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1 ,7(57(35$/68%67,787,$'7(/,48,',7<())(&7,$67,&.<35,&('(/ Ã J. Andrés (), J.D. López-Salido () and J. Vallés () () %DQFRGH(VSDxDDQG8QLYHUVLGDGGH9DOHQFLD () %DQFRGH(VSDxD5HVHDUFK'HSDUPHQ Sepember 999 $EVUDF The liquidiy effec, defined as a decrease in nominal ineres raes in response o a moneary expansion, is a major sylized fac of he business cycle. This paper seeks o undersand under wha condiions such an effec can be explained in a general equilibrium model wih sicky prices and capial adjusmen coss. The paper firs confirms ha, wih separable preferences, a low degree of ineremporal subsiuion in consumpion is a necessary condiion for he exisence of he liquidiy effec. Conrary o his resul, in a model wih non-separable preferences and capial accumulaion i akes an implausibly high degree of ineremporal subsiuion o produce a liquidiy effec. The robusness of hese resuls o alernaive degrees of nominal rigidiies, money demand properies and real rigidiies is also analyzed..h\zrugv ineremporal subsiuion, liquidiy effec, price and capial adjusmen coss. -(/&ODVVLILFDLRQ E3, E43. &RUUHVSRQGLQJ DXKRU Javier Vallés, Research Deparmen Banco de España, Alcalá 5, 84 Madrid, Spain. valles@bde.es e owe special hanks o Larry Chrisiano, Jordi Galí, Rober Kollmann and Sergio Rebelo for very helpful commens and suggesions. e also hank Isabel Correia, Olivier Jeanne, Alber Marce, Sephanie Smi-Grohé and Pedro Teles for insighful conversaions, and seminar paricipans a Banco de Porugal, CEPR 999 European Summer Symposium in Macroeconomics (Sinra), and he Sociey for Economic Dynamics 999 Annual Meeing (Alghero) for commens. The views expressed in his paper are he auhors' and do no necessarily reflec hose of he %DQFRGH(VSDxD.

2 ,QURGXFLRQ The negaive correlaion beween money growh and he nominal ineres rae is one of he mos salien feaures of he moneary ransmission mechanism. Mos researchers seek o idenify posiive moneary policy shocks as hose innovaions o money growh ha exer a posiive influence on oupu (RXSXHIIHF), a posiive one in prices (SULFHHIIHF) and reduce he nominal ineres rae on impac (OLTXLGL\HIIHF). In fac, he confidence of he profession in his scheme is such ha failure o produce any of hose effecs is aken as a puzzle ha calls ino quesion he idenificaion procedure. Our reading of his lieraure is ha he liquidiy effec is a nominal feaure ha any well-defined moneary model of he business cycle mus be able o produce under fairly general circumsances (see Chrisiano, Eichembaum and Evans (998)). Following he Lucas (98) program for business cycle research, wo broad classes of models aim o accoun for hose effecs emphasizing differen pars of he moneary policy ransmission mechanism. On he one hand, limied paricipaion models generae a liquidiy effec by allowing resricions in he adjusmen of agens porfolios which break down he ineremporal allocaion of consumpion (see, Fuers (99) and Chrisiano, Eichenbaum and Evans (997) -CEE, henceforh-). On he oher hand, an alernaive line of research has poined o he role of price rigidiies and capial adjusmen coss as he main facors deermining he behavior of nominal and real ineres raes afer a money supply shock (see, for insance, King and ason (996)). Conrary o he previous one, he liquidiy effec is far from graned in his class of models since i depends on how he nominal ineres rae is affeced by he ineremporal allocaion of wealh. In his paper we revisi he liquidiy issue wihin a general equilibrium model wih sicky prices and capial accumulaion in which money services provide uiliy o consumers. e show ha wih separable preferences, a posiive money shock induces a fall in ineres raes if he

