Linear-Quadratic Approximation to Optimal Policy: An Algorithm and Two Applications

Transcription

1 Linear-Quadraic Approximaion o Opimal Policy: An Algorihm and Two Applicaions Filippo Alissimo y European Cenral Bank and CEPR Diego Rodríguez Palenzuela x European Cenral Bank Sepember, 25 Vasco Cúrdia z Princeon Universiy Absrac This paper aims a bridging he gap beween recen heoreical progress in welfarebased opimal policy and is applicaion o models suiable for policy analysis. ih his purpose, he framework of Benigno and oodford (25b) wih a linear-quadraic (LQ) approximaion (in a imeless perspecive) o he opimal policy problem is applied o moneary policy in wo relaively sandard models. Furhermore he applicaions are performed hrough a sandardised algorihm ha is suiable for being implemened for a broad class of models. The auhors are graeful o Pierpaolo Benigno and Michael oodford for fruiful discussions and commens. e hanks Marco Del Negro and Frank Smes for sharing codes of he Smes and ouers model. Any errors or misakes are he responsibiliy only of he auhors. All views, conclusions and opinions expressed in he paper re ec solely hose of he auhors and no hose of he European Cenral Bank. y address: lippo.alissimo@ecb.in z address: vcurdia@princeon.edu x address: Diego.Rodriguez@ecb.in

2 Linear-Quadraic Approximaion o Opimal Policy Inroducion In recen years here has been considerable heoreical progress in deriving welfare-based opimal moneary policy in DSGE models. There have been however relaively few applicaions of hose resuls, probably due o compuaional di culies in applying hose resuls, in paricular o models which are o some exen suiable for policy analysis. This paper aims a parly lling his gap in wo ways. Firs, he paper presens an explici algorihm and a Malab funcion buil o implemen he heoreical resuls in Benigno and oodford (25b). This funcion is general enough so ha i can be readily implemened o a broad range of models. 2 The paper provides wo applicaions of an approximaion o opimal moneary policy, based on a linear-quadraic (LQ) approximaion o he opimal policy programme in a imeless perspecive. In doing his, we provide he descripion of he algorihm followed and show ha, in our applicaions, LQ opimal moneary policy can be compuaionally solved for in a relaively simple way. 3 The rs of our wo applicaion is o a sylised closed economy model wih moneary fricions. In his case he analysis of LQ opimal moneary policy akes hree (relaed) dimensions. The rs one is concerned wih he seady sae, he second enails he comparison o he cashless limiing case and he hird is he evaluaion of opimal policy vis-à-vis a sandard Taylor rule. In wha concerns he seady sae, under he cashless limiing case he in aion is always opimally se o zero. However, when moneary fricions are presen here is always some de aion in he opimal seady sae, a he same ime ha he nominal ineres rae falls shor of he discoun rae of he households. Under he presence of moneary fricions, he Friedman rule is always valid if prices are exible. I is moreover also valid under nominal rigidiy, if axes are no se o eliminae he monopolisic compeiion disorions and he level of price sickiness is very low. Under he second dimension of analysis, he responses o he di eren shocks are very similar in he cashless limiing case when compared o he moneary fricions case, wih exacly he same paern in boh cases and only minor di erences in magniude. Finally, he comparison beween he simple ineres rae rule o he opimal policy in his economy reveals ha, under increases o he ax rae or he price markup, he Taylor rule is no aggressive enough in reacing o in aion, compared o he opimal policy. Under a shock o he governmen expendiures he Taylor rule appears o be no as aggressive in reacing o he oupu changes as he opimal policy is. The responses o an oupu shock are however very di eren when we consider opimal policy and a simple Taylor rule. The opimal policy keeps he in aion close o he seady sae values by swiching he aggregae demand in he direcion of he change in he aggregae supply (hence in he considered case The Malab funcion uses he symbolic oolbox and herefore can be used only if ha oolbox is insalled. 2 In each applicaion we provide a small secion deailing he ransformaions of he basic models ha are needed in order o he srucure for he code. 3 For an addiional applicaion of he proposed algorihm o he linear-quadraic approximaion of opimal policy o a model of a currency union, see Alissimo, Benigno, and Rodriguez-Palenzuela (24). 2

3 Linear-Quadraic Approximaion o Opimal Policy of a posiive shock he opimal moneary policy is furher expansionary). The Taylor rule insead will reac o correc he increase in he oupu vis-à-vis he seady sae (he Taylor rule is no de ned in erms of oupu gap) and herefore i will be a conracionary policy, furher deepening he oupu gap in exisence. Our second applicaion of LQ opimal policy in a imeless prespecive is o a version of he sandard Neo-Keynesian model of a closed economy wih various srucural shocks, which follows closely Smes and ouers (23). This class of models is increasingly popular in empirical sudies as well as policy applicaions. One feaure ha is very ypical of his class of esimaed models is he assumpion of Taylor rules. I hen becomes especially imporan o invesigae he exen o which a Taylor rule diverges from opimal policy. The resuls presened here show ha he di erences beween he wo are no negligible boh qualiaively and quaniaively. In sum, he resuls sugges ha LQ policy di ereniaes o a larger exen is impac aking ino accoun he supply or demand naure of he shocks, in general exacerbaing he e ecs of he former and dampening he e ecs of he laer relaive o he Taylor case. hile resuls obained are preliminary, he work indicaes ha he relaively ligh compuaional burden for calculaing LQ opimal policy pus a premium o his approach when using i in he conex of policy simulaions. Moreover, he mehod proposed by Benigno and oodford (25b) and used in our algorihm is robus o any model ha can be pu in he form presened here, yielding a correc rs-order approximaion o he opimal policy problem. These properies promoe he fuure implemenaion of he algorihm in models wih high empirical conen, useful for policy analysis. The remainder of he paper is organized as follows. Secion 2 presens he srucure of he problem and he LQ soluion. The code used o implemen his soluion is hen discussed in secion 3. The wo applicaions, wih model descripion, implemenaion of he code and resuls, follow in secions 4 and 5. Secion 6 concludes. 2 The problem Following he framework of Benigno and oodford (25b), consider a general maximizaion problem of he form: V E X = U(x ; u ; X ; ); (2.) where U() is a funcional, < < ; X is a vecor of predeermined endogenous sae variables of dim(x ) = n X ; x is a vecor of non-predeermined endogenous variables of dim(x ) = n x and u is a vecor of conrol variables of dim(u ) = n u ; is a vecor of sochasic disurbances of dimension dim(x ) = n e. The maximizaion is subjec o he vecor of law of moion of dimension n X X + = F (x ; u ; X ; ) (2.2) 3

