Welfare Maximizing Operational Monetary and Tax Policy Rules

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1 Forhcomng n Macroeconomc Dynamcs (also publshed n 24 as CEPR DP 4782) Welfare Maxmzng Operaonal Moneary and Tax Polcy Rules Rober Kollmann (*) Deparmen of Economcs, Unversy of Pars XII 6, Av. du Général de Gaulle; F-94 Créél Cedex; France Cenre for Economc Polcy Research, UK Frs draf: March 7, 23 Ths draf: November 29, 24 Ths paper compues welfare maxmzng moneary and ax polcy feedback rules, n a calbraed dynamc general equlbrum model wh scky prces. The governmen makes exogenous fnal good purchases, leves a proporonal ncome ax, and ssues nomnal oneperod bonds. A quadrac approxmaon mehod s used o solve he model, and o compue household welfare. Opmzed polcy has a srong an-nflaon sance and mples perssen flucuaons of he ax rae and of publc deb. Very smple opmzed polcy rules, under whch he neres rae jus responds o nflaon and he ax rae jus responds o publc deb, yeld a welfare level very close o ha generaed by rcher rules. JEL classfcaon: E5, E6, H6. Keywords: Moneary polcy; Fscal polcy; Welfare (*) Tel.: ; E-mal: rober_kollmann@yahoo.com hp:// I hank A. Cukerman, M. Devereux, C. Erceg, H. Km, J. Km, T. Monacell, S. Schm- Grohé, P. Teles, M. Urbe and workshop parcpans a he Bank of Fnland, Unversy of Amserdam, European Unversy Insue, and a he Unversy of Copenhagen (Macroeconomc Dynamcs Conference, June 24) for useful suggesons and dscussons.

2 . Inroducon There has been much recen work on he effec of moneary polcy rules on welfare and busness cycles (see survey by McCallum (999)). Fscal polcy rules have receved less aenon; exsng sudes follow wo approaches: () dynamc exensons of Ramsey (927) ha deermne welfare maxmzng me pahs of fscal nsrumens; () analyses of he macroeconomc effecs of smple fscal feedback rules (e.g., Taylor (2)). The Ramsey approach s appealng as uses mcro-based models and focuses on household welfare as he creron for evaluang polcy. However, ha approach faces echncal dffcules, as Ramsey problems are generally no concave. 2 Furhermore, Ramseyype sudes ypcally use hghly sylzed models; also, Ramsey polcy rules are ofen complcaed whch may make dffcul o apply hem n pracce. By conras, mos sudes on smple feedback polcy rules use models ha are more realsc, bu ha are no fully mcro-based; ad hoc crera (such as he mpled volales of oupu and nflaon) are employed o evaluae polcy. Ths paper numercally compues welfare maxmzng operaonal feedback rules ha lnk moneary and fscal polcy o small ses of easly observable macro varables, for a calbraed busness cycle model wh saggered prce seng à la Calvo (983). The (poenal) non concavy of he Ramsey problem s nconsequenal for he approach here. The model has rgorous mcro-foundaons, bu s rcher han hose used n mos applcaons of he Ramsey approach. The economy feaures capal, varable labor supply, monopolsc compeon n goods markes, and exogenous shocks o producvy and o governmen purchases. The governmen leves a proporonal ncome ax, and ssues nomnal uncondonal one-perod bonds. Moneary polcy follows a Taylor-syle neres rae rule; he ax rae s se as a funcon of real publc deb, producvy and governmen purchases. The seady sae ax rae and he raos of deb and of governmen purchases o GDP are calbraed o OECD daa. I focus on polces characerzed by saonary flucuaons of real publc deb around s seady sae value. Under saggered prce seng (as assumed here), nflaon nduces neffcen dsperson of prces across frms (e.g., Erceg e al., 2); n an economy n whch prce sckness s he only dsoron, opmal moneary polcy enals full nflaon sablzaon, as ha polcy elmnaes neffcen cross-frm prce dsperson (e.g., Roemberg and Woodford E.g., Ayagar e al. (2), Bengno and Woodford (23), Char e al. (99), Correa e al. (2), Schm- Grohé and Urbe (24c), Su (24) (several of hese papers also deermne opmal moneary polcy). 2 Mos papers on Ramsey problems concenrae on he assocaed frs-order condons, whou esablshng ha second-order condons are me. 2

