1 (1+γ) w (1+γ) +1 f w +1. 2 w 1 f +1 (1+γ) w (1+γ) +1 f +1. w Since we assume complee markes an separable uiliy in labor (see Erceg e. al., 2000), we consier a symmeric equilibrium where c j = c, u j = u, k j 1 = k, x j = x, λ j = λ, q j = q,anwj = w. Therefore, he firs orer coniions associae o he consumer s problems are: (c hc 1 ) 1 hβe +1 (c hc ) 1 = λ λ = βe {λ +1 } Π +1 r = μ 1 a 0 [u ] ½ λ+1 q = βe (1 δ) q+1 + r +1 u +1 μ 1 λ +1a [u +1 ] ¾ x 1=q μ 1 S S 0 x x λ +1 + βe q μ x 1 x 1 x +1 S 0 x+1 1 λ x he buge consrain: c j + x j + m j p + b Z j+1 + p R q j+1, a j+1 ω j,+1, x+1 = w j l j + r u j μ 1 a [u j ] k j 1 + m j 1 b j + R 1 + a j + T + z p p an he laws of moion for f : f = 1 χ Π w 1 (w ) 1 λ w l + βθ w E w +1 Π +1 8 w 1 f +1 x 2.
an: f = ψ ϕ w w (1+γ) χ l 1+γ Π w + βθw E Π +1 (1+γ) w (1+γ) +1 f +1. w We nee boh laws of moion o be able, laer, o solve for all he relevan enogenous variables. Noe ha using he zero profis coniion for he labor supplier, w l = R 1 w 0 jl j j an he ne zero supply of all securiies, we have ha he aggregae buge consrain can be wrien as: c + x + R 1 0 m jj w l + r u μ 1 a [u ] k 1 + R 1 0 b j+1j + = p p R 1 m 0 j 1j + R 1 p R 1 b 0 jj + T + z p Thus, in a symmeric equilibrium, in every perio, a fracion 1 θ w of househols se w as heir wage, while he remaining fracion θ w parially inex heir price by pas inflaion. Consequenly, he real wage inex evolves: w 1 1 Π = θ w Π χ w 1 w 1 1 +(1 θ w ) w 1. i.e., as a geomeric average of pas real wage an he new opimal wage. This srucure is a irec consequence of he memoryless characerisic of Calvo pricing. 1.2. The Final Goo Proucer There is one final goo is prouce using inermeiae goos wih he following proucion funcion: Z 1 ε y = y ε 1 ε 1 ε i i. (3) 0 where ε is he elasiciy of subsiuion. Final goo proucers are perfecly compeiive an maximize profis subjec o he proucion funcion (3), aking as given all inermeiae goos prices p i an he final goo price p. As a consequence heir maximizaion problem is: max p y y i Z 1 0 p i y i i Following he same seps han for he wages,we fin he inpu eman funcions associ- 9
ae wih his problem are: y i = pi p ε y where y is he aggregae eman an he zero profi coniionp y = R 1 0 p iy i i o eliver: 1.3. Inermeiae Goo Proucers Z 1 p = 0 i, 1 p 1 ε 1 ε i i. There is a coninuum of inermeiae goos proucers. Each inermeiae goo proucer i has access o a echnology represene by a proucion funcion y i = A k α i 1 l i 1 α φz where k i 1 is he capial rene by he firm, li is he amoun of he packe labor inpu rene by he firm, an where A follows he following process: A = A 1 exp (Λ A + z A, ) where z A, = σ A ε A, an ε A, N (0, 1) The parameer φ, which correspons o he fixe cos of proucion, an z = A 1 1 α μ α 1 α guaranee ha economic profis are roughly equal o zero in he seay sae. We rule ou he enry an exi of inermeiae goo proucers. Since z = A 1 1 α μ α 1 α,wehaveha z = z 1 exp (Λ z + z z, ) where z z, = z A, + αz μ, 1 α an Λ z = Λ A + αλ μ 1 α. Inermeiaegoos proucerssolveawo-sages problem. Inhefirs sage, aken he inpu prices w an r as given, firms ren li an k i 1 in perfecly compeiive facor markes in orer o minimize real cos: min w l li,k i + r k i 1 i 1 subjec o heir supply curve: y i = ( A k α i 1 l i 1 α φz if A k α i 1 l i 1 α φz 0 oherwise 10
Assuming an inerior soluion, he firs orer coniions for his problem are: where % is he Lagrangian muliplier or: Therealcosishen: w = % (1 α) A ki 1 α l α i r = %αa ki 1 α 1 l 1 α i k i 1 = w l i + α w li 1 α r α 1 α w l i or: 1 w li 1 α Given ha he firm has consan reurns o scale, we can fin he real marginal cos mc by seing he level of labor an capial equal o he requiremens of proucing one uni of goo A ki 1 α l 1 α i =1or: A k α i 1 ha implies ha: Then: ha simplifies o: l 1 α α w i = A li 1 α r mc = mc = l i = α l i α α w 1 α r A 1 w 1 α 1 1 α α 1 α α w = A li =1 1 α r 1 α 1 α α α w 1 α r A α w 1 α r α A Noe ha he marginal cos oes no epen on i: all firmsreceivehesameechnology shocks an all firms ren inpus a he same price. In he secon sage, inermeiae goo proucers choose he price ha maximizes iscoune real profis. To o so, hey consier ha are uner he same pricing scheme han househols. In each perio, a fracion 1 θ p of firms can change heir prices. All oher firms can only inex heir prices by pas inflaion. Inexaion is conrolle by he parameer χ [0, 1], whereχ =0is no inexaion an χ =1is oal inexaion. 11
