The Ramsey Steady State under Optimal Monetary and Fiscal Policy for Small Open Economies. Angelo Marsiglia Fasolo


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1 The Ramsey Seay Sae uner Opimal Moneary an Fiscal Policy for Small Open Economies Angelo Marsiglia Fasolo July,
2 ISSN CGC / Working Paper Series Brasília n. 357 July 2014 p. 149
3 Working Paper Series Eie by Research Deparmen Depep Eior: Francisco Marcos Rorigues Figueireo Eiorial Assisan: Jane Sofia Moia Hea of Research Deparmen: Euaro José Araújo Lima The Banco Cenral o Brasil Working Papers are all evaluae in ouble blin referee process. Reproucion is permie only if source is sae as follows: Working Paper n Auhorize by Carlos Hamilon Vasconcelos Araújo, Depuy Governor for Economic Policy. General Conrol of Publicaions Banco Cenral o Brasil Comun/Dipiv/Coivi SBS Quara 3 Bloco B EifícioSee 14º anar Caixa Posal Brasília DF Brazil Phones: an Fax: The views expresse in his work are hose of he auhors an o no necessarily reflec hose of he Banco Cenral or is members. Alhough hese Working Papers ofen represen preliminary work, ciaion of source is require when use or reprouce. As opiniões expressas nese rabalho são exclusivamene os auores e não refleem, necessariamene, a visão o Banco Cenral o Brasil. Aina que ese arigo represene rabalho preliminar, é requeria a ciação a fone, mesmo quano reprouzio parcialmene. Ciizen Service Division Banco Cenral o Brasil Deai/Diae SBS Quara 3 Bloco B EifícioSee 2º subsolo Brasília DF Brazil Toll Free: Fax: Inerne: <hp//www.bcb.gov.br/?contactus>
4 The Ramsey Seay Sae uner Opimal Moneary an Fiscal Policy for Small Open Economies Angelo Marsiglia Fasolo The Working Papers shoul no be repore as represening he views of he Banco Cenral o Brasil. The views expresse in he papers are hose of he auhor an o no necessarily reflec hose of he Banco Cenral o Brasil. Absrac This paper escribes he seay sae allocaions an prices for small open economies uner opimal moneary an fiscal policy in a meiumscale DSGE moel. The moel encompasses he mos common nominal an real rigiiies normally foun in he lieraure in a single framework. The Ramsey soluion for he opimal moneary an fiscal policy is compue for a large space of he parameer se an for ifferen combinaions of fiscal policy insrumens. Resuls show ha, espie he large number of fricions in he moel, opimal fiscal policy follows he usual resuls in he lieraure, wih high axes over labor income an low axes subsiies on capial income. On he oher han, he choice of fiscal policy insrumens is criical o characerize opimal moneary policy. Fricions associae wih he small open economy framework o no play a criical role in characerizing he Ramsey planner s policy choices. JEL Coes: E52, E61, E63, F41, F42, F44 Keywors: Ramsey Policy; DSGE moels; Small Open Economies; Moneary an Fiscal Policy Research Deparmen, Banco Cenral o Brasil. This paper is base on he isseraion wrien as a requiremen o a PhD egree a Duke Universiy. The auhor woul like o hank all he paricipans of seminars a Duke Universiy an he Banco Cenral o Brasil an he Fulbrigh Commission an CAPES for financial suppor uring he PhD program. This paper, however, oes no necessarily represen he views an opinions of Banco Cenral o Brasil. 3
5 1 Inroucion Wha is he opimal combinaion of moneary an fiscal policy insrumens for a small open economy? How oes opimal policy changes when srucural parameers characerize an emerging economy, insea of a evelope economy? How o allocaions an prices uner opimal policy iffer in a small open economy if he insrumen se available for he benevolen planner changes? A lo of effor has been evoe o characerize opimal policy for evelope small open economies, wihou iscussing if he associae welfare ranking of policy recommenaions is sill he same for more volaile, less evelope economies. The high volailiy observe in emerging economies usually ranslaes in meiumscale ynamic sochasic general equilibrium DSGE moels o parameer values far from hose esimae o evelope economies. This paper escribes he allocaions uner opimal policy of a convenional meiumscale moel for small open economies. In his paper, he main ifferences beween Emerging Economies EMEs, henceforh an evelope, Small Open Economies SOEs, henceforh will be resrice o srucural parameers characerizing each economy. The exercise here focus on he escripion of he Ramsey policy seay sae, clarifying he raeoffs face by a benevolen cenral planner. The evelopmen of DSGE moels for close economies 1, comprising a large se of nominal an real rigiiies, change research on opimal moneary an fiscal policies, no only because of he eparure from analyically solving Ramsey s 1927[33] problem in racable moels, bu also from a heoreical poin of view. Moels wih such large number of fricions eman an equally large number of nonisorionary insrumens in orer o recover he firsbes allocaion as he equilibrium oucome of he opimal policy. In a framework consraine by he lack of such large se of insrumens, he lieraure focuse on he numerical characerizaion of he opimal policy s oucome in moels where he seay sae is isore as a consequence of he nominal an real rigiiies. In hese simulaions, he governmen chooses values for a se of insrumens in orer o maximize minimize an uiliy loss funcion. Few auhors in he lieraure provie a comprehensive iscussion abou he properies of he seay sae uner he Ramsey policy. Woofor 2003[43] provies a complee escripion of he seay sae policy of he basic New Keynesian moel for close economies. The auhor explores ifferences in he Ramsey policy oucome when imposing aiional resricions like hose