4 Linear-Quadraic Approximaion o Opimal Policy for he predeermined sae variables and o srucural equaions ha de ne he se of possible raional-expecaions equilibria and ha include a se of forward-looking relaions of he form E G(x ; u ; X ; ; x + ) = (2.3) where se of consrains is of dimension n G. In paricular n G < n x + n u : e also assume ha he shocks follow + = S + " + (2.4) where " should be a whie noise wih mean zero. I is well known ha he soluion of he problem in (2.) subjec o he consrains in (2.2, 2.3 and 2.4) implies ha he soluion is generally no ime-consisen. However if he policy is seleced so ha i ful ls some addiional consrain on he value of he non-predeermined endogenous variables in he iniial period, hen he resuling problem has a well de ned recursive srucure and he implied policy are ime invarian. Therefore he opimizaion problem is furher subjec o he consrain ha he economy s iniial evoluion be he one associaed wih he implemenaion of he policy in quesion. In he erminology of oodford (23), we are looking for he opimal from a imeless perspecive. Any such opimal policy will have o saisfy he addiional consrain ha he forward looking variables, in he iniial period, need o be equal o heir precommied values, x = x. For fuure purposes, le us also de ne: 2 y 4 x u X 3 5 z where dim(y ) = n y = n x + n u + n X and dim(z ) = n z = n y + n e. 2. The linear-quadraic approximaion of he problem hile general soluion of he problem of ineres involves he soluion of non-linear raional expecaion models associae o he s order condiion of he opimizaion problem in (2.) under he consrains (2.2, 2.3 and 2.4) and he proper iniial condiions, he aim here is o provide a proper ranking of policies which is correc in erms of welfare up o he second order in a proper neighbourhood of he deerminisic seady sae. To his end, Benigno and oodford (25b) consruc a second order approximaion of he objecive funcion by a discouned sum of purely quadraic erm in deviaion from he seady sae, up o erms ha are independen for he policy. Their approximaion begins by de ning seady-sae values (x; u; X) of he endogenous variables, and he seady-sae values of he Lagrange mulipliers ( ; '), associaed wih he problem of maximizing (2.) under he consrains (2.2) and (2.3). They hen consider second-order Taylor approximaions of he U, F, and G funcions, for he values of he endogenous variables near he seady-sae values. For x u X

5 Linear-Quadraic Approximaion o Opimal Policy example, for he I elemen of he F funcion, hey obain X I + = F I (x ; u ; X ; ) (2.5) = X + DF I (z z) + 2 (z z) D 2 F I (z z) + O(kk 3 ); where DF I and D 2 F I refer o he Jacobian and Hessian of he I-h elemen of F wih respec o z, respecively. By manipulaing hese local expansions and he FOCs of he consrained maximizaion problem a he seady sae values, hey obain a local approximaion of he objecive funcion (2.) of he form where V 2 E X = [(z z) Q(z z) + (z z) R(z z)] (2.6) J(x ; z ) + ip + O(kk 3 ) Q D 2 U + H; H J D 2 F J + ' j h D 2 G i + I x ^D 2 G i I x i ; R ' j I x ^DDG j ; and J and ' j are seady sae values of he Lagrange mulipliers associaed wih he J-h consrain (2.2) and he j-h consrain (2.3). 4 The second order approximaion of he law of moion for he predeermined variables, F, and he forward looking relaions, G; as well as heir values in seady saes were used o eliminae rs order erms from he second order approximaion of he objecive funcion. Furhermore if we consider allocaions which saisfy he iniial commimen ha x = x, hen he value of he J erm is nil and does no di er across policies considered. Therefore he approximaed objecive funcion delivers a proper ranking in erms of welfare, up o second order, of di eren policies which saisfy he proper iniial commimens ("imeless perspecive"). In he class of policies saisfying hose consrains, i follows ha a correc linear approximaion o opimal policy can be obained by maximizing he quadraic funcion of deviaion of variables from arges as: E X = [(z z) Q(z z) + (z z) R(z z)] (2.7) 4 Furher noice ha D 2 G i refers o he Hessian of he i-h elemen of G wih respec o z, ^D2 G i o he Hessian of he i-h elemen of G wih respec o x + and ^DDG j o he marix of cross derivaives beween z and x + of he j-h elemen of G, and I x is noaion for he selecion marix such ha I x z = x 5

6 Linear-Quadraic Approximaion o Opimal Policy subjec o X I + = X + DF I (z z) = DG i (z z) + ^DG i (x + x): The problem (2.7) has he sandard form of a linear-quadraic opimizaion and can be easily solved by sandard ools; furhermore his provides easy o check second order condiions. 3 The LQ code hile he soluion of he model in (2.7) is sandard all he inermediae seps necessary in order o ransform he general problem in (2.) ino is linear-quadraic equivalen are quie involved and lenghy, in paricular in he case of models of medium size. To his end, we designed he following Malab rouine: [A,B,C,Q,R,nr,nw,nd,labels] = LQ(x_,u_,X_,csi_,x_,X_,U,F,... G,BETA,S,x_ss,u_ss,X_ss,FLM_ss,... GLM_ss,Gcheck) which, uilizing he symbolic ools of Malab, nds for he linear-quadraic form associaed o a generic opimizaion problem as in (2.) under he consrains (2.2 and 2.3), solves for he opimal policy of he linear-quadraic problem and checks he second order condiions of such a problem. 3. Inpu in he programme The inpus needed for he rouine: lis of he endogenous variables x, X and u : [x_,u_,x_]; lis of he exogenous variables (shocks) ha saisfy (2.4) wih E [" ] = : [csi_]; lis of he endogenous variables nex period x + and X + : [x_,x_]; he funcional forms U (), F () and G (): [U,F,G] he parameer values (ineremporal discoun facor) and S (auocorrelaion of shocks): [BETA,S]; seady sae values (x; u; X; ; '): [x_ss,u_ss,x_ss,flm_ss,glm_ss]; Gcheck, which is a binary variable agging he opional esing for exisence and uniqueness of he soluion: [rue, false]. 6

