Graduate Macro Theory II: Extensions of Basic RBC Framework

Transcription

1 Graduae Macro Theory II: Exensions of Basic RBC Framework Eric Sims Universiy of Nore Dame Spring 25 Inroducion The basic RBC model which is jus a sochasic neoclassical growh model wih variable labor is he building block of almos all modern DSGE models. I fis he daa well on some dimensions, bu less well on ohers. In his se of noes we consider several exensions and modificaions of he basic framework. I call his se of noes RBC Exensions because all he exensions I assume here are real I do no ye deal wih nominal rigidiies, which one loosely hink of as New Keynesian. We will see hose laer. 2 Common Exensions This secion works hrough a number of exensions designed o make he RBC model i) more realisic and ii) a beer fi wih he daa. 2. Indivisible Labor One failure of he RBC model is ha i fails o generae sufficien volailiy in hours of work. I also models hours in a raher unrealisic way ha is a odds wih realiy all flucuaions in hours come from he inensive margin e.g. average hours worked) as opposed o he exensive margin he binary choice of wheher o work or no). In he real world mos people have a more or less fixed number of hours worked; i is flucuaions in bodies ha drive mos of he flucuaion in oal hours worked. In realiy, households face wo decisions: ) work or no and 2) condiional on working, how much o work. This is difficul o model because i inroduces disconinuiy ino he decisions household make. Hansen 985) and Rogerson 988) came up wih a convenien echnical fix. Suppose ha wihin period preferences of any household are: uc, n ) = ln c + θ n ) ξ ξ )

2 These preferences nes he basic specificaion ha we ve used already if ξ = which means we d have ln n ). γ = n n The Frisch labor supply elasiciy here would be given by ξγ), where so if ξ = we d be in he sandard case we sared wih). Bu playing wih he value of ξ would allow you o conrol he Frisch elasiciy and have i no simply depend on n as i does in he log case. Suppose ha he srucure of he world is as follows. There are a large number of idenical households. Households eiher work or hey do no. If hey work, hey work n hours, wih < n <. The amoun of work, n, can be inerpreed as a echnological consrain and is exogenous o he model. Each period, here is a probabiliy of working, τ, wih < τ <. This probabiliy is indexed by because i is a choice variable essenially he household can choose is probabiliy of working, bu no how much i does work if i does work. There is a loery such ha he household has a τ chance of being seleced o work; he rule of he game is here is perfec insurance in ha every household ges paid wheher hey work or no. Hence, in expecaion households will work n = τ n and hey will have idenical consumpion because of he implici assumpion of perfec insurance combined wih separabiliy across consumpion and leisure). We can wrie ou he households expeced flow uiliy funcion as: uc, n ) = ln c + τ θ n) ξ ξ + τ )θ ) ξ ξ We observe ha preferences are linear in τ. Collecing erms we ge: n) ξ ) uc, n ) = ln c + τ θ ) ξ + θ ) ξ ξ ξ ξ Now, from above we know ha τ = n n. Make his subsiuion: uc, n ) = ln c + n n) ξ ) n θ ) ξ + θ ) ξ ξ ξ ξ 2) 3) ) As long as ξ >, hen ξ ξ > n) ξ ξ. Hence, re-wrie his again as: uc, n ) = ln c n ) ξ n θ ) n) ξ + θ ) ξ ξ ξ ξ Le s define wo consans as: B = θ n ) ξ ξ ) n) ξ ξ D = θ ) ξ ξ We can acually jus drop D alogeher from he analysis adding a consan o he uiliy funcion won change he household s opimal choices. Then we can wrie he wihin period uiliy funcion as: 2

3 uc, n ) = ln c Bn 5) In oher words, uiliy effecively becomes linear in labor under his indivisible labor wih loeries framework. This holds for any value of ξ. Bu indeed, i is as if ξ =. In oher words, he aggregae labor supply elasiciy is infinie even if he micro labor supply is very small i.e. ξ very large). This is poenially very helpful one can generae more hours volailiy wih a higher Frisch elasiciy, and is poenially) no subjec o he criicisms ha he labor supply elasiciy is inconsisen wih micro evidence. The full model can hen be wrien as follows. I assume ha households own he capial sock and lease i o firms. Households also have access o one period bonds. I absrac from he presence of hese bonds in he firm firs order condiion because he quaniy of bonds ends up being indeerminae anyway. As such, he firm problem becomes saic. Households: Firms: max c,n,b +,k + E β ln c Bn ) = s.. c + k + δ)k + b + b = w n + R k + r b max n,k a k α n α w n R k In a compeiive equilibrium he firs order condiions hold and all budge consrains hold. This gives rise o he following characerizaion of equilibrium: = βe c ) R + + δ)) c + ) + r + ) c + c = βe 6) 7) B = c w 8) w = α)a k α n α 9) R = αa k α n α ) k + = I + δ)k ) y = c + I 2) y = a k α n α 3) ln a = ρ ln a + ε ) 3

4 We would like o come up wih a calibraion of his model ha is consisen wih our previous calibraions. We don acually need o calibrae anyhing ha goes ino B, jus he value of B. Bu how do we do ha? From 6), combined wih ), we can solve for he seady sae capial o labor raio ) k = n α β δ) ) α Now combine 8) wih 9) o solve for c in erms of he seady sae capial o labor raio: 5) c = ) k α B α) 6) Now go o he aggregae accouning ideniy, which can be wrien as: n ) k c = n α )) k δ n Since 6) and 7) mus boh hold, we can se hem equal o one anoher and solve for B, aking n = 3 as a given arge in he calibraion. We ge: B = n 7) α) ) k α n n k ) α n δ k ) ) 8) n If we use our now sandard calibraions of α =.33, β =.99, and δ =.25, hen we see ha k n = This implies ha B = I solve boh his model and he sandard RBC model wih log preferences over leisure and θ calibraed o guaranee n = /3). All oher parameers are he same across boh models. Below are impulse response o a echnology shock in each model:

5 x 3 A.2 Y 5 x 3 N x 3 C.6 I x 3 w x r Log Log Indivisible Labor 2 We see significanly more amplificaion in he indivisible labor case han in he log-log case, as shown by he differences beween he dashed and solid lines. Oupu and hours boh increase by significanly more on impac. As a resul of his, consumpion and invesmen boh go up by more iniially. Because he indivisible labor case is isomorphic o he Frisch labor supply being infinie, he labor supply curve is perfecly horizonal here. This is why we ge a bigger increase in labor hours and a smaller increase in wages) afer he produciviy shock. Quaniaively, indivisible labor improves he fi of he model along several dimensions. Firs, i provides greaer amplificaion I ge oupu volailiy of 2.2 percen wih indivisible labor, as opposed o.7 percen in he sandard case. This means ha I can mach he oupu volailiy in US daa wih smaller TFP shocks. In addiion, indivisible labor increases he relaive volailiy of hours subsanially. In he benchmark RBC case, he relaive volailiy of hours is.3. In he indivisible labor case i is.69. This is a large improvemen, hough i is sill quie far from he daa. Furhermore, indivisible labor makes wages somewha less volaile volailiy of.8 insead of.) and somewha less procyclical correlaion wih oupu of.92 insead of.99). 2.2 Non-Separabiliy in Preferences In our basic specificaion we have assumed wo kinds of separabiliy in preferences separabiliy beween leisure and consumpion inraemporal separabiliy) and separabiliy of boh leisure and consumpion across ime ineremporal separabiliy). We consider boh of hese in urn. 5

6 2.2. Inraemporal Non-Separabiliy: King, Plosser, and Rebelo 988) and Greenwood, Hercowiz, and Hoffman 988) The generic definiion of balanced growh pah is a siuaion in which all variables growh a a consan rae over ime hough his rae need no be he same across variables). A special case of a balanced growh pah is a seady sae, in which he growh rae of all variables is equal o zero. In our benchmark specificaion above here is no explici rend growh, hough we could fairly easily modify he model in such a way ha we ge essenially) he same firs order condiions in he redefined variables which are derended. In any balanced growh pah, feasibiliy requires ha hours no grow. The inuiion for his is sraighforward if hours were declining, we would evenually hi zero and have no oupu. If hours were growing, we would evenually hi, which is he upper bound on hours. I is sraighforward o show, under he assumpions abou echnology and producion we have made, consumpion and he real wage mus grow a he same rae along he balanced growh pah, irrespecive of he kinds of preferences. Consider a generic, possibly non-separable wihin-period uiliy funcion: uc, n ). The only assumpions are ha i is increasing and concave in is argumens. The generic saic labor supply condiion is as follows; u n c, n ) = u c c, n )w This is really jus an MRS = price raio condiion beween consumpion and leisure. To saisfy he condiions laid ou above namely ha consumpion and he wage grow a he same rae and hours no grow), i mus be he case ha his firs order condiion reduce o somehing like: fn ) = w c In oher words, he lef hand side mus be a funcion only of n, and he righ hand side mus feaure he wage over consumpion. Wih wages and consumpion growing a he same righ, he righ hand side will be consan along a balanced growh pah. Then wih he lef hand side only a funcion of n and, of course, parameers), here will be a unique soluion for n ha is no growing. Consider a sandard iso-elasic preference specificaion: uc, n ) = c σ σ + θ n ) ξ ξ The inraemporal firs order condiion for labor supply would hen be: θ n ) ξ = c σ w The only way ha we end up wih w c on he righ hand side is if σ = : in oher words, for hese preferences o be consisen wih balanced growh, i mus be he case ha uiliy over consumpion is log. This is poenially problemaic, because i imposes ha he coefficien of 6

7 relaive risk aversion is, which is much lower han wha is needed o explain hings like he equiy premium. King, Plosser, and Rebelo 988) show ha preferences mus ake he following form o be consisen wih balanced growh: uc, n ) = c v n )) σ σ if σ uc, n ) = ln c + ln v n ) if σ = The second sep follows from applicaion of L Hopial s rule. We require ha v n ) be an increasing and concave funcion. So as o make his all consisen wih our original specificaion, suppose ha v ) akes he following form: Wih his specificaion, if σ =, hen we ge: v n ) = exp θ n ) ξ ) ξ uc, n ) = ln c + θ n ) ξ ξ Then, if ξ =, by L Hopial s rule we would ge: uc, n ) = ln c + θ ln n ). If ξ =, we would ge uc, n ) = ln c + θ n ), which is essenially he indivisible labor model. Thus, we can nes all of hese specificaions in erms of his general funcional form. For he general case in which σ and ξ, we can verify ha hese preferences will be consisen wih consan labor hours in seady sae. Le s find he marginal uiliies: u c c, n ) = c exp θ n ) ξ ξ u n c, n ) = c exp θ n ) ξ ξ Then for he generic firs order condiion, we ge: u nc, n ) u c c, n ) = θ n ) ξ c )) σ exp θ n ) ξ ξ )) σ c exp θ n ) ξ ξ = w θ n ) ξ = w c ) ) n ) ξ In oher words, he saic firs order condiion for labor supply ends up looking exacly like i does in he case of log consumpion wih hese preferences. Hours will be saionary. θ and ξ will have exacly he same inerpreaions as in he basic model θ will deermine n and ξ will deermine he Frisch elasiciy). Noe ha you ll ofen see KPR preferences wrien wih v N ) = n ) θ or some variaion hereof, so ha uc, n ) = c n ) θ ) σ. When σ, his collapses o ln c σ + θ ln n ); his effecively imposes he log-log preference specificaion. By wriing he v ) funcion wih he exp operaor, I permi a more general specificaion of he uiliy from leisure. 7

8 Wha does σ govern? I is sill going o have he inerpreaion as he elasiciy of ineremporal subsiuion. The firs order condiions for he household side of he model for consumpion and bonds can be wrien: Log-linearize: λ = c σ exp θ n ) ξ )) σ ξ 9) λ = βe λ + + r )) 2) λ = βe λ + R + + δ)) 2) ln λ = ln β + ln λ + + ln + r ) λ λ λ = λ + λ λ + r r + r λ = E λ+ + β r The las line follows from he fac ha we define r as he acual deviaion from seady sae, no percenage deviaion. Now log-linearize he expression for λ: n ) ξ ) ln λ = σ ln c + σ)θ ξ λ λ λ = σ c c c + σ)θ n ) ξ n n ) λ = σ c + σ)θ n ) ξ n ñ Now combine hese wo expressions: Simplify: σ c + σ)θ n ) ξ n ñ = E σ c + + σ)θ n ) ξ n ñ + ) + β r E c + c ) = β ) σ) σ r + θ n ) ξ n E ñ + ñ )) 22) σ If we approximae β, hen his says ha he elasiciy of ineremporal subsiuions is σ, jus like in he case wih separable uiliy. There is jus an addiional erm now ha depends on expeced employmen growh, hough if σ = his erm drops ou and we are in he normal case. Wha his specificaion of preferences hus does is allow us o consider differen parameerizaions of σ differen from one while sill having preferences ha are consisen wih balanced growh. Loosely speaking, σ governs he household s desire o smooh consumpion. If σ is very large, he household will wan consumpion in expecaion) o be very smooh, whereas if σ is 8