3 (997)). The economy here has monopolsc compeon and ax dsorons s shown ha, neverheless, opmzed polcy under scky prces mples (almos) full nflaon sablzaon. In all model varans consdered here, opmzed polcy mples perssen flucuaons of he ax rae, and szable and perssen flucuaons of real publc deb; producvy shocks are much more mporan as a source of macroeconomc flucuaons han governmen purchases shocks. Very smple opmzed polcy rules--under whch he neres rae jus responds o nflaon, and he ax rae jus responds o publc deb--yeld a welfare level very close o ha generaed by rules ha spulae a response o addonal varables. As opmzed polcy n he baselne scky-prces model (wh nomnal publc deb) enals src nflaon sablzaon, real deb reurns are rskless, n ha seng; he behavor of he ax rae and of real acvy closely resembles ha generaed by a flexble-prces model wh ndexed (real) non-sae conngen deb. By conras, a flex-prces srucure wh nomnal deb mples a very dfferen opmzed ax behavor han he baselne scky-prces model. In such a srucure, nflaon does no cause neffcen prce dsperson across frms; when exogenous shocks occur, he governmen can mee s neremporal budge consran by alerng he real value of he nhered sock of (nomnal) publc deb va unancpaed nflaon changes. As a resul, opmzed polcy n a flex-prces-nomnal-deb srucure enals szable nflaon volaly, bu only small movemens of he ax rae. Char e al. (99) showed ha (opmal) monearyfscal Ramsey polcy mples hgh nflaon volaly n a flex-prces economy wh nomnal deb; Schm-Grohé and Urbe (24c) and Su (24) demonsraed ha Ramsey polcy enals much lower nflaon volaly when prces are scky. The paper here shows ha smlar predcons (effec of flex- vs. scky prces) hold when polcy s descrbed by smple opmzed rules. The model s solved usng Sms' (2) mehod ha s based on a second-order expanson of he equlbrum condons. In conras o he lnear, cerany-equvalen approxmaons ha are wdely used n macroeconomcs, hs mehod allows o capure he effec of rsk on agens' decson rules and s hus beer sued for welfare analyss. Compared o oher non-lnear mehods (see Judd (998)), hs echnque allows o easly solve models wh a rch srucure. The approach presened n hs paper mgh hus provde a racable way of compung opmzed polcy rules usng larger mcro-based smulaon models (such as hose currenly developed by,.a., he Fed and IMF). 3

4 Ths sudy bulds on my recen work ha compued welfare maxmzng smple moneary polcy rules, for calbraed New Keynesan models of open economes (see Kollmann (22, 24)); ha research absraced from fscal polcy The model A closed economy wh a represenave household, frms, and a governmen (moneary-fscal auhory) s consdered. 4 There s a sngle fnal good ha s produced by combnng a connuum of nermedae goods ndexed by s [,]. The fnal good s produced by perfecly compeve frms; can be consumed and used for nvesmen. There s monopolsc compeon n nermedae goods markes--each nermedae s produced by a sngle frm. Inermedae goods frms use capal and labor as npus. The household owns all frms and he capal sock, whch rens o frms. I also supples labor. The markes for renal capal and for labor are compeve. The governmen ses he nomnal neres rae, purchases an exogenous quany of he fnal good, leves a proporonal ncome ax and ssues nomnal non-sae conngen one-perod bonds. 2.. Fnal good producon The fnal good s produced usng he aggregae echnology ( ν )/ ν ν /( ν ) Q = { q ( s) ds}, wh ν >, where q () s s he quany of he ype s nermedae. Le p () s be he prce of ha good. Cos mnmzaon n fnal good producon gves: q() s = ( p()/ s P) ν Q, wh ν /( ν ) { ( ) }. P= p s ds The prce of he fnal good s P (s margnal cos) Inermedae goods frms The echnology of he frm ha produces nermedae good s s: ψ ψ y () s = K () s L () s, < ψ <. 3 Schm-Grohé and Urbe (24a) oo use a second-order approxmaon o compue opmzed smple moneary/fscal rules for a New Keynesan model, bu ha paper focuses on a seng wh lump sum axes. Second order approxmaons are also used by Bengno and Woodford (23) who analycally derve opmal moneary-fscal polcy, for a more sylzed economy (whou capal and wh a more resrcve srucure of shocks) and by Km and Km (2) who numercally compue opmzed ax polcy rules, for a wo-counry RBC model. 4 The srucure of preferences, echnologes and markes resembles ha n Kollmann (2, 22, 24). 4