The problem of he firmsishen: max E p i X τ=0 (βθ p ) τ λ +τ λ (à τy Π χ +s 1 p i p +τ mc +τ! y i+τ ) subjec o τ=0 à τy y i+τ = Π χ p i +s 1 p +τ Π χ p i +s 1 p +τ! ε y +τ, where he marginal value of a ollar o he househol, is reae as exogenous by he firm. Since we have complee markes in securiies an uiliy separable in consumpion, his marginal value is consan across househols an, consequenly, λ +τ /λ is he correc valuaion on fuure profis. Subsiuing he eman curve in he objecive funcion an he previous expression, we ge: à X max E (βθ p ) τ λ τy! 1 ε à τy! ε +τ mc p i λ +τ y+τ Π χ p i +s 1 p +τ or max E p i X τ=0 (βθ p ) τ λ +τ λ à τy Π χ +s 1 Π +s p i p! 1 ε à τy Π χ +s 1 Π +s p i p! ε mc +τ y+τ E whose soluion p i implies he firs orer coniion: X τ=0 or (βθ p ) τ λ +τ λ E X τ=0 à τy (1 ε) Π χ +s 1 Π +s à τy (βθ p ) τ λ +τ (1 ε) p i p Π χ +s 1 Π +s! 1 ε p 1 i + ε! 1 ε p i p + ε à τy à τy Π χ +s 1 Π +s Π χ +s 1 Π +s p i p! ε p 1 i mc +τ y+τ =0! ε mc +τ y+τ =0 where, in he secon sep, we have roppe irrelevan consans an we have use he fac ha we are in a symmeric equilibrium. Noe how his expression ness he usual resul in he fully flexible prices case θ p =0: p i = ε ε 1 p mc +τ 12
i.e., he price is equal o a mark-up over he nominal marginal cos. Since we only consier a symmeric equilibrium, we can wrie ha p i = p an: X à τy à τy! ε E (βθ p ) τ λ +τ (1 ε) mc +τ y+τ =0 an τ=0 Π χ +s 1 Π +s! 1 ε p p + ε Π χ +s 1 Π +s To express he previous firs orer coniion recursively, we efine: g 1 = E g 2 = E X τ=0 X τ=0 (βθ p ) τ λ +τ à τy (βθ p ) τ λ +τ à τy Π χ +s 1 Π +s Π χ +s 1 Π +s! ε mc +τy +τ! 1 ε p p y +τ an hen he firs orer coniion is εg 1 =(ε 1)g 2. As i was he case for f s ha we foun in he househol problem, we nee (βθ p ) τ λ +τ o go o zero sufficienly fas in relaion wih he rae of inflaion for g 1 an g 2 o be well efine an saionary. Then, we can wrie he g s recursively: Π χ ε g 1 = λ mc y + βθ p E g+1 1 Π +1 an Π χ 1 ε g 2 = λ Π y Π + βθ p E g Π +1 Π +1 2 +1 where: Π = p. p Given Calvo s pricing, he price inex evolves: p 1 ε = θ p Π χ 1 ε 1 p 1 ε 1 +(1 θ p ) p 1 ε or, iviing by p 1 ε, 1 Π 1=θ p Π χ 1 ε +(1 θ p ) Π 1 ε 13
1.4. The Governmen Problem The governmen ses he nominal ineres raes accoring o he Taylor rule: R R = R 1 R γr γπ Π Π y y 1 Λ y γ y 1 γr exp (m ) hrough open marke operaions ha are finance hrough lump-sum ransfers T. Those ransfers insure ha he efici are equal o zero: T = R 1 m R 1 0 jj m R 1 0 j 1j + b R 1 0 j+1j R b 0 jj 1 p p p p The variables Π represens he arge level of inflaion (equal o inflaioninheseaysae), R seay sae nominal gross reurn of capial, an Λ y he seay sae gross growh rae of y.theermm is a ranom shock o moneary policy ha follows m = σ m ε m where ε m is isribue accoring o N (0, 1). The presence of he previous perio ineres rae, R 1,is jusifie because we wan o mach he smooh profile ofheineresraeoverimeobserve in U.S. aa. Noe ha R is beyon he conrol of he moneary auhoriy, since i is equal o he seay sae real gross reurns of capial plus he arge level of inflaion. Applying he efiniion of ransfers above, he aggregae buge consrain of househols is equal o: c + x = w l + r u μ 1 a [u ] k 1 + z. 1.5. Aggregaion Firs, we erive an expression for aggregae eman: y = c + x + μ 1 a [u ] k 1 Wih his value, he eman for each inermeiae goo proucer is y i = c + x + μ 1 an using he proucion funcion is: A k α i 1 l i 1 α φz = c + x + μ 1 ε p i a [u ] k 1 i, p a [u ] k 1 p i p ε 14
Since all he firms have he same opimal capial-labor raio: k i 1 l i = α w 1 α r an by marke clearing Z 1 0 l ii = l an Z 1 k i 1 i = u k 1. imusbehecaseha: Then: A k α i 1 0 k i 1 l i l i 1 α = A ki 1 l i = u k 1. l α l i = A u k 1 l α l i Inegraing ou Z 1 α α Z u k 1 A l u k 1 1 0 l ii = A l l ii = A (u k 1 ) α l 0 an we have Define v p = R 1 0 we ge: A (u k 1 ) α l 1 α φz = c + x + μ 1 a [u ] k 1 Z 1 0 pi p 1 α ε i pi p ε i. By he properies of he inex uner Calvo s pricing 1 Π v p = θ p Π χ c + x + μ 1 ε v p 1 +(1 θ p ) Π ε. a [u ] k 1 = A (u k 1 ) α l 1 α φz Now, we erive an expression for aggregae labor eman. We know ha v p l j = wj w l 15
If we inegrae over all househols j, we ge Z 1 0 l j j = l = Z 1 wj where l is he aggregae labor supply of househols. Define Z 1 v w wj = j. Hence Also, as before: 2. Equilibrium v w = θ w w 1 w Π χw 1 Π 0 0 w l = 1 v w l. w jl v w 1 +(1 θ w )(Π w ). Aefiniion of equilibrium in his economy is sanar an he symmeric equilibrium policy funcions are eermine by he following equaions: The firs orer coniions of he househol (c hc 1 ) 1 hβe +1 (c +1 hc ) 1 = λ λ = βe {λ +1 } Π +1 r = μ 1 a 0 [u ] ½ λ+1 q = βe (1 δ) q+1 + r +1 u +1 μ 1 λ +1a [u +1 ] ¾ x 1=q μ 1 S S 0 x x λ +1 + βe q +1 μ x 1 x 1 x +1 S 0 x+1 1 λ x f = 1 χ Π w 1 (w ) 1 λ w l + βθ w E w 1 +1 f +1 R Π +1 f = ψ ϕ (Π w ) (1+γ) χ l 1+γ Π w + βθw E Π +1 w (1+γ) w +1 w 2 x+1 x (1+γ) f +1 16
The firms ha can change prices se hem o saisfy: Π χ g 1 = λ mc y + βθ p E Π χ g 2 = λ Π y + βθ p E Π +1 εg 1 =(ε 1)g 2 Π +1 ε g+1 1 g+1 2 1 ε Π Π +1 where hey ren inpus o saisfy heir saic minimizaion problem: The wages evolve as: mc = 1 Π 1=θ w Π χ w an he price level evolves: u k 1 l 1 1 α 1=θ p Π χ Governmen follow is Taylor rule R R = R 1 R 1 w 1 1 Π = α 1 α 1 α 1 α w γr γπ Π Π w r α w 1 α r α A 1 +(1 θ w )(Π w ) 1 1 ε +(1 θ p ) Π 1 ε y y 1 Λ y γ y 1 γr exp (m ) Markes clear: y = A (u k 1 ) α l 1 α φz v p y = c + x + μ 1 a [u ] k 1 17