9 quie small hen he household will be quie willing o allow consumpion o no be smooh again in expecaion). Below are impulse responses o a sandard echnology shock for differen values of σ. I fix all oher parameer values a heir baseline values. I consider he following values of σ:, 2, and..5 Y 8 x 3 C 6 x 3 N I 9 x 3 w 6 x r.3 8 σ = σ = 2 σ = As we migh expec, he iniial jump in consumpion is increasing in σ noe again ha large σ means you wan consumpion o be smooh in expecaion, no necessarily in response o a shock). This means ha he jump in labor is decreasing in σ. Why is ha? Think back he labor supply and demand curves. When TFP increases, labor demand shifs righ, he amoun by which is independen of σ. When consumpion increases, labor supply shifs lef. The bigger is he consumpion increase, he bigger is his inward shif in labor supply, and herefore he smaller is he hours response in equilibrium and he larger is he wage response. Tha s exacly wha we see in erms of he impulse responses: when σ is bigger, he hours jump is smaller, he oupu jump is smaller, and he wage jump is larger. This all suggess ha one way o make he model beer fi he daa is o make σ smaller we see here ha higher values of σ make he amplificaion problem in he model worse we ge less of a jump in labor because consumpion jumps more, which means oupu goes up by less when produciviy goes up. In paricular, we ge more employmen volailiy and hence more amplificaion for σ <. The problem wih his is ha mos micro evidence does no suppor such a claim here esimaes of σ are ypically far greaer han one. In paricular, Hall 988) says... supporing he srong conclusion ha he elasiciy of ineremporal subsiuion, he inverse of σ) is unlikely o be much above., and may well be zero. This would mean ha σ! A number 9

10 of papers in he asse pricing lieraure rely upon very large values of σ in order o generae he excess reurns on equiy over deb ha we see in he daa. If we ake values of σ much greaer han, he RBC model begins o fi he daa even worse han in he log case in erms of amplificaion and relaive volailiy of hours). I is worh menioning, however, ha mos of hese esimaes ha find very large values of σ are based on ime series daa. Gruber 26) finds a much smaller value of σ more like.5) using micro daa from looking a ax variaion. Greenwood, Hercowiz, and Hoffman 988) propose anoher popular uiliy specificaion ha feaures non-separabiliy beween consumpion and leisure/labor. Unlike KPR preferences, GHH preferences are no consisen wih balanced growh. Wha GHH preferences do is eliminae he wealh effec on labor supply as we will see in a momen, his means ha he FOC for labor gives labor as a funcion only of he wage no consumpion showing up). This means ha here is no wealh effec so, for example, when labor demand shifs ou because of a echnology shock, here is no inward shif of labor supply because of c increasing. This will resul in more amplificaion. I will wrie uiliy in erms of disuiliy from labor insead of uiliy from leisure. The GHH preference specificaion can be wrien: Uc, n ) = σ c θ n+χ + χ ) σ The marginal uiliies are: U c c, n ) = u n c, n ) = θn χ c θ n+χ + χ ) σ c θ n+χ + χ ) σ The generic firs order condiion for labor supply is: u n c, n ) = u c c, n )w Wih hese marginal uiliies, his works ou o: θn χ = w Here we see, as promised, ha he consumpion erm drops ou alogeher on he righ hand side labor is only a funcion of he wage. We can wrie he Euler equaions for bonds and capial by defining auxiliary variables λ = u c : λ = βe λ + R + + δ)) λ = βe λ + + r ) I solve he model using values σ = which corresponds o uiliy aking he form ln ) c θ n+χ +χ )

11 and χ =, using sandard values I ve been using. Below, I compare he impulse responses under sandard preferences uiliy akes he form here ln c θ n+χ +χ ); in each case I solve for θ such ha seady sae labor hours are.33. x 3 A.5 Y x 3 N x 3 C. I x 3 w x r Sandard GHH 2 3 We see here ha we ge much larger amplificaion wih GHH preferences oupu and hours rise by subsanially more and appear o be more persisen). Naurally, he wage rises by less his occurs because here is no inward shif of labor supply following he produciviy shock). Ineresingly, we see ha consumpion jumps up by more and acually invesmen increases by less. In erms of volailiies, under he sandard seup I ge oupu and labor volailiies HP Filered) of.5 and.5, for a relaive volailiy of abou /3. For he GHH specificaion, I ge oupu volailiy of.76 and labor volailiy of.9, for abou a relaive volailiy of.5. I could make hese numbers look even beer by lowering χ Ineremporal Non-Separabiliy: Habi Formaion Anoher imporan kind of non-separabiliy is non-separabiliy across ime. This usually goes by he name habi formaion, wih he idea ha people ge uiliy no from he level of consumpion, bu from he level of consumpion relaive o pas consumpion. The idea is ha one becomes accusomed o a cerain level of consumpion i.e. a habi ) and uiliy becomes relaive o ha. Habi formaion has been included in macro models for a variey of reasons. In paricular, habi formaion can help resolve some empirical failings of he PIH. For example, habi formaion can help resolve he excess smoohness puzzle because, he bigger is habi formaion, he smaller consumpion will jump in response o news abou permanen income. Anoher area where habi

12 formaion has gained ground is in asse pricing, in paricular wih regard o he equiy premium puzzle. A large degree of habi formaion, in essence, makes consumers behave as if hey are exremely risk averse, and can hereby help explain a large equiy premium wihou necessarily resoring o exremely large coefficiens of relaive risk aversion see he previous subsecion). Assume inraemporal separabiliy so ha uiliy from consumpion is logarihmic. Le he wihin period uiliy funcion be given by: uc, n ) = lnc φc ) + θ ln n ) φ is he habi persisence parameer; if φ = we are in he normal case, and as φ agens ge uiliy no from he level of consumpion, bu from he change in consumpion. For compuaional purposes we need o resric φ < if i is exacly hen marginal uiliy in he seady sae would be. Le s seup he household s problem using a Lagrangian. Assume ha households own he capial sock: L = E β lnc φc ) + θ ln n ) +... =... λ w n + R k + Π + + r )b c k + + δ)k b + )) The firs order condiions are: = λ = βφe c c φc c + φc 23) = θ = λ w n n 2) = λ = βe λ + R + + δ)) k + 25) = λ = βe λ + + r )) b + 26) The only firs order condiion ha is differen is he one ha defines λ ; if φ = we are back in he usual case. I is easies o solve his model by no subsiuing ou for he Lagrange muliplier jus rea i as anoher endogenous variable. Below are impulse responses using our oherwise sandard calibraion of a RBC model for comparing a value of φ = he sandard case) wih φ =.9 he habi formaion case). 2

