214s-43 Governmen Spending Mulipliers and he Zero Lower Bound in an Open Economy Charles Olivier Mao Takongmo Série Scienifique Scienific Series Monréal Novembre 214/November 214 214 Charles Olivier Mao Takongmo. Tous drois réservés. All righs reserved. Reproducion parielle permise avec ciaion du documen source, incluan la noice. Shor secions may be quoed wihou explici permission, if full credi, including noice, is given o he source.
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Governmen Spending Mulipliers and he Zero Lower Bound in an Open Economy * Charles Olivier Mao Takongmo Résumé/absrac Wha is he size of he governmen-spending muliplier in an open economy when he Zero Lower Bound (ZLB) on he nominal ineres rae is binding? Using a heoreical framework, in a closed economy, Chrisiano, Eichenbaum, and Rebelo (211), show ha, when he nominal ineres rae is binding, he governmen-spending muliplier can be close o four. Their heory helps us o undersand he governmen spending muliplier in ZLB, bu i is difficul o mach ha heory wih he daa. We propose a dynamic sochasic general equilibrium in open macroeconomics, wih marke imperfecions, wage and price rigidiies and endogenous smoohing moneary policy. We argue ha, in a closed economy and in he presence of ZLB, here is no crowding ou effec hrough ineres raes. We also argue ha in an open economy, here is anoher channel for he crowding ou effec via he real exchange rae. For an open economy, he muliplier falls o he levels usually observed in small, closed economies for which he ZLB is no binding. Mos clés/key words: Governmen-spending muliplier, zero lower bound, sicky price, sicky wages, Taylor rule. Codes JEL : E52, E62, F41, F44. * We hank Professor Jean-Marie Dufour, Professor Alain Paque, Professor Seven Ambler and Professor Alain Delacroix for useful discussions and commen. This work was suppored by McGill Universiy, CIRPEE (ESG- UQAM) and FARE (ESG-UQAM). Posdocoral fellow (Deparmen of Economics, McGill Universiy), Cenre ineruniversiaire de recherche en analyse des organisaions (CIRANO), Cenre ineruniversiaire sur le risque, les poliiques économiques e l'emploi (CIRPÉE). Mailing address: Leacock Building, Rm 442, 855 Sherbrooke Sree Wes, Monréal, Québec, Canada H3A 2T7. TEL: (514) 398 33; FAX: (514) 398 4938; e-mail: charles.maoakongmo@mcgill.ca; Url:hps://sies.google.com/sie/maoakongmocharles/
1 INTRODUCTION Wha is he pah followed by he scal muliplier in an open economy when he nominal ineres rae reaches he Zero Lower Bound (ZLB)? Using a heoreical framework, in a closed economy, Chrisiano, Eichenbaum, and Rebelo (211), show ha, when he nominal ineres rae is binding, he governmen-spending muliplier can be close o four. This heory helps us o undersand he dynamics of an economy in ZLB afer he increase in governmen spending, bu i is dicul o mach his heory wih he daa. For example, during he nancial crisis and he recession ha followed in 27, he ineres rae in he Unied Saes and in European counries, reached heir lowes levels. Many signican budge plans emerged; he American Recovery and Reinvesmen Ac (ARRA) in he Unied Saes (831 billion from 29 o 219) and he European Economic Recovery Plan (EERP) in he European Union ( 2 billion from 28 o 21). However, he raio of deb o GDP increased on average by 4.5% before 28 and by 8% afer 28 in he Unied Saes (see Boskin, 212). In his paper, we sugges ha he real exchange rae is a channel ha can explain why increasing governmen spending in ZLB, may no in some cases lead o a large governmen spending muliplier. We propose a heoreical open macroeconomics framework wih marke imperfecions, wage and price rigidiies and endogenous smoohing moneary policy. In our framework, we inroduce a shock on he discoun facor ha pushes he nominal ineres rae o is minimum level. We hen compue he pah followed by he scal muliplier due o increases of governmen spending in ZLB. We argue ha, in a closed economy and in he presence of ZLB, here is no crowding ou eec hrough ineres raes. We show, ha in an open economy, here is anoher channel for he crowding ou eec via he real exchange rae. For an open economy, he muliplier falls o he levels usually observed in small, closed economies for which he ZLB is no binding. We show ha increasing governmen spending increases aggregae demand, which leads o appreciaion of he real exchange rae ha is greaer han he appreciaion ha we would have had in he siuaion where he lower nominal ineres rae was no binding. The appreciaion of he real exchange rae hen reduces he scal muliplier. Our resuls are consisen wih hose of Peroi (24) which shows empirically ha he governmen spending eec on GDP ends o be lower for open economies. Our resuls also agree wih hose of Karras (212 ) which shows ha he scal muliplier decreases wih he degree of openness of he economy ( increased openness of he economy by 1% reduces he value of he muliplier of abou 5 % (daa - 62 2
counries from 1951 o 27)). Even if he analysis of Peroi (24) and hose of Karras (212 ) do no ake ino accoun he special case of ZLB, hey sill give a good picure on wha could happen in ZLB. 1.1 Lieraure review As Amano and Shukayev (21) argue, he ZLB consrains moneary policy. The real ineres rae aecs behavior of consumers and rms more han he nominal ineres rae. A low real ineres rae encourages more consumpion and invesmen. In he Zero Lower Bound (ZLB) case, moneary auhoriies and governmens mus nd anoher way o increase aggregae demand. The moneary auhoriies can, for example, convince agens ha prices will increase in he fuure, and he governmen can increase spending. Chrisiano, Eichenbaum, and Rebelo (211), using a heoreical model, nd ha he governmen-spending muliplier can be much larger han one (close o four) while he nominal ineres rae reaches he ZLB. However, he framework buil in a closed economy canno