3 ineremporal elasiciy of subsiuion of consumpion is low enough, hus generaing a large impac response of curren consumpion relaive o fuure consumpion. This resul has been poined ou by Jeanne (994) and CEE (997) among ohers. In addiion we show ha a low income elasiciy of money demand goes in he same direcion, making he liquidiy effec more likely wihou requiring a exremely high degree of risk aversion. The previous resuls are obained wih separable preferences beween consumpion and leisure. Alhough separable preferences have been used in many moneary models, hey can be hardly reconciled wih he balanced growh properies unless some parameric resricions are saisfied (see Chari, Kehoe ank McGraan (998)). In a model wih non-separable preferences hings are differen since a high ineremporal subsiuion is needed o produce he liquidiy effec. This is so because afer a moneary shock consumpion and leisure move in opposie direcions inducing couneracing effecs on he marginal uiliy of consumpion ha break down ou he negaive link beween ineremporal subsiuion and he impac response of he nominal ineres rae. Furhermore, if here is no capial accumulaion, he model does never produce a correcly signed liquidiy effec wihin he range of posiive risk aversion values. In order o obain a liquidiy effec we have o bring capial accumulaion in he economy. Bu, in doing so he ineres rae response o a posiive moneary innovaion is only negaive for very high values of he elasiciy of ineremporal subsiuion. The impac response of he nominal ineres rae also depends on oher parameers of he model like he degree of nominal (price) and real (capial adjusmen coss) ineria. Neverheless, hese parameers are of secondary imporance as compared wih ha of preferences as regards he abiliy of he model o generae he liquidiy effec. Then, in his kind of models a very high ineremporal subsiuion is a necessary condiion o generae a fall in he nominal ineres rae following a moneary shock. This is mos unforunae since i generaes implausibly large

4 impac responses in oupu, employmen and invesmen. Thus, we conclude ha accouning for he observed liquidiy effec sill remains as an unresolved puzzle for sicky price models. The res of he paper is organized as follows. Secion presens he model and defines he equilibrium. In secion 3 he model is calibraed o be compaible wih a well behaved seady sae. Secion 4 conains he main resuls of he paper. Secion 5 concludes wih some addiional remarks. 7KHRGHO ~ Households (indexed by L) maximise heir expeced lifeime uiliy 8 L, defined as he presen discouned value of he momenary uiliy 8 L condiional on he informaion available a They choose a join plan for consumpion (&), leisure (/) and end-of-period real balances (3), where he uiliy of real balances sems from he ransacion services provided by money: ~ 8 = (, L LR ( β 8 &,/ ) = L L L [] 3 Each household accumulaes capial and rens i o firms a he cos =. The accumulaion of capial is driven by,. L =, L ( δ ). L [] where is he rae of depreciaion,. is capial and, invesmen. Adjusing capial o is desired level is cosly for he households who own i; he funcion of adjusmen coss is chosen o produce non-zero coss in he seady sae:. $& L φ N, L, L. = L [3] These real adjusmen coss are paid hough he purchase of a CES baske of all he produced goods of he economy (see, Hairaul and Porier (993)). 3

5 wih φ. as he adjusmen cos scale parameer for capial. Households decide how o allocae savings beween money (), public secor deb (%) and capial. They receive dividends from heir share ( ) of profis ( ), nominal wage earnings (3:), income derived from rening capial, ineres paymens from bonds and ransfers (7) from he governmen. The budge consrain faced by each household can be wrien as follows: & L 3 L φ %. L 3, L 3 7 L 3 : / L 3 =. L, L =. L L U % L 3 - ω M = LMπ M [4] There are-firms indexed by M. An aggregaor ransforms heerogeneous goods (< M ) ino a composie good (<), hus generaing a demand schedule in erms of relaive prices. More formally, he problem faced by he aggregaor can be saed as follows: - D[ { < } 3< 3 < ; M M = M M ZKHUH θ /( θ ) /θ - ( θ ) < = - < M = θ M where is he elasiciy of subsiuion among he differen produced goods (< M ) and 3 M and 3 are he individual and aggregae oupu prices respecively. Since he elasiciy of subsiuion beween he differen goods is finie and higher han one, each firm has some monopoly power and cares abou is own price relaive o he aggregae one. The firs order condiions of his problem wih respec o < M yield he following demand schedule: /θ -< = M 3 M 3 [5] < using his resul and he zero profi condiion for he aggregaor yields: /( θ ) - 3 = (/ ) ( ) θ - 3 M M = 4