inclue here, as he imeless perspecive of he opimal policy problem. Sill in he close economy framework, bu now ealing wih variaions of moels similar in srucure o Chrisiano, Eichenbaum an Evans CEE, henceforh 2005[11], SchmiGrohé an Uribe SGU, henceforh 2005, 2006, 2007[36] [37] [38] explore he properies of he seay sae uner Ramsey opimal moneary an fiscal policies. These meiumscale moels o no have a close form soluion, like he basic srucures escribe in Woofor 2003[43]. Therefore, he only way o unersan an escribe opimal policy is by means of numerical simulaions. The resuls in erms of seay sae of prices usually poin ou for price sabiliy as he main oucome of he Ramsey planner, wih small variaions epening on he number of nominal an real rigiiies inclue in he moel. Despie he aopion of meiumscale moels for moneary policy analysis in some Cenral Banks, he research on opimal policy for SOEs in hese moels is sill very incipien. The main focus of he lieraure is on he evaluaion of opimal moneary policy in moels wih small eparures from he basic sicky price framework propose in Galí an Monacelli 2005[20] an Monacelli 2005[32]. Exensions ry o eal wih specific feaures of open economies: eviaions from he Law of One Price Kollmann 2002[25], Ambler, Dib an Rebei 2004[5]; incomplee foreign asse markes Ambler, Dib an Rebei 2004[5], Jusiniano an Preson 2009[24]; fiscal policy imension of he open economy framework Benigno an De Paoli 2009[8]. In he lieraure, no isincions are mae beween SOEs 1 The moels are usually some variaion of he framework in Chrisiano, Eichenbaum an Evans 2005[11]. See Schmi Grohé an Uribe 2006 an 2007[37][38] for opimal policy compuaion in hose moels. 4
6 an EMEs, resuling in he same policy recommenaions for boh ypes of counries espie large ifferences ocumene in he lieraure beween heir srucural parameers 2. The lieraure on EMEs focus on he escripion of hese economies, aing srucure over a basic moel in orer o capure isincive aspecs of aa. The higher volailiy in aa, when compare wih SOEs 3, brings aenion o opics like: foreign currency eman Felices an Tuesa 2007[19]; invesmen finance by foreign currency an balance shee effecs Devereux, Lane an Xu 2006[17] an Elekag an Tchakarov 2007[18], Baini, Levine an Pearlman 2009[7]; a commoiy secor, in orer o highligh he imporance of naural resources Laxon an Peseni 2003[26], Baini, Levine an Pearlman 2009[7]; househols heerogeneiy in crei marke access Baini, Levine an Pearlman 2009[7]. Mos of hese papers focus on he compuaion of opimal moneary policy rules, wih lile focus on fiscal policy or he srucural parameers 4, or, someimes, assuming a seay sae ha migh be ifferen from he Ramsey opimal soluion. Among he papers in his nonexhausive lis, Baini, Levine an Pearlman 2009[7] is he closes reference in erms of he heoreical framework aope here, as hey compue opimal moneary an fiscal policy rules in a moel wih several nominal an real rigiiies. However, he flexible price allocaion can always be recovere as an opimal oucome of he policy ue o he assumpion of a lump sum axaion as one of he fiscal policy insrumens. From a heoreical perspecive, he moel here epars from he lieraure as i oes no consier a se of lump sum mechanisms in orer o eliminae he isorions cause by nominal rigiiies: he Ramsey planner has access o isorionary consumpion, capial an labor income axes, besies he conrol of money supply an eb o balance he buge. As a consequence, he opimal policy allocaions are no necessarily equivalen o hose uner flexible prices, jus like he case for close economies escribe in SGU 2005 an 2006[36][37]. The compuaion of opimal policy is base on he soluion for he Ramsey problem, where a planner ries o maximize he iscoune expece uiliy of he represenaive househol. This approach iffers from oher suies where opimal policy is erive from he minimizaion of an arbirary loss funcion as a measure of welfare 5. The seay sae uner opimal policy here is characerize uner he same heoreical moel, bu fully exploring changes in parameer space, fiscal policy framework an nominal rigiiies in his case, rigiiies locae no only in omesic markes, bu also in he priceseing mechanism of impore an expore goos. One migh ask abou he imporance of generalizing resul on opimal policy uner commimen for EMEs, given he high volailiy an srucural changes characerizing hese economies. In realiy, mos of fiscal an moneary policy seings in EMEs are characerize by some ype of commimen. The aopion of conrolle exchange rae regimes in he early 90 s is an explici commimen o keep exchange rae flucuaions uner consrain. In he early 2000 s, several EMEs aope a combinaion of high fiscal surpluses an inflaion argeing regimes, while showing signs ha lile inerference will be mae in he exchange rae markes again, a new form of commimen. Even in abnormal perios, like suen sops episoes, governmens usually sign leers of inenions o insiuions like he IMF, commiing o a new macroeconomic arrangemen in orer o guaranee emergency loans. Thus, he policy problem of EMEs can be viewe as seing he righ commimen for hese economies, insea of a proposiion beween commimen versus iscreionary policies. In erms of resuls, price sabiliy seems o be he main goal of he Ramsey planner, given parameers 2 Silveira 2006[15] esimae he basic Galí an Monacelli 2005[20] moel using Brazilian aa. The poserior values for he elasiciy of he labor supply an he elasiciy of subsiuion beween impore an omesically prouce consumpion goos are ousie he bounaries foun in he lieraure. Anoher example is Elekag, Jusiniano an Tchakarov 2005[41], wih iscrepancies in he values for he ineremporal elasiciy of subsiuion an on he elasiciy of he labor supply, when compare o he calibraion use for close economies. 3 See Aguiar an Gopinah 2007[3]. 4 There is an effor o pu hese moels in an esimae framework. As an example, Elekag, Jusiniano an Tchakarov 2005[41] presen an esimaion of he moel publishe laer in Elekag an Tchakarov 2007[18]. 5 Some examples of he loss funcion approach are foun in Svensson 2000[40], Levin an Williams 2003[27], Laxon an Peseni 2003[26] an Jusiniano an Preson 2009[24] 5