ake ino accoun he eec of governmen spending on he real exchange rae, or is eec on he level of he rade balance deci. These eecs can have a real impac on he cos of impored goods and consumpion and herefore on he muliplier of public spending. The mechanism explaining he size of he governmen spending muliplier in a closed economy in ZLB is described by Chrisiano, Eichenbaum, and Rebelo (211) as follows: Following an increase in governmen spending, here is an increase in producion marginal cos and expeced inaion. This causes a decrease in he real ineres rae, and households consume more. The increase in household spending increases oupu, marginal cos and expeced inaion. This furher decreases he real ineres rae and so on, which in urn leads o a signican increase in producion. The heory may dier in an open economy. Concerning he heory, he basic framework is developed by Mundell (1963). The model predics ha in a small open economy wih exible exchange raes, a scal policy is ineecive 1 if capial mobiliy is perfec. Indeed, an increase in governmen spending nanced by borrowing, creaes an excess demand for goods, which ends o increase income. This increases he demand for money and he ineres rae, aracing foreign capial. The exchange rae hen appreciaes, which in urn leads o an equivalen decrease 2 in income hrough a rade imbalance. Even if he Mundell 1 Following an increase in public spending, he IS curve shifs o he righ. As he cenral bank does no inervene, he LM curve does no shif. The ineres rae increases and he real exchange rae appreciaes. The appreciaion of he real exchange rae penalizes expors and simulaes impors, which heoreically re-shifs he IS curve o is iniial posiion. 2 According o he Mundell (1963) model, income canno change when he money supply and 3
(1963) framework is very resricive 3, he resuls described in he model are simple and undersandable. Concerning empirical analysis, using dieren empirical mehodologies, many auhors nd governmen spending mulipliers o be more or less close o one. The size of he muliplier depends on he mehod, period, and on he governmen spending indicaor aken ino accoun. However i is clear from empirical lieraure ha he muliplier urns ou o approximaely one. Fisher and Ryan (21) uses as an indicaor, of governmen spending, he impac on income of he larges companies wih governmen conracs in he miliary secor. They nd a muliplier of governmen spending equal o 1.5 over a period of 5 years. Fisher and Ryan (21) shows ha a posiive shock o governmen spending is associaed wih an increase in oupu, hours worked and consumpion. They nd ha wages decrease iniially and hen increase one year afer he shock. The narraive approach 4 developed by Ramey and Shapiro (1998) idenies he response of he economy due o a susained and unpredicable increase of exogenous governmen spending. The narraive approach beer approximaes he period corresponding o he spending shock. Miliary spending is no heoreically explained by economic hisory. The narraive approach appears beer han VAR for predicing periods of exogenous shocks and susained miliary spending. Ramey and Shapiro (1998) shows ha he governmen spending muliplier is secor-specic. Ramey and Shapiro (1998) nds ha producion and consumpion decline following a governmen spending shock 5. ineres rae are consan (LM does no change because he cenral bank does no inervene and i = i does no change in he model since our counry is small and here is perfec capial mobiliy). Therefore naional income remains unchanged. As savings and axes do no change because of he balance of he propery marke, he increase in public spending is exacly equal o he impor surplus. 3 As noed by Andrew (2), Mundell's (1963) model has a lack of realism: domesics and foreign capial are perfecly subsiuable. Sicky prices and aggregae supply are no modeled. There are no microeconomics foundaions in he model. The model is saic and here is neiher wealh nor capial accumulaion. Domesic and foreign ineres raes are assumed o be idenical. 4 The Narraive approach uses newspaper informaion o idenify periods of hisorical shocks. Increased miliary expendiure is hen included as a dummy variable in an AR model o esimae he response of he economy. 5 However, some reservaions should be made when considering hese resuls: he shocks, as specied, are only posiive. The resuls imply good precision when esimaing he period of susained increase in public expendiure. 4
2 Mehodology To ake ino accoun he specics of he open economy, he framework is as follow: he nal goods are produced by compeiive rms using a quaniy of naional aggregae goods and a quaniy of impored aggregae goods. The naional aggregae good is produced by compeiive rms using a coninuum of diereniaed naional inermediae goods produced by domesic rms in monopolisic compeiion. The aggregae impored good is produced by naional compeiive rms using a coninuum of diereniaed impored goods, produced by foreign rms in monopolisic compeiion. The aggregae naional inermediae goods and aggregaed impored inermediae goods are imperfecly subsiuable. One par of he naional aggregaed good is expored and he res is combined wih he aggregaed impored good for he producion of he nal good. The aggregae impored good canno be consumed direcly; his impored aggregae good is used only in he producion process of he naional nal good. Households are characerized by dieren ypes of work, and ac in monopolisic compeiion in he labor marke. The nal good is used for consumpion, governmen spending and is also used as inpu in naional producion of inermediae goods. There is nominal rigidiy of wages and prices: prices and wages are sicky in he sense of Calvo (1983). Prices of inermediae goods (domesic and foreign) and wages are se in advance. There is a coninuum of ypes of job oers wih consan elasiciy of subsiuion. The labour is oered by