13 x 3 A x 3 Y x 3 N x 3 C.6 I x 3 w x r Sandard Habi 2 2 We observe ha he main difference is ha consumpion jumps up by less on impac he bigger is φ. The inuiion for his is high consumpion oday lowers uiliy omorrow, oher hings being equal, he bigger is φ. Hence people will behave cauiously in essence by no adjusing consumpion by much. The oher impulse responses are reasonably similar across parameerizaions, hough hours don jump up by much, he real wage jumps up by a lo, and oupu doesn jump up by much i.e. his is no going o improve he fi of he model along hose dimensions). The main dimension along which he inclusion of habi formaion does help he model mach he daa is no in erms of uncondiional momens, bu raher in erms of condiional impulse response funcions. Mos esimaed impulse responses o idenified shocks say, moneary policy shocks) show hump-shaped responses of consumpion. This is difficul o generae wihou habi formaion. Noe ha you can combine his kind of habi formaion wih differen preferences specificaions e.g. I could embed his ino he non-separable KPR or GHH preferences; I would jus replace c wih c φc ) everywhere in boh cases. Anoher form of habi formaion is someimes wha is called exernal habi formaion or Caching Up wih he Joneses Abel, 99). Here he idea is ha uiliy from consumpion depends no on consumpion relaive o own lagged consumpion, bu raher on consumpion relaive o lagged aggregae consumpion he idea being ha you care abou your consumpion relaive o ha of your neighbor. Now, of course, in a represenaive agen framework own and aggregae end up being he same. The difference is ha exernal habi formaion simplifies he problem, because he consumer does no ake ino accoun he effec of curren consumpion decisions on he habi sock essenially he second erm in he expression for λ above drops ou). 3

14 2.3 Capial and Invesmen Adjusmen Coss The sandard RBC model has he implicaion ha he price of capial goods relaive o consumpion is e.g. Hayashi s q is always ), which in urn says ha he value of a firm is jus he physical capial sock. The sandard RBC model also ypically doesn generae hump-shaped impulse responses o shocks, which seems o be a feaure of he daa. In his subsecion we inroduce convex coss o adjusing he capial sock. The idea is ha here is some cos o adjusing he capial sock or he level of invesmen) relaive o wha is normal. This will have he effec of breaking he q = resul in he basic model and will also impar some ineresing dynamics in he model. I ll begin wih wha I ll call capial adjusmen coss because hey follow he form se by Hayashi 982). In paricular, assume ha he capial accumulaion equaion can be wrien: K + = I φ 2 ) 2 I δ K + δ)k K When φ =, his is he sandard accumulaion equaion. If φ >, hen doing invesmen differen han seady sae in seady sae I K = δ) resuls in a cos which essenially makes your capial depreciae faser. Noe ha his cos is i) denominaed in unis of curren physical capial and ii) is symmeric so doing oo lile invesmen relaive o seady sae also coss you some capial, which may seem a bi funny). Household preferences are sandard. Insead of combining he flow budge consrain wih he accumulaion, le s rea hem as separae his urns ou o be easier). The household problem is: max C,I,N,K +,B + E ) β ln C θ N +χ + χ = s.. C + I + B + w N + R K + Π + + r )B K + = I φ ) 2 I δ K + δ)k 2 K Se up a Lagrangian wih wo consrains: { L = E β ln C θ N +χ + χ + λ w N + R K + Π + + r )B C I B + ) +... = + µ I φ ) 2 I δ K + δ)k K +)} 2 K The firs order condiions are: C = C = λ

15 = θn χ N I = λ = µ φ = λ w )) I δ K [ ) φ 2 ) I+ I+ I+ = µ = βe R + λ + µ + δ + µ +φ δ + µ + δ) K + 2 K + K + K + Le s define q µ λ. B + = λ = βe λ + + r ) µ is he marginal uiliy of having come exra insalled capial e.g. K + ), and λ is he marginal uiliy of having some exra consumpion. The raio is hen how much consumpion you would give up o have some exra fuure capial e.g. i is he relaive price of capial in erms of consumpion. If φ =, we see ha λ = µ, so q = always. If φ >, q can be differen from. Using his formulaion, we can wrie hese FOC as: q = βe C C + q = [R + + q + δ) + φ )) I φ δ K I+ K + δ ) I+ K + φ 2 ))] I+ δ K + The firs expression esablishes ha invesmen o capial, I K, is an increasing funcion of q : for his o be bigger han δ, q mus exceed one. The second is funcionally a difference equaion in q : curren q is a discouned value of he fuure marginal produc of capial, fuure adjusmen coss, and fuure q +, where he discouning is by he household s sochasic discoun facor, βe C +. I solve he model using my sandard parameer values here χ = ) for hree differen values of φ:, 2, and. The impulse responses are below C ] 5

16 2 x 3 Y x 3 C x 3 N I x 3 w x r x 3 q. K 2.5 φ = φ = 2 φ = As prediced, if φ = hen q = always and he responses are he same as in he basic RBC model. As φ ges bigger, q rises more in response o he produciviy shock. Quie naurally, we observe ha invesmen goes up by less, and capial accumulaes more slowly, he bigger is φ. Because i is cosly o adjus he capial sock quickly, invesmen doesn jump as much, which means consumpion jumps more; his mechanically resuls in a smaller increase in employmen han we would ge because labor supply shifs in more. We also see ha he real ineres rae rises by less significanly so). φ breaks he igh connecion beween he real ineres rae and he marginal produc of capial; his is a good hing from he perspecive of he model, since we d really raher r no rise when hi wih a produciviy shock, since in he daa he real ineres rae is essenially acyclical. An alernaive adjusmen cos specificaion is based on Chrisiano, Eichenbaum, and Evans 25). I refer o his is an invesmen adjusmen cos as opposed o a capial adjusmen cos). I call i an invesmen adjusmen cos because i) he adjusmen cos is measured in unis of invesmen, no unis of capial as above, and ii) he adjusmen cos doesn depend on he size of invesmen relaive o he capial sock, bu raher on he growh rae of invesmen. Le he capial accumulaion equaion be: In seady sae I I K + = [ φ 2 ) ] 2 I I + δ)k I =, so his revers o he sandard accumulaion equaion. As we can see, he cos depends on he growh rae of invesmen and is measured in unis of invesmen raher han unis of capial. 6