7 use in calibraion. However, he number of available axes for he governmen plays a key role in seing he opimal axes an ineres raes uner ifferen assumpions on nominal an real rigiiies. To be more specific, he inclusion of consumpion axes as one of he axes available in he moel usually resuls in price sabiliy as he opimal oucome, eliminaing almos all raeoffs relae o he combinaion of nominal an real rigiiies. This resul confirms, for a moel esigne for small open economies, he proposiions in Correia, Nicolini an Teles 2008[14] regaring he role of a ax over he final goo of he economy, visavis a ax over inermeiae inpus. The classical resul from Ju 2002[23], of a high subsiy o capial relaive o he reurns of labor, is robus for fiscal policy, irrespecive o he se of insrumens available o he benevolen governmen. This paper is organize as follows. Chaper 2 presens he DSGE moel, wih focus on he equaions characerizing he equilibrium an he connecions beween he srucural fricions an he lieraure on DSGE moels, finishing wih he efiniions of he compeiive an he Ramsey equilibria. Chaper 3 shows he baseline calibraion of he moel. Chaper 4 presens a eaile analysis of he seay sae of he moel, assuming srucural parameers normally observe in he lieraure. Chaper 5 conclues. 2 A MeiumScale Moel for Small Open Economies In his secion, a brief escripion of he moel is provie wih he characerizaion of he househol an he firms problem, he policy rules for he governmen in a compeiive equilibrium, he foreign secor an aggregaion 6. Afer ha, he efiniions of boh a compeiive an Ramsey equilibria are presene. The moel is an exension for a small open economy of he close economy moel for moneary policy analysis propose in CEE 2005[11] an Alig, Chrisiano, Eichenbaun an Liné 2005[4]. Similar moels are use in Aolfson, Laseén, Liné an Villani 2007[2] an in Chrisiano, Traban an Walenin 2007[12]. These moels combine he sicky price framework of Galí an Monacelli 2005[20] an Monacelli 2005[32] o a nominal an real fricions base in CEE 2005[11]. From he househols perspecive, he moel presens exernal habi persisence in consumpion, ajusmen coss for invesmen, porfolio an capial uilizaion. Househols objecive is o maximize he iscoune value of expece uiliy. They own capial, eman money o buy consumpion goos an se wages afer observing he eman for his specific ype of labor. Househols in each perio buy boh omesically prouce an impore goos for consumpion, sell labor o saisfy he eman by firms afer he accepance of he propose wage an se he rae of capial uilizaion. In orer o ransfer wealh across perios, househols rae bons omesically an in inernaional markes an accumulae capial buil from boh omesically prouce an impore goos. They also face a cashinavance consrain, requiring omesic currency o buy a share of oal consumpion goos. Firms in he raable an nonraable secors of he omesic economy ren capial an labor from househols o prouce goos. They se prices in a Calvo syle, wih an exogenous probabiliy of opimizing prices in perio. Firms from he raable secor have o compee wih impore goos reailers. These reail firms buy goos prouce abroa an sell hem omesically, also ajusing prices in omesic currency in a Calvo syle. Also, firms from he raable secor can sell goos for he expore goos reailers. These firms buy omesically prouce goos an sell hem abroa, seing price in foreign currency in a Calvo syle hus, local currency pricing in boh omesic an foreign markes jusifies pricingomarke iscriminaion, as commonly seen in he lieraure 7. A eman for foreign currency is jusifie by a working capial consrain for impore goos reailers, wih hose firms selling bons o obain foreign currency o finance he acquisiion of foreign inpus. 6 Appenix A liss he final se of equilibrium coniions. 7 Some moels wih a leas parial local currency pricing are Kollmann 2002[25], Ambler, Dib an Rebei 2004[5], Devereux, Lane an Xu 2006[17], Chrisiano, Traban an Walenin 2007[12] an Jusiniano an Preson 2009[24]. 6
8 The governmen in a compeiive equilibrium ses nominal ineres raes accoring o a Taylor rule base on curren inflaion, in orer o mach an exogenous, imevarying inflaion arge. In erms of fiscal policy, he governmen has hree insrumens available o finance an exogenous sream of consumpion: money, bons sol omesically, an isorionary axes. The governmen migh ax in ifferen raes consumpion an he income from capial, labor an profis. In he compeiive equilibrium, axes on labor are se accoring o a simple policy rule base on oal governmen liabiliies. Taxes on consumpion, capial an on profis, as well as governmen spening, are exogenous. The foreign secor is escribe by a VAR incluing he foreign currency supply, oupu, inflaion, ineres raes an he risk premium. The moel has 16 shocks, wih five from he foreign secor, plus he following: one on he price of impore goos in foreign currency; wo saionary secorial prouciviy shocks; a nonsaionary aggregae prouciviy shock; a nonsaionary, invesmenspecific shock; governmen spening; hree ax shocks; moneary policy shock an a inflaion arge shock. 