a coninuum of households in monopolisic compeiion on wages. This framework is common in he lieraure of open economy (see Ambler, Dib and Rebei 24, Gali and Monacelli 25). We allow our model o generae a ime-varian discoun facor ha will help us o push ineres rae o is lowes level. To allow he scal muliplier o be greaer han one, we consider a non separable uiliy funcion so ha he marginal uiliy of consumpion will depend posiively on hours worked. We also consider an endogenous moneary policy. The policy saes ha, due o he shock, he moneary auhoriies se he nominal ineres rae such ha i converges smoohly o he lowes level, bu remains posiive and diereniable a all poins. This is a modied version of moneary policy used by Chrisiano, Eichenbaum and Rebelo (211) and i will help us o use an exising program o solve he model. When he nominal ineres rae reaches he lowes level, he governmen increases spending in order o simulae he economic aciviy. We hen compue he pah followed by he governmen spending muliplier. 5
2.1 The household The populaion is represened by a coninuum of agens on a uni inerval indexed by j. The uiliy funcion of household j is dened as follows: U(j) = E = ( (β ) u C (j), M ) (j), h (j), G P The discoun facor (β ) is ime-varian. This is he only ype of shock in he absence of capial and a risk premium on capial ha could push he ineres rae o is lower level (see Amano and Shukayev 29). Since we do no have capial in our analyical framework, we have o consider he ime variable discoun facor. In his work, i is he shock on discoun facor ha will push he economy o ZLB. For each period, household j chooses he amoun of money o hold, consumpion, he amoun of domesic and foreign asses, and he salary level if required o maximize is iner-emporal uiliy funcion, aking ino accoun heir budge consrain, he demand of labor ype j and he ransversal condiion 6 on asses. The insananeous uiliy funcion is: u(.) = ([ ( C (j) γ 1 γ + b 1 γ ( M(j) P ) γ 1 ) ] γ α ) 1 σ γ 1 γ [1 h (j)] 1 α 1 1 σ In he closed economy analysis, Chrisiano, Eichenbaum, and Rebelo (211) shows (specifying dieren ypes of uiliy funcions) ha o have a scal muliplier greaer han one, i is necessary o consider a uiliy funcion for which he marginal uiliy of consumpion depends posiively on hours worked. I is herefore necessary o have a non separable uiliy funcion. E is he expecaion operaor condiional on he ime, C (j) he household consumpion a he end of period, and M (j) he ne amoun of currency held by he agen a he end of period. P is he price index a ime, h he number of hours worked by he household a ime and G he governmen spending. α (, 1), γ >, σ > and u is a concave funcion. b is he shock on money demand. This shock evolves according o he following AR(1) process: 6 Among he possible soluions, we choose he one for which he amoun (in value) of asses held by he agen, a he end of he period, is zero. I would be subopimal o nish wih a posiive sock in asse value since more consumpion improves well-being. (1) (2) 6
log(b ) = (1 ρ b ) log(b) + ρ b log(b 1 ) + ɛ b (3) wih ɛ b i.i.d. The household's budge consrain is given by: P C (j) + M (j) + Dg (j) R + e B (j) κ R = (1 τ )W (j)h (j) + M 1 (j) + D g 1 + e B 1(j) + T + D d + D m (4) where W (j) is he nominal wage se by he household. τ is he labor ax. D g is domesic obligaion purchased by household a ime, which is used by he governmen o nance is deci. B represens foreign bonds, purchased by a household a ime, and e is he nominal exchange rae. R and R are respecively he domesic and foreign nominal ineres raes beween ime and + 1. D d is he nominal pro received by domesic rms and D m is he nominal pro received by rms ha impor inermediae goods. T is he lump-sum ransfer from he governmen. κ is he risk premium ha adjuss he uncovered ineres rae pariy. κ correcs he problem of he random walk followed by consumpion around he equilibrium when domesic and foreign ineres raes are assumed equal. Ambler, Dib and Rebei (24) dene he risk premium as depending on he raio of ne foreign asses and domesic producion. [ ( ) ] e B log(κ ) = ϕ exp 1 (5) P d Y where P d is he domesic price index. The foreign ineres rae R follows he following AR(1) process: log(r ) = (1 ρ R ) log(r ) + ρ R log(r 1) + ɛ R, (6) ɛ R is i.i.d wih zero mean and variance σ R Noe σ h, he elasiciy of subsiuion beween dieren ypes of work, aggregae labor is dened by: (ˆ 1 h = ) σ h h (j) σ h 1 σ h 1 σ h dj The demand for labor of ype j is herefore 7 (7) 7 See Dixi, Sigliz (1977) for more deails 7
( ) σh W (j) h (j) = h (8) W where he aggregae wages W is given by: (ˆ 1 W = ) 1 W (j) 1 σ 1 σ h h dj (9) The rs order condiions 8 of household j s problem, concerning consumpion, money, purchases of naional obligaion and purchases of foreign bonds are wrien as: α (1 h (j)) (1 α)(1 σ) C (j) 1 γ [( ( C (j) γ 1 γ + b 1 γ M (j) P ) γ 1 γ )] αγ(1 σ) γ 1 1 = (j) P P d (1) α (1 h (j)) (1 α)(1 σ) b 1 γ ( M (j) P ) 1 γ P d P [( C (j) γ 1 γ + b 1 γ [ ] P d = (j) β E P+1 d +1 (j) ] (j) R (j) κ R = β E [ P d P+1 d +1 (j) [ P d = β E e +1 P+1 d +1 (j) e ] ( M (j) P ) γ 1 γ )] αγ(1 σ) 1 γ 1 Consider he following noaion: p = P /P d, m = M /P, p m = P m /P d, p d = P d /P d, π = P /P 1, π d = P d /P 1 d, w = W /P d, π = P /P 1, s = e P /P d, r = T /P d 8 I is consisen o divide he wo sides of budge consrain by domesic price index p d when wriing he Lagrangian (11) (12) (13) 8