6 The represenaive firm produces a a marginal cos ha is increasing in aggregae oupu. ih flexible prices, hese funcional forms imply ha, in seady sae, firms charge a markup of price over marginal cos equal o. The represenaive firm chooses a plan for producion, labour demand and capial as o maximise he expeced presen value of is profis: π M = ρ 3 = ( π M, where is a pricing kernel represening he marginal uiliy value o he represenaive household of an addiional uni of profis accrued in period ρ = β Λ. Profis and echnology are given by he following expressions: 3 π M = 3 3 / 3 =. 3 $& < M < M : M M M [6] α ( α ) < M = $. M / M Φ [7] where$ Ã describes he economy-wide sae of echnology a period and represens a fixed cos; is exisence makes i possible for a firm o earn zero profi in he long run. The monopolisic compeiion environmen makes i possible o incorporae sicky prices ino he model. e inroduce nominal price rigidiy following Roemberg (98), by assuming ha firms face convex coss of adjusing prices. Specifically, hese coss are expressed as follows: 3 M $& < φ \ M < 3 = µ M [8] here Λ represens he marginal uiliy of consumpion. 5

7 where φ \ measures he degree o which firms dislike o deviae in heir price seing EHKDYLRUIURPKHFRQVDQLQIODLRQUDH 3 The public secor budge consrain is given by he following equaion: (% U % ) = 3 7 [9] The governmen derives revenue from issuing money and deb, which i uses o make ransfers o he households and o pay ineres on ousanding deb. The moneary policy can be described by he following exogenous process for he growh of money: / = µµ ; where represens he seady sae money growh. A shif in moneary policy akes he form of an unexpeced ρµ permanen rise in money: µ = µ exp{ ε }, where µ ε µ is a normally disribued LLG. zero mean shock wih sandard deviaion σ µ. Finally, he fiscal policy reacion funcion has no sochasic componen. e specify he following rule in erms of he ransfers and real bonds: 3 7 = τ % ; where is a posiive consan. Thus, ransfers are deermined o mainain dynamic sabiliy of he model. This specificaion is in he spiri of Leeper (99) and guaranees a Ricardian regime characerised by a combinaion of acive moneary policy and passive fiscal policy. e define a V\PPHULFPRQRSROLVLFFRPSHLLRQHTXLOLEULXP as he se of decision rules of household L and firm M such ha: DThe se of quaniies:< M & L, L / L. L L % L maximise he consrained presen value sream of uiliy of he represenaive household and he consrained presen value of profis earned by he represenaive firm, E The se of prices (3 : = U ) clear he goods markes, he labor marke and he money, bonds and capial markes. 3 As noed by oodford (996) his model leads o a Phillips rade-off in which fuure inflaion expecaions play a crucial role in he join dynamics of inflaion and oupu. 6

8 An exensive represenaion of he symmeric equilibrium is obained from he firs order condiions of boh he L K household and he M K firm. Aggregaing over L and M yields a se of equaions which define he symmeric equilibrium of he economy (see Appendix for deails). Nominal variables grow a he rae. To solve he model we firs wrie he equilibrium ~ equaions in erms of saionary variables ( ; µ ). Second, since an exac expression for ; he equilibrium canno be found analyically, we approximae he soluion by log-linearizing he equilibrium around he seady sae. Then, following Sims (995), we wrie he sysem of ransformed equaions as: Γ [ ζ ; where [ is he vecor of percenage =Γ [ Γ ε µ Γ 3 deviaions of endogenous variables wih respec o heir seady sae; µ is he moneary policy shock, and he las erm ζ [ ( [ defines expecaional errors. 4 The parameer marices L (i=,,,3) are non-linear ransformaions of he srucural parameers. 6SHFLILFDLRQRI3UHIHUHQFHVDQG&DOLEUDLRQ 3UHIHUHQFHV e consider hree differen momenary uiliy funcions ha have been exensively used in he business cycle lieraure and which differ from each oher in he wihin period separabiliy beween consumpion and leisure. Alhough separable preferences have been advocaed in many moneary models wih money in he uiliy funcion, such preferences can hardly be reconciled wih balanced growh properies unlike non-separable preferences (King, Plosser and Rebelo (988)). As will become clear laer, he liquidiy effec depends heavily on he cross derivaive 8 &/ once we allow for he presence of significan subsiuion and income effecs in he labor supply. Thus, he following hree alernaive specificaions of equaion [] have been considered: 4 This soluion mehod is based on generalized eigenvalue decomposiion and i exends he one described by Blanchard and Khan (98). 7