2.1 Househols There is a coninuum of infinielylive househols i i [0, 1] populaing he omesic economy, each of hem wih an enowmen of labor ype i, h i. There is no populaion growh an labor can no be sol for firms in he res of he worl. In he ineremporal problem, househols maximize iscoune uiliy choosing consumpion, capial uilizaion an invesmen for each secor, wages, hours worke an seing he money eman, nex perio s foreign an omesic bon holings an physical capial sock. The general ineremporal househol problem, given he nonponzi games consrains, is: s.. : max E 0 =0 β [1 γ log C i ζc 1 + γ log 1 h i] P 1 + τ c C i + Υ 1 P I x, i + In,i + P M i + R 1 B h, i +S R f 1 IB i + W φ w2 2 W i π χw W µi 1i + ψ 1 Bh,+1 i 2 Y Y B h Y 2 + ψ2 2 Y SIB +1i P Y + 1 τ φ P Φ i + 1 τ k 2 rer IB Y = P 1 M 1 i + 1 τ h W ih i [ P R k n, µ n, Υ 1 a µ n, K n, i + Rx,µ k x, Υ 1 a µ x, K x, i ] + B h,+1 i + S IB +1 i K j,+1 i = 1 δ K j, i + I j,i Υ +1 ℵ 1 ℵ I j, i a µ j, = θ 1 µ j, 1 + θ 2 2 µ j, 1 2 I i, I i, 1 = φi 2 I j, 1 i K j, = µ j, K j, 2 I i, µ I j = {x, n} I i, 1 = µ Υ +1 = 1 ρ Υ µ Υ + ρ Υ µ Υ + ɛ Υ Υ +1; ɛ Υ N 0, σ Υ ϖ W i h i = h W M i ν m 1 + τ c C i 7
9 In his problem, β is he ineremporal iscoun facor of he uiliy funcion. The uiliy funcion is logseparable in erms of consumpion an labor, wih consumpion ajuse by exernal habi persisence 8. The egree of habi persisence is efine by he parameer ζ [0, 1. Househols accumulae physical capial, K j,, for j = {x, n} represening he secors of he economy, buying from he firms invesmen goos ha epreciae a a rae δ. Define Υ 1 as he nonsaionary inverse of he relaive price of invesmen in erms of consumpion goos. The relaive price of invesmen goos can also be inerpree as a echnology shock affecing he linear proucion funcion available o househols o ransform consumpion goos in invesmen goos 9. Invesmen is subjec o an ajusmen cos ℵ., in he same fashion as in CEE 2005[11] an Alig, Chrisiano, Eichenbaun an Line 2005[4] such ha ℵ 1 = 0, ℵ 1 = 0, ℵ 1 > The funcional form follows SGU 2006[37], wih µ I efining he seay sae growh of invesmen. Househols ren capial for firms afer seing he rae of capial uilizaion for each secor µ j,, paying a cos given by a µ j, o change he uilizaion level in each perio an in each secor. The aferax privae reurn of capial in each secor is efine, hus, as 1 τ k P R k j, µ j, Υ 1 a µ j, K j, i. The supply of labor is ecie by each househol aking as given he aggregae wage an eman for labor of he economy, W an h, an he ajusmen cos for wages. As a monopolis of a specific ype of labor, househol chooses he nominal wage W i an supplies all he emane labor h i given he accepance of W i. The elasiciy of subsiuion across ifferen ypes of labor is given by ϖ > 1. The nominal wage ajusmen cos funcion allows for parial inexaion base on curren inflaion, eermine by χ w χ w [0, 1]. The presence of sicky wages resuls in an aiional isorion, efine by mcw, which is he markup impose over real wages as househols supply a specific ype of labor. The quaraic ajusmen cos 11 is consisen wih he absence of lump sum insrumens o correc for wealh ispersion across househols. Wageseing processes base on Calvo moel creae ispersion in wage income across househols, wih he represenaive househol recovere by lump sum subsiy schemes or an asse marke srucure capable of insuring househols agains wage ispersion. Boh insrumens woul be conroversial wih he evaluaion of opimal policy assuming ha governmen oes access lump sum schemes o suppor agens. Anoher alernaive is o assume a cenralize union coorinaing he supply of labor, as in SGU 2006[37]. The assumpion of a labor union wih such marke power, however, oes no seem reasonable for evelope small open economies ousie a few European counries 12. Sill in he buge consrain, househols allocae wealh over ime buying oneperio, nonsae coningen nominal bons from he governmen, B h,+1 i, or from he res of he worl, IB +1 i. In he laer case, bons are price in foreign currency, an S is he nominal exchange rae. In orer o ajus porfolio, an o inuce saionariy in he moel, househols incurs in coss base on he variance of he sock of bons as a proporion of he GDP 13. Househols also receive iviens from firms Φ i. Finally, following SGU 2007[38], househols eman money, M i in orer o pay for a share ν m 0 of consumpion. The sequence of evens in each perio is he same as in CEE 2005[11], wih he househols firs eciing consumpion an capial allocaion, hen eciing, in sequence, he financial porfolio, wages an he labor supply, an he final composiion of porfolio beween bons an money. 8 In erms of noaion, he general variable x i represens he choice of househol i on perio abou x. The variable x is he aggregae value of x i for he economy. 9 See Greenwoo, Hercowiz an Krusell 2000[22] an SGU 2006[37]. 10 Ajusmen coss an he srucure of shocks, espie no affecing he eerminisic seay sae of he moel, are kep in he escripion of he moel for he sake of compleeness. I is lef, also, as a suggese srucure for reference in orer o suy he ynamic properies of he Ramsey equilibrium. 11 See, for insance, Chugh 2006[13] an GarcíaCicco 2009[21]. 12 Accoring o aa from OECD 2004, only Denmark, Sween, Finlan, Icelan an Belgium presene a seay increase in raeunion ensiy from 1960 o In Lain American economies, he raeunion ensiy is no only lower, compare o Scaninavian counries, bu also eclining since he 1990 s see Visser Marin Tergeis, 2008[42]. 13 See SGU 2003b[35]. The use of he raio o GDP in he funcional form aope here makes i easier o obain he saionary form of he moel. 8