2.2 Discoun facor shock and ZLB As we said, he discoun facor is he only ype of shock in he absence of capial and risk premium on capial ha could push he ineres rae o is lower bound (see Amano and Shukayev 29). Since we do no have capial in our analyical framework, we have o consider he ime-varian discoun facor. In his work, i is he shock on he discoun facor ha will push he economy o ZLB. The discoun facor shock increases he propensiy o save. The pracical mechanism is relaively he same as he one presened by Chrisiano, Eichenbaum, and Rebelo (211). I is as follows: Iniially (ime -1) he economy is in he seady sae, driven by he Taylor rule and he macroeconomic framework presened above (β ( 1) = 1 ). Then here is a posiive R shock on he discoun facor (a ime (β = 1)). Subsequenly, he discoun facor gradually reurns o is equilibrium value. Le R be he ineres rae a ime induced by he shock o he discoun facor a ime. For simpliciy, afer he shock on he discoun facor, he ineres rae may remain a is hreshold level (R l ) wih probabiliy (p r ), or i may reurn o is seady sae (R) wih probabiliy (1 p r ); in he laer case i remains a he saionary level forever. The sochasic process describing he behavior of ineres raes afer he shock can be as follows (see Chrisiano, Eichenbaum, and Rebelo 211): P r [ R +1 = R l R = R l] = p r, P r [ R +1 = R R = R l] = 1 p r, P r [ R +1 = R l R = R ] =, P r [R +1 = R R = R] = 1, We can easily wrie he discoun facor process as an AR (1): (14) β = p r β 1 + (1 p r )β + ɛ β (15) The parameer β is he seady sae value of he discoun facor. I is calibraed o he sandard value of.99. 2.3 Salaries Salary is dened in Calvo framework (see Calvo 1983). Wih probabiliy (1 d w ), household j is allowed o adjus he salary. Oherwise he previous period salary remains in place. When considering all households, a proporion (1 d w ) of households re-opimize heir wages, and he oher proporion d w mainains he previous salary. 9
The aggregae wage index can be wrien as (ˆ 1 W = and rearranged as: ) 1 1 σw ( W (j) 1 σw dj = 1 σw (1 d w ) W ) 1 + d w W 1 σw 1 σw 1 (16) W 1 σw = (1 d w ) W 1 σw + d w W 1 σw 1 (17) The Salary is he value ha maximizes he expeced uiliy of he household under he budge consrain for he expeced ime period where wages remain xed. This salary will remain valid unil he nex auhorizaion of wage readjusmen. The Lagrangian associaed wih he wage problem is as follows: L = max W ([ ( C +l(j) γ 1 γ (β d w ) l l= + b 1 γ +l ( M+l (j) P +l ) γ 1 ) ] γ α )1 σ γ 1 γ [1 h +l(j)] 1 α 1 σ 1 ( (β d w) l +l P d l= +l ( ) [ + (β d w) l +l P d P +l C (j) + M +l (j) + Dg +l (j) B+l + e (j) ] R l= +l +l κ +l R+l ) [ (1 τ +l) W ] (j)h +l(j) + M +l 1(j) + D g +l 1 + e +lb+l 1 (j) + T +l + D+l d + Dm +l (18) Throughou he period of xed wage, he household is subjec o he following consrain: h +l (j) = ( ) σh W (j) h +l (19) W +l When he household is allowed o adjus he salary, he opimal level is as follows. σ h W (j) = (1 α) σ h 1 E l= (β d w) l C +l (j) αγ(1 σ) 1 ( ) γ 1 γ 1 γ M+l (j) γ + b h +l P +l (j) [ 1 h +l (j) ] (1 α)(1 σ) 1 +l ( ) E l= (β d w) l +l P +l d (1 τ +l )h +l (j) (2) γ 1 γ 1
2.4 Producion of naional inermediae goods Firms producing inermediae goods use nal goods as inpus. The producion funcion of rm i producing inermediae goods is: Y (i) = X (i) φ (A h (i)) 1 φ (21) where h (i) is labor, X (i) he quaniy of nal good used by rm i, and A he echnology shock ha follows he auo-regressive process below: log(a ) = (1 ρ A ) log(a) + ρ A log(a 1 ) + ɛ A (22) where ɛ A is i.i.d wih zero mean and variance σ A. Prices are se by Calvo, rm i re-opimizes is price P d (i) wih probabiliy (1 d p ), and chooses he quaniies of nal goods and labor ha maximize is expeced pro. This is represened by he value of sock shares i issues. The price se in period remains for l period wih probabiliy (d p ) l. Le represen he marginal uiliy of wealh, ha is he Lagrange muliplier of he household problem. Le P represen he price of he nal good and P d he price index of naional inermediae goods. A rm producing he inermediae good solves he following problem: max E {X (i),h (i), P d(i)} l= (β d p ) l ( +l subjec o he following producion funcion: ) ( P d (i)y +l (i) W +l h +l (i) P +l X +l (i) P d +l ), (23) Y (i) = X (i) φ (A h (i)) 1 φ (24) and subjec o he demand for he inermediae good i in he producion of he nal good. Le ( θ) represen he demand elasiciy for he inermediae good. Demand for he inermediae good i is given by: ( ) θ P d Y +l (i) = +l (i) Y +l, (25) P+l d Les ξ (i) denoe he Lagrange muliplier associaed wih producion funcion consrain. The rs order condiions are given by: 11
W P d = ξ (i)(1 φ)a (X (i)) φ (A h (i)) φ = ξ (i)(1 φ) Y (i) h (i) P P d = ξ (i)φ (X (i)) φ 1 (A h (i)) 1 φ = ξ (i)φ Y (i) X (i) (26) (27) ( ) ( ) θ E P d l= (β d p ) l +l ξ (i) = +l (i)y +l (i) θ 1 ( ) ( ) (28) E l= (β d p ) l +l Y+l (i) P d +l 2.5 Impored inermediae good There is monopolisic compeiion on impored goods. There is a coninuum of rms and each rm impors a diereniaed good in uni inervals. These impored goods are imperfecly subsiuable, and are used in he producion of he composie good impored, noed Y m, and produced by a represenaive rm. Wih probabiliy (1 d m ) he rm ha impored he inermediae good reopimizes is price P m so as o maximize is expeced weighed pro under is demand consrain. The problem can be wrien as: max { P m (i)} E l= (β d m ) l ( +l ) ( m P (i) e +l P+l P+l d ) ( P m ) ϑ (i) Y m, P m +l P is he price index of impored goods in foreign currency, and ϑ he elasiciy of demand of impored goods i. As before, he rs-order condiion gives: ( ) ( ) ϑ E P m l= (β d m ) l +l (i) = ϑ 1 ( E l= (β d m ) l +l ( Y+l m (i)e P +l +l P+l ) ( d Y m +l (i) P+l d ) +l ) (29) 2.6 Aggregaed naional good The naional good is produced by a represenaive rm from domesic inermediae goods. 12