17 The household problem is oherwise he same as before. As we did earlier, form a Lagrangian wih wo consrains: { L = E β ln C θ N +χ + χ + λ w N + R K + Π + + r )B C I B + ) +... = [ + µ φ ) ] )} 2 I I + δ)k K + 2 I The firs order condiions are: = λ = µ φ I I 2 I C = C = λ = θn χ N ) 2 φ = λ w ) I I I I K + = µ = βe λ + R + + δ)µ + ) ) B + = λ = βe λ + + r ) We can again define q = µ λ. Doing so, we can wrie: = q φ 2 C q = βe R + + δ)q + ) C + ) ) 2 ) I I I φ I I I ) I+ I+ + βe µ + φ I C I+ + βe q + φ C + I ) I+ Above I have made use of he fac ha λ = C. Relaive o he capial adjusmen cos case earlier, he FOC for K + is much simpler and defines q is he presen discouned value of he renal rae on capial; he FOC for I is subsanially more complicaed. Bu you Can see ha if I is large relaive o seady sae, hen q will be greaer han. I solve he model quaniaively for his specificaion of adjusmen coss for differen values of he adjusmen cos parameer φ. The impulse responses are below: I I ) 2 ) 2 7

18 2 x 3 Y x 3 C x 3 N I x 3 w 5 x r x 3 q. K φ = φ = 2 φ = In erms of he dynamics of q and K, his specificaion of adjusmen coss plays a fairly similar role o he capial adjusmen cos specificaion q rises more and K accumulaes more slowly he bigger is φ. Bu he oher dynamics are quie differen. Firs, noe ha for posiive values of φ boh he invesmen and oupu impulse responses are hump-shaped. This means ha he growh raes of invesmen and oupu are auocorrelaed, which is acually a feaure of he daa ha he basic RBC model is incapable of maching. You can see why you ge his in erms of invesmen growh raes jus from he specificaion of he adjusmen cos. Since here is a convex cos in he invesmen growh rae, you wan o slowly adjus invesmen growh his gives you he hump-shaped invesmen response, which parially carries over ino oupu. Because invesmen jumps up so lile, consumpion responds more o he produciviy shock wih bigger adjusmen coss, which mechanically feeds ino a smaller response of employmen. Wha is perhaps mos marked in hese responses is he response of he real ineres rae. Wih hese adjusmen coss, he real ineres rae acually declines afer he produciviy shock raher han increasing. As wih he capial adjusmen cos his invesmen adjusmen cos breaks he connecion beween he real ineres rae and he marginal produc of capial here R increases bu r decreases. Wih hese adjusmen coss he breaking of his connecion is much sronger han in he capial adjusmen cos specificaion. This is useful because a major failure of he basic RBC model is he srong procylicaliy of he real ineres rae prediced by he model, whereas in he daa real ineres raes are eiher acyclical or mildly counercyclical. In DSGE models feauring many of hese addiions, i is now mos common o include he invesmen adjusmen cos specificaion in place of he Hayashi syle capial adjusmen cos specificaion, precisely because i can generae hump-shaped impulse responses posiive auocorrelaion 8

19 of growh raes) and works o brake he procyclicaliy of he real ineres rae. 2. Non-Saionary Technology In he specificaions we have hus far looked a, we have implicily, mos of he ime) assumed ha he non-saionary series of he model are rend saionary, because we assumed ha labor augmening echnology followed a deerminisic, linear ime rend, while he oher produciviy erm followed a saionary AR) process: y = a k α z n ) α z = + g z ) z ln a = ρ ln a + ε When wriing down he model wih no growh, as I have been doing, I ve been impliciy seing g z = and z =. Allowing g z > requires re-wriing he variables, bu would only very mildly affec he equilibrium condiions in essence, would have a small effec on he discoun facor). A priori, some people objec o he noion of emporary produciviy shocks if one hinks of a as represening knowledge, does i make sense o forge hings we ve learned. Le s insead suppose ha echnology follows a sochasic rend. We can ge rid of z alogeher and wrie he model as: y = a k α n α ln a = ρ a )g z + ρ a ln a + ε ln a = ln a ln a Here wha I ve assumed is ha a follows an AR) process in he growh rae, wih ρ a <. This means ha shocks, ε, will have permanen effecs on he level of a. g z is he mean growh rae; if I se ρ a =, hen ln a would follow a random walk wih drif. We could wrie his process in he levels as: a a ) ) ρa a = expg z ) ρa expε ) a 2 If you ake logs of his you ge back he process wrien above. Because we only wan one period of leads/lags in wriing down he equilibrium condiions, i is useful o inroduce a new variable, call i g a a. We can hen wrie he process above as: Or in logs: g = expg z ) ρa g ρa expε ) 9

20 ln g = ρ a )g z + ρ a ln g + ε Le s figure ou how o ransform he variables of his model. Sar wih he producion funcion, hen ake logs, and hen firs difference so as o ge in growh rae form: y = a k α n α ln y = ln a + α ln k + α) ln n ln y ln y ) = ln a ln a ) + αln k ln k ) + α)ln n ln n ) Along a balanced growh pah hours will no grow and capial will grow a he same rae as oupu capial and oupu mus grow a he same rae because he real ineres rae is consan along a balanced growh pah, and he real ineres rae in he long run is ied o he capial/oupu raio). Using hese facs, we have: ln y ln y ) = ln a ln a ) + αln y ln y ) ln y ln y = α ln a ln a ) This says ha, along a balanced growh pah, oupu will grow a α he rae of echnological progress if we had wrien his as labor augmening echnological progress, as opposed o neural, hey would grow a he same rae... hese wo seups are equivalen provided we re-define he rend growh rae appropriaely). Play around wih he above: ) y a ln = ln y y y = a a a a y α ) α ) α = y α a In oher words, along he balanced growh pah oupu divided by a α does no grow i.e. i is saionary. Hence, we can induce saionariy ino he model by dividing hrough by his. Define he following saionariy inducing ransformaions: 2