9 ( σ ) / σ / ε σ Γ ( ) / L 8 = κ L L & E [] L 3 σ / ε D( σ ) / σ σ Γ ψ ( σ ) 8 = ( ) L L & E / L [] L 3 σ σ / σ σ Γ = Y 8 L L & E ψ / [] L 3 L The usual resricions imposed on he parameers ensure ha he uiliy is concave, & and 3 are normal goods, and he ineres elasiciy of money demand is sricly negaive. 5 e can hink of he insan uiliy funcion as depending on a composie good, which is a flexible CES aggregaor of consumpion and real balances. This allows us o make our exercises across alernaive preferences comparable in erms of he specificaion of he money demand leaving he money demand properies unresriced. In paricular, he log-linear approximaion of he firs order condiions yields he following expression for he money demand: P S = εf F εu U, where ε F = ( σ ) /( Γ), ε = /( Γ)( U), and U is he seady sae nominal ineres U rae. Finally, noice ha seing Γ = σ yields a uni income elasiciy of money demand. Equaion [] represens separable preferences as he ones recenly used by CEE (997) where! characerizes risk aversion and is he labor supply elasiciy. Equaion [] is a general form of wo alernaive preference specificaions. On he one hand, if we se D, we obain he sandard Cobb-Douglas preference specificaion (CD, henceforh). On he oher hand, if D we obain he preferences used by Chari, Kehoe and McGraan (996) 5 hen Γ = σ we ge he usual CES aggregaor. The sign of U cm equals he sign of - (where m=(m/p)). 8

10 from which i is easy o compue he labor supply elasiciy. The hird class of uiliy funcions (expression []) is he one advocaed by Greenwood, Hercowiz and Huffman (988) (GHH, henceforh). These preferences have wo properies ha may be relevan o undersanding he liquidiy effec: (i) firs, he elasiciy of ineremporal subsiuion of leisure is zero; and (ii) he number of hours worked (L) is a funcion of he curren wage, and so here is no income effec on labor supply. The elasiciy of labor supply implied by hese preferences is (wih Y>). Finally, he parameers κ and will be chosen so ha he oal hours worked by agens are a given proporion of heir ime endowmen. &DOLEUDLRQ This secion describes he benchmark values used o compue he response of he economy o moneary shocks. These parameers ogeher wih he seady sae of he economy are repored in Table. e se he discoun parameer o which implies ha he real ineres rae is equal o percen per annum. The nominal ineres raes, U, and he inflaion rae (money growh rae, ) were se a and, respecively. Our benchmark value for he risk aversion parameer ( ) is equal o. ih separable and GHH preferences he elasiciy of labor supply wih respec o real wages was se equal o. hen we use non-separable preferences as in [], he share parameer (parameer Dor ) is se as o ensure ha agens work percen of heir ime endowmen. Finally, he benchmark parameers in he money demand funcion are F and U. The former is essenially he long run elasiciy esimae by Lucas (988). Neverheless, 9

11 following King and ason (996) we choose a much lower value for he ineres rae elasiciy. 6 Nex, we consider he echnology and capial adjusmen cos parameer. The labor income share ( ) is equal o, and he annual depreciaion rae is equal o percen. The capial adjusmen cos parameer, φ., is se equal o. This value implies ha he insallaion of capial involves a percen cos in erms of invesmen, and a capial-oupu raio of. These values are consisen wih microeconomeric esimaes (see, for insance, hied 99). Neverheless, we will also analyze he effecs of changing he capial adjusmen coss in erms of he liquidiy effec. e urn now o he consideraion of he parameers, and φ \. The elasiciy of demand is chosen ( such ha he markup in a flexible price economy is percen (say, ). Assuming zero profis in he seady sae, he previous assumpion is equivalen o making he value of < $N / also equal o percen. Alhough hese values are convenional in he lieraure, our resuls do no change when a lower markup is used (say, 5 percen) as suggesed by he evidence in Basu and Fernald (997). The price adjusmen cos parameer (φ \ ) canno be calibraed using seady sae informaion. To choose a value we follow recen esimaes of he new-keynesian Phillips rade-off by Sbordone (998). In paricular, we se φ \, which implies ha firms change prices every monhs. 7 Finally, we se corresponding o he esimaed value for M growh in he US (Cooley and Hansen (995)). 6 This low value is consisen wih a low degree of subsiuion in money demand over he business cycle. Moreover, he empirical evidence repored by Goldfeld and Sichel (99) is consisen wih such a value. Indeed, hose auhors advocaed a lower value for he income elasiciy. e will analyze how our resuls depend on seing F equal o. as an alernaive o uni elasiciy. 7 As can be seen from he previous expression, holding he ime beween price adjusmens consan, a lower markup (higher ) implies a much higher cos of adjusing prices.

