Optimal Monetary Policy in the New Keynesian Model

Transcription

1 Opimal Moneary Policy in he New Keynesian Model Eric Sims Universiy of Nore Dame Spring Inroducion These noes describe opimal moneary policy in he basic New Keynesian model. 2 Re-wriing he Basic Model The basic NK model can be characerized by wo main (log-linear) equaions: he Phillips Curve and he Euler/IS equaion. Here I have aken he libery of seing he elasiciy of ineremporal subsiuion o uniy: r f π = γ(ỹ Ỹ f ) + βe π +1 Ỹ = E Ỹ +1 r To economize on noaion, le s define X Ỹ Ỹ f as he oupu gap. Similarly, le s define as he flexible price real ineres rae (or naural real ineres rae ) as ha rae ha would obain if prices were fully flexible. We can solve for his by looking a he Euler equaion: r f = E Ỹ f +1 Ỹ f Because of he assumpion on preferences, in paricular ha he coefficien of relaive risk aversion is 1, we know ha labor hours would be consan if prices were flexible, so we know ha he flexible price level of oupu evolves exogenously in line wih he level of echnology, wih ỹ f = ã. If we assume ha echnology obeys an AR(1), hen we can model he flexible price equilibrium level of oupu as following he same AR(1). Ỹ f = ρỹ f 1 + ε This means ha we can solve for he flexible price real ineres rae as: r f = (ρ 1)Ỹ f 1

2 Plugging in he process for Ỹ f and simplifying we ge a process for he naural rae of ineres: r f = (ρ 1)ρỸ f 1 + (ρ 1)ε r f = ρ rf 1 + (ρ 1)ε We can hen summarize he main equaions of he model as follows: π = γ X + βe π +1 (1) X = E X+1 ( r r f ) (2) r f = ρ rf 1 + (ρ 1)ε (3) In he background here is also (i) a money demand relaionship and (ii) a Fisher relaionship. For now, we can hink abou he cenral bank effecively being able o choose r, given a choice of ĩ and an implied pah for E π +1. Given ha, as well as r f (which is he exogenous driving force), X and π will be deermined. 3 Disorions and Welfare There are wo welfare-reducing disorions in he NK model, one of which is essenially long run and he oher which is shor run. The long run disorion is ha he flexible price level of oupu will be lower han wha would obain in he firs bes. This is because, in he flexible price version of he model, firms will se price equal o a markup over marginal cos. Hence here will be oo lile employmen. The shor run disorion is due o price sickiness, and leads o non-opimal flucuaions in relaive prices. We assume ha he cenral bank is concerned wih he shor run disorion and ha he long run disorion has been aken care of via some kind of Pigouvian ax. This works ou o a subsidy for labor, equal o he inverse price markup. This means we can inerpre ỹ f as he opimal equilibrium value of oupu from he perspecive of he cenral bank. This means ha, oher hings being equal, he cenral bank would like o eliminae oupu gaps. 4 Opimal Policy In addiion o disliking oupu gaps, we also assume ha he cenral bank dislikes inflaion. We assume ha welfare of he cenral bank is a presen discouned value of a quadraic loss funcion in inflaion and he oupu gap. This loss funcion can acually be derived from aking a quadraic approximaion o household welfare, while using he linearized equilibrium condiions (see Gali or Woodford s exbooks for a formal derivaion. You may wonder why he cenral bank cares abou inflaion over and above he oupu gap (which, via he logic above, he cenral bank would like 2