where l = v w l v p = θ p Π χ 1 Π v w = θ w w 1 w ε v p 1 +(1 θ p ) Π ε Π χw 1 v w Π 1 +(1 θ w )(Π w ) an k (1 δ) k 1 μ 1 S x x 1 x =0. 3. Saionary Equilibrium Sincewehavegrowhinhismoelinucebyechnological change, mos of he variables are growing in average. To solve he moel, we nee o make variables saionary. 3.1. Manipulaing Equilibrium Coniions Firs, we work on he firs orer coniions of he househol c h c 1 1 z 1 c+1 hβe +1 z z 1 z q μ = βe ½ λ+1 z +1 λ z f = 1 1=q μ Ã1 S λ z = βe {λ +1 z +1 μ r = a 0 [u ] z +1 z +1 z z +1 z R Π +1 } h c 1 = λ z z z μ (1 δ) q+1 μ z +1 μ +1 + μ +1 r +1 u +1 a (u +1 ) " +1 x # " x # x! z z x 1 S 0 z z z z x z 1 z 1 x 1 z 1 z 1 1 z 1 z 1 +βe q +1 μ +1 λ +1 λ S 0 " x+1 x z z z +1 z +1 w 1 χ w Π w λ z l + βθ w E z z #Ã x+1 z +1 z +1 x z z 1 Π +1 f = ψ ϕ (Π w ) (1+γ) χ l 1+γ Π w (1+γ) + βθw E Π +1! 2 w +1 z +1 z +1 w z z w +1 z +1 z +1 w z z 1 (1+γ) ¾ f +1 f +1 18
The firms ha can change prices se hem o saisfy: g 1 y = λ z mc + βθ p E z g 2 = λ z Π y z + βθ p E Π χ Π +1 εg 1 =(ε 1)g 2 Π χ Π +1 ε g+1 1 g+1 2 1 ε Π Π +1 where hey ren inpus o saisfy heir saic minimizaion problem: The wages evolve as: mc = u l 1=θ w Π χ w an he price level evolves: k 1 z 1 μ 1 = α w 1 1 α z r μ z μ z 1 μ 1 1 α 1 α α w 1 1 (r z μ ) α z1 α μ α 1 α α A 1 Π 1 Ã w 1 z 1 w z 1 Π 1=θ p Π χ Governmen follow is Taylor rule R R = R 1 R γr Π z 1 z! 1 +(1 θ w )(Π w ) 1 1 ε +(1 θ p ) Π 1 ε γπ Π y z y 1 z 1 Λ y z z 1 γ y 1 γ R exp (m ) Markes clear: y μ α 1z 1 α A z = z bu since μ α 1z α 1 = z 1 A 1,wehave y z = z 1 A A 1 z u u k 1 z 1 μ 1 α l 1 α φ k 1 v p z 1 μ 1 α l 1 α φ 19 v p
where y = c + x + z 1 μ 1 a [u ] z z z z μ k 1 z 1 μ 1 l = v w l v p = θ p Π χ 1 Π v w = θ w à w 1 z 1 z 1 w z z ε v p 1 +(1 θ p ) Π ε! v w χ Π w 1 Π 1 +(1 θ w )(Π w ) an k z μ z μ k 1 (1 δ) z 1 μ 1 z 1 μ 1 μ z μ 1 z 1 à 1 S " x #! z z x x 1 =0. z 1 z 1 z 3.2. Change of Variables We now reefine ha variables o obain a sysem on saionary variables ha we can easily manipulae. Hence, we efine ec = c z, λ e = λ z, er = r μ, eq = q μ, ex = x z, ew = w z, ew = w z, e k = k z,aney μ = y z. Then, he se of equilibrium coniions are: The firs orer coniions of he househol: 1 1 z 1 z +1 ec hec 1 hβe +1 ec +1 hec = λ e z eλ = βe { λ e z R +1 } z +1 Π +1 er = a 0 [u ] ( ) eλ+1 z μ eq = βe ((1 δ) eq +1 + er +1 u +1 a (u +1 )) eλ z +1 μ +1 ex z 1=eq 1 S S 0 ex z ex z f = 1 +βe eq +1 e λ+1 eλ ex 1 z 1 z z +1 S 0 ex+1 ex z +1 z χ ( ew ) 1 λ e Π w ( ew ) l + βθ w E z ex 1 z 1 ex 1 z 1 2 ex+1 Π +1 f = ψ ϕ (Π w ) (1+γ) χ l 1+γ Π w + βθw E Π +1 z +1 ex z 1 ew +1 ew (1+γ) ew +1 ew 1 z +1 f +1 z z +1 z (1+γ) f +1 20
The firms ha can change prices se hem o saisfy: g 1 = λ e Π χ mc ey + βθ p E g 2 = λ e Π χ Π ey + βθ p E Π +1 εg 1 =(ε 1)g 2 Π +1 ε g+1 1 g+1 2 1 ε Π Π +1 where hey ren inpus o saisfy heir saic minimizaion problem: The wages evolve as: mc = 1 Π 1=θ w Π χ w an he price level evolves: u e k 1 l 1 1 α 1 ew 1 1=θ p Π χ Governmen follow is Taylor rule: R R = R 1 R = α ew 1 α er 1 α 1 α ew 1 Π z 1 z γr γπ Π Π z μ z 1 μ 1 α ( ew ) 1 α er α 1 +(1 θ w )(Π w ) 1 1 ε +(1 θ p ) Π 1 ε ey ey 1 z z 1 Λ y γ y 1 γr exp (m ) Markes clear: ey = ec + ex + z 1 z ey = μ 1 a [u ] μ e k 1 A A 1 z 1 z u e k 1 α l 1 α φ v p 21
where l = v w l v p = θ p Π χ 1 Π v w = θ w ew 1 ew ε v p 1 +(1 θ p ) Π ε Π χw 1 v w z 1 z Π 1 +(1 θ w )(Π w ) an e z k z 1 μ μ 1 (1 δ) e k 1 z μ z 1 μ 1 ex 1 S ex 1 z z 1 ex =0. 4. Solving he Moel We will solve he moel by loglinearizing he equilibrium coniions an applying sanar echniques. Before loglinearizing, we nee o fin he seay-sae of he moel. We will solve he normalize moel efine in he las secion. Noe ha laer, when we bring he moel o he aa, we will nee o uno he normalizaion. 4.1. The Seay-Sae Now, we will finheeerminisicseay-saeofhemoel. Firs,leez = exp(λ z ), eμ =exp(λ μ ),an ea =exp(λ A ). Also, given he efiniion of ec, ex, ew, ew,aney,wehave ha Λ c = Λ x = Λ w = Λ w = Λ y = Λ z. Then, in seay-sae, he firs orer coniions of he househol can be wrien as: 1 ec hec hβ 1 ecez hec = λ e ez 1=β 1 R ez Π er = a 0 [1] eq = β 1 ((1 δ) eq + eru a [1]) ezeμ 1=eq (1 S [ez] S 0 [ez] ez)+β eq ez S0 [ez] ez 2 f = 1 ( ew ) 1 λ e Π ew l χ w 1 + βθ w ez 1 f Π f = ψϕ (1+γ) Π w l 1+γ Π χ w (1+γ) + βθw ez (1+γ) f Π 22
he firs orer coniions of he firm as: g 1 = λmcey e Π χ ε + βθ p g 1 Π g 2 = λπ e Π ey χ 1 ε + βθ p g 2 Π mc = he law of moion for wages an prices as: εg 1 =(ε 1)g 2 u e k l = α ew 1 α er ezeμ 1 α α 1 1 ew 1 α er α 1 α α an he marke clearing coniions as: Π χ w 1 1=θ w ez (1 ) +(1 θ w ) 1 Π w Π Π χ 1 ε 1=θ p +(1 θ p ) Π 1 ε Π ec + ex = ey v p ey = A z (ue k) α (l ) 1 α φ l = v w l Π v p χ ε = θ p v p +(1 θ p ) Π ε Π Π v w χ w = θ w ez v w +(1 θ w )(Π w ) Π e kezeμ (1 δ) e k ezeμ (1 S [z]) ex =0. To fin he seay-sae, we nee o choose funcional forms for a [ ] an S [ ]. Fora [u] we pick: a [u] =γ 1 (u 1)+ γ 2 2 (u 1) 2. Since in he seay sae weh havei u =1,hener = a 0 [1] = x γ 1 an a [1] = 0. The invesmen ajusmen cos funcion is S x 1 = κ( x 2 x 1 Λ x ) 2. Then, along he balance growh pah, S [Λ x ]=S 0 [Λ x ]=0. Using his wo expressions, we can 23