5 y() s s he frm's oupu a dae. s an exogenous producvy parameer (common o all nermedaes' producers). K( s ) [ Ls ( )] s he capal [labor] used by he frm. Is margnal cos s: ψ ψ ψ ψ = (/ ) ψ ( ψ) MC R W, where R [ W ] s he renal rae of capal [wage rae]. The frm's prof s: π ( p ( s)) = ( p( s) MC)( p( s)/ P) ν Q. There s saggered prce seng, à la Calvo (983), n he nermedae goods secor: frms n ha secor canno change prces, unless hey receve a random "prce-change sgnal." The probably of recevng hs sgnal n any parcular perod s -d, a consan. Followng Erceg e al. (2) I assume ha when a frm does no receve a "prce-change sgnal," s prce s auomacally ncreased a Π, he seady sae growh facor of he fnal good prce. (Throughou hs paper, he erm "seady sae" refers o he deermnsc seady sae.) Frms are assumed o mee all demand a posed prces. They maxmze he value of her prof sream, ne of he ncome ax pad by he household on profs. Consder an nermedae good producer ha, a me, ses a new prce, p,. If no j "prce-change sgnal" s receved beween and + j, he prce s p, Π a + j.the frm ses j j, = ρ j, + j + j π+ j Π = + j p Arg Max d E{ ( ) ( p )/ P }, where j p + s he ncome ax rae a + j; ρ, + s a prcng kernel (for valung dae + j pay-offs) ha equals he household's margnal rae of subsuon beween consumpon a and + j (see below). Le Ξ ρ The soluon of he decson problem regardng p, s: ν, + j, + j( P+ j) Q+ j. ν j ν j { j= } { j= + + } p = (/( ν ν )) ( d Π ) E Ξ ( ) MC ( d Π ) E Ξ ( ).,, j j j, j j The fnal good prce P evolves accordng o ( P ) = d( P Π ) + ( d)( p ). ν ν ν, 2.3. The represenave household Household preferences are descrbed by: = = E β U( C, L ), wh < β <. () C [ L ] s consumpon [labor effor]. U s a uly funcon gven by: UC (, L) = ln( C) L. The household accumulaes physcal capal, subjec o he law of moon K = K( + δ ) + I, wh < δ <, (2) where I s gross nvesmen ( δ : deprecaon rae of capal). The household holds nomnal one-perod bonds. Is budge consran s: A ( ) ( ) ( ( )) + + P C + I = A + + WL + π p s ds+ RK T. (3) 5

6 A s a sock of nomnal one-perod bonds ha maure n perod ; s he neres rae on hese bonds. T ( WL+ π( p( s)) ds+ RK δpk ) s he ax pad by he household. The household chooses a sraegy { A,,, } + K+ C L = o maxmze (), subjec o (2),(3). The followng equaons are frs-order condons of hs problem: = ( + ) { ( / )} E ρ, + P P +, wh ρ, + = β / +, CC, = E{ ρ ([ R / P δ]( ) + )}, (4) ( ) W/ P= C. (5) 2.4. The governmen budge consran The governmen budge consran s PG + D( + ) = D+ + T, where G are (exogenous) governmen fnal good purchases; maures n. D s he sock of nomnal one-perod publc deb ha 2.5. Marke clearng condons Markes for nermedaes clear as nermedae goods frms mee all demand a posed prces. Marke clearng n fnal good, labor, and renal capal markes requres: Y= C+ I+ G, L = Lsds, () K = K() s ds. Bond marke clearng requres: A = D Polcy rules Much recen research has focused on moneary polcy rules ha spulae a response of he neres rae o nflaon (e.g., Taylor (999a)). The baselne neres rae rule consdered here s: π = +Γ Π, (6) wh Π=Π Π ( )/ Π, where Π= PP / s he fnal good gross nflaon rae. s he seady sae nomnal neres rae. Throughou he paper, varables whou me subscrps denoe seady sae values, and xˆ = ( x x)/ x s he relave devaon of a varable x from s seady sae value, x. π Γ s a polcy parameer. The ax rae s se as a funcon of real publc deb, and of he exogenous varables: B G = + Γ ( B B) +Γ +Γ G, (7) 6