10 Thus, omesic currency is expresse as an enofperio aggregae. Define λ /P, λ q j,, λ m λ an λ 1 τ h W / P mcw he Lagrange mulipliers on he buge consrain, on he capial accumulaion equaions, on he cashinavance consrain an on he labor eman funcion, respecively. Noing ha, in he symmeric equilibrium, C i = C an W i = W, he final se of equilibrium coniions of he ineremporal problem of househols is given by: 1 τ h W 1 + τ c C ζc 1 = γ 1 γ 1 γ C ζc 1 = 1 + τ c λ λ [1 ψ 1 Bh,+1 Y λ [1 ψ 2 S IB +1 P Y B ] h Y ] rer IB Y mcw 1 + ν m R 1 R 1 h 1 + ν m R 1 R = βr E λ+1 π +1 = βr f E S+1 S P P +1 λ+1 λ q x, = βe { λ+1 [ 1 τ k +1 R k x,+1 µ x,+1 Υ 1 +1 a µ x,+1 + q x,+1 1 δ ]} 5 λ q n, = βe { λ+1 [ 1 τ k +1 R k n,+1 µ n,+1 Υ 1 +1 a µ n,+1 + q n,+1 1 δ ]} 6 K n, = µ n, K n, 7 K x, = µ x, K x, θ 1 + θ 2 µ n, 1 = Rk n, Υ 1 9 θ 1 + θ 2 µ x, 1 = Rk x, Υ 1 10 R = 1 r,+1 11 R = R 1 ψ 1 Bh,+1 Y λ Υ 1 = λ q x, [1 Ψ +βe I x, I x, 1 λ+1 q x,+1 I x,+1 I x, B h Y I x, I x, Ψ I x,+1 Ψ I x, I x, I x, 1 ] 13 λ Υ 1 = λ q n, [1 Ψ +βe I n, I n, 1 λ+1 q n,+1 I n,+1 I n, I n, I n, 1 2 Ψ I n,+1 Ψ I n, I n, I n, 1 ] 14 I K x,+1 i = 1 δ K x, i + Ix,i x, i 1 ℵ Ix, 1 i I K n,+1 i = 1 δ K n, i + In,i n, i 1 ℵ In, 1 i
11 1 ϖ ϖ + 1 mcw +βe ϖh 1 τ h = λ +1 φ w λ π χw 1 +1 W+1 W φ w π χw 1 W W 1 2 W+1 µ I π χw 1 +1 W W µ I W 1 17 π χw 1 From he firs orer coniions, noice ha he uncovere ineres pariy UIP coniion beween omesic an foreign ineres raes can be recovere afer linearizing equaions 3 an 4. The UIP coniion hols in is sric sense only in he seay sae, since he nonlinear ynamics is also influence by he presence of omesic an foreign porfolio ajusmen coss. This is a eparure from oher suies, like Aolfson, Laseén, Liné an Villani 2007[2], where he only source of iscrepancy beween he omesic an foreign ineres raes from he UIP coniion is he ebelasic foreign ineres rae. As he escripion of he foreign block of he moel will make clear, he UIP coniion combines he ebelasic foreign ineres rae an he porfolio ajusmen cos propose in SGU 2003b[35]. The househol also solves a sequence of minimizaion problems consraine by he CES funcion in orer o choose he composiion of he consumpion an invesmen goos. Expressing firs he consumpion problem, househols ecie beween impore an omesically prouce goos in he raable goos baske, an hen chooses he opimal composiion of raable an nonraable goos. For simpliciy, assume ha porfolio ajusmen coss are pai wih a share of he consumpion goos acquire by he househols. As a consequence, he cos minimizaion problem is given by: min C n,,c,,c m,,c x, P n, C n, + P, C, C + P AC b, + P AC ib, = [ C, = 1 κ 1 ϱ C [1 ω 1 ε C ε 1 ε n, ϱ 1 ϱ x, + κ 1 ϱ 1 ϱ C ϱ m, + ω 1 ε 1 ] ε ε 1 ε C ε, ] ϱ ϱ P AC b, = ψ 1 2 Y Bh,+1 Y B 2 h Y P AC ib, = ψ 2 2 Y S IB +1 IB P Y Y Combine he firs orer coniions o obain he eman for each ype of raable goo: 2 C m, = κ C x, = 1 κ Pm, P, Px, ϱ C, 20 P, ϱ C, 21 By analogy, he opimal ecision beween raable an nonraable goos is given by: C, = ω C n, = 1 ω P, P Pn, ε C + P AC b, + P AC ib, 22 P ε C + P AC b, + P AC ib, 23 Househols solve a similar problem seing he invesmen goo in each secor. For simpliciy, assume ha he weighs an he elasiciies of subsiuion among ifferen invesmen goos are he same as hose for consumpion goos. Ajusmen coss in capial uilizaion are pai in erms of 10
12 aggregae invesmen. The eman for home prouce an impore invesmen goos become: Υ 1 I = Υ 1 I n, + a µ n, K n, + Ix, + a µ x, K x, 24 I m, = κ I x, = 1 κ I, = ω I n, = 1 ω Pm, P, P, Px, ϱ I, 25 P, P Pn, P ϱ I, 26 ε Υ 1 I 27 ε Υ 1 I Firms There are four secors in he economy, each secor compose by a coninuum of firms in a monopolisic compeiive framework. Firms in nonraable n an raable x secors eman labor an capial o prouce. Firms in he impore expore goos secor, m xp, buy he final goo an sell i in he omesic economy res of he worl. Firms chooses he amoun of inpu an se prices base on a probabiliy α i, i = {n, x, m, xp}, ha is inepenen across secors an firms. Firms no allowe o opimize prices in perio change prices accoring o an inexaion rule base on pas inflaion. Impore goos firms mus finance inpu acquisiion using foreign currency. There is no firm enry ino or exi ou of secor i. Equaions escribing price ynamics woul resul, in a loglinearize moel aroun price sabiliy, in equaions like he New Keynesian Phillips curve. However, since price sabiliy migh no be opimal for Ramsey planner, he recursive formulaion for he firs orer coniion in erms of prices escribe in SGU 2006[37] is aope Domesic nonraable goos proucers problem: Firms in he nonraable secor prouce goos use for consumpion, invesmen an spen by he governmen. Seing real profis as Φ n, i n, he problem of proucers of ype i n prouc i n [0, 1] is o maximize he expece iscoune sream of profis, subjec o he eman for goo i n, he proucion echnology an he aggregae eman for nonraable goos. Firms choose in each perio he eman for labor, capial an, wih probabiliy 1 α n, hey opimize prices. The problem is given by: max E 0 =0 Pn, i n r 0, P n, D n, i n W P n, s.. : D n, i n = P n,, i n P ηn Pn,i n Yn, P n, R k P n,k n, i n n, Y n, = C n, + G n, + Υ 1 P P n, I n, a n, K n, i n θ z, i n z χ n D n, i n Υ θ = z z z +1 z = µ z +1 = 1 ρ z µ z + ρ z µ z + ɛ z +1; ρ z [0, 1 ; ɛ z N 0, σ z log a n,+1 = ρ n log a n, + ɛ n +1; ρ n [0, 1 ; ɛ n N 0, σ n In his problem, a n, is a saionary, secorspecific echnology shock, z is a laboraugmening, non 11