3 o eliminae). If you go back o he CES aggregaor over inermediae goods, you will noe ha i is concave, meaning ha households (or he final goods firm, if you like) would like o smooh over inermediae inpus. In a flexible price world, all inermediae producers would choose he same price (e.g. hey all desire a relaive price of 1). If aggregae inflaion is differen from zero, wih price sickiness relaive prices a he inermediae firm level ge disored (e.g. here is price dispersion). This leads o a non-smooh allocaion of inermediaes, which resuls in a welfare loss. Le ω denoe he relaive weigh ha he cenral bank places on he oupu gap. In he formal derivaion, you can show ha his is equal o γ ɛ, where γ is he slope of he Phillips Curve and ɛ is he price elasiciy of demand. The cenral bank would herefore like o minimize he following: min ( 1 2 E ( π β 2 + ω X ) ) 2 = The 1/2 on he ouside is jus a scaling erm ha doesn affec he opimum bu simplifies hings a bi. As noed above, we can hink abou he cenral bank as choosing inflaion and he oupu gap, given is choice of ĩ, which hen deermines r given a pah of E π +1. This mus be done subjec o he consrain of he Phillips Curve, however. We consider wo cases. In he firs, called discreion, he cenral bank solves he one period problem each period. In he oher, called commimen, he cenral bank solves he enire problem a he beginning of ime and commis o is policy. We sar firs wih he discreion case. The problem can be wrien: Se he problem up as a Lagrangian: L = 1 2 The firs order condiions are: 1 ( min π 2 + ω π, x 2 X ) 2 s.. π = γ X + βe π +1 ( π 2 + ω X ) 2 + λ ( π γ X ) βe π +1 L = π = λ π L = ω x X = λγ Combing FOCs so as o eliminae he Lagrange muliplier, we ge: X = γ ω π (4) 3

4 Loosely speaking, his firs order condiion can be inerpreed as a lean agains he wind policy. If he oupu gap is posiive, he Fed will wan o pursue a policy in which i lowers inflaion (and vice-versa). Nex, consider he problem under commimen. Here, he objecive of he cenral bank is no jus he curren objecive, bu he presen discouned value of he flow objecive funcions. Lagrangian is: L = E = ( β 1 ( π 2 + ω 2 X ) 2 + λ ( π γ X ) ) βe π +1 The ime -1 expecaion of inflaion a ime, E 1 π, is aken as given. Hence, oher ha π, π shows up wice in he period objecive, he period consrain, and he period 1 consrain (i.e. as E 1 π ). Le s work hrough he FOCs: A L = π = λ π L = β 1 λ 1 β E 1 β π + E 1 β λ = E 1 π = E 1 λ λ 1 > π We can combine hese firs order condiions o ge: L X = X = γ ω λ X = γ ω π E X +1 = X γ ω E π +1 Sar his a he beginning of ime and subsiue forward: X = γ ω π E X1 = X γ ω E π 1 = γ ω π γ ω E π 1 E X = E X1 γ ω E π 2 = γ ω π 2 γ ω E π 1 γ ω E π 2. E X = γ ω Now, wha is he sum of he inflaion raes beween period and period? I is he (log) price level minus he price level in he period before period. Then he firs order condiion becomes: j= π j 4

5 E X = γ ω E ( P P 1 ) (5) Since his mus hold in expecaion, and here are no disurbances ha show up here eiher, i mus also hold ex-pos. This means we can ge rid of he expecaions operaor in he FOC. To simplify maers, we can also normalize he iniial price level o, so he FOC becomes: X = γ ω P (6) This looks similar o he firs order condiion under commimen, bu i feaures he price level as opposed o price inflaion. As such, his kind of rule is called a price level argeing rule. As, X, which means ha lim E P = P 1. This means ha he policy under commimen implies ha he price level always reurns o rend. Now, le s give a lile hough o he soluions under discreion and commimen. From he FOC for discreion, noe ha a policy of X = π = is consisen wih he FOC holding. Bu is i consisen wih he oher equaions of he model holding? I urns ou ha i is. If you plug hese ino he Phillips Curve, you can see ha he Phillips Curve can hold a all imes wih π = X =. From he IS equaion, his implies ha r = r f a all imes. Wih inflaion always equal o zero, his implies ha he nominal ineres rae should rack he naural rae of ineres a all imes, ĩ = r f. Blanchard and Gali (27) erm his resul he Divine Coincidence. Wha hey mean by his erm is ha, in his model, here is no radeoff beween he wo objecives in he Fed s dual mandae. In he simples NK model, a policy of sabilizing inflaion compleely will auomaically sabilize he oupu gap, and vice versa. Ineresingly, his also means ha here is no gain from commimen over discreion. If he discreion soluion always has boh inflaion and he gap equal o zero, i achieves he global minimum of he objecive funcion. There is also no relevan radeoff beween real (gap) and nominal (inflaion) sabilizaion: he cenral bank can achieve boh. There are a couple of ways o implemen his policy in a Dynare code. The easies is o simply replace a Taylor rule (or money supply equaion) wih he firs order condiion in he lis of equilibrium condiions. This will spi ou a ime pah for he nominal ineres rae and ime pahs for inflaion and he gap. Below I show impulse responses for a quaniaive version of he model. I se γ =.3, β =.99, ρ =.95, and ɛ = 1, which implies a weigh on he oupu gap of.3. For ease of comparison, I compare he opimal discreion/commimen responses (as noed above, hese are he same) wih he response ha would obain wih a Taylor rule of he form ĩ = ρ i ĩ 1 + (1 ρ i )φ π π, where I se ρ i =.8 and φ π = 1.5. These are impulse responses o a produciviy shock (which manifess iself as a shock o he naural rae of ineres): 5