7 B where B D /( PY) s real publc deb, normalzed by seady sae real GDP, Y. Γ, Γ and are polcy parameers. Seng B Γ a a suffcenly hgh posve value ensures governmen solvency. Ayagar e al. (22) presen analycal resuls abou opmal (Ramsey) fscal polcy n an nfnely-lved economy wh a proporonal ncome ax and non-sae-conngen real publc bonds; 5 Ayagar e al. show ha Ramsey polcy may enal ha he long run values of deb and axes dffer grealy from he values observed n realy. E.g., under ceran assumpons, a governmen ha nally has posve deb runs fscal surpluses, unl owns a sock of asses whose neres ncome covers all subsequen governmen purchases (a ha pon, he ax rae s se o zero). Ths predcon s a odds wh he daa: hroughou modern hsory governmens have been ne debors. To rule ou unrealsc long run behavor of publc deb, I mpose he resrcon ha he uncondonal mean value of real deb has o be close o s seady sae value B: 6 G Γ EB B <. (8) I se B,, and he seady sae rao of G o GDP a B=.5, =.25 and G/Y=.2, respecvely; hese values are n he range of pos-wwii fscal daa for OECD economes (see Kollmann (998) for daa on axes and governmen purchases n G7 economes). (8) s also used for a echncal reason: he soluon mehod used here s based on a second order Taylor expanson of he model, around a gven seady sae (see below); hs mehod s no sued for he analyss of "large" changes n sae varables. (8) ensures ha real deb (and oher sae varables) say n he neghborhood of he seady sae. 7 B Γ, π A an nal dae =, he governmen makes a commmen o se he parameers Γ, Γ and Γ G a me-nvaran values ha maxmze he condonal expeced value of household lfe-me uly (), subjec o he laws of moon of he endogenous varables mpled by household decsons, and subjec o (8). I assume ha a = he predeermned 5 The Ayagar e al. model s more sylzed han he srucure consdered here ( assumes a perfecly compeve, real economy, whou capal accumulaon or producvy shocks). 6 Ayagar e al. sugges an alernave way of rulng ou unrealsc deb behavor: an exogenous posve lower bound d on governmen deb: d D / P. The numercal algorhm used here canno handle nequaly + consrans. Insead of posulang exogenous deb lms, an alernave approach (lef for fuure research) mgh be o assume ha publc deb provdes lqudy servces o households; see, e.g., Woodford (99); f hese servces are suffcenly srong, hen a benevolen governmen would always selec a posve level of deb. 7 When (8) s no mposed, hen welfare maxmzaon wh respec o polcy parameers (see below) selecs parameers for whch he approxmaon s very poor and resuls are nonsenscal (asronomcal welfare gans, assocaed wh long run values of deb and axes ha are markedly below seady sae values). 7

8 sae varables equal her (deermnsc) seady sae values, and ha he exogenous varables a = equal he uncondonal means of hese varables Parameers and soluon mehod The model s calbraed o quarerly daa. The seady sae real neres rae r s se a r =., a value ha corresponds roughly o he long-run average (quarerly) reurn on capal. Thus, β = /. s used ( β ( + r) = holds n seady sae). The seady sae prce-margnal cos markup facor for nermedae goods s se a ν/( ν ) =.2, conssen wh he fndngs of Marns e al. (996) for OECD counres. The echnology parameer ψ s se a ψ =.24, whch enals a 6% seady sae labor ncome/gdp rao, conssen wh OECD daa. Aggregae daa sugges a quarerly capal deprecaon rae of abou 2.5%; hus, δ =.25 s used. Emprcally, he mean prce-change nerval s abou 4 quarers (Taylor (999b)). Hence, I se d=.75. The seady sae growh facor of prces s se a Π= (Π has no real effecs, because of ndexng); hus he seady sae nomnal neres rae s =.. As saed above, I se B=.5, =.25, G/Y=.2. The exogenous varables follow AR() processes: ε are = ρ + ε, ρ <, G = ρ G G G G + ε, ρ <. (9) γ ε are ndependen whe noses wh sandard devaon σ and σ G, respecvely. I se ρ =.95, σ =.; hese (or very smlar) values are wdely used n he RBC leraure, and are conssen wh emprcal evdence on he me seres behavor of aggregae producvy (e.g., Presco (986)). Fng (9) o quarerly governmen purchases n G7 counres gves esmaes of ρ G and σ G n he range of.95 and., respecvely; 8 hus, I se ρ G =.95, σ G =.. The model s solved usng Sms' (2) algorhm/compuer code ha s based on second-order Taylor expansons of he equlbrum condons, around a (deermnsc) seady sae. 9 I numercally solve he governmen's opmzaon problem; aenon s resrced o values of he polcy parameers (see (6), (7)) for whch a unque saonary equlbrum exss. 8 Esmaon perod: 97Q-24Q (usng G : relave devaon of 8 G from fed geomerc rend); e.g. for US and Japan: ρ G,US =.95, σ G,US =.76 ; ρ G,JA =.95, σ G,JA =.6. 9 Judd and Guu (993), Collard and Jullard (2), Schm-Grohé and Urbe (24b).a. also develop soluons of dynamc models based on second-order expansons.