The naional good is an aggregae of a coninuum of inermediae goods Y (i) produced locally. The producion funcion is a consan elasiciy of subsiuion echnology funcion: (ˆ 1 Y = ) θ Y (i) θ 1 θ 1 θ di The rm producing he naional good solves he following problem: (3) max P d Y {Y (i)} The rs order condiion gives: (ˆ 1 P d (i) = (Y (i)) 1 θ ˆ 1 P d (i)y (i)di (31) ) 1 Y (i) θ 1 θ 1 θ di P d = The demand funcion for inermediae good is: ( Y (i) Y ) 1 θ P d (32) ( ) P d θ Y (i) = (i) Y P d (33) By aggregaing boh sides of demand equaion, we obain he price index for domesic goods. (ˆ 1 ) 1 1 θ P d = P d (i) 1 θ di. (34) (1 d p ) is dened as he proporion of rms ha readjus heir prices. We can herefore spli domesic rms ino wo groups: hose which are allowed o adjus heir prices in period and hose which coninue o apply he price of he previous period ( 1). The price index in period can be wrien as: [ P d = d p (P 1) d 1 θ + (1 d p )( P ] 1 d ) 1 θ 1 θ where P d is he price index se by rms ha have he righ o readjus heir prices in period. (35) 13
2.7 Aggregae expored good One par of he naional aggregaed good is expored 9, while he oher 1 is combined wih impored goods o produce naional nal goods. We have: Y = (1 α x )Y + α x Y (36) Y = Y d + Y x (37) and Y d = (1 α x )Y Y x = α x Y α x > is he proporion of he naional good ha is expored. The expored good is par of a coninuum of imperfecly subsiuable goods ha are used by a foreign represenaive company. The elasiciy of subsiuion beween goods is noed ι. The foreign demand of he naional good is represened by he following equaion: we deduce ha Y = ( ) P d ι Y e P Y x ( ) P d ι = α x Y e P (38) As we assume, our economy is small, i herefore does no inuence he foreign price index or aggregae foreign producion. The foreign price and foreign producion follow he following processes: log ( P P 1 ) = (1 ρ π ) log(π ) + ρ π log( P 1 ) + ɛ P 2 π (39) log (Y ) = (1 ρ y ) log(y ) + ρ y log(y 1) + ɛ y (4) where π and Y are respecively inaion and foreign producion in he seady sae. 9 Y x 1 Y d = α x Y = (1 α x )Y 14
2.8 Aggregae impored good Wih a consan elasiciy of subsiuion producion funcion, a represenaive rm produces a good using a coninuum of impored goods Y m (i). The producion funcion is: (ˆ 1 Y m = Y m (i) ϑ 1 ϑ di ) ϑ ϑ 1 The represenaive rm solves he following problem: (41) max {Y m P m Y m (i)} ˆ 1 P m (i)y m (i)di (42) The rs order condiion gives he demand equaion for he impored inermediae good i: Y m (i) = ( P m P m (i) ) ϑ Y m (43) We can deduce he price index for impored inermediae goods: (ˆ 1 P m = ) 1 P m (i) 1 ϑ 1 ϑ di A proporion (1 d m ) rms imporing he inermediae good re-opimize heir price a ime, while he oher porion d m keep he price of previous period ( 1). We can rewrie he price index of impored goods as follows: [ ( ) ] 1 (1 ϑ) P m = d m (P 1) m (1 ϑ) 1 ϑ + (1 d m ) P m where P m is he price index of rms ha re-opimize heir prices in period. 2.9 The nal good The nal good Z is produced by a represenaive rm ha uses he aggregaed naional good as well as he aggregaed impored good. The echnology used o produce he nal good is he consan elasiciy of subsiuion producion funcion: Z = [ α 1 v d (Y d ) v 1 v + α 1 v m (Y m ) v 1 v ] v v 1 (44) (45) (46) 15
The rm producing he nal good solves he following problem: max {Y d,y m } The rs order condiions gives: P Z P d Y d P m Y m (47) and Y d ( ) P d v = α d Z (48) P Y m hen he price of he nal good can be wrien as: ( ) P m v = α m Z (49) P P = [ α d (P d ) 1 v + α m (P m ) 1 v] 1 1 v (5) As a reminder, he nal good is used for consumpion, in he process o produce inermediae goods, and is also used for governmen spending. Z = C + X + G (51) 2.1 Trade balance equilibrium Ne pros of asses purchased abroad in local currency are equal o he ne value of goods purchased abroad in local currency. Income from foreign asses purchased in a preceding period minus asses purchased in he curren period are equal o impored goods minus expored goods. The rade balance can be summarized by he following equaion B e B 1 e = e κ R P Y m P d Y x (52) where P s = e ; b P d = B P The balance of paymens can be represened by he following equaion: b κr b 1 π = Y x s Y m (53) 16
2.11 Transformaion of variables To solve he model, i is necessary o use saionary variables. We have he following noaions: p = P /P d, m = M /P, p m = P m /P d, p d = P d /P d, π = P /P 1, π d = P d /P 1 d, w = W /P d, π = P /P 1, s = e P /P d, d g = D g /P d 2.12 New Phillips curves The Phillips curve denes inaion dynamics as a funcion of fuure inaion and real marginal coss. The Phillips curve can also describe curren inaion as a funcion of he expeced inaion and he oupu gap. We will have one Phillips curve for wages on he labor marke, one for he price of he domesic inermediae goods marke, one for price in impored inermediae goods marke, and from hese we can hen easily derive a Phillips curve for price in he nal goods marke. We nd he opimal price levels when rms are allowed o change heir prices. Firms allowed o re-opimize heir prices ake ino accoun he probabiliy of fuure price changes. The discoun rae considered by rms akes ino accoun he valuaion of fuure consumpion by households. Firms also ake ino accoun he real marginal coss and demand for goods hey produce. We have found ha he curren price index level is a weighed average of he price index of he previous period and he price index se by rms allowed o opimize heir price. Previous versions of wages and prices conain innie summaions. In order o solve he model we have o make anoher ransformaion. We will combine he wages and prices ha we obained previously wih he dynamics of wages and prices obained in he Calvo framework. Furhermore, we will ransform our variables, such ha each new variable will be in percenage deviaion from he seady sae of he variable i represens. Thus, when he iniial variables converge o he seady sae, he new variables converge o zero. Afer ha we can have a Taylor expansion series around zero ( see Brook Taylor 11 1715 also known as Mac-Lauren developmens series) for all ransformed variables. To illusrae he model, he Taylor expansion is in rs order. Bu for beer accuracy, in he compuaion algorihm we will exend he Taylor approximaion o order 3. ˆ For he domesic inermediae good: 11 If a funcion f is diereniable a x up o order n, we can wrie he polynomial funcion as follows: f(x) = n f (k) (x ) k= k! (x x ) k + ɛ n (x) wih lim ɛ n (x) = x x 17