rearrange he sysem of equaions ha eermine he seay-sae as: 1 (1 hβez) 1 h ez R = Πez β 1 ec = e λ er = γ 1 er = 1 βezeμ (1 δ) / β ezeμ 1 βθw ez ( 1) Π (1 χ )(1 ) w f = 1 w λ(π e w ) l 1 βθw ez (1+γ) Π (1 χ )(1+γ) w f = ψ (1+γ) Π w l 1+γ 1 βθp Π (1 χ)ε g 1 = e λmcey 1 βθp Π (1 χ)(1 ε) g 2 = e λπ ey mc = εg 1 =(ε 1)g 2 e k l = α ew 1 α er ezeμ 1 α α 1 1 ew 1 α er α 1 α α 1 θ w Π (1 χ w )(1 ) ez (1 ) 1 θ w = Π w 1 1 θ p Π (1 χ)(1 ε) 1 θ p = Π 1 ε ec + ex = ey v p ey A = e ez (e k) α (l ) 1 α φ l = v w l 1 θ p Π (1 χ)ε v p = Π ε 1 θ p 1 θw ez Π (1 χ ) w v w =(Π w ) 1 θ w e k = ezeμ ezeμ (1 δ) ex. Firs, noice ha here is some resricions on γ 1 r = 1 β (1 δ) ezeμ = γ β 1 ezeμ 24
an ha he nominal ineres rae is: R = Πez β The relaionship beween inflaion an opimal relaive prices is: 1 Π θp Π (1 ε)(1 χ) = 1 θ p expression from which we obain he following wo resuls: 1. Ifhereiszeropriceinflaion (Π =1), hen Π =1. 2. Ifhereisfullpriceinexaion(χ =1), hen Π =1. 1 1 ε From he opimal price seing equaions, we ge ha he marginal cos is: mc = ε 1 ε 1 βθ p Π (1 χ)ε Π 1 βθ p Π (1 χ)(1 ε) The relaionship beween inflaion an opimal relaive wage is: 1 θw Π (1 χ Π w w )(1 ) ez (1 ) 1 1 =. 1 θ w Then we fin ha: an he opimal wage evolves: α α 1 1 α ew =(1 α) mc er ew = ewπ w Again, we can noe he following: 1. Ifhereiszeropriceinflaion (Π =1) an zero growh rae (ez =1), hen Π w =1. 2. Ifhereisfullwageinexaion(χ w =1) an zero growh rae (ez =1), hen Π w =1. We have he wo following equaions for he wage househol ecision: 1 βθw ez 1 Π (1 χ )(1 ) w f = 1 w Π w λl e an 1 βθw ez (1+γ) Π (1 χ w )(1+γ) f = ψ Π w (1+γ) l 1+γ. 25
Diviing he secon by he firs elivers: 1 βθ w ez (1+γ) Π (1 χ w )(1+γ) 1 βθ w ez ( 1) Π = ψ Π w γ l γ (1 χ w )(1 ) 1 w λ which efines l as a funcion of λ. e Now, we will look for anoher relaionship beween l an eλ o have wo equaions wih wo unknowns. Before oing so, we highligh ha, again, here are wo ineresing cases: 1. If here is zero price inflaion (Π =1) an zero growh rae (ez =1), hen we ge he saic coniion ha real wages are a markup over he marginal rae of subsiuion 1 beween consumpion an leisure. 2. Ifhereisfullwageinexaion(χ w =1) an zero growh rae (ez =1), hen we ge he saic coniion ha real wages are a markup over he marginal rae of subsiuion 1 beween consumpion an leisure. The expression for he ispersion of prices is given by: v p = 1 θ p Π ε 1 θ p Π (1 χ)ε where no price inflaion or full price inexaion elivers no price ispersion in seay-sae. Using he expression for Π w we fin ha he wage ispersion in seay-sae is: v w = 1 θ w 1 θ w Π (1 χ w ) ez (Πw ) where no price inflaion or full wage inexaion combine wih zero growh elivers no wage ispersion in in seay-sae. The relaionship beween labor eman an labor supply is: l = v w l. Noe now ha ey = ea ez (e k) α (l ) 1 α φ v p. Bu, since in seay-sae e k = ezeμ ex, i is he case ha: ezeμ (1 δ) ec + ezeμ (1 δ) e k = ey = ezeμ ea ez (e k) α (l ) 1 α φ v p 26
ha allows us o fin e k as funcion of l : Then: e k l = Ω = α ew 1 α er ezeμ e k = Ωl. ec = = ea ez Ωα l φ v à p A e ez (vp ) 1 Ω α ezeμ (1 δ) Ωl ezeμ ezeμ (1 δ) Ω ezeμ! l (v p ) 1 φ Now, we can express he marginal uiliy of consumpion in erms of hours, an ge anoher relaionship beween l as a funcion of λ: e (1 hβ)ez 1 h Ãà 1 A e ez ez (vp ) 1 Ω α Usingbohrelaionshipwecansolveforl an ge: 1 βθ w ez (1+γ) Π (1 χ w )(1+γ) 1 βθ w ez ( 1) Π = (1 χ w )(1 ) ψ γ Π w l γ 1 w (1 hβ)ez 1 h 1 ea ez ez (vp ) 1 Ω α ezeμ (1 δ) Ω ezeμ! 1 ezeμ (1 δ) Ω l (v p ) φ! 1 = λ ezeμ e l (v p ) 1 1. φ Noe ha his is nonlinear equaion. Therefore we will use a roo finer o fin l. Once we have l, we can solve for capial, invesmen, oupu, an consumpion as follows: e k = Ωl ex = ey = ec = 4.2. Loglinear approximaions ezeμ (1 δ) e k ezeμ ea (e k) α (l ) 1 α φ ez v p à e A ez (vp ) 1 Ω α! ezeμ (1 δ) Ω l (v p ) 1 φ ezeμ For each variable var,weefine var =logvar log var, wherevar is he seay-sae value for he variable var. Then, we can wrie var = var exp var. 27
We sar by log linearizing he marginal uiliy of consumpion: 1 1 z 1 z +1 ec hec 1 hβe +1 ec +1 hec = λ z z e. (4) 1. z I is helpful o efine he auxiliary variable aux = ec hec 1 1 z Then, we have ha: aux hβe ez +1 aux +1 = e λ where ez +1 = z +1 z. Then, (4) can be wrien as: aux exp aux hβzauxe exp aux +1+ b ez +1 = e λe b e λ which can be loglinearize as: aux aux hβze (aux +1 + b ez +1 ) = λ eb λ e. (5) Using he following wo seay sae relaionship ha aux(1 hβez) = λ e an ha E b ez+1 =0, we can wrie: aux hβze aux +1 =(1 hβz) b λ e. Nex, we loglinearize he auxiliary variable: auxe aux = exp b ec exp b ec h ez ec expb ec 1 b ez 1. The loglinear approximaion is: auxaux = ec h 1 ez ec b ec h 2 ez ec ec b ec + ec h 2 ez ec hec bec 1 ez b ez. Making use of he fac ha aux = ec h ez ec 1,wefin: auxaux = aux à b ec h 1 ez ec ec b ec + ec h 1 ez ec hec bec 1 ez b ez!, 28