13 saionary echnology shock, affecing firms using labor as inpu. In orer o guaranee zero profis in he seay sae, z χ n inrouces a fixe cos proporional o he evoluion of he nonsaionary shock, following CEE 2005[11] an SGU 2005, 2006[36][37]. Parameer η n is he elasiciy of subsiuion across varieies of nonraable goos. From he firs orer coniions in erms of, i n an K n, i n, i is possible o prove ha he capiallabor raio an he marginal cos are he same across firms in he nonraable secor. Seing mc n, as he Lagrange muliplier on he firm s eman consrain, he equilibrium coniions are: P W = mc n, 1 θ a n, z P n, R k n, P P n, = mc n, θa n, Kn, Kn, z, θ 29 z, θ 1 30 Prices are forme in a Calvo syle wih inexaion, where α n is he probabiliy ha firm i n is no allowe o opimally ajus is price in perio. In he case firms are no allowe o opimize prices, hey follow he rule P n, i n = π κn n, 1 P n, 1 i n, for 0 κ n 1 an π n,+1 = Pn,+1 P n,. Seing he Lagrangean of he problem, consiering only he relevan erms for price eerminaion: L n = E s=0 1 ηn s αnr s,+s P n,+s Pn, i n π κ n P n,+s k=1 mc n,+s n,+k 1 π n,+k 1 η n Y n,+s ηn s Pn, i n π κ n P n,+s k=1 n,+k 1 π n,+k η n Y n,+s In his problem, r,+s is he sochasic iscoun facor beween perios an + s, an P n, i n is he new price se by firms allowe o ajus prices in perio. The firs orer coniion for firms is: E s=0 ηn s Pn, αnr s i n,+s Y n,+s P n,+s P n,+s k=1 π κ n n,+k 1 π n,+k η n η n 1 η n Pn, i n P n,+s s π κ n k=1 n,+k 1 π n,+k mc n,+s Since markup over prices is he same across firms, he symmeric equilibrium is characerize by all firms in secor n allowe o ajus prices in perio seing he same price: P n, i n = P n,. Following SGU 2006[37], spli he firs orer coniion in wo pars, X 1 o obain a recursive soluion for he priceseing problem: an X 2, an efine p n, = P n, P n, = 0 in orer 1 ηn X 1 pn, π κ n = Y n, p 1 ηn n, n, mc n, + α n r,+1 E p n,+1 π 1+ηn/ηn n,+1 ηn X ηn X 2 η = Y n, p ηn n 1 pn, π κ 1 ηn n n, n, + α n r,+1 E X η n p n,+1 π ηn/ηn n,+1 X 1 = X Traable goos proucers problem: A raable goos proucer i x i x [0, 1] solves he same problem as he nonraable proucer, using labor an capial as proucion inpus. The proucion of he raable goo is ivie beween omesic 12
14 absorpion consumpion, invesmen an governmen spening an he eman of a coninuum of i xp exporing firms D xp,. The raable goos firm problem is given by: max E 0 =0 r 0, P x, s.. : D x, i x = P x, i x D x, i x W h x, i x P R k P x, P x, P x,k x, i x x, ηx Px,i x Yx, P x, Y x, = C x, + G, + Υ 1 P P x, I x, + D xp, a x, K x, i x θ z h x, i x z χ x D x, i r Υ θ = z z z +1 z = µ z +1 = 1 ρ z µ z + ρ z µ z + ɛ z +1; ρ z [0, 1 ; ɛ z N 0, σ z log a x,+1 = ρ x log a x, + ɛ x +1; ρ x [0, 1 ; ɛ x N 0, σ x χ x is a fixe cos relae o he nonsaionary shock ensuring zero profis in seay sae. Parameer η x is he elasiciy of subsiuion across ypes of raable goos. Seing mc x, as he Lagrange muliplier, he cos minimizaion problem, noing ha capiallabor raio is he same across firms, implies: θ P Kx, W = mc x, 1 θ a x, z P x, z h x, 34 θ 1 Rx, k P Kx, = mc x, θa x, P x, z h x, 35 Similar o he nonraable secor, price ajusmen is base on a Calvo mechanism wih inexaion, wih 0 κ x 1 efining he egree of inexaion in he raable secor. Taking he firs orer coniions in erms of P x, i x, an efining π x,+1 = Px,+1 P x,, he opimal price se by each firm solves a recursive problem ha can be spli in wo equaions for Z 1, Z 2, such ha Z 1 = Z 2, an p x, = P x, P x, : 1 ηx Z 1 px, π κ ηx x = p 1 ηx x, x, Y x, mc x, + α x r,+1 E Z p x,+1 π 1+ηx/ηx x,+1 ηx Z 2 η = p ηx x 1 px, π κ 1 ηx x x, x, Y x, + α x r,+1 E Z η x p x,+1 π ηx/ηx x,+1 Z 1 = Z Impore goos firms problem: Following Lubik an Schorfheie 2006[29], eviaions from he Law of One price arises as a consequence of price sickiness in impore an expore goos. An impore goos firm i m i m [0, 1] buys a bunle of he inernaional homogeneous goo 14 an label i as an impore goo ype i m. In orer o buy inpu from he res of he worl, he firm nees o pay using foreign currency. The firm sells inraperio bons in foreign markes in orer o ge foreign currency, bu i oes no ransfer financial 14 In he moel, one counry buys a combinaion of final goos from ifferen counries, generaing a gap beween he worl s CPI P an he price of he bunle impore by a given counry P m,. 13
15 wealh over ime. As a consequence, firms face only an increase in he marginal cos of proucion. As a iming convenion, rae bons o no reflec in he en of perio balance of paymens. The same framework is aope in Chrisiano, Traban an Walenin 2007[12] an Menoza an Yue 2008[31]. The buge consrain of he imporing firm i m, expresse in erms of omesic prices, is given by: S P P M m,i m + S P B m,+1i m = S P P 1M m, 1i m + S P R f 1 B m,i m + Pm, i m S P m, P D m, i m z χ m Φ m, i m where χ m is a fixe cos associae wih he nonsaionary shock in orer o guaranee zero profis in seay sae. Following assumpions ha firms o no keep any financial wealh across perios an ha all profis are isribue o he househols, obain he expression for real profis: P Mm,i m + R f Bm,+1i m = 0, Pm, i m S P m, = Φ m, i m = D m, i m z χ m S P R f 1 M P P m,i m The impore goos firm problem becomes: max E 0 P m,i m =0 r 0, [ Pm, i m S P m, P D m, i m z χ m S P m, P m, P R f R f 1 ηm Pm, i m s.. : D m, i m = C m, + Υ 1 P I m, P m, M m, i m P m, P D m, i m R f P P m, M m,i m where P m, is he price of he impore goo bough by he omesic economy, quoe in foreign prices. Define η m as he elasiciy of subsiuion across varieies of impore goos, α m as he probabiliy ha firm i m is no allowe o ajus prices in perio, π m,+1 = Pm,+1 P m, an 0 κ m 1 he egree of inexaion in he impore goos secor. Taking he firs orer coniions in erms of P m, i m, an as a consequence of he same markup over prices across firms given by he real exchange rae base on he impor price level, he symmeric equilibrium is characerize by P m, i m = P m,. The recursive soluion for he problem is obaine afer efining Y 1 an Y 2 such ha Y 1 = Y 2, an p m, = P m, P m, : Y 1 Y 2 = p 1 ηm m, = p ηm m, C m, + Υ 1 C m, + Υ 1 P I m, P m, P I m, P m, S P m, P m, 1 + Rf 1 R f + α m r,+1 E pm, p m,+1 ηm 1 + α m r,+1 E η m pm, p m,+1 1 ηm π κm m, π 1+ηm/ηm m,+1 ] ηm Y ηm ηm πm, κm Y π ηm/ηm m,+1 Y 1 = Y Expore goos firms problem: An expore goos firm i xp i xp [0, 1] buys a share of he final raable goo in he omesic economy an sell i o he res of he worl, seing prices in foreign currency in a Calvo syle hus, prices are 14