21 ŷ a k a Î a ŵ a ĉ a y α k α I α w α c α There is one very sligh modificaion due o a iming assumpion we need o divide by k by a. Inuiively, his is because k is chosen a, no. We need o wrie i his way so ha k is a predeermined sae variable in he model; scaling k by a insead would also render he model saionary, bu would induce issues wih he soluion. We can use hese ransformaions o aler he firs order condiions of he basic model as needed. Begin wih he producion funcion, dividing boh sides by he scaling facor a α : ŷ = a α α k α n α dae. Now, muliply and divide by a α α ŷ = a o ge he capial sock in he correc erms: α α ŷ = k α a a a Using our change of variables, we can wrie his as: ŷ = g α α α a α α n α ) α α kα n α kα n α Nex go o he capial accumulaion equaion, and divide boh sides by he scaling facor a 2

22 k + α a = I a α k+ = Î + δ) k a α + δ) k a a α a ) α We can simplify his furher by noing our change of variable: k+ = Î + δ)g The accouning ideniy is he same in erms of he ransformed variables as always: ŷ = ĉ +Î. Nex, consider expressions for facor prices: α k Transform hese: w = α)a k α n α R = αa k α n α For he renal rae, we have: a w α ŵ = α)a α = α)a α α a α α k a α α ŵ = α)g α α k α n α α n α kα n α α α R = αa a R = α k α a a a ) α n α kα n α R = αg kα n α Now wha abou he Euler equaions? Sar wih he one for bonds: 22

23 c = βe c + + r ) α α a a = βe + r ) c c + α a a = βe c α + α + a = βe g ĉ α + α a + r ) c + ĉ + + r ) Since he renal rae on capial is saionary, he Euler equaion for capial is going o look he same in erms of ransformed variables: = βe g ĉ α + ĉ + R + + δ)) The labor supply condiion is sraighforward o re-wrie in erms of ransformed variables since w c shows up on he righ hand side, and we divide boh variables by he same scaling facor, we can simply wrie: θ = ŵ n ĉ Then he full se of equilibrium condiions are: = βe g ĉ = βe g ĉ α + This is nine equaions in nine variables. α + ĉ + + r ) ĉ + R + + δ)) θ = ŵ n ĉ ŵ = α)g α α kα n α R = αg kα n α ŷ = ĉ + Î k+ = Î + δ)g ŷ = g α α kα n α α k ln g = ρ a )g z + ρ a ln g + ε I solve he model using our sandard parameer values. Dynare will produce impulse responses 23

24 of he derended variables. To consruc impulse responses of he regular variables, I need o ransform he responses produced by Dynare. Firs, I can find he response of he log level of a by cumulaing he response of he log growh rae, ln g. Then once I have he impulse response of a, I can add back α ln a o he impulse responses of he oher variables o ge hem back in log level form recall ha Dynare will produce he impulse response of he logs, so ln x. Since x x, I can add α ln a o ln x o recover he log level of he variable of ineres; I only α a need o do his for variables ha are non-saionary so no for he real ineres rae, he renal rae on capial, or he level of employmen). I compue impulse responses under some differen parameer configuraions in order o make a few poins. Firs, I show IRFs under a saionary shock wih ρ a =.979. Second, I show wo differen permanen shocks, one wih ρ a = random walk) and he oher wih ρ a =.33 so some posiive auocorrelaion in he growh rae). I se he size of he ransiory shock o. so produciviy increases by one percen); in he case of he wo permanen shocks, I se he size of he shocks such ha he long run effec on produciviy is also. so one percen); his means ha he size of he shock is. in he random walk case, bu ρ a ). more generally. So for he shock ha is persisen in growh raes, I m acually hiing he economy wih a smaller iniial shock. Below, he solid black lines show he responses o he ransiory produciviy shock; he dashed black line shows he responses o he random walk produciviy shock; he blue solid line shows he responses o he persisen growh rae shock. 2 x 3 a.5 y.5 c x 3 n.6 I. w x r 2 Saionary, ρ a =.979 ρ a = ρ a = The main ake-away from his picure is ha he impac effecs on oupu, hours, and invesmen 2

25 are smaller he more persisen is he produciviy shock. For he shock ha is auocorrelaed in growh raes, we see ha labor hours essenially do no reac. Also worh noing is ha afer abou periods he responses wih he random walk shock and he persisen growh rae shock are very similar. Wha is going on here? I s he same inuiion from labor demand-supply. If he produciviy shock is permanen, you ge a much bigger iniial jump in consumpion han you do in he ransiory shock case; his ranslaes o a bigger inward shif of he labor supply curve, which for a given labor demand shif resuls in a smaller increase in n. A smaller increase in n means ha oupu goes up by less on impac; his combined wih a bigger increase in c means a smaller increase in invesmen. One hing ha is ineresing o noe is ha he impac increase in c is acually smaller in he case of he auocorrelaed shock han in he case of he random walk shock. Wha is going on here? Consumpion is very forward-looking; by he way I consrained he size of he shocks, he long run effecs are idenical in each case. Because you ge more of he exra produciviy sooner in he random walk case han laer in he persisen growh rae case, he wealh effec is acually bigger in he random walk case, so c jumps by more alhough no by a lo). Why hen do we observe a smaller increase in n in he auocorrelaed growh raes case han he random walk case if he wealh effec / jump in c is smaller? I s because he ouward shif in labor demand is also smaller given ha I consrained he long run effec on produciviy o be he same in boh cases; when he shock is auocorrelaed, he immediae effec on produciviy and hence labor demand) is smaller. This combines o resul in a much smaller increase in n, a smaller iniial increase in y, and a much smaller iniial increase in i. The boom line here is ha i is relaively) sraighforward o augmen he model o allow for permanen produciviy shocks, bu his will only make he model fi he daa worse along he dimensions of he relaive volailiy of hours as well as he overall lack of amplificaion. 2.5 Variable Capial Uilizaion In his secion we consider adding anoher amplificaion mechanism o he model variable capial uilizaion. The idea here is ha while he capial sock may be predeermined wihin period, bu we can work he capial more inensively or no) depending on condiions. One could also hink abou variable labor uilizaion, bu for his o work we d need o make labor a leas parially predeermined you ll only ge a unique value of uilizaion if you can compleely change he facor capial in our main case, in his case labor). There are differen ways o model capial uilizaion and he coss associaed wih i. Wha I m going o do here assumes i) he households own he capial sock, ii) he households choose he level of uilizaion, and iii) he household leases capial services he produc of uilizaion and physical capial) o firms a renal rae R. The cos of uilizaion is faser depreciaion. Define u o be uilizaion and K u K as capial services K is he physical capial sock). The depreciaion rae is now a funcion of capial uilizaion, which we wan o normalize o be one in seady sae. In paricular: 25