12 7KH(IIHFVRI3HUPDQHQ8QDQLFLSDHGRQH\6XSSO\6KRFNV In his secion we assess he role of preferences, capial accumulaion and oher feaures of our model in shaping he response of ineres raes o money shocks, or he OLTXLGL\ HIIHF. e proceed sep by sep, firs working wih a version of he model wih separable preferences, hen inroducing non-separable preferences and finally adding capial ino he model. e show ha for a given exogenous money growh rule he exisence of a liquidiy effec depends criically on he ineracion of hese elemens wih he res of he model. ih he complee model we analyze he properies of he parameerizaions ha generae he liquidiy effec, discussing he imporance of he real and nominal rigidiies and of he persisence of he money growh shock for obaining he liquidiy effec. (FRQRP\ZLKRXFDSLDO6HSDUDEOH3UHIHUHQFHV Le us consider a simple version of he model presened in secion, where preferences are separable as expressed by [], and wihou capial, so ha he producion funcion is defined as ( α ) < = $ Φ. Figure A shows ha posiive and persisen money shocks generae a / M M posiive and persisen response of consumpion, real balances and oupu. Since prices are sicky, a posiive nominal shock increases nominal demand and marginal coss bu lowers he markup. Labor demand, oupu and consumpion increase. Following he rise in real wages, labor supply also increases, pushing he economy closer o he compeiive equilibrium. Evenually, he price level adjuss o is new level, resoring he iniial reducion in he markup and reurning he economy o is seady sae. Neverheless, hese real effecs occur along wih a posiive impac on he nominal ineres rae. This impac effec is characerized by wo equilibrium equaions represening he ineremporal allocaion of consumpion and he demand of money balances: 8 8 These are derived from expression [A], [A3] and [A4] in Appendix.

13 8 3 β U ( ( 8 / ) [3], / = F, F = P, / F, [4] U where equaion [3] represens he ineremporal Euler equaion of consumpion, and equaion [4] deermines he opimal allocaion beween consumpion and real balances wihin period (i.e. i represens a money demand equaion). CEE (997) combines equaion [3] wih a cash-in-advance consrain obaining an expression of he ype: [ & 8 ) /( / )] & β. They show ha he expecaional erm becomes a 8 F, = U ( ( F, consan (say, A) in he even ha prices are se one period in advance and money growh follows an LLG process. Under hese assumpions, afer a posiive money growh shock, he only way o generae a liquidiy effec is hrough a reducion in & 8 F. Under separabiliy beween consumpion and leisure, hen σ U = & $ β. Thus, afer an increase in he money supply ha / leads o a rise in C, he nominal ineres rae will fall if and only if he risk aversion parameer is greaer han one (!). Unforunaely, his resul canno be so easily obained once we relax some of he CEE s assumpions regarding he price seing and he money growh process. Neverheless, we can sill log-linearize he above expressions [3] and [4] around he seady sae o obain: 9 U ( S ( ( F F ) = σ [3a] P S = εf F εu U [4a] 9 Noice ha o ge his expression we are also assuming ha he parameer b in expressions []-[] approaches o zero so real he balances disappear of [3a]. e recall ha his is no a bad approximaion if we calibrae he model by looking a he average velociy for M in US (see Chari e al. (996) and Kim (998)). Neverheless, in all he simulaions we eliminae his assumpion.

14 where lower case leers represen deviaions from heir seady sae value. Solving now for he nominal ineres rae in hese wo expressions and assuming ha money demand only responds o movemens in consumpion ε U : U σ ( P ( S ε F σ ε [5] F = For a given degree of money and price persisence he moneary shock generaes an impac increase in money and prices ha is expeced o coninue in he fuure. Under hese fairly general circunsances wo resuls follow from he above expression. The firs is ha, for given values of ε F and, high risk aversion (i.e. a low ineremporal elasiciy of subsiuion (σ )) is needed for he exisence of he liquidiy effec. The inuiion behind his resul can also be cas in erms of he impac effec on he righ hand side of expression [3a], i.e. he real ineres rae. Afer a posiive money shock, consumpion rises a ime and rises furher in o decline from onwards (see Figure A). Thus, he impac effec on he real ineres rae is posiive and i is given by σ = imes he expeced increase in consumpion. This increase is furher reinforced by he rise in expeced inflaion. A lower degree of ineremporal subsiuion of consumpion induces a lower expeced rise in consumpion from o, which will evenually become negaive for very large values of σ. Thus, his produces a subsanial reducion in he real ineres rae (by an amoun equals o σ ( F F ) ) which, if srong enough, migh compensae he increase in inflaion ( hen ε r is differen from zero we solve forward for he nominal ineres rae and obain: U = I Seing ε r = yields expression [5] ha i is easier o inerpre. In our simulaions we will consider ha ε r =.. 3 M Σ δ ( P ( S where δ = ε M σ ε M M = F F κ κ σ ; κ = ε U ; I = ε F κ σ.