16 sicky in foreign currency. The firm problem is given by: max P x, ixp E 0 [ ] S P x, i xp P x, R 1 r 0, D xp, i xp M xp, i xp z χ xp P R P ηxp x, i xp s.. : D xp, i xp = X =0 where χ xp is a fixe cos associae wih he nonsaionary shock in orer o guaranee zero profis in seay sae. Define η xp as he foreign elasiciy of subsiuion across varieies of omesic expore goos, πx,+1 = P x,+1 P, 0 κ x, xp 1 as he egree of inexaion in he expore goos secor, α xp as he probabiliy ha firm i x is no allowe o opimize prices in perio, an Px, as he price of he raable goo quoe in foreign currency. Taking firs orer coniions in erms of P x, i xp, an given ha he symmeric equilibrium is characerize by P x, i x = P x,, he recursive soluion for he pricing problem of he exporing firms is obaine afer properly efining U 1 an U 2 an p x, = P x, P : x, U 1 U 2 = p 1 ηxp P x, p x, x, X S Px, + α xp r,+1 E p x,+1 P x, 1 ηxp = p ηxp ηxp η xp 1 p x, x, X + α xp r,+1 E η xp p x,+1 π x, κxp π x,+1 1+ηxp ηxp π x, κxp π x,+1 ηxp ηxp 1 ηxp U ηxp U U 1 = U Governmen In a compeiive equilibrium, he governmen follows simple rules o se moneary an fiscal policy. For he sake of his paper, as he seay sae of he moel is no affece by ineres rae rules, assume ha he governmen follows a sanar Taylor rule, base on he eviaions of inflaion from an exogenous, auocorrelae inflaion arge an an auoregressive componen 15 : log R+1 R R = ρ R log + α π log R π+1 π o +1 π o +1 = 1 ρ π oπ o + ρ π oπ o + ɛ πo +1; + ɛ R +1; ɛ R N 0, σ R 45 ɛ πo N 0, σ π o 46 The governmen, in orer o finance is exogenous expeniures, G, collecs isorionary axes on consumpion, labor, capial an profis income τ c, τ h, τ k an τ φ, sells bons omesically, B g, an conrols he money supply, M. The governmen buge consrain is given by: P G + R 1 B g, = P T + P M + B g,+1 P 1 M 1 G = z g g = 1 ρ g g + ρ g g 1 + ɛ g ; ɛ g N 0, σ g 47 T = τ c C + τ h W h + τ φ Φ + τ k [ R k n, µ n, Υ 1 a µ n, K n, + Rx,µ k x, Υ 1 a µ x, ] K x, 48 Rewriing he governmen buge consrain as a funcion of oal real governmen liabiliies L : L 1 M 1 + R 1 P 1 B g, To be more specific, any Taylor rule ha ensures a nonzero, saionary inflaion aroun he seay sae saisfy he requiremens of he compeiive equilibrium. 15
17 = L = R π L 1 + R G T R 1 M 50 To close he ynamics of fiscal policy, assume ha governmen follows a policy rule for labor income axaion base on real liabiliies as a funcion of GDP an he oupu gap. Taxes on capial an profis are exogenous. The assumpion of a policy rule for labor income axes is an arbirary choice ha oes no affec he seay sae of he moel. For simpliciy, assume also ha axaion on profis is consan over ime. Noice ha axes on profis are lump sum ransfers from househols o he governmen. In his sense, i oes no inerfere wih ynamics uner he compeiive equilibrium, where profis are zero. τ h τ h L = ψ li l + ψ y y y + ɛ τh ; ɛ τh N 0, σ τh 51 Y y τ k = 1 ρ τk τ k + ρ τk τ k 1 + ɛ τk ; ɛ τk N 0, σ τk 52 τ φ = τ φ 53 τ c = 1 ρ c τ c + ρ c τ c 1 + ɛ τc ; ɛ τc N 0, σ c 54 Aiionally, he governmen solves he same problem as househols o eermine he consumpion of raable an nonraable goos. By assumpion, he governmen oes no consume impore goos 16 : G n, = 1 ω G, = ω P, Pn, P P ε G 55 ε G Inernaional Financial Markes an Worl s Economy The ransmission of shocks from inernaional financial markes assumes a marke capable of pricing counryspecific risk on bons rae ousie omesic economy. A mechanism o inuce saionariy as in SGU 2003b[35] is use o eermine he risk premium of bons as a funcion of he ne foreign posiion of he economy. The inernaional ineres rae is given by: R f = R S IB 1 + ξ κ1 +1 / IB P Y Y κ2 57 In his equaion, R is he nominal ineres rae on a riskfree bon; ξ is an exogenous shock wih expece value equal o he long run risk premium of he omesic economy, ξ ; he las erm is he gap of exernal eb as a proporion of GDP. The worl s economy is se as a saionary VAR wih oupu, y, inflaion, π, ineres raes, R, growh of money supply, M, an he risk premium, ξ, proviing five shocks for he moel. As he VAR oes no influence he seay sae, i is no necessary o be more specific wih respec o aiional srucure. In he sysem, A is a 5 by 5 marix of coefficiens, is a 5 by 5 covariance marix of shocks. M M ξ ξ R R π π y y = A M 1 M ξ 1 ξ R 1 R π 1 π y 1 y + ɛ m ɛ ξ ɛ R ɛ π ɛ y ɛ m ɛ ξ ɛ R ɛ π ɛ y ii 0, 58 Two assumpions close he link beween prices an quaniies of goos beween he omesic economy an he res of worl. Firs, assume ha foreign househols solve an expeniure minimizaion 16 The same assumpion is use in Lubik an Schorfheie 2006[29]. 16