6 1 x 1 3 Oupu Gap 1 x 1 3 Inflaion Taylor Rule Discreion 3 x 1 4 Ineres Rae x 1 3 Price Level As prediced via our discussion above, he opimal policy resuls in no movemen of he gap or inflaion a any horizon. This also hen shows up as a consan price level. In conras, he Taylor rule resuls in a negaive oupu gap (oupu rises by less han he flexible price level), disinflaion, and a prey large fall in he price level. The Taylor rule also resuls in a larger and more persisen decline in he nominal ineres rae han does he opimal policy under discreion. Would i be possible o implemen he opimal discreionary/commimen policy (again, hey are he same here) using a rule for he ineres rae or money growh? A leas in his circumsance, he answer is yes. Firs, consider a money rule. Suppose ha he demand for real balances is given by: m = 1 ν Ỹ κĩ The soluion discussed above means ha Ỹ = Ỹ f and ĩ = r f. Plug his in: From above, we have r f Now we know ha Ỹ f m = 1 ν Ỹ f m = 1 ν Ỹ f κ r f f = (ρ 1)Ỹ. Plug his in: m = 1 ν Ỹ f ( κ(ρ 1)Ỹ f 1 = (1 κ(ρ 1)) ν = ρỹ f 1. Hence: ( κ(ρ 1)Ỹ f 1 = (1 + κ(1 ρ)) ν Ierae his back one period o eliminae Ỹ f : ( ) 1 m = ρ m 1 + (1 + κ(1 ρ)) ν ) ) Ỹ f ρỹ f 1 + ( 1 ν (1 + κ(1 ρ)) ) ε Now le s wrie his in erms of he nominal money supply, which he cenral bank conrols, by noing ha m = M P, wih π = P P 1 : ε 6