26 earlier: δu ) = δ + φ u ) + φ 2 2 u ) 2 The household problem is oherwise sandard. We can wrie i as: max C,N,K +,u E β ln C + θ ln N )) = s.. C + K + δu ))K + B + w N + R u K + Π + + r )B The firs order condiions for bonds/consumpion and labor work ou o be he same as we had = βe + r ) C C + θ = w N C The firs order condiion for uilizaion works ou o: Or: δ u ) = R The Euler equaion for capial is: The firm produces oupu using: φ + φ 2 u ) = R C = βe C + R + u + + δu + ))) Y = A Kα N α Noe ha he firm acs as hough i ges o choose capial services a renal rae R, even hough household can separaely choose uilizaion and capial. The firs order condiions of he firm problem are oherwise sandard: w = α)a Kα N α R = αa Kα N α The aggregae resource consrain is sandard and we assume ha produciviy follows an AR) in he log. The full se of equilibrium condiions can be wrien geing ride of K by replacing i 26

27 wih u K ): = βe + r ) C C + θ = w N C φ + φ 2 u ) = R C = βe C + R + u + + δu + ))) w = α)a u α K α N α R = αa u α K α N α Y = A u α K α N α Y = C + I K + = I + δu ))K δu ) = δ + φ u ) + φ 2 2 u ) 2 ln A = ρ ln A + ε This is equaions in variables. The parameers of he uilizaion cos funcion are no enirely free. To normalize u = in seady sae, we mus have: R = β δ ). Bu from he FOC for uilizaion his requires: β δ ) = φ In oher words, φ mus be se o pin down seady sae uilizaion a ; φ 2 is a free parameer, and δ is he seady sae depreciaion rae, which we can calibrae o mach he seady sae invesmen-oupu raio as before. We can hink abou φ 2 as fixing u =, which pus us back in he RBC model. This may no be obvious, because one would be emped o look a he above expression and hink his fixed R a is seady sae value, β δ). This is no so, because imes is somehing finie. 27

28 .3 Y x 3 N.2 u x 3 C 5 2. I. 2 2 x 3 w 6 2 x r φ 2 = φ 2 =. φ 2 = We can see ha here is significanly more amplificaion of he produciviy shock when he coss of uilizaion are lower e.g. when φ 2 is smaller): oupu, employmen, consumpion, and invesmen rise by significanly more wih variable uilizaion han wihou. Wihou variable uilizaion φ 2 = ), HP filered oupu volailiy is abou.7; as I decrease φ 2, I ge oupu volailiy of.9 and.27 φ 2 =. and φ 2 =., respecively). This means ha, in principle, we d need smaller produciviy shocks o generae he observed oupu volailiy we observe in he daa, which many people find aracive. We also see ha he inclusion of variable uilizaion slighly) increases he relaive volailiy of employmen: is volailiy relaive o oupu goes from.3 in he no uilizaion case o.9 wih φ 2 =.. Finally, and imporanly, i is worh poining ou ha variable capial uilizaion invalidaes sandard growh accouning echniques o ge measured TFP. Measured TFP is defined as ln  = ln Y α ln K α) ln N. In his model, his is equal o ln A + α ln u. Hence, o he exen o which uilizaion moves around, he volailiy of measured TFP will be an oversaemen of he volailiy of he model counerpar, ln A. We see his in he quaniaive simulaions here. The volailiy of acual ln A is.7; wih no uilizaion he volailiy of measured TFP is he same, wih φ 2 =. i is.3, and wih φ 2 =. i is.8. This is a big difference. Because he model is only driven by one shock he produciviy shock), he inclusion of variable capial uilizaion doesn do much o change he cyclicaliy of measured TFP versus rue produciviy hey are boh very posiively correlaed wih oupu. Suppose here is some shock which raises N like a governmen spending shock, or perhaps a preference shock). Because labor and capial services are complemenary in he model, anyime N goes up you d like o also increase u. Bu from above, his means ha measured TFP will rise along wih oupu) even hough acual 28

29 ln A won change. Thus, variable capial uilizaion will change boh he volailiy and he cyclicaliy of measured TFP measured TFP will be boh more volaile and more posiively correlaed wih oupu wih variable capial uilizaion han he rue exogenous produciviy variable, ln A, is. 2.6 Preference Shocks The basic RBC model focuses on he produciviy shock, bu i is possible and fruiful o consider oher sources of shocks. I is popular o wrie down models where here are preference shocks shocks o how agens ge uiliy from consumpion and/or leisure/labor. I will wrie down he sandard model wih wo such shocks an ineremporal preference shock ha governs how you weigh curren uiliy relaive o fuure uiliy, and an inraemporal preference shock ha governs how you value uiliy from labor/leisure. Consider he following household problem: max C,K +,N,B + E ) β ψ ln C ν θ N +χ + χ = s.. C + K + δ)k + B + B w N + R K + Π + r B I assume ha boh ψ and ν follow mean zero AR)s in he log so ha he non-sochasic levels are uniy): ln ψ = ρ ψ ln ψ + ε ψ, ln ν = ρ ν ln ν + ε ν, The exogenous variable ψ is an ineremporal preference shock i doesn impac you value uiliy from consumpion versus uiliy from leisure, bu raher how you value uiliy oday versus uiliy in he fuure. If ψ increases, for example you place relaively more weigh on presen uiliy han he fuure. ν is an inraemporal preference shock i affecs how you value uiliy from consumpion relaive o disuiliy from labor or uiliy from leisure). The firs order condiions of he model can be wrien: [ ] ψ+ = βe R + + δ)) C ψ C + C = βe [ ψ+ ψ C + + r ) ] ν θn χ = C w Two hings o poin ou. Firs, ψ does no show up in he labor supply condiion: higher ψ increases he marginal uiliy of boh consumpion and he marginal disuiliy of labor, bu hese 29