7 Linear-Quadraic Approximaion o Opimal Policy The rouine requires as inpus he seady sae values of he endogenous variables as well as of he mulipliers associaed wih he opimal policy problem. 5 To his end he main rouine, before calling he LQ funcion, is required o solve he non-linear sysem of equaions as described in Benigno and oodford (25b). However solving such a sysem is no a rivial ask, in paricular for large dimensional sysem and i requires careful aenion in he choice of iniial condiions of he soluion. Therefore we preferred o keep he seady sae compuaion as separae and o rea seady sae values as an inpu wih respec o he LQ rouine.i is imporan o noice ha i is no advisable o use numerical minimizaion algorihms for he soluion of he sysem due o heir signi can inaccuracy. Insead mehods aimed a he soluion of sysems of non-linear equaions yield much more precise soluions for he seady sae values. This is all he more imporan given ha hese values will ener he marices presened in (2.7). 6 In wha follows, he seps performed by he code are described.. Variables in log deviaions wih respec o he seady saes The rs sep of he rouine expresses all variables z in log erms performing he following ransformaion: ^zj; = log z ^z j; : j; ) if z j 6= ^z j; = z j; if z j = : 2. Quadraic objecive funcion This rs sep forms he quadraic objecive funcion of he above policy problem which has a form V = X (bz bz) Q(bz bz) + 2(bz bz) R(bz bz) (3.) 2 = where bz = log zj if z j 6= z j if z j = ; Q D 2 U + H; H J D 2 ~ F J + ' j [D 2 G j + I x;z ^D 2 G j I x;z ]; R ' j I x;z ^DDG j ; where D 2 U, D 2 ~ F J and D 2 G j are he Hessian of U, ~ F J, G j wih respec o ^z ; ^D2 G j is he Hessian of G j wih respec o ^x + ; and ^DDG j is a marix of cross derivaives 5 For furher deails on he exac sysem of equaions ha needs o be solved refer o Benigno and oodford (25b). 6 A mehod for solving he sysem is provided in he codes used o generae he resuls of his paper, for he wo applicaions. The algorihm used is based on he csolve.m Malab funcion creaed by Chrisopher Sims. These codes can be obained, upon reques. 7

8 Linear-Quadraic Approximaion o Opimal Policy beween ^x + and ^z. The code uses he seady-sae values for he Lagrangian mulipliers; compues analyically he above Hessians and evaluaes hem imposing he seady-sae condiion and using he seady-sae values obained as inpu. I hen compues he marices Q, H and R. Furhermore, we needed o use he log of F because he consrain mus be sais ed given ha we log ransformed he variables and so we de ne: ~F (^z ) : ~F J (^z ) = log F J (^z ) if X J 6= ~F J (^z ) = F J (^z ) if X J =. Finally noe ha we use he noaion I x;z for he selecion marix such ha I x;z ^z = ^x. 3. Solve he LQ problem This hird sep maximizes (3.) under he log-linear approximaion o he consrains ^X + X b = D F ~ (bz bz) h i DG (bz bz) + E ^DG(bx+ bx) = where again he derivaives D F ~, DG, ^DG are wih respec o ^z, ^z, ^x +, respecively. The code compues analyically hese derivaives ha are evaluaed imposing he seady-sae condiion and a seady-sae values. By assuming a process for he shocks as + = S + " + where " + is a vecor of whie-noise shocks, he se of condiions ha is used o deermine he opimal pah is = I u;z QIy;z ^y I u;z R I y;z by + I u;z QI;z + I u;z RIy;z bw bw + by + I u;z RI;z + I u;z R I;zS (3.2) E ^y + +I u;z D ~ F + I u;z DG ' = I x;z QI y;z ^y +I x;z R I y;z by E ^y + + I x;z QI;z + I x;z RIy;z bw bw by + I x;z RI;z + (3.3) I x;z R I ;zs + I x;z D ~ F + I x;z DG ' + ^DG d = I X;z QIy;z E ^y + +I X;z R I y;z E r + by + I X;z QI;zS + I X;z RIy;z ^y by by + I X;z RI;z + I X;z R I;zS 2 (3.4) + I X;z D F ~ E + + I X;z DG E ' + = I X;y E ^y + by + D F ~ Iy;z ^y by + D F ~ I;z (3.5) 8

9 Linear-Quadraic Approximaion o Opimal Policy = DGI y;z ^y by + DGI ;z + ^DGI x;y E ^y + by (3.6) e can wrie more compacly he above sysem as where = w + ^y (3.7) = d + ' (3.8) = E ^y + r (3.9) = S " (3.) AE k + = Bk + C" (3.) 2 k 6 4 r ^y w ' by d The code inpus he above en condiions and nds represenaion (3.). Then sandard packages as REDS-SOLDS can be used o nd he opimal pah Check for Second Order Condiions I is possible o evaluae second order condiions of he soluion of he linear-quadraic problem and he code prins a message wih a con rmaion of heir veri caion or, insead, a warning, if hese condiions are no sais ed. 3.2 Oupu of he programme The code provides he following oupus:. afer solving he LQ problem he code yields marices A, B and C for he soluion in (3.); 2. provides he Q and R marices characerizing he approximaed objecive funcion; 3. prins a message abou he saus of he second order condiions; 4. if Gcheck is rue, prins a message regarding he exisence and uniqueness of he soluion; 5. reurns he number of ari cial variables creaed and he enire lis of labels for he k vecor. 7 The seup is no immediaely in he form of he GENSYS rouine bu can be adaped as he code iself does for he Gcheck opion. 9 by by 3 7 5

10 Linear-Quadraic Approximaion o Opimal Policy 4 Opimal policy in a model wih moneary fricions In his secion we presen a model ha follows closely Benigno and oodford (25a) wih he change ha we inroduce moneary fricions as suggesed in Schmi-Grohé and Uribe (24). Given ha he model is no di eren in any oher respec, here we presen only is main characerisics. 4. The model The represenaive household maximizes X U = ~u (C ; ) = Z ~v (H (j) ; ) dj ; where C is a Dixi-Sigliz aggregae of consumpion of each of a coninuum of di ereniaed goods: Z C c (i) di ; wih he elasiciy of subsiuion equal o >, and h (j) is he supply of labour of ype j. The funcional forms for ~u and ~v are he following ones: ~u (C ; ) ~v (H ; ) ~ C C ~ ~ ; + H+ H + : e follow Schmi-Grohé and Uribe (24) in modelling moneary fricions by assuming ha money faciliaes consumpion purchases. In paricular, we impose a proporional ransacion cos o consumpion purchases, s (n ), ha depends on households consumpionbased money velociy, n P C =M. The exac speci caion for s () is given by s (n) = an + b n 2 p ab: The budge consrain is herefore ( + s (n )) P C + M + B = + i m M + ( + i ) B (4.) + Z (j) H (j) dj + Z (i) di + T ; where is a shock o he ransacion coss wih mean one, i is he ineres rae on bonds and i m is he ineres rae paid on money balances and se by he moneary auhoriies. e furher have P C Z p (i) c (i) di;