7 ( ) M = ρ M 1 ρ π (1 ρ) P 1 + (1 + κ(1 ρ)) ε (7) ν In oher words, he cenral bank needs o se he money supply according o an AR(1) in he level, wih AR coefficien equal o he AR coefficien on he echnology process. As long as ρ < 1, i will raise he money supply in response o posiive echnology shocks. There is also a sligh adjusmen for lagged inflaion and he price level. The cenral bank can also accomplish is goal wih a modified Taylor rule. In paricular, suppose i ses he nominal ineres rae according o he following rule. ĩ = r f + φ π π + φ x X (8) This Taylor rule looks similar o he Taylor rules we ve looked a, bu wih wo differences: here is no smoohing parameer, and here is a sochasic inercep equal o he naural rae of ineres. A policy rule of ĩ = r f would resul in equilibrium indeerminacy for an ineres rae rule o work, as we have seen, here needs o be a sufficienly srong reacion o endogenous variables like inflaion and/or he oupu gap. Wha is kind of ineresing here, however, is ha in equilibrium inflaion and he oupu gap would always be zero wih his ineres rae rule (provided φ π and/or φ x are sufficienly large). This means ha one would observe ĩ = r f, bu if he cenral bank announced ha as he policy rule, i would resul in indeerminacy. The cenral bank in essence has o promise o move ineres raes sufficienly in response o inflaion and he oupu gap o preven hose from ever occurring. 5 Cos-Push Shocks and an Oupu-Inflaion Tradeoff Le s modify he Phillips curve o conain an addiional erm: π = γ X + βe π +1 + ũ (9) Here ũ is referred o as he cos-push erm. I is a shock which, in a sense, changes he oupu-inflaion radeoff. I is exogenous and he cenral bank akes i as given. The srucural inerpreaion of his erm is no always very clear one specific inerpreaion is ime-variaion in ɛ, which means here are ime-varying desired markups. More generally, he cos-push erm can be hough of as somehing which drives a wedge beween marginal cos and he oupu gap. In realiy i is a convenien shorcu o make he cenral bank s problem more ineresing. Assume ha i follows an AR(1): ũ = ρ u ũ 1 + ε u, (1) Why does he inclusion of he cos-push erm make he cenral bank s problem more ineresing? The opimizaion problem of he cenral bank is idenical o before, and resuls in he same firs order condiions under discreion and commimen: 7

8 X = γ ω π (11) X = γ ω P (12) The reason he cos-push shock makes hings more ineresing is ha i is in general no going o be feasible o implemen a no gap / no inflaion equilibrium. If ũ, hen X = π is no consisen wih he Phillips Curve holding. In oher words, he inclusion of he cos-push shock (i) generaes a non-rivial radeoff for he cenral bank, as i canno achieve a zero inflaion / zero gap oucome, and (ii) opens up he door for welfare gains from commimen. Of course, condiional on a produciviy shock (so ha ũ = ), here is no real radeoff so i would be opimal for he cenral bank o compleely sabilize boh inflaion and he oupu gap in response o produciviy shocks, wih no resuling welfare gain from commimen. Below I compue impulse responses o a cos-push shock under various differen moneary policy rules. Given he lineariy of he model, he responses o he produciviy shock are idenical o wha I showed above, and under eiher commimen or discreion boh he oupu gap and inflaion are compleely sabilized in response o he produciviy shock. Regardless of he kind of policy (eiher of he opimal policies or he Taylor rule), he cos-push shock causes he oupu gap o decline (oupu falls) and inflaion rises. The firs se of plos shows he responses under he baseline Taylor rule as well as hose under he opimal policy under discreion. Ineresingly, he simple Taylor rule appears o do beer in he sense of a small decline in he oupu gap and a smaller increase in he price level (smaller increase in inflaion over mos horizons). Under eiher scenario, here is a permanen increase in he price level following he cos-push shock..2 Oupu Gap.1 Inflaion x 1 3 Ineres Rae Price Level Taylor Rule Discreion 2 Nex, I compare he impulse responses from he opimal policy under discreion wih he opimal policy under commimen. Here we see some sark differences. Firs, he oupu gap response is smaller (on impac) under commimen han under discreion. Second, he inflaion response is much smaller under commimen and much less persisen, so much so ha he price level reurns o is original value under commimen, whereas he price level permanenly rises under discreion. Third, here is a very differen behavior of he nominal ineres rae: under commimen he nominal 8