ha simplifies o: auxaux = aux à b 1 h 1 bec + 1 h 1 h bec 1 ez ez ez b ez! aux = b Puing he (5) an (6) ogeher: 1 h Ã! 1 b ec hb ec 1 + hb ez. (6) ez ez ez b 1 h à 1 b ec hb ec 1 + hb ez ez ez ez (1 hβz) b λ e =! ( hβze b +1 Afer some algebra, we arrive o he final expression: 1 h Ã!) 1 b ec+1 hb ec ez ez + hb ez +1. ez (1 hβz) b λ e = b hβze +1 b 1+h2 β b ec + 1 h ez h ez b ec 1 + βhz E 1 h ez 1 h b ec+1 ez h ez 1 h ez b ez This expression helps o unersan he role of he habi persisence parameer h. If we se h =0(i.e., no habi), we woul ge: beλ = b b ec where he lags an forwar erms rop. Now, we loglinearize he Euler equaion: To o so, we wrie he expression as eλ = βe { e λ +1 z z +1 R Π +1 }. eλe b λ e = βe { λe e b R λ+1 e 1 Re b } eze z z, ΠeΠ b +1 By using he fac ha we simplify o: R = Πez β e b λ e = E {e b R λ+1 e 1 e b } e z z, eπ b +1 29
Now, i is easy o show ha: beλ = E { b e λ+1 + b R b Π +1 }. (9) Le us now consier: Firs, we wrie: er = a 0 [u ]. ere b er = a 0 u exp bu, where he loglinear approximaion elivers: er b er = a 00 [u] ubu. Since er = a 0 [u], hen: b er = a00 [u] u bu a 0, [u] or b er = γ 2 û. (10) γ 1 The nex equaion o consier relaes he shaow price of capial o he reurn on invesmen: ( ) eλ+1 1 1 eq = βe ((1 δ) eq +1 + er +1 u +1 a [u +1 ]) eλ ez +1 eμ +1 where eμ +1 = μ +1. We can we wrie his expression as μ ( eqe b eq β = ezeμ E exp beλ +1 b ez +1 b eμ +1 (1 δ) eqe b eq +1+ eru exp b er +1 +bu +1 a u exp bu +1 ). Loglinearizaion elivers: eq b eq = β ezeμ E + β ezeμ 4 b e λ+1 b ez +1 b eμ +1 ((1 δ) eq + eru a [u]) E (1 δ) eq b eq +1 + eru ber+1 + bu +1 a [u] ubu +1. Making use of he following seay-sae relaionships: u =1,a[u] =0, er = a 0 [u], eq =1 30
an 1= β er + β (1 δ), ane ezeμ ezeμ b ez +1 b eμ +1 =0, he previous expression simplifies o: ha implies: b eq = ((1 δ)+er) β ezeμ E b e λ+1 + b eq = E b e λ+1 + The nex equaion o loglinearize is: ex 1=eq 1 S z ex 1 z 1 S 0 ex ex 1 which can be rearrange as: β (1 δ) E b eq+1 + 1 ezeμ z z 1 ex z ex 1 z 1 β (1 δ) E b eq+1 + β ezeμ ezeμ erue b er +1, +βe eq +1 e λ+1 eλ β(1 δ) E b er+1. (11) ezeμ 2 z S 0 ex+1 z +1 ex+1 z +1 z +1 ex z ex z i i 1 = eq exp b eq 1 S hez exp b ex + b ez S hez 0 exp b ex + b ez ez exp b ex + b ez + +β q ez E exp b eq b ez +1 +4 b h i λ+1 e S 00 ez exp b ex +1 + b ez +1 ez 2 exp 2( b ex +1 + b ez +1). Taking he loglinear approximaion (an using he fac ha eq =1)wege: 0= b eq S 00 [ez] ez 2 b ex + b ez + β [ez] ez 3 E z S00 b ex +1 + b ez +1. Reorganizing: κez 2 b ex + b ez = b eq + βκez 2 E b ex +1 (12) where κ comes from he ajusmen cos funcion. We move now ino loglinearizing he equaions ha escribe he law of moion of f an f = 1 χ ( ew ) 1 λ e Π w 1 ( ew ) l + βθ w E ew +1 Π +1 ew f = ψ ϕ (Π w ) (1+γ) χ l 1+γ Π w (1+γ) + βθw E ew +1 Π +1 ew 1 z +1 f +1 z (1+γ) z +1 f +1. z 31
The firs equaion can be wrien as: f f exp b = 1 ( ew ) 1 λ e ew l exp (1 )b ew + b λ e + b ew + b l + βθ w Π (1 )(1 χ w ) ez 1 f fe exp b (1 )( Π b +1 χ w bπ + b ew +1+ b ez +1). Since E b ez+1 =0, we can loglinearize he previous expression as: ff b = 1 ( ew ) 1 λ e ew l (1 ) b ew + b λ e + b ew + b l + βθ w Π (1 )(1 χ w ) ez 1 fe bf+1 (1 ) (bπ +1 χ w bπ )+ b ew +1. Since in he seay-sae we have ha 1 βθ w ez 1 Π (1 )(1 χ w ) = 1 (w ) 1 λw l, f we can wrie ha: bf = 1 βθ w ez 1 Π (1 )(1 χ ) w (1 ) b ew + b λ e + b ew + b l + βθ w Π (1 )(1 χ w ) ez 1 E bf+1 (1 ) Π b+1 χ w bπ + b ew +1 (13) Le us know consier he secon equaion escribing he behavior of f. Firswecanwriei as: f f exp b = ψϕ(l ) 1+γ exp b +bϕ +(1+γ)( b ew b ew )+(1+γ) b l + +βθ w ez (1+γ) Π (1+γ)(1 χ w ) f fe exp b +1 +(1+γ)( Π b +1 χ w bπ + b ew +1 +b ez +1). I can be shown ha in seay-sae: 1 βθ w ez (1+γ) Π (1+γ)(1 χ w ) = ψϕ(l ) 1+γ f an since E b ez+1 =0, ha implies bf = 1 βθ w ez (1+γ) Π (1+γ)(1 χ ) w b + bϕ + (1 + γ) bew b ew +(1+γ) b l +βθ w ez (1+γ) Π (1+γ)(1 χ w ) E bf+1 + (1 + γ) Π b +1 χ w bπ + b ew. (14) +1 32
Le us loglinearize he law of moion for g 1 an g 2. Firs consier ha can be rewrien as: g 1 = λ e Π χ ε mc ey + βθ p E g+1 1 Π +1 g 1 exp bg1 = e λmcey exp b e λ + cmc + b ey +βθp g 1 Π ε(1 χ) E exp ε( bπ +1 χbπ )+bg 1 +1. If we loglinearize ha las expression, we ge: g 1 bg 1 = λmcey e beλ + cmc + b ey + βθ p g 1 Π ε(1 χ) E ε( Π b +1 χπ b )+bg +1 1. I can shown ha in seay sae herefore, bg 1 = 1 βθ p Π ε(1 χ) b eλ + cmc + b ey 1 βθ p Π ε(1 χ) = e λmcey g 1 + βθ p Π ε(1 χ) E ε(bπ +1 χbπ )+bg +1 1. (15) Le us now consier: g 2 = λ e Π χ 1 ε Π ey Π + βθ p E g Π +1 Π +1, 2 +1 ha can rewrien as: g 2 exp bg2 = e λπ ey exp b e λ +bπ +b ey +βθp Π (1 ε)(1 χ) g 2 E exp (1 ε) ( b Π +1 χbπ ) ( b Π +1 bπ )+bg 2 +1. If we loglinearize ha las expression, we ge: g 2 bg 2 = λπ e ey beλ + Π b + b ey +βθ p Π (1 ε)(1 χ) g 2 E (1 ε) bπ+1 χπ b bπ +1 Π b + bg +1 2. I can be shown ha: 1 βθ p Π (1 ε)(1 χ) = e λπ ey g 2, 33