11 Linear-Quadraic Approximaion o Opimal Policy wih p (i) as he price of good i and P is he Dixi-Sigliz aggregae price index, Z P p (i) di : (4.2) Using o denoe he Lagrangian muliplier of he budge consrain in he consumer s problem, we ge he following equaions: ~u (C ; ) = [ + P s (n ) + s (n ) n ] ; (4.3) p (i) c (i) = C ; (4.4) P = ( + i ) E + ; (4.5) s (n ) n 2 = i i m + i : (4.6) This las equaion conveys he money demand of his economy. Noice ha i depends on he spread beween he ineres rae on bonds relaive o he ineres rae on money balances. This is crucial because i allows us o disinguish beween he moneary fricions model and he cashless limiing case as in oodford (998). In he case wih moneary fricions, which will be our baseline case, he moneary auhoriy simply ses he ineres rae on money balances o zero and hen uses he money supply o manage he money marke so as o in uence he ineres rae in he bonds marke. Therefore we can hink of he operaional arge as i and he way o achieve ha lef o operaions deparmen of he moneary auhoriy. In he cashless limiing case hen we assume ha money is no an issue in he economy and herefore money saiaion is sais ed, meaning ha we mus have he wo ineres raes equal. And so again we can proceed as if he cenral bank acually conrols i. In erms of modelling his and allowing he model o ow smoohly ino he compuer codes ha we are proposing in his paper we se wih and (4.6) becomes i m = i ; (4.7) s (n ) n 2 = ( ) i + i : (4.8) e hen assume ha = as he benchmark case and, in he opposie case, =, we recover he cashless limiing case, in which n = n and s (n ) =, for all. Labour marke condiions allow he households o charge an exogenous markup so ha we have he wage deerminaion by w (j) = ~v (H (j) ; ) : (4.9)

12 Linear-Quadraic Approximaion o Opimal Policy Each di ereniaed good is supplied by a monopoly. The echnology is common for all goods: y (i) = A f (h (i)) = A h (i) = ; (4.) where >. In equilibrium oupu is equal o spending by he households and by he governmen hence Y = ( + s (n )) C + G ; (4.) where he real governmen expendiures are exogenous. Hence we can wrie consumpion as Noice ha he real money balances are deermined by C = Y G + s (n ) : (4.2) m Y G n ( + s (n )) ; (4.3) which is a residual equaion wih he only purpose of deermining m M =P. Each rm ses prices a a given period wih probabiliy ( ), wih <. Hence here is a fracion of prices (by he law of large numbers) ha remain unchanged. The sochasic discoun facor is in equilibrium equal o ;T = T T ; (4.4) so ha he rms maximize he ne presen value of heir pro s: X E T = where he afer-ax pro funcion is T ;T p (i) ; p j T ; P T ; Y T ; T ; p (i) ; p j ; P ; Y ; ( p (i) ) p (i) Y h (i) w (i) : P The opimal pricing can be expressed as (p ) +! = E P T = T P (+!) T A (+!) T ;T T Y +! T P E T = T ;T ( T ) Y T PT H T = T Bu, in order o be able o apply he compuer rouines presened wih his paper we need o have all expressions in a recursive way. In order o ge ha we can wrie hen p P = K F 2 +! ; (4.5) :

13 Linear-Quadraic Approximaion o Opimal Policy and de ne F E X T = K E X T = () T PT ( T ) T P T Y T ; (4.6) P (+!) PT : (4.7) () T T y (Y T ; T ; T ) Y T T P hence The price index evolves hrough he following law of moion: h P = Now inser (4.5) ino his equaion o ge P = " ( ) ( ) (p ) + P F K = F K +! P + P where P =P. Finally he desired recursive represenaions of F and K are and K = i : (4.8) # ; +! ; (4.9) F = ( ) P Y + E + F + ; (4.2) where is a measure of he price dispersion, de ned as: y (Y ; ; ) Y + E h (+!) + K + i ; (4.2) Z (+!) p (i) di: For he law of moion of he price dispersion, from is de niion, P P (+!) = P (+!) + ( ) (p ) (+!) ; because we know ha for he indusries in which prices do no change heir relaive prices are sill he same. ih some more manipulaion we obain = (+!) (+!) + ( ) : (4.22) 3

14 Linear-Quadraic Approximaion o Opimal Policy wih Finally, we rewrie he uiliy funcion in a more convenien way, 8 U = X = [u (Y ; n ; ) (Y ; ; )] ; (4.23) C u (Y ; n ; ) = ~ ~ Y G ~ ; (4.24) + s (n ) (Y ; ; ) = + Y +! A +! H ; (4.25) where! ( + ). 9 Given ha we will need o consider explicily he value of he Lagrangian muliplier i is convenien o consider i normalised in such a way ha i converges o a consan value in seady sae. Therefore we de ne ~ P. Taking his ransformaion ino accoun we can summarise all he relevan equaions in he economy as follows: + ~ = s (n ) u + s (n ) + s Y (Y ; n ; (n ) n ) ; (4.26) " # ~+ ~ = ( + i ) E ; (4.27) s (n ) n 2 = ( ) + i + i ; (4.28) m Y G ( + s (n )) n ; (4.29) F K = +! ; (4.3) F = ( ) ~ Y + E + F + ; (4.3) K = y (Y ; ; ) Y h i + E (+!) + K + ; (4.32) = (+!) (+!) + ( ) : (4.33) 8 For he derivaion of hese expressions, refer o Appendix A.2. 9 As menioned in Benigno and oodford (25a), de ned as is,! refers o he elasiciy of real marginal cos wih respec o a rm s own oupu. 4

15 Linear-Quadraic Approximaion o Opimal Policy Finally we can summarise all he shocks in his economy by making explici he vecor. This vecor includes he shocks already normalized so ha hey have zero mean, he assumed value in seady sae. e can hen wrie h i ^ ^ ^G C b H b ^A ^ : 4.2 Model in he LQ form In order o use he code described in secion 3 we need o mach he srucure presened in secion 2. In he curren seup we can de ne F ; K ; ; ~ ; x u (i ; Y ; n ; m ) ; X ; and he vecor of consrains G is de ned in ( ) and he predeermined consrain, wih he implici de niion of F () is presened in (4.33). e mus have he funcional U () wih he argumens fx ; u ; X ; g bu, in his model, inside i we have (Y ; ; ) hence U () depends on which is one elemen of he X + vecor. In order o solve for his issue we can plug F () ino U () so ha i will depend on, i.e. X. This subsiuion mus be done only afer he consrains are creaed, in paricular (4.32). This shows ha he framework necessary for applying he proposed compuer rouines is no so resricive as i migh look a a rs glance. 4.3 Opimal policy In his secion we compare he impac of he moneary fricions in he model, saring wih he impac in he opimal seady sae. e hen analyse how he opimal moneary policy di ers in he case wih moneary fricions and in he cashless limiing case. For he parameerizaion of he model we will follow he baseline parameers proposed by Benigno and oodford (25a) for easier comparison and for he moneary fricions we use he values esimaed in Schmi-Grohé and Uribe (24). All of he parameers are shown in Table Opimal seady sae The opimal seady sae is a focal poin of he enire analysis as he approximaions are all derived around hese numbers. Given is imporance, we make here a horough analysis of i in he mos relevan scenarios considered. These normalizaions are he following ones: = ( ) exp ( ^ ); = exp (^ ); G = ^G if G = and G = G exp ^G oherwise; C = C exp C b ; H = H exp H b ; A = A exp ^A ; and = exp (^ ), in which we assume =. Noice ha,, G, C, H, A and refer o he seady sae values of he respecive variables. In he calculaion of he seady sae no approximaion is made. 5