9 3. Resuls Smulaon resuls are repored n Tables -2. The varables are quarerly. Y s (real) GDP. Def = ( D D )/( PY) s he real (secondary) fscal defc, normalzed by seady sae GDP. In + he Tables, he sascs for Def, he neres rae ( ), he ax rae ( ) and real deb ( B ) refer o dfferences of hese varables from seady sae values ( s a quarerly rae expressed n fraconal uns), whle sascs for he remanng varables refer o relave devaons from seady sae values. All sascs are expressed n percenage erms. I express welfare as he permanen relave change n consumpon (compared o seady sae), ζ, ha yelds expeced = lfeme uly: ( β) U(( + ζ) CL, ) = E β U( C, L). = m Noe ha he polcy maker maxmzes ζ. I decompose ζ no componens, denoed ζ and v ζ, ha reflec he condonal expeced values of consumpon and hours { EC, EL }, and he condonal varances of consumpon { V ( C )}, respecvely. Cols. -4 n Table show resuls under scky prces; Cols. -2 peran o he baselne model (wh polcy rules (6)-(7)); Cols. 3-4 assume alernave polcy rules. Cols. 5-6 assume flexble prces. Cols. 7-8 peran o he frs-bes allocaon,.e. o he soluon of a socal plannng problem n whch household welfare s maxmzed subjec o he economy's resource consrans (akng { G } as exogenously gven). Table 2 shows dynamc responses o shocks. 3.. Scky prces In he baselne model (scky prces, nomnal deb), he opmzed polcy parameers are Γ= π B , Γ= 9.9. Thus, opmzed moneary polcy has a src an-nflaon sance (an ncrease n nflaon rggers a sharp ncrease n he nomnal neres rae); as a resul, he sandard devaon of he nflaon rae s (essenally) zero. Under saggered prce seng, nflaon nduces neffcen dsperson of prces across nermedae goods producers (e.g., Erceg e al., 2). In an economy n whch prce sckness s he only dsoron (so ha he flex-prces equlbrum would be effcen), Up o a 2nd-order approx.: U (( ) C, L ) U ( C, L ) ( ) [ E ( C L ) V 2 ( C m v + ζ + β β L )]. ζ, ζ are defned by (( m ), ) (, ) ( ) [ ] v U + ζ C L = U C L + β β EC LE L, U(( + ζ ) C, L) = U( C, L) ( β) V 2 β ( C ). I can be m v verfed ha ( + ζ) = ( + ζ )( + ζ ). π The opmzed Γ π parameer s very large. Welfare s a very fla funcon of Γ. Imposng a moderae bound π on s value (e.g. Γ ) does no affec he resuls. 9

10 opmal moneary polcy fully sablzes nflaon, as ha polcy elmnaes neffcen crossfrm prce dsperson (e.g., Roemberg and Woodford, 997). The economy here has oher dsorons (monopolsc compeon, ncome ax)--neverheless, opmzed polcy under scky prces enals (almos) full nflaon sablzaon. Opmzed fscal polcy mples ha an ncrease n he sock of deb by an amoun equal o (quarerly) seady sae GDP ( Y ), rases he ax rae by 9.9 percenage pons (roughly one-hrd of he seady sae ax rae). Expermens wh alernave values of B Γ π G B (holdng consan Γ, Γ, Γ ) show ha he model has a saonary soluon when Γ.62. Values of Γ B ha are smaller han he opmzed coeffcens (bu larger han.62) enal volaons of he long run consran on real deb (8). Real deb flucuaons are hghly perssen (auocorrelaon:.999) and volale: he sandard devaon of real deb (normalzed by Y ) s 7.2%--whch s markedly larger han he sandard devaon of (quarerly) GDP and consumpon (6.58% and 5.58%, respecvely), bu smaller han he sandard devaons of nvesmen (2.65%) and of he capal sock (normalzed by Y ), 49.3%. The ax rae undergoes non-neglgble, counercyclcal flucuaons (sandard devaon of :.7%; correlaon wh GDP: -.8). Because he ax rae s a funcon of he sock of real deb, oo s hghly perssen (auocorrelaon:.998). The uncondonal mean value of he capal sock exceeds s seady sae value (by.49%)--whch can be vewed as reflecng precauonary savng (n he sochasc economy). Mean consumpon s slghly above seady sae (by.5%), whle mean hours are below seady sae (-.7%). Condonal welfare s lower han seady sae welfare, ζ =.72%. m The "mean componen" of he welfare measure s posve, ζ =.5% ; however, ha v effec s domnaed by he negave effec of consumpon varance on welfare: ζ =.73%. These predcons are based on he assumpon ha he economy s smulaneously subjeced o shocks o producvy and o governmen purchases; see Col. n Table. Col. 2 shows predcons for he case where here are jus governmen purchases shocks. These shocks explan only 6% of he varances of he ax rae, and less han 3% of he varances of consumpon and oupu (ha are generaed when here are smulaneous producvy and governmen purchases shocks). G shocks also have a markedly smaller effec on welfare han producvy shocks ( ζ =.4% when here are jus G shocks).