9 rae iniially falls, whereas under discreion i rises..2.3 Oupu Gap 8 x 1 3 Inflaion Ineres Rae.1 Price Level.5.8 Commimen Discreion Wha is he source of he gains from commimen? As you will recall from looking a he firs order condiions, he difference beween commimen and discreion boils down o an implici price level (commimen) versus inflaion (discreion) arge. The price level arge has he effec of beer anchoring expeced inflaion, E π +1, because agens know ha he cenral bank will always enac policy so as o reurn he price level o is arge. Beer anchored inflaion expecaions (e.g. less volaile expeced inflaion) improves he available radeoff beween curren inflaion and he oupu gap ha he cenral bank can achieve. Mechanically from he Phillips Curve, he more E π +1 moves around, he more eiher inflaion or he oupu gap will have o move around (or boh), eiher of which reduce welfare. Hence, a cenral message ha comes ou of his exercise is ha commimen and expecaions are of cenral imporance for good moneary policy. 6 Opimal Policy Numerically Using Second Order Approximaions In many conexs, moneary policy being one, we can use sraigh up linearizaion abou a nonsochasic seady sae o hink abou opimal policy. 1 Why is his? In a linearizaion, expeced values of variables are equal o heir seady sae values, and in many conexs policy doesn affec he seady sae, and hence doesn affec means. Concreely, in a linearizaion he expeced value of uiliy (or welfare, aken o mean he presen discouned value of flow uiliy) is independen of policy parameers like wha show up in a Taylor rule. Therefore, he naural objecive funcion for a policy-maker (expeced welfare) isn impaced by he policy parameers, and he problem is degenerae. Define welfare as he presen discouned value of he flow uiliy of a represenaive agen. This can be wrien recursively as: 1 Noe ha in differen conexs where policy choices affec he seady sae his is no he case. For example, he level of rend inflaion will affec he non-sochasic seady sae in he basic New Keynesian model. Similarly, ax insrumens will affec he seady sae of a model wih fiscal policy. 9

10 V = U(C, N ) + βe V +1 (13) Noe ha his looks like a Bellman equaion, bu here is no max operaor, because I assume ha C and N have been chosen opimally. The objecive of a policy maker will be o pick policies o maximize he uncondiional expecaion of welfare, e.g. E(V ). In a firs order approximaion, E(V ) = V = 1 1 β U(C, N ), which in his case is independen of he sance of moneary policy. To hink abou policy, we herefore need o use a higher order approximaion o welfare. There are wo approaches one can ake here. In he one described above, you linearize he equilibrium condiions of he model, bu ake a second order approximaion o he recursive represenaion of welfare. Doing ha and simplifying yields he quadraic loss funcion I showed above. In he oher approach, you can ake a second order approximaion o all he equilibrium condiions, including he recursive represenaion of welfare. This isn as nea in he sense ha i s no possible o derive simple quadraic loss funcions ha are clean, bu one can numerically calculae he expeced values of welfare for differen policy specificaions and compare hose o one anoher. I ll do a concree example. I ake he basic NK model where policy is characerized by an exogenous money supply rule. I solve he model in Dynare using a second order approximaion (raher han firs), which is Dynare s defaul anyway. Below is he ex of my.mod file: 1 var Y C in infl inflr N w mc A vp x1 x2 m dm r Yf gap V; 2 varexo ea em; 3 4 parameers psi bea phi sigma ea epsi hea rhoa rhom sigea sigem pisar; 5 6 load parameer nk money; 7 8 se param value('psi',psi); 9 se param value('phi',phi); 1 se param value('ea',ea); 11 se param value('sigma',sigma); 12 se param value('hea',hea); 13 se param value('rhoa',rhoa); 14 se param value('sigea',sigea); 15 se param value('sigem',sigem); 16 se param value('pisar',pisar); 17 se param value('rhom',rhom); 18 se param value('bea',bea); 19 se param value('epsi',epsi); 2 21 model; % (1) Euler equaion 24 exp(c)ˆ( sigma) = bea*exp(c(+1))ˆ( sigma)*(1+in)*(1+infl(+1))ˆ( 1); % (2) Labor supply 1