We will see laer ha ˆπ d = ˆp d ˆp d 1 ; ˆp d = pd pd p d ˆπ d = β ˆπ d +1 + (1 β d p )(1 d p ) d p ˆξ (54) ˆξ = (1 φ)ŵ + φˆp (1 φ)â (55) and ˆξ = ξ ξ. As a reminder, ξ ξ is he Lagrange muliplier associaed wih he producion funcion consrain. I herefore represens he real marginal cos of producing one uni of domesic inermediae good. ˆ For he impored inermediae good ˆπ m = β ˆπ m +1 + (1 β d m )(1 d m ) d m ŝ (56) wih ˆπ m = ˆp m ˆp m 1; ˆp m = pm pm p m e is he exchange rae. This is he value of a uni of foreign currency in erms of domesic currency. P is he price index of impored goods in foreign currency is he real exchange rae and ŝ = s s ; s s = e P /P d ˆ For he wages + (1 β d w )(1 d w ) d w + (1 β d w )(1 d w ) d w ˆπ w = β ˆπ w +1 [ h (1 (1 α)(1 σ)) + τ ] 1 hĥ 1 τ ˆτ ˆ ŵ ( ) [ αγ(1 σ)/(1 γ) γ 1 c γ 1 γ + b 1 γ 1 γ m γ γ c γ 1 γ ĉ + 1 γ b 1 γ 1 γ m γ ˆb + γ 1 γ b 1 γ m γ 1 γ ˆm ] (57) 2.13 The IS-dynamic equaion We can wrie curren aggregae producion in erms of expeced fuure aggregae producion, curren and expeced fuure ineres rae, as well as presen and fuure governmen spending. To do his, we will use he aggregae resource consrain, he opimal condiions for he household problem, and he opimal condiions of rms producing he inermediae naional goods. We hen have 18
[ [(α ασ 1)z + c(1 ] α)(1 σ) + x(α ασ 1)] (ẑ +1 ẑ ) (αγ(1 σ)+(1 γ)) = b 1 γ 1 γ m γ ( ˆR+1 ˆR ) ) + G(α ασ 1) (Ĝ+1 1 R Ĝ c γ 1 γ +b 1γ m γ 1 γ (1 φ) [ c(1 α)(1 σ) h + x(α ασ 1)] (â 1 h +1 â ) + [ c(1 α)(1 σ) h φ + x(α ασ 1)(1 φ)] ˆπ w 1 h +1 + [ ( ) c(1 α)(1 σ) h (v φ) + x(α ασ 1)(v + φ 1) + 1] α m(p m ) 1 v ˆπ m 1 h p ˆκ ˆR ŝ +1 + ŝ (58) wih ˆπ = α m(p m ) 1 v p ˆπ m (59) 3 Moneary and governmen spending policies Moneary policy follows he Taylor rule (see Taylor 1993 ) when he ineres rae is sricly posiive. The cenral bank ses he nominal ineres rae in response o shor-erm ucuaions in inaion(π = P P 1 ), ucuaions in he money (µ = M M 1 ), P ucuaions in producion (Y ) and ucuaions in real exchange rae (s = e P ) ( see Ambler and al. 23). When he Taylor rule implies a negaive ineres rae (r aylor ), he moneary auhoriies sysemaically x he nominal ineres rae o he lowes level (see Chrisiano, Eichenbaum and Rebelo 211). Noe R aylor = 1 + r aylor, and R = r + 1 Moneary policy can hen be summarized in he following equaion (see Amano and Ambler 212) wih log(r ) = max {log(r aylor ), } (6) log (R aylor ) = log(β)+φ π log(π /π)+φ y log(y /Y )+ρ µ log(µ /µ)+ρ s log(s /s)+ɛ R (61) where Y, π, µ, and s denoe he saionary sae value of producion, saionary sae value of inaion, saionary growh rae value of money supply and saionary value of real exchange rae. ɛ R is i.i.d shocks o moneary policy wih zero mean 19
and variance σ R. The cenral bank can only indirecly conrol shor-erm ineres raes, by seing he Bank Rae. The error erm reecs developmens in money and nancial markes ha are no explicily capured by our model. Pracical case The funcion dening he level of nominal ineres raes (log(r ) = max {log(r aylor ), }) in our analyical framework is no diereniable around he inersecion beween he Taylor rule funcion and he lowes nominal ineres rae (ZLB). We will herefore smooh he previous ineres rae funcion around he ZLB so ha he new ineres rae funcion can be diereniable a every poin. As in Amano and Ambler (212), we will wrie: log(r ) = max {log(r aylor ), } (log(r aylor )) 2 + a 2 + log(r aylor ) 2 (62) a is a smoohing parameer ha denes he curvaure of he new moneary ineres rae policy funcion around he ZLB. The new moneary policy Due o he shock, he moneary auhoriies se he nominal ineres rae such ha i converges smoohly o he lowes level, bu remains posiive and diereniable a all poins 3.1 Governmen response o demand shock The governmen budge consrain is as follows: P G + T + D g 1 = τ W h + M M 1 + Dg R (63) To obain his mahemaically, we jus rewrie he household's budge consrain, aking ino accoun he rade balance, replacing dividends by rms' pros, and using he equaion characerizing how he nal resource is used in he economy (Z = C + X + G). The equaion above shows ha governmen spending is well represened by he purchases, ransfers and deb repaymens. Governmen revenues are represened by axes on wages, by money creaion and by new borrowing. 2
There are no Ponzi games, i.e: This imply ha: ( ) P G + T 1 i= R + D g = i =1 D g lim i= R = i ( τ W h + M M 1 1 i= R i =1 This means ha he presen value of governmen spending over he original deb is equal o he presen value of governmen revenues. Discouned ax revenues are equal o discouned loans. A posiive shock o he discoun facor (β ) leads he nominal ineres rae o he ZLB, hen, he governmen reacs immediaely or afer a delay o his demand shock in order o simulae economic aciviy. The governmen's response o he demand shock caused by a posiive shock on he discoun facor is dicaed by he following rule: G = (1 ρ g ).G + ρ g.g 1 + T ρ gβi (β i β) + ɛ g (64) i is he number of periods afer impac. ρ gβi is he governmen's response a ime o he demand shock ha occurs a ime i. G is he level of public expendiure in he seady sae. ɛ g is a zero-mean, serially uncorrelaed governmen policy shock wih sandard deviaion σ g, ɛ g is a shock ha represened he unpredicable governmen spending. This specicaion will idenify he governmen spending muliplier as a funcion of governmen ime reacion. The level of axes follows an AR (1): 4 Analysis and resuls 4.1 Resoluion mehod i= log(τ ) = (1 ρ τ ) log(τ) + ρ τ log(τ 1 ) + ɛ τ (65) We follow he Dynare algorihm, based on he perurbaion mehod (see Collard and Juillard 21, Schmi-Ghore and Marín Uribe 24), which allows for Taylor 21 )