11 Dynamc responses Under opmzed polcy, a posve producvy shock rggers a rse n oupu, consumpon, nvesmen and hours worked; he prce level remans (essenally) consan, and he ax rae falls. Inally, ax revenues rse; real deb falls, wh a one-perod delay (n he long run, real deb revers o s pre-shock level). On mpac, a posve shock o governmen purchases rases oupu, nvesmen and hours worked, and lowers consumpon; ax revenues rse, and real deb ncreases. (See Panels (a.), (b.) n Table 2.) Rcher/smpler polcy rules Expermens wh rcher polcy rules (ha perm a drec response of he polcy nsrumens o seleced addonal varables) only yeld very small welfare gans, compared o he baselne rules (6),(7). For example, Col. 3 (Table ) assumes ha s se as a funcon of nflaon, GDP and he real defc, whle s a funcon of real deb, producvy, governmen purchases π and nflaon: Y Def =+ΓΠ+Γ Y +Γ Def ; G + B B ( B) G π = Γ +Γ +Γ +Γ Π. Some promnen crcs of he ECB s sngle-mnded pursu of prce sably advocae a responsveness of moneary polcy o broader macroeconomc/fscal condons (e.g., French Presden Chrac (24)); he neres rae rule shown above perms such a responsveness. The opmzed neres rae rule exhbs a posve [negave] response coeffcens for GDP [he defc], whle he opmzed ax rae rule has a negave response π B coeffcen for nflaon. However, he opmzed coeffcens Γ, Γ, Γ and G Γ, as well as predced behavor and welfare are essenally he same as under he baselne rules--he welfare gan from usng he rcher rules (nsead of (6),(7)) corresponds o a (permanen) consumpon ncrease of merely.%. Col. 4 assumes ha he ax rae jus responds o real deb (whle he neres rae jus π responds o nflaon, as under (6)); here, he opmzed polcy parameers are: Γ= 9.53, Γ = 8.56 ; agan, predced sascs are close o hose under he baselne specfcaon (e.g., B sandard devaons of GDP, ax rae and nflaon: 6.23%,.69% and.%, respecvely); he welfare loss from usng he smpler ax rae rule (nsead of (7)) s.76%. Comparson wh frs-bes economy A comparson beween he dsored scky-prces economy and he frs-bes (undsored) economy (Cols. 7-8) shows: () The levels of economc acvy and welfare are noceably

12 lower n he dsored equlbrum--for example, seady sae consumpon and hours worked are 43% and 34% lower, respecvely (han n he frs-bes economy); he welfare dfference s equvalen o a permanen 2.46% consumpon loss (no repored n Table ). 2 () Responses o shocks are qualavely smlar, across he wo srucures. In he sckyprces economy, economy, oupu, hours worked and nvesmen respond less srongly--on mpac--o producvy shocks, and more srongly o governmen purchases shocks, han n he frs-bes economy. (See Panels (a.3), (b.3) n Table 2.) 3.2. Flexble prces Nomnal deb Opmzed polcy under flexble prces (and nomnal deb) dffers markedly from opmzed polcy under scky prces: wh flex prces, he opmzed nflaon and deb coeffcens are B Γ=.97, Γ=.57 (see Col. 5). In a flex-prces-nomnal-deb srucure, nflaon does no π cause neffcen prce dsperson across frms; when exogenous shocks occur, he governmen can mee s neremporal budge consran by alerng he real value of he nhered sock of (nomnal) publc deb va unancpaed nflaon changes. As a resul, opmzed polcy enals szable nflaon volaly, and much smaller movemens of he ax rae han under scky prces (sandard devaons of Π and under flexble prces: 3.2% and.%, respecvely). As n he baselne scky-prces srucure, he real value of he sock of deb s hghly volale, and deb and he ax rae are hghly serally correlaed. Table 2 (Panels (a.2), (b.2)) shows ha, n he flex-prces-nomnal-deb model varan, he governmen responds o a posve producvy nnovaon and he ensung rse n ax revenues by nducng an unancpaed fall n he prce level, and hus a rse n real publc deb (hereafer, real deb revers o s seady sae value); a posve shock o governmen purchases rggers an unancpaed rse n he prce level, and hus a fall n real deb. Char e al. (99) showed ha (opmal) moneary/fscal Ramsey polcy mples hgh nflaon volaly n a flex-prces economy wh nomnal deb; Schm-Grohé and Urbe (24c) and Su (24) demonsraed ha Ramsey polcy enals much lower nflaon volaly when prces are scky. The paper here shows ha smlar predcons (effec of flex- vs. scky prces) hold when polcy s descrbed by smple opmzed rules. 2 The welfare fgures (, m v ζ ζ, ζ ) for he frs-bes economy n Cols. 7-8 are % equvalen varaons n consumpon, relave o he seady sae of he frs-bes economy. 2