16 Linear-Quadraic Approximaion o Opimal Policy The rs resul worh menioning is ha under he cashless limiing case, =, in he opimal seady sae we nd ha in aion is always se o zero and herefore he nominal (and real) ineres rae is hen se o equal he discoun rae of he households, roughly 4% in annual erms. hen we consider he presence of moneary fricions, however, i is always he case ha we ge some de aion in he opimal seady sae, a he same ime ha he nominal ineres rae falls shor of he discoun rae of he households. The degree of opimal seady sae de aion depends, among oher hings, on he price sickiness of he economy and on he ax rae. To help undersand he opimal seing of boh he nominal ineres and in aion raes in seady sae, Figure presens hese agains di eren levels of price rigidiy, and for wo di eren levels of he ax rae. In panel A we have he baseline scenario, in which he ax rae is se o.2. In panel B we presen he resuls under he level of axes ha would eliminae he disorions generaed by he monopolisic compeiion. 2 The rs resul is ha he degree of opimal de aion in seady sae decreases wih price rigidiy () and, in he limiing case, as he probabiliy of no adjusing prices converges o uniy he de aion level gradually converges o zero. Therefore, on his basis, he divergence beween he moneary fricions and cashless limiing cases opimal de aion increases wih price exibiliy. This evoluion of seady sae in aion rae is also presen in Khan, King, and olman (23) and Schmi-Grohé and Uribe (24). hen commening on he Friedman rule Khan e al. (23) menion ha due o he Keynesian fricions he nominal ineres rae is kep above zero. Figure is in accordance wih his indeed bu if hose fricions are weak enough hen in our model he nominal ineres rae begin o fall below zero. Because ha is no operaionally possible hen he ineres rae his he so called "lower bound" and is se o be zero, as in he Friedman rule. Therefore he de aion akes place, in order o pu he real ineres rae in line wih he ime discoun rae. However his happens only if he Calvo probabiliy of no changing prices,, falls o 6% or below, which are raher low numbers for his parameer. If, insead, he ax rae is se o eliminae he disorions creaed by he monopolisic compeiion hen he gure shows ha he nominal ineres rae is always posiive, converging o zero, bu always from above. ha happens in our model is ha he moneary fricions are u-shaped as a funcion of he money velociy of consumpion, no being resriced o be equal o zero afer he minimum. Therefore if he disorions are eliminaed he e cien opimum for oupu and real money balances can hen be achieved. hen, insead, he monopolisic disorions are no o se, reducing real money balances reduces oupu and hen his is no he direcion o go. So real money balances should be increased o he possible exen and his will raise seady sae oupu bu no up o he e cien level. Because he moneary fricions may have a negaive slope, for values of he velociy below he e cien one, hen he ineres rae is pushed o he lower bound. Noice ha Schmi-Grohé and Uribe (24) use exacly he same fricions in a model very similar o ours and hey o no ge his resul (hey never hi he lower bound of he nominal ineres rae). The reason for ha is ha hey are insead 2 e acually should refer o i as subsidy rae: in order o eliminae he monopolisic compeiion disorions he ax rae mus be se equal o = = ( ) and considering > implies a negaive ax rae. 6

17 Linear-Quadraic Approximaion o Opimal Policy opimizing on boh moneary and scal policy simulaneously. e can neverheless reach he conclusion ha under he exisence of moneary fricions he Friedman rule is always valid in he exible prices case. I is also valid if axes are no se o eliminae he monopolisic compeiion disorions and he level of price sickiness is very low. To conclude he opimal seady sae analysis we repor in Table 2 he values for some variables under di eren scenarios, in order o allow a clear comparison beween he cashless limiing case and he moneary fricions one. Regarding ineres raes and in aion rae he mos relevan conclusions were already menioned before. The only hing ha we can add is ha he level of price sickiness will no in uence he opimal seady sae levels of he ineres rae or in aion in he case of he cashless limiing case, precisely because as repored already in Benigno and oodford (25a), i is always opimal o se in aion o zero. More ineresing, in he moneary fricions case, even hough he nominal ineres rae and in aion rae do change quie signi canly across di eren levels of price rigidiy, he level of price dispersion barely changes. Also noice ha he velociy of money, n, does no change much. Moreover, he value of he velociy is abou 6.3% above he saiaion level, he one prevailing in he cashless limiing case. Only when we ge very close o exible prices does he velociy begin e ecive convergence o ha level. Finally i is imporan o verify he impac of he moneary fricions in he real side of he economy, namely, in he oupu level. In his respec we noice ha he level of oupu is fairly consan across di eren levels of price sickiness and always lower under he case of moneary fricions relaive o he cashless limiing case. However he di erence in oupu is no very signi can, abou.6% of he cashless limiing level. So on his measure we can say ha he moneary fricions do no have a signi can impac on he real side of he economy Opimal responses o shocks In his secion we show he behaviour of opimal policy in face of he various shocks presen in he economy. Noice however ha we do no show he responses o shocks o he labour supply jus because hey are qualiaively exacly he same as a echnology shock in an economy in which labour is he only inpu. The only di erences are numerical due o di eren scales bu ha is no relevan in he curren analysis. 3 In he following analysis he verical axis in all he gures should be inerpreed as percenage poins and he responses of he ineres rae and in aion are annualized. In he baseline case we use a persisence level corresponding o a.7 coe cien for AR() process describing he logs of he shocks. e rs compare he wo main cases of our model: he fricions and cashless economies. The responses under he wo cases are shown in Figures 2 hrough 7. The main resuls can be summarised as follows: rs, he responses are very similar in he wo scenarios considered, wih exacly he same paern in boh cases and only minor di erences in magniude; second, 3 Indeed he wo shocks appear always ogeher in he reduced se of equaions for he economy wih he form A +! and ha is why he qualiaive e ecs are exacly he same and he scales di eren. H 7