expansion in any higher order. In our analysis, we have a smoohing ineres rae policy ha should be approached wih precision. A second order polynomial approximaion is beer o smooh our moneary policy. For his reason we chose he perurbaion mehod wih second order Taylor expansion. Resuls in order 3 are he same. We herefore limied our Taylor expansion o he order wo. Wih he noaion of Collard and Juillard (21) and hose of Schmi-Ghore and Marín Uribe (24), he equilibrium condiions of our raional expecaions model can be summarized as follows: E [f(y +1, y, x +1, x, u, u +1 )]= f denes he se of equilibrium equaions, y is he vecor dening he se of variables o predic, x is he se of predeermined variables and u is he shock vecor. A ime, he agens know he value of predeermined variables of ime 1 and observe he shock a ime. Their decisions are based on beliefs ha relae o variables y +1, all he curren variables y and x. The soluion o his problem is a se of relaionships beween curren variables, he predeermined variables and shocks ha saisfy he original equaion sysem (dening he equilibrium condiions of our model). Solving his problem is he same as nding wo funcions g and h such ha : y = g(x ) x = h(x 1, u ) We can herefore rewrie he equilibrium condiion in he form: F (x ) = E [f (g(h(x, u +1 )), g(x ), h(x, u +1 ), x, u, u +1 )]= The sraegy is o wrie he Taylor expansion for g and for h in he chosen order n, around he seady sae, and hen o nd he coeciens of he nh-order polynomials considered. We also have o undersand ha F and is derivaives in any order are zero a all poins (see Collard and Juillard 21, Schmi-Ghore and Marín Uribe 24, for deails). 4.2 Indicaors of he scal muliplier We can have dieren indicaors for he governmen spending muliplier. These indicaors canno necessarily have idenical values. Like any indicaor, he goal is o make a comparison beween wo dieren siuaions. 22
In our framework we would like o know he pah followed by he governmen spending muliplier indicaor, following a demand shock. Several auhors use he impac muliplier (see Chrisiano and al. 211, Monford and Uhlig 29). This allows one o evaluae he change in oupu in a given period due o a change in governmen spending in he curren period. This indicaor is dened as follows: Impac muliplier(k) = Y +k G (66) Monford and Uhlig (29) and oher works use he presen value muliplier a lag k dened as follows: presen value muliplier a lag(k) = k = k = (1 + i) Y (1 + i) G (67) This indicaor is used o updae he impac of scal policy on k periods. To idenify he evoluion of he public expendiure muliplier, we considered he wo following indicaors: Y +k G dy divdg(k) = Z +k G dzdivdg(k) = We can also use he following indicaor Z +k G dz(k)divdg = ( ) Y+k Y +k 1 ( G ) (68) G 1 ( ) Z+k Z +k 1 ( ) (69) G G 1 ( ) Z+k Z ( ) (7) G G 1 By logarihmic ransformaion, our indicaors are exacly he impac mulipliers used by Chrisiano and al (211). Our indicaors can be direcly implemened in our algorihm in order o see heir changes over ime. 23
4.3 Calibraion Tables 1, 2 and 3 presen calibraion values. The saionary discoun value, β, is sandard and leads o a saionary annual real ineres rae value equal o 4 %. The elasiciy of subsiuion beween dieren kinds of labor is also sandard and is se a σ h = 6. The probabiliies of no adjusing wages are se so ha wages adjus afer six quarers and prices afer wo quarers, approximaely. The parameers θ and ϑ ha dene elasiciies of subsiuion beween goods are se so ha he markup in he marginal cos is approximaely 14 %. Table 1: The household Parameers Descripion β =, 99 Saionary discoun parameer ( Ambler and al., 24). 1 α =, 71 Leisure share in he uiliy funcion (Chrisiano and al., 211). γ =, 3561 Weighed real-consumpion cash ow (Ambler and al., 24). σ = 2 Parameer relaed o risk aversion (Chrisiano and al., 211). ρ b =, 645 AR(1) parameer for shock on money demand (Ambler and al., 24). σ h = 6 Elasiciy of subsiuion beween dieren labor ypes (Ambler and al., 24). d w =, 8257 Probabiliy of no adjusing wages (Ambler and al., 24). Table 2: Naional inermediae goods Parameers Descripion φ =, 3788 Share of nal goods in he producion of inermediae goods (Ambler & al 24 ) ρ A =, 8795 Coecien of lagged variable in AR (1) of produciviy d p =, 4398 Probabiliy of no adjusing prices (Ambler & al 24) θ = 2, 95 Elasiciy of demand for inermediae goods 24
Parameers Table 3: Ohers parameers Descripion d m =.558 Probabiliy of no adjusing prices of rms ha impor inermediae goods. (Ambler & al 24) ϑ = 2.95 Elasiciy of subsiuion beween impored inermediae goods(ambler & al 24) ρ G =.8 AR(1) parameer relaed o public spending (Chrisiano & al 211) ρ Gβ =.8 ρ y =.8835 ρ A =.8795 AR (1) Parameer of public spending for discoun facor Coecien of lagged variable in AR (1) of foreign producion Coecien of lagged variable in AR (1) of produciviy ρ τ =.432 Coecien of lagged variable in AR (1) of ax rae (Ambler & al 24) Y =.1 R = 1.15 π = 1 A = 1 Saionary value of foreign producion Saionary value of foreign ineres rae Saionary value of foreign inaion rae Saionary value of naional produciviy G =.562 Saionary level of Governmen spending (Ambler & al 24 ) τ =.29 Saionary level of ax (Ambler & al 24 ) p r =.5 Auo-regressive parameer of he discoun facor afer he shock ι =.5962 Elasiciy of subsiuion beween impored goods (Ambler & al 24) α x =.74 Share of expored domesic good produc (Ambler & al 24) α m =.3594 Share of impored goods in he producion of naional nal good (Ambler & al 24 ) ν =.5962 Elasiciy of subsiuion beween aggregae impored goods and aggregaed domesic goods(ambler & al 24 ) φ π =.73 Inaion parameer in Taylor rule (Ambler & al 23 ) φ µ =.559 Parameer associaed wih growh rae of real cash in he Taylor rule (Ambler & al 23 ) φ s = Parameer associaed wih real exchange rae in he Taylor rule (Ambler & al 23 ) φ y = Parameer associaed o producion in he Taylor rule (Ambler & al 23 ) υ =.36 Parameer relaed o producion funcion for he nal good σ G =.16 Sandard deviaion of governmen spending shock(ambler & al 24 ) σ β =.4 Sandard deviaion of discoun facor 4.4 Resuls In he beginning, he nominal ineres rae is binding due o he shock on he nominal discoun facor. When he governmen reacs insananeously, he aggregae demand increases, leading o appreciaion of real exchange rae. The appreciaion of he real exchange rae reduces he governmen spending muliplier. This resul is no sensiive o he governmen spending muliplier indicaor aken ino accoun. The resul is 25