13 Indexed deb Col. 6 consders a flex-prces economy wh ndexed deb. The governmen canno use unancpaed nflaon changes o mee s solvency condon, n ha seng (as here moneary polcy has no real effecs). Welfare ( ζ =.72%) s lower han n he flex-prces srucure wh nomnal deb. The opmzed fscal polcy rule, and he mpled behavor of real varables are very smlar o hose generaed n he scky-prces economy wh nomnal deb. Inuvely, hs s due o he fac ha he src nflaon sablzaon enaled by opmzed polcy n he scky-prces srucure mples: () all frms se dencal prces (as s he case under flexble prces); () nomnal bonds are rskless, n real erms (as under ndexed deb). 3 3 The nomnal neres rae canno be deermned unquely from welfare maxmzaon, n he flex-prcesndexed-deb srucure; hus no predcons for nomnal varables (and Γ π ) are repored n Col. 6. 3

14 References Ayagar, R., Marce, A., Sargen, T., Seppala, J., 22. Opmal Taxaon Whou Sae- Conngen Deb, Journal of Polcal Economy, Bengno, P., Woodford, M., 23. Opmal Targeng Rules for Moneary and Fscal Polcy, NBER Macroeconomcs Annual (forhcomng). Calvo, G., 983. Saggered Prces n a Uly Maxmzng Framework, Journal of Moneary Economcs 2, Chrac, J., 24. Televson nervew gven on Baslle Day, July 4 ( Char, V., Chrsano, L., Kehoe, P., 99. Opmal Fscal and Moneary Polcy: Some Recen Resuls, Journal of Money, Cred, and Bankng 23, Collard, F., Jullard, M., 2. Perurbaon Mehods for Raonal Expecaons Models, WP. Correa, I., Ncoln, J., Teles, P., 2. Opmal Fscal and Moneary Polcy: Equvalence Resuls, Workng Paper, Bank of Porugal. Erceg, C., Henderson, D., Levn, A., 2. Opmal Moneary Polcy Wh Saggered Wage and Prce Conracs, Journal of Moneary Economcs 46, Judd, K., Guu, S., 993. Perurbaon Soluon Mehods for Economc Growh Models, n: Varan, H., ed., Economc an Fnancal Modelng wh Mahemaca (Sprnger Verlag, NY), 8-3. Judd, K., 998. Numercal Mehods n Economcs (MIT Press, Cambrdge, MA). Km, J. Km, S., 2. Welfare Improvng Tax Polces n a Two-Counry Model, Manuscrp. Kollmann, R., 998. U.S. Trade Balance Dynamcs, Journal of Inernaonal Money and Fnance 7, Kollmann, R., 2. The Exchange Rae n a Dynamc-Opmzng Busness Cycle Model Wh Nomnal Rgdes, Journal of Inernaonal Economcs 55, Kollmann, R., 22. Moneary Polcy Rules n he Open Economy: Effecs on Welfare and Busness Cycles, Journal of Moneary Economcs 49, Kollmann, R., 24. Welfare Effecs of a Moneary Unon: he Role of Trade Openness, Journal of he European Economc Assocaon 2, Marns, J., Scarpea, S., Pla, D., 996. Mark-Up Prcng, Marke Srucure and he Busness Cycle, OECD Economc Sudes 27, 7-5. McCallum, B., 999. Issues n he Desgn of Moneary Polcy Rules, n: Taylor, J., Woodford, M., eds. Handbook of Macroeconomcs, Vol. C (Elsever, Amserdam), Presco, E., 986. Theory ahead of busness cycle measuremen, Federal Reserve Bank of Mnneapols Quarerly Revew, Ramsey, F., 927. A Conrbuon o he Theory of Taxaon, Economc Journal 37, Roemberg, J., Woodford, M., 997. An Opmzaon-Based Economerc Framework for he Evaluaon of Moneary Polcy, NBER Macroeconomcs Annual 2, Schm-Grohé, S., M. Urbe, 24a. Opmal Smple and Implemenable Moneary and Fscal Rules. CEPR DP Schm-Grohé, S., M. Urbe, 24b. Solvng Dynamc General Equlbrum Models Usng a Second-Order Approxmaon, Journal of Economc Dynamcs and Conrol 28, Schm-Grohé, S., M. Urbe, 24c. Opmal Fscal and Moneary Polcy Under Scky Prces, Journal of Economc Theory 4, Sms, C., 2. Second Order Accurae Soluon of Dscree Tme Dynamc Equlbrum Models, Workng Paper, Prnceon Unversy ( ~ sms). Su, H., 24. Opmal Fscal and Moneary Polcy wh Scky Prces, Journal of Moneary Economcs 5, Taylor, J., 999a. Moneary Polcy Rules (Unversy of Chcago Press, Chcago). Taylor, J., 999b. Saggered Prce and Wage Seng n Macroeconomcs, n: Taylor, J., Woodford, M., eds., Handbook of Macroeconomcs (Elsever, Amserdam), 9-5. Taylor, J., 2. The Polcy Rule Mx: a Macroeconomc Polcy Evaluaon, n: G. Calvo, R. Dornbusch, M. Obsfeld, eds., Rober Mundell Fesschrf (MIT Press), Woodford, M., 99. Publc Deb as Prvae Lqudy, Amercan Economc Revew 8,