30 ψ cancel ou. Second, since ψ is mean revering, when ψ is high we will have E + π be relaively low. This means ha an increase in ψ is isomorphic o a emporary reducion in β i means you are relaively more impaien. Third, ν shows up in he labor supply condiion in a way analagous o a disorionary ax rae on labor income if you divide boh sides by ν you see ha an increase in ν is funcionally equivalen o an increase in τ n side of he condiion). since τ n ) would show up on he righ hand I solve he model quaniaively using a firs order log-linear approximaion using our sandard parameer values a Frisch elasiciy of, χ = ). I assume ha he AR coefficiens on he wo preference shocks are.9 e.g. ρ ψ = ρ ν =.9) and ha he sandard deviaions of each shock are one percen e.g..). I m no rying o parameerize hese in any serious way; and for impulse response analysis in a firs order approximaion, he size of he shocks is irrelevan for he shape of he IRFs, hough i may have large effecs on uncondiional momens. Below are he responses o he ineremporal preference shock, ψ. We see ha oupu, hours, invesmen, and he real ineres rae all decline immediaely, while consumpion and he real wage increase. Wha is going on here is he following. The increase in ψ is effecively like a decrease in he discoun facor households value curren uiliy relaively more han fuure uiliy. This means hey wan o consume more in he presen and work less hence he increase in C and decline in N in a mechanical sense from he FOC for labor he increase in C shifs he labor supply curve in). The inward shif of labor supply along a sable labor demand curve leads o an increase in he wage. Falling hours wih no immediae change in A or K means ha Y mus fall. Oupu falling wih consumpion increasing means ha invesmen mus fall. The real ineres rae mus fall immediaely. There are wo ways o see his. Firs, since consumpion is high and falling, he real ineres rae mus fall for he Euler equaion o hold. Second, he fall in N lowers he marginal produc of capial, and wih capial fixed his means ha he renal rae on capial mus decline, and wihou adjusmen coss r moves in he same direcion as R. Overall, while his shock produces ineresing dynamics, i does no produce posiive co-movemen beween consumpion and oupu, and hence canno be he primary driving force of he business cycle. 3

31 Y 6 x 3 C.2 I x 3 N x 3 w x r Nex, consider he inraemporal preference shock. This leads o a reducion in C, N, Y, and I, wih an increase in w. The increase in ψ means ha people dislike labor relaively more his means naurally ha hey wan o work less. This shifs he labor supply curve in; along a sable labor demand curve, his means ha he wage mus rise. Lower employmen means lower oupu. Consumpion also falls his occurs naurally because household income declines. Invesmen also falls. 3

32 Y x 3 C.2 I N 8 x 3 w 2 x r We see here ha, unlike he ineremporal preference shock, he inraemporal preference shock can produce co-movemen beween consumpion and oupu and employmen. I ve said before ha he FOC for labor makes i difficul for consumpion and employmen o move ogeher unless A changes. In a mechanical sense here, an increase in ν funcionally plays a similar role alhough in a differen way). Wha you need o ge co-movemen beween C and N is for eiher he labor demand or supply curves o shif for a reason oher han pure wealh effec of C shifing he labor supply curve. ν will do he rick, as would a change in he ax rae on labor income see discussion above). 2.7 Invesmen Shocks The sandard RBC model assumes ha shocks o neural produciviy are he primary or sole) driver of business cycle flucuaions. Anoher kind of disurbance ha has recenly received a good deal of aenion is a shock o he marginal efficiency of invesmen MEI), or jus invesmen shock for shor. The invesmen shock makes he economy more producive a ransforming invesmen ino new physical capial in a way somewha analogous o how he produciviy shock, A, makes you more producive a ransforming capial and labor ino oupu). Le Z denoe he invesmen shock. I eners he capial accumulaion equaion as follows: K + = Z I + δ)k Here, an increase in Z means you ge more K + for a given amoun of I e.g. his shock increases he efficiency of invesmen. Some auhors have argued ha his shock is a reduced form proxy for 32

33 modeling he healh of he financial sysem he financial sysem essenially urns invesmen ino capial, so he higher or lower) Z is, he beer or worse) he financial sysem is. Le s again assume ha he household owns he capial sock and leases i o firms. household problem can be wrien wih wo consrains: max C,N,K +,B + E { } β ln C θ N +χ + χ = s.. The C + I + B + B w N + R K + Π + r B These consrains can be combined ino one: K + = Z I + δ)k C + K + Z δ) K Z + B + B w N + R K + Π + r B The firs order condiions for bonds, labor, and capial can be wrien: ) = βe + r ) C C + θn χ = C w [ ] Z = βe Z + R + + δ)) C C + Z + The res of he equilibrium condiions are sandard: Y = C + I K + = Z I + δ)k w = αa K α N α R = α)a K α N α Y = A K α N α I assume ha boh A and Z follow mean zero AR) processes in he log: ln A = ρ a ln A + ε a, ln Z = ρ z ln Z + ε z, I use a sandard parameerizaion of he model. Here I assume ha ρ z =.9 and ha he sandard deviaion of he invesmen shock is.. The impulse responses o he invesmen shock 33

34 are shown below..5 Y x 3 C. I x 3 N x 3 w x r We observe ha he invesmen shock leads o a sizeable increase in oupu, hours, and invesmen, wih reducions in consumpion and he real wage. The inuiion for wha is going on is as follows. When Z increases, since urning invesmen ino capial is more efficien, i makes sense o save more hrough capial. Hence, consumpion jumps down. This resuls in an inward shif of he labor supply curve along an iniially sable labor demand curve), which leads an increase in N and a reducion in w. The increase in N resuls in an increase in Y, which combined wih he reducion in C means I is higher. The real ineres rae rises because he marginal produc of capial is iniially higher. As we go furher ou, we sar o accumulae more capial, and he Z shock fades away, consumpion beings o increase, which shifs labor supply back in, driving down N and he wage up. We do no generae co-movemen beween C and N and Y and I ) here for reasons ha have been menioned above: wih our sandard labor supply relaionship, absen a change in A consumpion and hours canno move ogeher his holds he capial sock fixed, which is a safe approximaion a shor forecas horizons). Hence, while he invesmen shock produces ineresing dynamics here, in he model as currenly presened i canno be a major source of business cycle flucuaions as i does no ge he co-movemen righ. In difference seups where he labor supply condiion is broken his is no necessarily he case. A final cavea is in order here. The invesmen shock as presened is closely relaed o a differen kind of shock ha is ofen called invesmen-specific echnology or IST). This is a shock which affecs he ransformaion of consumpion goods ino invesmen goods, whereas he invesmen shock laid ou here impacs he ransformaion of invesmen goods ino capial goods. Since 3