18 Linear-Quadraic Approximaion o Opimal Policy he consumpion based velociy of money (n ) says consan in he cashless case and mimics he paern of he ineres rae in he fricions case and his is he case in all scenarios and shocks excep for he shock o he ransacion coss; and, hird, he shock o he ransacion coss can be considered negligible in heir impac on he oher variables. To be more speci c abou some of hese issues we analyse now he responses o each of he shocks. In Figure 2 we can observe ha an increase of one percenage poin in he ax rae leads rms o reduce producion and increase prices. In response o he shock he moneary auhoriy (opimally) ses higher ineres raes o bring he in aion under conrol, so ha he nal impac on he in aion rae is only.8% in boh scenarios and quickly is brough o levels very close o zero. Indeed he pah of in aion is he same in he wo cases. However here are sligh di erences in he policy ha led o his oucome. Ineres raes should increase 3 basis poins more on impac in he moneary fricions case, which hen leads o a minor worsening of he recession in ha scenario. The responses o an increase of he price markup of % is shown in Figure 3. Once more he e ecs in he variables are he expeced ones. The increase in he price markup leads o in aionary pressures ha lead he moneary auhoriies o use conracionary policy hrough he increase of he ineres rae in 5 basis poins. This in urn will slow down he economy, leading o a fall in oupu of he order of.8% and conrol in aionary pressures. This case is in fac very similar o he case of an increase in he ax rae. In he laer, however, he similariies beween he wo scenarios, are even more pressing, being hard o disinguish he wo. An increase in he governmen expendiures of he order of % of he GDP is shown in Figure 4. The increase in governmen expendiures is an aggregae demand (AD) expansion ha generaes in aionary gap, promply closed by he moneary auhoriies wih a subsanial increase in he ineres rae. The auhoriies reac so srong ha he in aion no only is brough under conrol bu i acually goes o zero from below, on impac. Due o ha policy he impac on oupu is raher small and he muliplier e ec simply is erased, so ha a % increase in governmen expendiures led o only.2% increase in oupu. Again he di erences beween he moneary fricions and he cashless limiing cases are very small, consising of an increase in he ineres rae of 2 basis poins in he fricions scenario and abou 22 basis poins in he cashless case. On oupu here seems o be some di erence in magniude bu no signi can a all. A similar paern ensues in he case of a posiive shock o household preferences for consumpion, which also enails an expansion in AD, as shown in Figure 5. The responses o an increase in % in he produciviy of labour are presened in Figure 6. As should be expeced he oupu increases wih he produciviy. However he magniude of he increase is remarkable, of he order of 2.5%, compared o a % shock. The reason for his is ha as he oupu has some endency o increase on impac, due o sluggish price adjusmen, no all rms can adap fully and so here is sill a recessionary gap ha is promply closed by he moneary auhoriies wih a very aggressive ineres rae policy. Indeed he ineres rae falls by 6 basis poins in boh scenarios considered. This closes he gap and prevens he prices from falling (hey acually marginally increase on impac). 8

19 Linear-Quadraic Approximaion o Opimal Policy Again he responses are idenical in he wo scenarios. The in aion response seems a bi more di eren bu given he scale ha disincion can be considered o negligible. Finally, he responses o he ransacions cos shock are depiced in Figure 7. The only hing worh menioning is ha all he responses have scales ha make he pahs meaningless and basically we can say ha all relevan variables are essenially se o zero. The excepion is he velociy, n, which falls on impac under he fricions scenario, and hen gradually recovers. This is he only deparure from he he idea ha he velociy is a scaled version of he pah of he ineres rae. This is also consisen wih he fac ha moneary fricions have very lile impac in he economy. Acually i can be inerpreed as he resul ha he ransacion coss are very close o zero in he opimal seady sae. e now bring he analysis furher and connec i o he discussion in he previous subsecion relaing he wo cases for axes: one in which he ax rae is simply se a 2% in seady sae (and herefore he monopolisic disorions are no o se) and a second case in which he ax rae is se so as o o se hese disorions, becoming acually a subsidy. The resuling responses in he case of moneary fricions under disored seady sae and no disorions are presened in Figures 8 hrough 2. The main conclusion ha we can ake is ha he ineres rae is less volaile in he non-disored scenario relaive o he disored one. In paricular, in response o an increase in he ax rae, an increase in he price markup or an increase in he governmen expendiures, he ineres rae increases always less han in he disored case, or even decreases on impac, allowing he in aion deviaions from seady sae o slighly be higher a he expense of higher oupu levels. In he case of a posiive echnological shock he ineres rae is decreased in boh cases bu less so in he disored one. This leaves in aion essenially a zero in boh cases even if he pahs are symmeric around he seady saes. However oupu is slighly higher, relaive o he respecive seady sae, in he disored case. In he case of a consumpion preferences shock, here is absoluely no change wha so ever. Finally, in order o close he analysis we compare he opimal policy o an alernaive policy. Tha alernaive policy could be de ned in muliple ways bu jus as an illusraive example we use a simple Taylor rule wih he following form: + i = ( + {) y Y 4 Y where and y have he usual inerpreaions and he division by four of he laer is mean o keep he inerpreaion of y in annual erms (given ha each period in his model is considered o be a quarer). e use, jus for illusraive purposes, coe ciens equal o.5 and.5 respecively. Noice ha hese were no opimized in any respec. This example shows ha i is also easy in he framework proposed o evaluae a simple rule and compare i o he opimal policy responses, as presened in Figures 3 hrough 8. The comparison beween he simple ineres rae rule o he opimal policy in his economy reveals ha under increases o he ax rae (Figure 3) or he price markup (Figure 4) oupu, in aion and nominal ineres raes are higher on impac under he alernaive Taylor rule compared o he opimal policy. The reason for his is ha in aion is relaively 9

20 Linear-Quadraic Approximaion o Opimal Policy higher han he ineres rae so ha he real ineres rae is acually lower under he alernaive policy. This implies ha probably he reacion of he nominal ineres rae o in aion is no as aggressive as i should opimally be. In he case of he increase of governmen expendiures (Figure 5) we can observe under he alernaive rule higher oupu level and lower ineres rae, wih he in aion rae very close o he seady sae. This migh imply ha he ineres rae is acually no aggressive enough o oupu changes. In he case of he consumpion preferences shock nohing changes, hough. The responses o a posiive produciviy shock (Figure 7) under he alernaive rule yields radically di eren responses han under he opimal policy. Now he oupu expansion is subsanially rimmed down due a much higher ineres rae. This leads o a signi can level of de aion in he shor run (.25%). This shows ha he ineres rae falls in response o de aion bu essenially real ineres raes are lef a he seady sae level and so oupu does no expand as much as before. This shows ha he moneary policy is raher passive, no reacing o he recessionary gap ha formed (he poenial oupu expanded much more han he oupu did). This is he resul of he ineres rae rule aking ino accoun he oupu deviaions from seady sae insead of he acual oupu gap. In ligh of his inerpreaion we can review he oher ones and conclude ha his seems like a likely cause of he previous di erences from opimal policy as well. So i could be ineresing o incorporae an ineres rae rule in erms of oupu gap, insead of jus oupu deviaions from seady sae. 5 Opimal moneary policy in a Neo-Keynesian model (Smes-ouers (23)) In his secion we discuss resuls on he Linear-Quadraic opimal moneary policy in a slighly simpli ed version of he closed economy model in Smes and ouers (23). e rs lay ou he model equaions and he equilibrium condiions de ning he LQ programme, as well as he calibraion used. e hen describe he impulse responses of seleced variables o he shocks in he model, boh under LQ opimal policy and under a sandard moneary policy rule. 5. The model 5.. Consumers A ime, he uiliy funcion of he rapresenaive agen is U = E " X s s U C +s ; C +s ; E B V L h +s; E L ; E B # ; (5.) 2