herefore: bg 2 = 1 βθ p Π (1 ε)(1 χ) b eλ + Π b + b ey +βθ p Π (1 ε)(1 χ) E (1 ε) bπ+1 χπ b Noe ha is easy o show ha εg 1 =(ε 1)g 2 loglinearizes o: bπ +1 Π b + bg +1 2. (16) bg 1 = bg 2. (17) Now, le us loglinearize he relaionship beween he capial-labor raio an he real wagereal ineres rae u e k 1 = α ew z μ. l 1 α er z 1 μ 1 I is easy o show ha bu + b e k 1 b l = b ew b er + b ez + b eμ. (18) Le us loglinearize he marginal cos mc = 1 α α 1 1 ( ew ) 1 α er α 1 α α o ge cmc =(1 α) b ew + α b er. (19) Leusnowconcenraeonheaggregaewagelawofmoion: 1 Π 1=θ w Π χ w 1 w 1 he above expression can be rewrien as: w z 1 z 1 +(1 θ w ) 1 Π w 1=θ w Π (1 χ w )(1 ) ez (1 ) exp (1 )( bπ χ w bπ 1 +bπ w +ez ) +(1 θ w )(Π w ) 1 exp (1 ) bπ w, an loglinearize o: θ w Π (1 χ w )(1 ) ez (1 ) ( b Π χ w b Π 1 + b Π w + ez )=(1 θ w )(Π w ) 1 b Π w, 34
ha we can wrie as: θ w Π (1 χ w )(1 ) ez (1 ) (1 θ w )(Π w ) 1 (bπ χ w bπ 1 + bπ w + ez )= b ew b ew. (20) Now we concenrae on he aggregae price law of moion: ha can be rewrien as: 1 Π 1=θ p Π χ 1 ε +(1 θ p ) Π 1 ε 1=θ p Π (1 ε)(1 χ) exp (1 ε)(b Π χ b Π 1 ) +(1 θ p )(Π ) (1 ε) exp (1 ε)b Π ha loglinearizes o: θ p Π (1 ε)(1 χ) (1 θ p )(Π ) ( bπ (1 ε) χbπ 1 )=bπ. (21) The Taylor rule loglinnearizes o: br = γ R br 1 +(1 γ R ) γ Π bπ + γ y (4 b ey + b ez ) + bm. (22) The marke clearing coniions: an can be wrien as: an v p ey exp bvp +b ey ea = ez respecively, where A e +1 = A +1 A. l = v w l, ec + ex + a [u ] e k 1 eμ ez = ey, v p ey = e A ez u e k 1 α l 1 α φ l exp b l = v w l exp bvw +b l, h i a eu exp b eu ek b ek 1 exp ec exp b ec +ex exp b ex + ezeμ exp b eμ + b ez = ey exp b ey, u e α b ea k el 1 α b ez +α bu b + e k 1 +(1 α) e l exp φ 35
Loglinearizing we ge: b l = bv w + b l, (23) an c b ec + x b ex + γ 1 e k ezeμ bu =ỹ b ey, (24) (ỹ v p ) bv p + b ey A = e u ez e α 1 α k el be A b ez + α bu + b e k 1 +(1 α) e l. (25) Le us consier now 1 Π v p = θ p Π χ ε v p 1 +(1 θ p ) Π ε ha can be wrien as v p exp bvp = θp Π ε(1 χ) v p exp ε(b Π χ b Π 1 )+bv p 1 +(1 θp ) Π ε exp εb Π. Using 1 θp Π ε(1 χ) v p =(1 θ p ) Π ε we ge: bv p = θ p Π ε( ε(1 χ) Π b χπ b 1 )+bv p 1 1 θ p Π ε(1 χ) επ b. (26) Le us know consier he law of moion of he wage ispersion: v w χ Π w = θ 1 1 w 1 w v 1 w +(1 θ w )(Π w ). Π ez w ha can be wrien v w e bvw = θ w Π (1 χ w ) ez v w e (b Π χ w b Π 1 + b ew b ew 1 + b ez )+bv w 1 +(1 θw )(Π w ) e (b ew b ew ). Using 1 θw Π (1 χ w ) ez v w =(1 θ w )(Π w ) we ge: bv w = θ w Π (1 χ w ) ez Π b χ w bπ 1 + b ew b ew 1 + b ez + bv 1 w 1 θ w Π (1 χ w ) ez ( b ew b ew ). (27) Finally, le us loglinearize he law of moion of capial 36
k ez eμ =(1 δ) k x 1 + eμ ez 1 S ez x. x 1 If we rearrange erms, we ge: i kezeμ exp b k + b ez + b eμ =(1 δ) k exp b k 1 +ezeμ exp b ez + b eμ 1 S hze b ez +4 b ex x exp b ex. Loglinearizing: kezeμ b k + b ez + b eμ =(1 δ) k b k 1 + ezeμ x( b ez + b eμ + b ex ). Noe ha, using k = ezeμ x, we can rearrange he previous expression o ge: ezeμ (1 δ) b k + b ez + b eμ = (1 δ) b k 1 + ezeμ ezeμ (1 δ) ( ezeμ b ez + b eμ + b ex ) or b k = (1 δ) b k 1 + ezeμ ezeμ (1 δ) b ex 1 δ bez + ezeμ ezeμ b eμ. (28) 4.3. Sysem of Linear Sochasic Difference Equaions We now presen he equaions in he sysem as orere in Uhlig algorihm moel2fun.m Equaion 1 The firs equaion is θ w Π (1 χ w (1 ) )(1 ) z (1 θ w )(Π w ) 1 (bπ χ w bπ 1 + bπ w + ez )= b ew b ew We make use of he fac ha bπ w = b ew b ew 1,anefine he auxiliary parameer a 1 = )(1 ) z (1 ) θ wπ (1 χ w θwπ (1 χw)(1 ) z (1 ). In he Uhlig coe his expression appears as a (1 θ w )(Π w ) 1 1 =, (1 θ w ){exp[log(π w )]} 1 because when we solve for he seay sae of Π w, we express i in log-levels. Then, he equaion boils own o: a 1 bπ χ w a 1 bπ 1 + a 1 b ew a 1 b ew 1 + a 1 ez = b ew b ew an rearranging: a 1 bπ χ w a 1 bπ 1 +(1+a 1 ) b ew a 1 b ew 1 + a 1 ez b ew =0 (29) 37
Equaion 2 The secon equaion is: θ p Π (1 ε)(1 χ) (1 θ p )(Π ) (1 ε) (b Π χ b Π 1 )= b Π In orer o make noaion for compac, efine a 2 = θ pπ (1 ε)(1 χ) (1 θ p. Noehainhefile his )(Π ) (1 ε) θ expression appears as a 2 = p Π (1 ε)(1 χ), because when we solve for he seay sae (1 θ p){exp[log(π )]} (1 ε) value Π, we express i in log-levels. Subsiuing for a 2 : a 2 (bπ χbπ 1 )=bπ Rearranging: a 2 bπ a 2 χbπ 1 bπ =0 (30) Equaion 3 The hir equaion is b er = φ u û, where φ u = γ 2 /γ 1.Then, b er + φ u û =0 (31) Equaion 4 The fourh equaion is bg 1 = bg 2 Rearranging: bg 1 bg 2 =0 (32) Equaion 5 The fifh equaion is: bu + b e k 1 b l = b ew b er + b ez + b eμ Rearranging: Equaion 6 The sixh equaion is: bu + b er + b e k 1 b l b ew b ez b eμ (33) cmc =(1 α) b ew + α b er Rearranging: (1 α) b ew + α b er cmc =0 (34) 38