11 27 psi*exp(n)ˆ(ea) = exp(c)ˆ( sigma)*exp(w); % (3) Money demand 3 exp(m) = hea*((1+in)/in)*exp(c)ˆ(sigma); % (4) Marginal cos 33 exp(mc) = exp(w)/exp(a); % (5) Resource consrain 36 exp(c) = exp(y); % (6) Producion funcion 39 exp(y) = exp(a)*exp(n)/exp(vp); 4 41 % (7) Price dispersion 42 exp(vp) = (1 phi)*(1+inflr)ˆ( epsi)*(1+infl)ˆ(epsi) + (1+infl)ˆ(epsi)*phi*exp(vp( 1)); % (8) Price evoluion 45 (1+infl)ˆ(1 epsi) = (1 phi)*(1+inflr)ˆ(1 epsi) + phi; % (9) Rese price inflr = (epsi/(epsi 1))*(1+infl)*exp(x1)/exp(x2); 49 5 % (1) x1 51 exp(x1) = exp(c)ˆ( sigma)*exp(mc)*exp(y) + phi*bea*(1+infl(+1))ˆ(epsi)*exp(x1(+1)); % (11) x2 54 exp(x2) = exp(c)ˆ( sigma)*exp(y) + phi*bea*(1+infl(+1))ˆ(epsi 1)*exp(x2(+1)); % (12) Produciviy 57 A = rhoa*a( 1) + ea; % (13) Real balance growh 6 dm = (1 rhom)*pisar infl + rhom*infl( 1) + rhom*dm( 1) + em; % (14) Real balance growh definiion 63 dm = m m( 1); % (15) Real ineres rae (Fisher relaionship) 66 r = in infl(+1); % (16) Flexible price oupu 69 exp(yf) = ((1/psi)*(epsi 1)/epsi)ˆ(1/(sigma + ea))*exp(a)ˆ((1+ea)/(sigma + ea)); 7 71 % (17) Oupu gap 72 gap = Y Yf; % (18) Welfare 75 V = C psi*exp(n)ˆ(1+ea)/(1+ea) + bea*v; 11

12 76 77 end; seady; 8 81 shocks; 82 var ea = sigeaˆ2; 83 var em = sigemˆ2; 84 end; soch simul(order=2,irf=2,ar=,nocorr,nograph); The above is he same as my earlier Dynare file, excep for he fac ha I used a second order approximaion and ha I included a recursive represenaion of welfare as an equilibrium condiion. In wriing ha, since I assume ha σ = 1, I wrie flow uiliy as log over consumpion; since he model is log-linearized and Dynare inerpres C as he naural log of consumpion, ha s why consumpion appears linear in wriing down V bu noe ha i s acually he log. Below is he non-sochasic seady sae of he model: STEADY-STATE RESULTS: Y C in.111 infl inflr N w mc A vp x x m dm r.111 Yf gap V Below are he momens, including he expeced values of he variables: APROXIMATED THEORETICAL MOMENTS 12

13 VARIABLE MEAN STD. DEV. VARIANCE Y C in.1.. infl..6. inflr N w mc A vp.22.. x x m dm r Yf gap V We observe here ha he mean / expeced value of welfare is lower (-5.52) han seady sae welfare (-5.268). The reason why average welfare is lower han seady sae welfare is because agens don like volailiy because of concaviy in preferences. In a firs order approximaion his wouldn show up, bu in he second order approximaion i does. We can hen solve he model under an alernaive moneary policy rule. For example, suppose ha we have a money growh rule, bu we se he sandard deviaions of moneary shocks o. This in effec means ha he money supply would be consan. Below are he seady sae values and expeced values: STEADY-STATE RESULTS: Y C in.111 infl inflr N w mc A 13

14 vp x x m dm r.111 Yf gap V APROXIMATED THEORETICAL MOMENTS VARIABLE MEAN STD. DEV. VARIANCE Y C in.11.. infl..39. inflr N w mc A vp.9.. x x m dm..39. r Yf gap V Geing rid of he shock o he money growh rae doesn affec he seady sae value of welfare, bu i does increase he mean value of welfare ( vs ). The unis of welfare are no paricularly inerpreable, so we ofen wan o express he differences in consumpion equivalen unis. The hough experimen is o say How much consumpion would one be willing o give up (each period) under one policy o have he same welfare as under a differen policy. I is arbirary which policy one akes o be he baseline here I m going o assume i s he policy wih no moneary shock. Le λ denoe he fracion of consumpion one would be willing o give up in each period in one economy, call his economy. Wih log uiliy over consumpion, we have: 14