21 Linear-Quadraic Approximaion o Opimal Policy wih U C +s ; C +s ; E B E B (C +s C +s ) C ; C Z +L V L h +s; E L ; E B E B L Lh +s E L dh: + L where i is undersood ha households obain uiliy from consumpion of an aggregae index C ; relaive o an inernal habi depending on pas aggregae consumpion, while receiving disuiliy from labour L h. Uiliy also incorporaes a consumpion preference shock E B and a labour supply shock E L. Each household h maximizes is uiliy funcion under he following budgeary consrain: B P ( + i ) + I + C = B R + ( ;) h L h dh + A + T T + R k CU K (CU ) K ; P P where B is a nominal bond, h is he wage, A h is a sream of income coming from sae coningen securiies, T T h and ; are governmen ransfers and ime-varying labour ax respecively, and R k CU K (CU ) K represens he reurn on he real capial sock minus he cos associaed wih variaions in he degree of capial uilizaion. As in Chrisiano, Eichenbaum, and Evans (25), he income from rening ou capial services depends on he level of capial augmened for is uilizaion rae and he cos of capaciy uilizaion is zero a full capaciy (() = ). Separabiliy of preferences and complee nancial markes ensure ha Households have idenical consumpion plans. The rs order condiion relaed o consumpion expendiures is given by = U C; + E U C2;+ (5.2) where is he Lagrangian muliplier associaed wih he budge consrain. The rs order condiions corresponding o he quaniy of coningen bond is: P = ( + i )E (5.3) 5..2 Labour supply and wage seing + P + Each household is a monopoly supplier of a di ereniaed labour service. For he sake of simpliciy, we assume ha he sells his services o a perfecly compeiive rm (Labour Packers) which ransforms i ino an aggregae labour inpu using he following echnology: Z L = L h dh : The household faces a labour demand curve wih consan elasiciy of subsiuion: L h h = L ; 2

22 Linear-Quadraic Approximaion o Opimal Policy where he aggregae wage is given by Z = h dh : The real wage seing equaions can be wrien in he following recursive form (see Appendix B.2 for a derivaion): Z ; = L ew Z ; (+ L ) ( ) = ( ) P w Z 2; Z 2; = ( (+ L ) R; L + L 5..3 Invesmen decisions = R; R; E L E B + ;+ ; ) R; L + E Z 2;+ + + (5.4)! ; (5.5) (+ L ) A ; (5.6)! : (5.7) The capial is owned by households and rened ou o he inermediae rms a a renal rae R k. Households choose he capial sock, invesmen and he capaciy uilisaion rae in order o maximize heir ineremporal uiliy funcion subjec o he ineremporal budge consrain and he capial accumulaion equaion given by: K + = ( )K + E I S I I I ; (5.8) where is he depreciaion rae and S() he adjusmen cos funcion, where i is assumed ha I S = S( I ) = i I I 2 ( I ) 2 : I Firs-order condiions resul in he following equaions for he real value of capial, invesmen and he capaciy uilisaion rae: Q = E + Q + ( ) + R +CU k + (CU + ) E Q (5.9) 22

23 Linear-Quadraic Approximaion o Opimal Policy I = Q S I " + +E Q + I I+ I S I I 2 S I+ I I E I E I + # (5.) R k = (CU ) (5.) 5..4 Final goods secor Final producers are in perfec compeiion and aggregae a coninuum of di ereniaed inermediae producs. The elemenary di ereniaed goods are imperfec subsiues wih elasiciy of subsiuion denoed, such ha Z Y = The aggregae price index is de ned as Z P = p(h) " Y (h) " " " dh : " " dh ; and domesic demand is allocaed across he di ereniaed goods as follows " p(h) 8h 2 [; ] Y (h) = Y: P 5..5 Inermediae rms Inermediae goods are produced wih a Cobb-Douglas echnology as follows: 8h 2 [; ], Y (h) = E A (CU (h)k (h)) L (h) ; where E A is an exogenous echnology shock and is a xed cos ensuring ha pro s are zero in he seady sae. Firms are monopolisic compeiors and produce di ereniaed producs. In each period, a rm h faces a consan probabiliy, P, of being able o reopimize is nominal price. This probabiliy is independen across rms and ime. The average duraion of a rigidiy period is P. If a rm canno reopimize is price, he price evolves according o he following simple rule: H P p (h) = H p (h): P 2 Therefore, rm h chooses ~p (h) o maximize is ineremporal pro " X E j P P +j +j p ;+j Y +j (h) ( )~p (h) P j= 23 P MC +j P +j!# ;

24 Linear-Quadraic Approximaion o Opimal Policy where ;+j = j +j P P +j is he marginal value of one uni of money o he household, MC +j is he real marginal cos, is a ime-varying ax on rm s revenue. Due o our assumpions on he labour marke and he renal rae of capial, he real marginal cos is idenical across producers, MC = ( ) E A ( ) R k : ( ) In our model, all rms ha can reopimize heir price a ime choose he same level. The rs order condiion associaed wih he rm s choice of P ~ (h) is " X E j ;+j Y +j (h)p +j ( ) ~p!# (h) P +j P +j " P +j P P " MC +j = : j= hen he probabiliy of being able o change prices ends owards uniy, his implies ha he rm ses is price equal o a consan markup (wih = ) over marginal cos as in he exible-price model. Oherwise he rm imposes his markup o he weighed-average of marginal coss over ime. Only a fracion p of producers can reopimize is price, each period. So he aggregae producer-price-index has he following dynamic: " P " P = p P + ( p ) ~p " (h) : P 2 This price seing scheme is easily rewrien in he following recursive form: Z ; ( p ) = p Z 2; ; (5.2) " # + Z ; = MC Y + p E Z;+ ; (5.3) " # Z 2; = E P + Y + p E Z2;+ : (5.4) Capial labour raio is equalized across rms and linked o he relaive cos of facors: L R k CU K = : 24