Equaion 7 The sevenh equaion is: br = γ RR 1 b +(1 γ R ) γ ΠΠ b + γ y (4 b ey + b ez ) + bm Rearranging: b R + γ R b R 1 +(1 γ R )γ Π b Π +(1 γ R )γ y b ez +(1 γ R )γ y b ey (1 γ R )γ y b ey 1 + bm =0 (35) Equaion 8 The eighh equaion is: c b ec + x b ex + γ e 1k ezeμ bu =ỹ b ey Noe ha in he coe, because we have solve for he log-seay sae, consans ener as follows: c = exp [log ( c)], x = exp[log( x)], k h i = exp log k, ỹ = exp log ỹ. Rearranging: c b ec + x b ex + γ e 1k ezeμ bu ỹ b ey =0 (36) Equaion 9 The ninh equaion is: (ỹ v p ) bv p + b ey A = e u ez e α 1 α k el bea b ez + α bu + b e k 1 +(1 α) e l A Define he parameer prouc = e u e k α el 1 α. ez Noe ha in erms of Uhlig noaion, his n h A is efine as prouc = e exp [log(u)] exp log( e io α n h 1 α. k) exp log( e l )io ez Also noe ha in erms of he coe, we have ha ỹ =exp log ỹ, v p = exp [log (v p )]. Subsiuing: (ỹ v p ) bv p + b ey be = prouc A b ez + α bu + b e k 1 +(1 α) e l Rearranging: (ỹ v p )bv p +(ỹ v p ) b ey (prouc) b e A +(prouc) b ez (α)(prouc)bu (α)(prouc) b e k 1 (1 α)(prouc) e l =0 (37) 39
Equaion 10 The enh equaion is: bv p = θ p Π ε( ε(1 χ) Π b χπ b 1 )+bv p 1 1 θ p Π ε(1 χ) επ b Define a 3 = βθ p Π ε(1 χ). Then, θ p Π ε(1 χ) = a 3 β. This ype of parameer efiniion will become clearer when we analyze he price seing equaions. Then, subsiuing: bv p = a 3 β ε bπ χε a 3 Π β b 1 + a 3 β bvp 1 1 a 3 β εbπ An rearranging: a 3 ε Π β b a 3εχ Π β b 1 + a 3 β bvp 1 1 a 3 β ε b Π bv p =0 (38) Equaion 11 The elevenh equaion is: bv w = θ w Π (1 χ w ) ez bπ χ wπ 1 b + b ew b ew 1 + b ez + bv 1 w 1 θ w Π (1 χ w ) ez ( b ew b ew ) Firs, le s efine a 4 = θ w Π (1 χ w ) ez. Subsiuing: bv w = a 4 Π b χ w bπ 1 + b ew b ew 1 + b ez + bv 1 w (1 a 4 ) ( b ew b ew ) Rearranging: bv w = a 4 bπ χ w a 4 bπ 1 + a 4 b ew a 4 b ew 1 + a 4 b ez + a 4 bv w 1 (1 a 4 ) b ew +(1 a 4 ) b ew Then, a 4 bπ χ w a 4 bπ 1 + b ew a 4 b ew 1 + a 4 b ez + a 4 bv w 1 (1 a 4 ) b ew bv w =0 (39) Equaion 12 The welfh equaion is: b l = bv w + b l Whichwerearrangeinhecoeoappearas: b l bv w b l =0 (40) 40
Equaion 13 The hireenh equaion is: b k = (1 δ) b k 1 + ezeμ ezeμ (1 δ) b ex 1 δ bez + ezeμ ezeμ b eμ whichwerearrangeo: (1 δ) b k 1 + 1 ezeμ (1 δ) b ex 1 δ bez + ezeμ ezeμ b eμ b k =0 (41) Equaion 14 The foureenh equaion is: b ez = b e A + α b eμ 1 α whichwerearrangeo: 1 b A e + α 1 α 1 α b eμ b ez =0 (42) Equaion 15 The fifeenh equaion is: (1 bβez) b λ e = b bβeze +1 b 1+b2 β b ec + 1 b ez Rearranging: b ez b ec 1 + βb z E 1 b ez 1 b b ec+1 ez b ez 1 b ez b ez b bβeze +1 b 1+b2 β b b ec + 1 b ez b ec 1 + βb z E 1 b ez ez 1 b b b ec+1 ez b ez (1 bβez) b λ e =0 (43) 1 b ez ez Equaion 16 The sixeenh equaion is: beλ = E { b e λ+1 + b R b Π +1 } whichwerearrangeo: E { b e λ+1 b e λ + b R b Π +1 } =0 (44) Equaion 17 The seveneenh equaion is: b eq = E b e λ+1 + β (1 δ) E b eq+1 + 1 ezeμ β(1 δ) E b er+1 ezeμ 41
whichwerearrangeo: E b eλ+1 b e λ + β (1 δ) E b eq+1 + 1 ezeμ β(1 δ) E b er+1 ezeμ b eq =0 (45) Equaion 18 The eigheenh equaion is: κez 2 b ex + b ez = b eq + βκez 2 E b ex +1 which we rearrange by unoing he firs-ifference operaor: b eq + βκez 2 E b ex+1 (1 + β)κez 2 b ex + κez 2 b ex 1 κez 2 b ez =0 (46) Equaion 19 The nineeenh equaion is: bf = 1 βθ w ez 1 Π (1 )(1 χ ) w (1 ) b ew + b λ e + b ew + b l + βθ w Π (1 )(1 χ w ) ez 1 E bf+1 (1 ) bπ+1 χ wπ b + b ew +1 efine a 5 = βθ w ez 1 Π (1 )(1 χ w ). Subsiuing: bf =(1 a 5 ) (1 ) b ew + b λ e + b ew + b h l + a 5 E bf+1 (1 ) Π b+1 χ w bπ + b i ew +1 Rearranging: (1 ) b ew +(1 a 5 ) b e λ +(1 a 5 ) b ew +(1 a 5 ) b l + a 5 E b f+1 +( 1)a 5 E bπ +1 ( 1)a 5 χ w bπ (1 )a 5 E b ew +1 b f =0 (47) Equaion 20 The wenieh equaion is: bf = 1 βθ w ez (1+γ) Π (1+γ)(1 χ ) w b + bϕ + (1 + γ) bew b ew +(1+γ) b l +βθ w ez (1+γ) Π (1+γ)(1 χ w ) E bf+1 + (1 + γ) bπ+1 χ wπ b + b ew Define a 6 = βθ w ez (1+γ) Π (1+γ)(1 χ w ). Then, bf = (1 a 6 ) b + bϕ + (1 + γ) bew b ew +(1+γ) b l +a 6 E bf+1 + (1 + γ) Π b+1 χ w bπ + b ew 42 +1 +1
Rearranging, (1 a 6 ) b +(1 a 6 ) bϕ + (1 + γ)(1 a 6 ) b ew (1 + γ) b ew +(1+γ)(1 a 6 ) b l b f +a 6 E b f+1 + a 6 (1 + γ) E b Π+1 a 6 (1 + γ) χ w b Π + a 6 (1 + γ) E b ew +1 =0 (48) Equaion 21 The weny firs equaion is bg 1 = 1 βθ p Π ε(1 χ) b eλ + cmc + b ey + βθ p Π ε(1 χ) E ε( Π b +1 χπ b )+bg +1 1 As before, efine a 3 = βθ p Π ε(1 χ). Then: (1 a 3 ) b e λ +(1 a 3 ) cmc +(1 a 3 ) b ey + εa 3 E b Π+1 χεa 3 b Π + a 3 E bg 1 +1 bg 1 =0 (49) Equaion 22 The weny secon equaion is bg 2 = 1 βθ p Π (1 ε)(1 χ) b eλ + Π b + b ey +βθ p Π (1 ε)(1 χ) E (1 ε) Π b+1 χbπ Define: a 7 = βθ p Π (1 ε)(1 χ). Subsiuing: Π b +1 bπ + bg +1 2 (1 a 7 ) b e λ + b Π +(1 a 7 ) b ey +(ε 1)a 7 E b Π+1 χ(ε 1)a 7 b Π a 7 E b Π +1 + a 7 E bg 2 +1 bg 2 =0 Shocks Thepreferenceshockshashefollowingsrucure: b = ρ b 1 + ε, bϕ = ρ ϕ bϕ 1 + ε ϕ, Noe ha Uhlig oes no allow o wrie somehing like b = ρ b 1 + σ ε,. Weeclarehe variance-covariance marix laer. The following shocks are no efine because of heir i.i.. naure. beμ = z μ, b e A = z A, m = σ m ε m, 43