15 E(V (λ)) = E [ ] ln(c (1 + λ)) θ N 1+χ 1 + χ + βe V+1(λ) Since he λ shows up every period and is deerminisic, his reduces o: The erm in brackes is jus: [ ] E(V (λ)) = 1 ln(1 + λ) + E ln C θ N 1+χ 1 β 1 + χ + βe V+1 E(V (λ)) = 1 ln(1 + λ) + E(V ) 1 β Then we wan o find he λ ha equaes his wih expeced welfare in he alernaive economy, call i economy 1. We have: Or: E(V (λ)) = E(V 1 ) Solving for λ: 1 ln(1 + λ) + E(V ) = E(V 1 ) 1 β λ = exp ( (1 β)(e(v 1 ) E(V )) ) 1 If I ake he baseline economy o have higher welfare, hen he erm inside he exp is negaive, which means λ <. This makes sense you would be willing o give up consumpion in he high welfare economy o have he same welfare as anoher economy governed by a policy resuling in lower welfare. Using he differences in expeced welfare for he wo economies in quesion here, and a value of β =.99, I ge a value of λ =.13. This means ha an agen would be willing o forfei abou.1 percen of consumpion each period in he economy wih no moneary shocks o avoid going o he economy wih moneary shocks. This number may no seem large bu we ypically find prey small welfare differences under differen macro policies, so i s no abnormally low. I solved he model assuming zero rend inflaion. Suppose I solve he model where seady sae inflaion is insead se o.5 (approximaely 2 percen a an annualized frequency). Below are he seady saes and he means: STEADY-STATE RESULTS: Y C in

16 infl.5 inflr N w mc A vp x x m dm r Yf gap V APROXIMATED THEORETICAL MOMENTS VARIABLE MEAN STD. DEV. VARIANCE Y C in infl inflr N w mc A vp.5.3. x x m dm r Yf gap V Unlike he sandard deviaion of he moneary policy shock, he level of rend inflaion does affec he seady sae we can see ha seady sae welfare is lower wih higher rend inflaion ( vs ). We also see ha expeced welfare is lower, a (compared o -5.52). 16

17 We could calculae consumpion equivalen welfare differences based boh on seady sae welfare or expeced welfare. Based on seady saes, we would ge λ ss =.19; based on he means, we would ge λ ss =.29. Hence, we observe ha he welfare loss based on means is higher han he welfare loss based on he seady sae. This is because rend inflaion does wo hings: firs, i disors he seady sae by effecively increasing he seady sae markup and increasing seady sae price dispersion. Bu i also ineracs wih he shocks o have a bigger effec on means: wih posiive rend inflaion, price dispersion is firs order, which makes sochasic shocks more cosly. Now, le s compare he welfare of he money growh rule (baseline case wih no rend inflaion and he moneary shock urned off) o a Taylor rule wih coefficiens ρ i =.8, φ π = 1.5, and φ x =.5 (again, wih no moneary shock). Below are he seady saes and means: STEADY-STATE RESULTS: Y C in.1119 infl e-7 inflr e-6 N w mc A vp e-9 x x m r.111 Yf gap e-7 V APROXIMATED THEORETICAL MOMENTS VARIABLE MEAN STD. DEV. VARIANCE Y C in infl inflr N w

18 mc A vp.2.. x x m r Yf gap V Here, we observe ha expeced welfare in he Taylor rule economy is (versus in he economy wih a money growh rule and no shock o he money growh rae). Calculaing he consumpion equivalen difference yields , which means ha agens would be willing o give up roughly.8 percen of consumpion each period in he Taylor rule economy o avoid being subjeced o he money growh rae economy. We could alernaively see wha welfare is in an economy wih inflaion argeing. Tha is, I replace he Taylor rule in my code wih a condiion ha inflaion always equals is seady sae value. This doesn affec seady saes bu affecs means. The mean value of welfare is This is slighly preferred o he Taylor rule, which is again slighly preferred o he money growh rule, hough again he welfare differences end o be quie small. 18