25 Linear-Quadraic Approximaion o Opimal Policy 5..6 Governmen Public expendiures are subjec o random shocks E G. The governmen nances public spending wih labour ax, produc ax and lump-sum ransfers, P E G ; L P Y P T T = : Specifying he ineres rae rules followed by he moneary auhoriies nally closes he model. In he case of he Taylor rule he exac expression is, much like in he previous model, + i + { = 2 r + i + { 4 Y Y 3 r y 4 5 : This is no exacly he same policy rule ha Smes and ouers (23) consider bu i is one rs approximaion o i. In order o make i close o heirs we consider r = :85, = :5 and y = :8 (hence y =4 = :2, similar o heir esimaes). One big di erence is ha we are no considering he oupu gap bu only deviaions from seady sae oupu. This can have an imporan impac in policy and herefore shall be invesigaed in fuure research Marke clearing condiions Aggregae producions are obained using he CES aggregaor Z dz ; and labour demands are given by he following relaions: as = ( by Y D = E A (CU K ) (L ) ( ) Y ; ) Y implies ha pro s are zero in seady sae. Price dispersion is de ned as D = R D = ( p ) Z ; Z 2; (h) (+L) p(h) P dh, and follows he law of moion given + p D ; (5.5) while D ; = R dh and he derivaion of he relaed equaions follows. Aggregae demand is given by Y = C + I + E G + (CU ) K 25

26 Linear-Quadraic Approximaion o Opimal Policy 5.2 Summary of model equaions The se of srucural equaions is given by: = E B (C hc ) C E E+ B (C + hc ) C (g) P = ( + i )E (g2) + P + Q = E + (Q + ( ) + (CU + ) CU + (CU + )) I = Q S I " + +E Q + I I+ I S I I 2 S I+ I I E I E I + # E Q (g3) (g4) R k = (CU ) (g5) L = CU (CU ) K (g6) Y = C + I + E G + (CU ) K (g7) D C + I + E G + (CU ) K = E A (CU K ) (L ) ( ) Y (g8) Z ; ( Z 2; p ) = p 2 Z ; = MC Y + p E 4 + +!! Z ;+ " # Z 2; = E P + Y + p E Z2;+ 3 (g9) 5 (g) (g) Z ; = E L LL + L 2 + E (+ L ) 4 + (+ L ) Z ;+ 3 5 (g2) Z 2; = E L + E " + Z 2;+ # (g3) 26

27 Linear-Quadraic Approximaion o Opimal Policy Z ; ( ) w Z 2; (+ L ) = K = ( )K + E I D = ( p ) Z ; Z 2; D ; = ( ) S I I + p D (+ L ) w Z ; Z 2; I (+ L ) (+ L ) (+ L ) + D ; knowing ha he following de niion holds: and ha and MC = 5.3 The model in LQ form ( ) (CU + ) E A ( ) ( ) : S( I ) = i I 2 ( I ) 2 I (CU ) = rc Rk ss exp(rc (CU )): (g4) e now need o mach he srucure presened in secion 2 one more ime. Firs of all noice ha in he model here are wo elemens of + showing up in he G-ype consrains, which would violae he framework for he LQ funcion. In order o overcome his issue we de ne wo new variables o include in he x vecor, F B and F I. In his way hey will show up in x + and no violae he framework. In order o connec hese o he rue shocks we inser wo exra equaions o he G-ype consrains, E B = F B ; and E I = F I : e can hen de ne: x C ; ; Q ; I ; CU ; R k ; ; Z ; ; Z 2; ; Z ; ; Z 2; ; ; F B ; F I ; u (Y ; L ; i ) ; X (K ; D ; D ; ; I ; C ; ; ; i ) ; (f) (f2) (f3) (g5) (g6) and he vecor of consrains G is de ned in (g-g4), added by (g5-g6) and he predeermined consrain, wih he implici de niion of F () is presened in (f-f3). 27

28 Linear-Quadraic Approximaion o Opimal Policy 5.4 Opimal responses o shocks In his secion we compare he behaviour of opimal policy in face of he various shocks presen in he economy, using he behaviour under a sandard Taylor rule as he comparaive benchmark. 4 The calibraion used for he parameers values is presened in Table 3. 5 Resuls are shown in Figures (9-24). 6 In he case of produciviy shocks (Figure 9) opimal moneary policy exacerbaes he e ec on oupu and consumpion componens of he produciviy shock, relaive o he Taylor rule benchmark. Under LQ opimal policy oupu, consumpion and invesmen are hus signi canly higher han under he Taylor rule case, in paricular on impac. This leads o seadily higher pah for in aion and real wages under LQ policy. Conversely, LQ policy implies lower nominal ineres raes relaive o he Taylor case during he rs quarers. A possible inerpreaion of hese resuls is ha LQ policy aims a maerisalising he e ciency gains ha are hindered by nominal rigidiies. By conras, he Taylor rule, by muing he favourable impac of he posiive produciviy shock, is over-resricive relaive o LQ. For he preference shock (Figure 2), by conras o he previous case, moneary policy largely couners he in aionary e ecs of he preference shock on oupu ha akes place under he Taylor rule policy. In order o sabilise in aion LQ opimal policy couners he expansion in oupu and consumpion ha would ake place, o he exen of acually reversing he e ec and generaing a conracion in hese variables. This is achieved by a higher pah for nominal ineres raes in he rs quarers. In exchange of his, LQ policy delivers a lower pah for labour e or and more sable prices. In he case of he governmen purchase shock (Figure 2), opimal moneary policy is again resricive relaive o he Taylor case. However, in his case he e ec on oupu componens is purely in composiion, wih he pah for aggregae oupu being similar under LQ and Taylor policies. Indeed, he LQ policy diminishes overall he exen of invesmen crowding-ou, while i has a more ambiguous e ec in erms of consumpion crowding-ou relaive o Taylor. The negaive impac on wages is diminished under LQ compared o Taylor. These e ecs are, again, achieved wih a relaively conracionary moneary policy up-fron, relaive o Taylor policy. In boh cases in aion is sabilised around seady sae levels under boh policies. Figure 22 depics he case of a (negaive) labour supply shock. In his case he deparure beween LQ and Taylor policies is relaively large compared o he oher shocks. Under Taylor all variables excep wages are a eced o a raher limied exen, i.e. under Taylor he 4 Unlike he case of he model wih moneary fricions discussed above, in his case he calculaion of he model s seady-sae is sraighforward. In his case he seady-sae is disored as seady-sae price and wage mark-ups are assumed o be above one and heir e ec on welfare is no compensaed wih lumpsum ransfers. Therefore, in his case, he focus is he comparison of model dynamics under opimal and non-opimal moneary policies, around a disored seady-sae. 5 The calibraion used ensures ha he impulse respond funcions under his speci caion for he Taylor rule moneary policy case are qualiaively similar o he ones in Smes and ouers (23). 6 The responses of he ineres and in aion raes were annualized so ha we can read hem in erms of annual percenage poins. 28