4.4. Solving he Moel Now, le sae = bπ, b ew, bg 1, bg 2, b e k, b R, b ey, b ec, bv p, bv w, b eq, b e f, b ex, b e λ, b ez 0, nsae = ber, bu, bπ, b l, cmc, b l, b ew 0, exo = z μ,, b, bϕ,z A,,m 0, an ε =(ε μ,, ε,, ε ϕ,, ε A,, ε m, ) 0. Then, we nee o wrie he sysem efine above in Uhlig s forma, i.e.: an 0=AA sae + BB sae 1 + CC nsae + DD exo, Ã FF sae +1 + GG sae + HH sae 1 0=E +JJ nsae +1 + KK nsae + LL exo +1 + MM exo exo +1 = NN exo + Σ 1/2 ε +1 wih E ε +1 =0.!, 4.4.1. Wriing he Moel in Uhlig s Form Firs, noe ha from his secion on, an in he coes, we efine he variables in erms of heir loglinear eviaion from seay-sae. The marices in Uhlig s noaion are as following 44
AA = a 1 1+a 1 0 0 0 0 0 0 0 0 0 0 0 0 a 1 a 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 α 0 0 0 0 0 0 0 0 0 0 0 0 0 (1 γ R )γ Π 0 0 0 0 1 (1 γ R )γ y 0 0 0 0 0 0 0 (1 γ R )γ y 0 0 0 0 0 0 ỹ c 0 0 0 0 x 0 0 0 0 0 0 0 0 ỹ v p 0 ỹ v p 0 0 0 0 0 prouc a 3 ε/β 0 0 0 0 0 0 0 1 0 0 0 0 0 0 a 4 0 0 0 0 0 0 0 1 0 0 0 0 a 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 (1 δ) 0 (1 δ) z μ z μ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 χ w a 1 a 1 0 0 0 0 0 0 0 0 0 0 0 0 0 χa 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ BB = R (1 γ R )γ y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α(prouc) 0 0 0 0 0 0 0 0 0 0 a 3 εχ 0 0 0 0 0 0 0 a 3 β β 0 0 0 0 0 0 χ w a 4 a 4 0 0 0 0 0 0 0 a 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 δ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z μ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45
CC = 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 φ u 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 α 0 0 0 1 0 0 0 0 0 0 0 0 0 γ 0 1 k 0 0 0 0 0 z μ 0 α(prouc) 0 (1 α)(prouc) 0 0 0 0 0 (1 a 3 β )ε 0 0 0 0 0 0 0 0 0 0 (1 a 4 ) 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 DD = 0 0 0 0 0 0 0 0 prouc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 δ 0 0 0 0 z μ α 1 0 0 0 1 α 1 α 46
FF = 0 0 0 0 0 0 0 βb z 1 b z 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 β(1 δ) z μ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 βκ z 2 0 0 a 5 ( 1) 0 0 0 0 0 0 0 0 0 0 a 5 0 0 0 a 6 (1 + γ) 0 0 0 0 0 0 0 0 0 0 a 6 0 0 0 a 3 ε 0 a 3 0 0 0 0 0 0 0 0 0 0 0 0 a 7 (ε 1) 0 0 a 7 0 0 0 0 0 0 0 0 0 0 0 GG = 0 0 0 0 0 0 0 GG 1,8 0 0 0 0 0 (1 bβ z) GG 1,15 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 GG 4,13 0 κ z 2 a 5 χ w ( 1) (1 a 5 ) 0 0 0 0 0 0 0 0 0 1 0 1 a 5 0 a 6 χ w (1 + γ) GG 6,2 0 0 0 0 0 0 0 0 0 1 0 0 0 a 3 εχ 0 1 0 0 0 1 a 3 0 0 0 0 0 0 1 a 3 0 a 7 (ε 1)χ 0 0 1 0 0 1 a 7 0 0 0 0 0 0 1 a 7 0 where GG 1,8 = 1+βb2 1 b z GG 1,15 = b z(1 b z ) GG 6,2 = (1 a 6 )(1 + γ) GG 4,13 = (1 + β)κ z 2 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z(1 b z ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 HH = 0 0 0 0 0 0 0 0 0 0 0 0 κ z 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 β(1 δ) 0 0 0 0 0 0 z μ JJ = 0 0 0 0 0 0 0 0 0 0 0 0 0 a 5 (1 ) 0 0 0 0 0 0 a 6 (1 + γ) 0 0 0 0 0 0 0 0 0 a 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 KK = 0 0 0 0 0 0 0 0 0 0 1 a 5 0 0 1 0 0 0 (1 a 6 )(1 + γ) 0 0 (1 + γ) 0 0 0 0 1 a 3 0 0 0 0 1 0 0 0 0 0 βb z 0 0 0 0 0 0 0 0 0 0 0 0 0 LL = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 MM = 0 0 0 0 0 0 0 0 0 0 0 1 a 6 1 a 6 0 0 0 0 0 0 0 0 0 0 0 0 48
0 0 0 0 0 0 ρ 0 0 0 NN = 0 0 ρ ϕ 0 0 0 0 0 0 0 0 0 0 0 0 σ 2 μ 0 0 0 0 0 σ 2 0 0 0 Σ = 0 0 σ 2 ϕ 0 0 0 0 0 σ 2 A 0 0 0 0 0 σ 2 m 4.4.2. Wriing he Likelihoo funcion Uhlig s soluion meho gives us he following soluion: sae = PP sae 1 + QQ exo an nsae = RR sae 1 + SS exo. We observe obs =(logπ, log R, 4 log w, 4 log y ) 0. Therefore, o wrie he likelihoo funcion, we nee o wrie he moel in he following sae space form: S = A S 1 + B ε obs = C S 1 + D ε where S 1 =(1, sae,sae 1,exo 1 ). 4.4.3. Builing he A an B marices Hence, A = 1 0 0 0 0 PP 0 QQ NN 0 I 0 0 0 0 0 NN 49
an B = 0 QQ 0 I Σ1/2. 4.4.4. Builing he C an D marices Define he observables vecor: obs =(logπ, log R, 4 log w, 4 log y ) 0. Noe ha: bπ = PP (1, :) sae 1 + QQ (1, :) exo = PP (1, :) sae 1 + QQ (1, :) NN exo 1 + QQ (1, :) Σ 1/2 ε, br = PP (6, :) sae 1 + QQ (6, :) exo = PP (6, :) sae 1 + QQ (6, :) NN exo 1 + QQ (6, :) Σ 1/2 ε, b ew b ew 1 = PP (2, :) sae 1 + QQ (2, :) exo PP (2, :) sae 2 QQ (2, :) exo 1 = PP (2, :) sae 1 PP (2, :) sae 2 +QQ (2, :) NN exo 1 QQ (2, :) exo 1 + QQ (2, :) Σ 1/2 ε, an b ey b ey 1 = PP (7, :) sae 1 + QQ (7, :) exo PP (7, :) sae 2 QQ (7, :) exo 1 = PP (7, :) sae 1 PP (7, :) sae 2 +QQ (7, :) NN exo 1 QQ (7, :) exo 1 + QQ (7, :) Σ 1/2 ε, Also, remember ha bπ =logπ log Π ss, br =logr log R ss, b ew b ew 1 = 4 log ew = 4 log w 4 log z,an b ey b ey 1 = 4 log ey = 4 log y 4 log z. Since 4 log z = αλμ+λ A + 1 α αz μ, +z A,,wehave: 1 α obs = C S 1 + D ε. 50
where an C = log Π ss PP (1, :) 0 QQ (1, :) NN log R ss PP (6, :) 0 QQ (6, :) NN αλ μ +Λ A 1 α PP (2, :) PP (2, :) QQ (2, :) (NN I) αλ μ +Λ A 1 α PP (7, :) PP (7, :) QQ (7, :) (NN I) QQ (1, :) QQ (6, :) D = α 0 0 1 0 QQ (2, :) + 1 α α 0 0 1 0 QQ (7, :) + 1 α Σ 1/2 51