18 problem o se he eman for home prouce raable goos. The soluion of his problem is given by: P x, X = P η z y 59 Finally, he erms of rae are efine as he raio beween he expore goos an he impore goos price levels, boh in foreign currency. The ynamics of he price of impore goos is given by an errorcorrecion moel ha ensure he erms of rae becomes saionary, as in GarcíaCicco 2009[21]: [ wih X = M M π m π m ξ ξ o = π x, πm, o 1 60 = υ π m 1 1 o π + υ 1 m 2 o + X 1 + ɛ πm R f R π π ] y. y ɛ πm N 0, σ πm Aggregaion an Relaive Prices In orer o fin an expression for he aggregae consrain of he economy, sar from he eman face by a nonraable proucer firm an inegrae i over all i n firms, noing ha, = 1 0, i n i n, an ha capiallabor raio is consan across all he firms. Define s n, = 1 ηn, Pn,i n in o obain: 0 P n, a n, Kn, θ z, z χ n = s n, C n, + G n, + Υ 1 P I n, P n, Obain he recursive form of s n, : s n, = 1 0 ηn Pn, i n i = s n, = 1 α n p ηn n, + α n P n, Also, from he efiniion of he nonraable goos price inex: [ 1 P n, = 0 ] 1 1 ηn P n, i n 1 ηn i = 1 = 1 αn p 1 ηn n, π n, π κn n, 1 + α n π κ n 62 ηn s n, 1 63 n, 1 π n, 1 ηn 64 Similar expressions can be wrien for resource consrain, price ispersion an price inex of impore an omesically prouce raable goos an he prices of expore goos in foreign currency: D m, z χ m = s m, C m, + Υ 1 P I m, 65 P m, ηm π s m, = 1 α m p ηm m, m, + α m s m, 1 66 π κm m, 1 π κ m 1 ηm 1 = 1 α m p 1 ηm m, 1 m, + α m 67 π m, a x, Kx, θ z h x, z χ x = s x, C x, + G, + Υ 1 P I x, + D xp, 68 P x, ηx π s x, = 1 α x p ηx x, x, + α x s x, = 1 α x p 1 ηx x, π κx x, 1 + α x π κ x x, 1 π x, 1 ηx 70 17
19 D xp, z χ xp = s xp, X 71 s xp, = 1 α xp p ηxp π ηxp xp, x, + αxp π κxp s xp, 1 72 xp, 1 1 = 1 α xp p 1 ηxp xp, π κxp 1 ηxp x, 1 + α xp 73 The oal amoun of work hours supplie by he omesic househols is given by: π x, h x, +, = h 74 The exernal equilibrium assumes ha ne foreign posiion of omesic househols is proporional o rae balance in seay sae. Again, noice ha he exernal equilibrium in bon markes oes no inclue bons issue by impore goos firms, as hey are negoiae an liquiae insie each perio: ] R P x, X P m, D m, [1 f 1 + = S R f 1 P IB S P+1IB R f In orer o se he marke clearing coniions for omesic bons an money marke, assume ha foreign househols an omesic firms o no eman home governmen bons. As a consequence: B g, + B h, = 0 76 Finally, he gross omesic prouc an aggregae profis are given by: Y = C + ψ 1 2 Y B+1 B 2 + ψ 2 Y Y 2 Y S IB +1 P Y 2 rer IB Y + Υ 1 I + G + P x, X P m, P P D m, [ ] R f R f 77 Φ = Y W h R k n,µ n, K n, R k x,µ x, K x, Relaive prices The complee se of relaive prices in he moel is given by: p = P, P pn = P n, P = π, π = π n, π P, 1 P 1 79 P n, 1 P 1 80 px = P x, P, = π x, π, P x, 1 P, 1 81 pm = P m, P, = π m, π, P m, 1 P, 1 82 pm = P m, P = π m, π P m, 1 P 1 83 rer = S P P 84 18
20 2.7 Saionary Form an Equilibrium The objecive of his secion is o characerize he saionary compeiive an Ramsey Equilibria. Define saionary allocaions such ha, for a generic variable X an he appropriae ren Ž, he saionary variable is given by x X /Ž. The moel in saionary form is escribe by he following variables: prices: π, π n,, π x,, π,, π m,, w, rx,, k rn,, k r,+1, mcw, mc n,, mc x,, rer, π, πx,, π m, p n,, p x,, p m,, p x,, p, pn, px, pm, pm, o ; ineres raes: R, R, R, R f ; allocaions: c, c,, c n,, c m,, c x,, i, i,, i n,, i m,, i x,, x, m,, xp,, µ x,, µ n,, i x,, i n,, y, k x,, k n,, k x,, k n,, h,,, h x,, x 1, x 2, z 1, z 2, y 1, y 2, u 1, u 2, ib, b h,, ξ, M, y, s n,, s m,, s x,, s xp,, λ, q x,, q n,, g,, g n,, m, φ ; governmen policies: τ h, l,, b g, ; omesic shocks: g, τ k, τ c, τ φ, a x,, a n,, µ z, µ Υ, π o. The equaions escribing he law of moion of he variables are given by a se of equilibrium coniions for he househol equaions 128, firms responsible for omesic proucion equaions 2938, exporing an imporing firms equaions 3944, governmen equaions 4556, foreign secor equaions 5761, aggregaion an price inexes equaions an relaive prices equaions Aiionally, here are 4 exogenous processes for secorial prouciviy an aggregae prouciviy growh a x,, a n,, µ z, µ Υ. Alernaively, here are 83 equaions for enogenous variables 17 an 9 omesic exogenous sochasic processes for a oal of 92 variables in he moel. Prices an shocks are saionary, bu allocaions mus be normalize in orer o ensure saionariy. The se of variables given by { K n,+1, K n,+1, K x,+1, K x,+1, I, I,, I n,, I m,, I x,, I x,, I n,} mus be normalize by z Υ, while he se given by {Y, C, C,, C n,, C m,, C x,, W, X, D m,, D xp,, B h,+1, } B g,+1, IB +1, M, X 1, X 2, Z 1, Z 2, Y 1, Y 2, U 1, U 2, G, G,, G n,, L, T mus be ajuse by z. Finally, renal rae of capial { } Rx,, k Rn, k an he shaow prices of invesmen { qx,, q n, } are ivie by Υ 1, while he Lagrange muliplier of consumpion, λ, is normalize by z 1 o obain λ Compeiive Equilibrium { } Definiion 1: Given exogenous pahs for shocks g, τ k, τ φ, τ c, a x,, a n,, µ z, µ Υ, π o, foreign secor } { variables { M, ξ, R, π, y, πm,, policy processes for ineres raes R, R }, R f an axes τ h, } an iniial values for prices {π 1, π n, 1, π x, 1, π, 1, π m, 1, w 1, p 1, pn 1, px 1, pm 1, pm 1, o 1 } an allocaions {c 1, i x, 1, i n, 1, k x,0, k n,0, b h, 1, b g, 1, ib 1, s n, 1, s m, 1, s x, 1, s xp, 1, l 1, a saionary compeiive equilibrium is a se of processes for prices { π, π n,, π x,, π,, π m,, w, rx,, k rn,, k r,+1, mcw, } mc n,, mc x,, rer, πx,, p n,, p x,, p m,, p x,, p, pn, px, pm, pm, o an allocaions {c, c,, c n,, c m,, c x,, i, i,, i n,, i m,, i x,, x, m,, xp,, µ x,, µ n,, i x,, i n,, y, k x,, k n,, k x,, k n,, h,,, h x,, x 1, x 2, z 1, z 2, } y 1, y 2, u 1, u 2, ib, b h,, b g,, s n,, s m,, s x,, s xp,, λ, m, q x,, q n,, g,, g n,,, l, φ such ha, afer saionary ransformaions of he respecive equaions: a househols maximize uiliy; b firms maximize profis; c governmen balances is buge; markes clear Ramsey Equilibrium The Ramsey equilibrium is evaluae from he imeless perspecive escribe in Woofor 2003[43], where he governmen can no change is policy from he ime when he Ramsey policy is implemene 17 Noe ha equaion 58 is a 5variable VAR. 19
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