15 expecaions. The real ineres rae moves o mach he expeced change in he marginal uiliy beween period and. Since consumpion is he only argumen of marginal uiliy, i is he expeced change in consumpion wha maers for he liquidiy effec. Thus, a subsanial decrease in F Ã as compared wih F is needed o obain a decline in he real ineres rae; his is only possible if households are willing o smooh consumpion over ime, i.e. if (σ ) is low enough. The second resul is ha for sufficienly low money demand elasiciy wih respec o consumpion (ε c ), he liquidiy effec may also be obained wihou requiring a high value for risk aversion. In our model, money is inroduced as an argumen in he uiliy funcion and he shor run consumpion elasiciy is lef free so ha a low value of F favors an impac fall in he nominal rae. The lower he income elasiciy of money demand he smaller he subsiuion away from bonds and hus he more likely he impac fall in he ineres rae following he iniial increase in he supply of real money balances. To pu some numbers o hese resuls, we simulae in Figure B he impac effec on he nominal ineres rae and real balances of a money growh shock under differen values of he risk aversion parameer. This Figure confirms he previous surmise. Thus, in sicky price models if preferences are separable, he higher he risk aversion he more likely is he liquidiy effec. Noice ha he exisence of price and money growh persisence implies an implausibly large degree of risk aversion in order o generae he liquidiy effec. hen he income elasiciy of real balances is less han one a lower value of he risk aversion will generae ha effec. In a cash-in-advance economy wihou capial Jeanne (994) also ges his resul. 4

16 (FRQRP\ZLKRXFDSLDORQ6HSDUDEOH3UHIHUHQFHV hen preferences are non-separable, hings are differen. In paricular, expression [5] no longer represens he ineres rae response since he ineremporal allocaion of consumpion depends on oher feaures of he model. Figure A shows he response of he main variables in he model o a persisen money shock. The resuls correspond boh o Cobb-Douglas preferences and o GHH preferences. A firs glance, he impulse response funcions look very similar in his economy as compared wih hose in a world wih separable preferences (i.e. hose depiced in Figure A): a posiive moneary shock increases oupu, consumpion and real balances, while he nominal ineres rae sill increases on impac. Neverheless, a closer look a he resuls shows an ineresing deparure from he general proposiion enunciaed above. As he sensiiviy analysis in Figure B makes clear, a higher risk aversion is no longer a necessary condiion for he exisence of he liquidiy effec. Indeed he opposie is rue. To give some inuiion for his laer resul, we consider he non-separable preferences given by expression [], when D. Proceeding a in he previous secion, he log-linear equaion for he ineremporal consumpion allocaion [3] akes now he following form: U ( S / ψ = / ( σ )(( l ) (( F F ) l [3b] σ This expression saes ha he real ineres rae moves as o ensure he equalizaion of he marginal uiliy of consumpion beween and, bu now he raio of marginal uiliies no only depends on he expeced change in consumpion bu also on he expeced response of labor supply. As he elasiciy of ineremporal subsiuion, (σ ), falls boh ( F and ( l become smaller and evenually negaive, exering opposie effecs upon he raio of marginal uiliies; he leisure effec dominaes making he liquidiy effec less raher han more likely for given inflaion expecaions. 5