also no sensiive o he ime a which he governmen reacs o he discoun facor shock. Figure 1 displays he impulse response funcion of real exchange rae (s), he governmen spending muliplier (DydivdG) and oher endogenous variables. The gure shows ha afer he increase in governmen spending, he real exchange rae appreciaes, reaching approximaely he level.27, relaive o is saionary value. Figure 2 displays he impulse response funcion ha proves ha he increase of governmen spending leads o less appreciaion of he real exchange rae compared o he siuaion where he nominal ineres rae is binding. When we compare gures 1 and 2, i is easy o observe ha he appreciaion of he real exchange rae is more imporan when he ZLB on he nominal ineres rae is binding compared wih he siuaion where i is no (.27 compared o.2). However he dierence beween he wo values is no oo high, and economeric mehods are needed o es wheher his dierence is signican or no. Noneheless his small appreciaion is sucien o reduce he governmen spending muliplier. The maximum governmen spending muliplier is abou 1.1. This value is clearly less han 4, which represens he maximum muliplier found by Chrisiano, Eichenbaum and Rebelo (211). Figure 3 displays he governmen spending muliplier pah when considering he indicaor in order 2 ( Y +2 G ). Our resul is no sensiive o he indicaor ha we use 12. Figure 4 displays resuls obain when he governmen reacs afer 5 periods. I conrms he fac ha he appreciaion of real exchange rae arises in period 5, exacly when he governmen reacs. I also shows he considerable reducion of governmen spending muliplier in period 5. 12 ( ) Z+2 Y +2 Z +1 dzdivdg(2) = ( ) (71) G G G 1 26
Figure 1: Discoun facor shock coupled wih increases of governmen spending (indicaor Y G dy divdg()).6 bea 3 x 1 3 G 2 x 1 3 Y.4.2 2 1 1 1 1 2 3 4 1 2 3 4 2 1 2 3 4 4 x 1 3 Z.4 s.2 dydivdg 2.2.2 2.2.4 4.4 1 2 3 4 1 2 3 4.6 1 2 3 4.2 dzdivdg 4 x 1 3 c 4 x 1 3 h 2 2.2.4 2 2.6 1 2 3 4 4 1 2 3 4 4 1 2 3 4 27
Figure 2: Increases of governmen spending wihou any discoun facor shock (indicaor Y G dy divdg()) 2 x G 1 3 1.5 1.5 15 x Y 1 4 1 5 3 x Z 1 3 2 1 1 2 3 4 5 1 2 3 4 1 1 2 3 4.1 s.1 dydivdg.2 dzdivdg.1.2.1.1.1.3.2 1 2 3 4 1 2 3 4.2 1 2 3 4 2 x 1 3 c 2 x 1 3 h 5 x 1 3 p 1 1 5 1 1 2 3 4 1 1 2 3 4 1 1 2 3 4 28
Figure 3: Discoun facor shock coupled wih increases of governmen spending (indicaor Y +2 G dy divdg(2)).6 bea 3 x 1 3 G 2 x 1 3 Y.4.2 2 1 1 1 1 2 3 4 1 2 3 4 2 1 2 3 4 4 x 1 3 Z.4 s.1 dydivdg 2.2.1 2.2.2 4.4 1 2 3 4 1 2 3 4.3 1 2 3 4.1 dzdivdg 4 x 1 3 c 4 x 1 3 h.1 2 2 2 2.2 1 2 3 4 4 1 2 3 4 4 1 2 3 4 29
Figure 4: Discoun facor shock coupled wih increases of governmen spending afer 5 periods.6 bea 3 x 1 3 G 4 x 1 3 Y.4.2 2 1 2 2 1 2 3 4 1 2 3 4 4 1 2 3 4 5 x 1 3 Z.5 s.2 dydivdg.2 5.5 1 2 3 4 1 2 3 4.4 1 2 3 4.2 dzdivdg 4 x 1 3 c 4 x 1 3 h.1 2 2.1 2 2.2 1 2 3 4 4 1 2 3 4 4 1 2 3 4 3
5 Conclusion The aim of his paper was o idenify he pah followed by he governmen spending muliplier in an open economy when he Zero Lower Bound (ZLB) on nominal ineres rae is binding. We have shown ha increasing governmen spending in a ZLB resuls in appreciaion of he real exchange rae, and a depreciaion of he governmen spending muliplier. In fac, he increase in governmen spending increases aggregae demand, which leads o he appreciaion of real exchange rae. The appreciaion of real exchange rae reduces he governmen spending muliplier. When we compare our economy in wo dieren siuaions i.e. when he nominal ineres rae is binding versus when i is no, we nd ha he appreciaion of he real exchange rae is greaer in he siuaion where he zero lower bound is binding. Our resul is no sensiive o he indicaor used o evaluae he muliplier and i is also no sensiive o he dae a which he governmen reacs o he discoun facor shock. Despie he use of a non-separable uiliy funcion (as required by Monacelli & Peroi 27 and Chrisiano & al. 211 and ohers), i was no possible o have a governmen spending muliplier larger han one due o he role played by he real exchange rae. In conclusion, he ZLB removes he crowding ou eec ha passes hrough he ineres rae 13. In an open economy, here is anoher channel for he crowding ou eec ha passes hrough he real exchange rae. Thus, he muliplier drops o he normal level of he small closed economy when he nominal ineres rae is no binding. Our resul is similar o he one obain by Peroi (24), ha shows empirically ha he eec of governmen spending on Gross Domesic Produc (GDP) ends o be lower for open economies. Our resul is also similar o he one found by Karras (212) ha proved ha he muliplier ends o decrease wih he degree of an economy's openness ((X + M)/P IB). Using daa for 62 counries, for he period of 1951 o 27, hey found ha an increase of openness by 1 % reduces he value of muliplier by 5 %). Peroi (24) and Karras (212) sudied periods when he nominal ineres rae was no binding. An exension of our resuls can be o quanify empirically hose resuls in ZLB (despie he shor period of daa). 13 As he ineres rae dened by he Taylor rule, is below he hreshold ineres rae, he higher ineres raes caused by he crowding ou eec may remain below he hreshold and he ineres rae applied by he cenral bank remains he hreshold ineres rae. This simply means ha here is no crowding ou eec. 31
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