15 Table. Opmzed polcy rules and frs bes oucome Scky prces Flexble prces Baselne Rules Rcher Smpler Rules Rule Nomnal Indexed Deb Deb Frs-bes,G G,G,G,G,G,G G () (2) (3) (4) (5) (6) (7) (8) Sandard devaons (n %) Y C I Π G B Def Correlaons wh GDP G Auocorrelaons Y B Means (n %) Y C L K B Welfare (% equvalen permanen varaon n consumpon) ζ m ζ v ζ Polcy parameers Γ π Y Γ Def Γ B Γ Γ G Γ Γ π Noes--Cols. Labeled ", G" : smulaneous producvy & governmen purchases shocks; Cols. labeled " G ": jus gov purchases shocks. Y: GDP; C: consumpon; I: physcal nvesmen. Π : gross nflaon rae; : nomnal neres rae; G: governmen purchases; : ax rae; B: real publc deb; m v Def: real budge surplus; L: hours worked; K: capal sock. ζ, ζ, ζ : welfare measures. Momens of,, B, Def refer o dfferences from seady sae values. Momens of remanng varables: relave devaons from seady sae values. All sascs have been expressed n percenage erms. 5

16 Table 2. Dynamc responses o shocks Exogenous Y C I L K P Taxrev. B varable (a) % responses o % producvy nnovaon (a.) Scky prces j = j = j = j = (a.2) Flexble prces (nomnal deb) j = j = j = j = (a.3) Frs-bes j = j = j = j = (b) % responses o % governmen purchases nnovaon (b.) Scky prces G j = j = j = j= (b.2) Flexble prces (nomnal deb) G j = j = j = j = (b.3) Frs-bes G j = j = j = j = Noes: j : perods afer shock. Cols. labeled Y, C ec. show responses of he correspondng varables. Taxrev.: real ax revenue, normalzed by seady sae real GDP. (See Table for defnons of oher varables.) Responses of capal peran o end-of-perod socks ( K j + ). Responses are generaed as follows. A some dae T all sae varables are se a seady sae values. A "baselne" pahs for endogenous varables s compued by seng all exogenous nnovaons o zero a T. Then responses o one-me % nnovaons a T are compued; he Table repors devaons (mulpled by ) of hese responses from he baselne pahs (responses of neres rae ( j ), ax rae ( j ), Taxrev., and real deb ( B j Dj/( PY j )) : dfferences from baselne pahs; responses of oher varables: relave devaons from baselne pahs). 6