17 An addiional resul is ha in his case he liquidiy effec is never obained for posiive values of σ. This resul also relies heavily on he fac ha in a model wihou capial consumpion is proporional o labor (i.e. F αl ). Thus, for he benchmark values of he parameers he expeced increase in consumpion and leisure roughly compensae each oher in expression [3b] leaving he nominal ineres rae almos unchanged, for a given expeced inflaion, as we move owards higher values of he risk aversion. As noed in secion 3, he CD preferences impose srong income and subsiuion effecs on leisure afer a posiive money shocks. Thus, we now analyze how allowing for a zero ineremporal subsiuion effec on leisure and a zero income effec on labor supply affec our previous resul. e use he GHH preferences ha imply he following (log-linear) expression for he inerermporal consumpion allocaion: η Y [(( F F ) ( l l )] ( U ( S ) = σ η ( [3c] where he seady sae soluion imposes F\ hih hese preferences households smooh ( F ηl ) over ime insead of F Ã and, unlike wih CD preferences, he parameer is independen of he ineremporal subsiuion. The absence of an income effec on he labor supply makes he movemens in consumpion and leisure roughly proporional and such proporion is independen of he degree of ineremporal subsiuion. Moreover, increases in he risk aversion (σ ) affec he nominal ineres rae more significanly han in he case of CD preferences bu always in he opposie direcion ha required for a liquidiy effec. Since his is a mos unforunae feaure of he model we may ask a his poin wha does i ake o obain a proper liquidiy effec in a sicky price model wih non-separable preferences. In general his parameer is close o one in marke economies. 6

18 ha we need is a mechanism ha drives he response of labor supply and consumpion significanly apar so ha he real ineres rae could fall subsanially on impac. (FRQRP\ZLKFDSLDORQVHSDUDEOH3UHIHUHQFHV Capial accumulaion is a key feaure of he model ha is expeced o have a major effec on he way some endogenous variables respond o a moneary shock. In paricular, curren consumpion can be made less responsive o he money shock for a given oupu response, since households devoe par of heir income o inves in capial goods (in addiion o bonds and real balances). e will show ha in order o generae he liquidiy effec he mere presence of capial is no enough, wha is needed is srong incenive o accumulae i. This can be obained in he model previously discussed by means of high degree of ineremporal subsiuion (say, /σ ) and low adjusmen coss of capial (φ N ). The reason why we need a high ineremporal subsiuion sems from he fac ha wih low ineremporal subsiuion he response of consumpion afer a persisen money shock is similar o ha in oupu so ha he paern of impulse-responses resemble very much ha of an economy wihou capial. 3 The impulse responses in Figure 3A of an economy wih capial accumulaion show ha he previous surmise is correc. Sill, in he case of high risk aversion values here are no incenives o save and accumulae capial, and herefore he allocaion of real balances and consumpion has no undergone a significan change wih respec o he model wihou capial. Thus he money marke equilibrium generaes he same pah in erms of ineres raes: an increase on impac followed by a smooh decrease over ime. Neverheless, he money shock simulaion when σ is low generaes a large subsiuion effec ha is refleced in a huge oupu and labor responses on impac bu only a small change in consumpion. For σ he ineres 3 A similar reasoning applies o a high value of he capial adjusmen coss (φ N ). 7

19 rae response generaes a liquidiy effec. e can explain he impac response of ineres raes in erms of he expression [3b]: if he risk aversion parameer is less han one, hen here is a large expeced fall in labor afer he firs period ha moves he curren ineres raes down. 4 The impulse responses in Figure 3A also highligh he role of capial in his model. Since σ (/.75) is large, consumers have he incenive o pospone consumpion which, unlike he model wihou capial, does no necessarily implies posponing producion (and so employmen) oo. On he conrary, as he real rae falls he iniial jump in he expeced shadow price of capial leads o a sharp increase in he demand for invesmen. Since, given he exisence of sicky prices, oupu is demand deermined he increase in invesmen ranslaes ino oupu so increasing labor demand alongside. 5 As can be seen in expression [3b], now he sharp expeced reducion in labor ( ( l ) more han compensaes he expeced consumpion increase (( F o produce a srong fall in he real ineres rae. hy do households pospone consumpion and work harder oday despie he fac ha he real rae has fallen? The reason is ha here is a new asse, capial, which makes very profiable any addiional amoun of resources devoed o accumulae i. As σ ges smaller all hese effecs become smaller oo and he economy resembles very much he one wihou capial in which consumpion and labor balance each oher is effec on he marginal uiliy of he household so leaving no room for movemens in he real ineres rae. 4 In Appendix we show he implicaions of such a risk aversion parameer in erms of he labor supply elasiciy. 5 Noice ha he addiional oupu demand in period mus be enirely me by a rise in employmen since. is predeermined. 8

20 To sum up, when preferences are non-separable and here is capial accumulaion he liquidiy effec is more likely he smaller he risk aversion parameer, jus he opposie of wha happens in an economy wih separable preferences. Neverheless he range of values for which such an effec exiss is very small and is always for risk aversion values lower han one (see Figure 3B). Tha generaes implausibly large movemens in oupu and labor whereas consumpion movemens remain very low afer a moneary shock. To ge a more complee picure of he feaures shaping he iniial response of he nominal ineres rae in his model, we check he robusness of our resuls wih respec o he remaining parameers of he model. Figure 4 depics he resuls of sensiiviy analyses wih regard o nominal and real rigidiies as well as o he money shock persisence. ihin he low risk aversion region (doed line) he liquidiy effec is more likely he lower he adjusmen cos of capial, he lower persisence of he money growh shock and he higher he price sickiness. The cos of adjusing capial is borne by households and he lower his cos he higher he incenive o accumulae capial, leading o he sronger oupu and employmen responses needed o guaranee he liquidiy effec. The impac response of he nominal ineres rae is always higher, and ha of real balances lower, he higher he value of ρ µ. Similarly, as expeced, real balances rise and he ineres rae falls more, he higher he value of φ <, i.e. he liquidiy effec is more likely he higher he degree of price ineria. ihin he region of higher risk aversion (coninuous), he sensiiviy of he ineres raes and real balances o his parameer is much lower. Therefore, under non-separabiliy and capial accumulaion he risk aversion parameer is he crucial one, as compared wih hose reflecing he inensiy of price ineria and capial adjusmen coss, in shaping ineres rae movemens. 9

21 &RQFOXVLRQV Alhough neiher he oupu nor he price effecs of a moneary expansion need o be preceded by a fall in he nominal ineres rae, he liquidiy effec is viewed by many economiss as one of he well esablished empirical facs in moneary economics. General equilibrium models aimed a represening such mechanism face he challenge of reproducing he liquidiy effec along wih oher business cycle feaures of marke economies. In his paper we have discussed under wha condiions a general equilibrium model wih coss of adjusing prices and capial is capable of generaing a downwards movemen of he nominal rae following a posiive moneary innovaion. In a world wihou capial and separable preferences, he logic of he ineremporal allocaion of wealh leads o he following resul: i akes a low elasiciy of ineremporal subsiuion for he nominal rae o fall on impac. e show ha i is also possible o ge he same resul by reducing he income elasiciy of money demand over he business cycle. This is so because he marginal uiliy of consumpion is jus driven by he dynamics of consumpion. hen preferences are no separable he previous resul does no apply. Now, he impac response of nominal ineres rae is no sensiive o changes in he ineremporal subsiuion. The reason is also sraighforward: now consumpion and labor supply movemens balance each oher is effec on he marginal uiliy of he household so leaving no room for movemens in he real ineres rae. In an economy wih non-separable preferences and capial accumulaion he liquidiy effec is no achieved unless he ineremporal subsiuion is very high and he capial adjusmen coss are no high. In such a case, his ineremporal subsiuion effecs leads households o pospone consumpion invesing in capial and so increasing curren oupu and labor demand by

22 an amoun large enough o reduce he real ineres rae by more han he expeced increase in prices. The logic of his resul can be analyzed in he ligh of he moneary ransmission mechanism of he simples limied paricipaion model. In his kind of models, resricions on he adjusmen of consumer s porfolios make i possible ha he addiional liquidiy of he sysem is no necessarily devoed o consumpion. The addiional savings mus be devoed o increase he demand of producive facors, which firms are willing o do if he ineres rae falls. In sicky price models where he marginal uiliy of consumpion depends upon consumpion and leisure we can ge a similar resul imposing a srong bias owards he ineremporal subsiuion and low adjusmen coss of capial. Neverheless, in his laer model his comes a he cos of generaing implausibly large impac responses in oupu, employmen and invesmen. Finally, he impac response of he nominal ineres rae also depends on oher parameers of he model like he degree of nominal (price) and real (capial adjusmen coss) ineria. Neverheless, hese parameers are of secondary imporance as compared wih ha of preferences as regards he abiliy of he model o generae he liquidiy effec. Thus, we conclude ha accouning for he observed liquidiy effec sill remains as an unresolved puzzle for sicky price models of he moneary ransmission mechanism.

23 $SSHQGL[$QH[HQVLYHUHSUHVHQDLRQRIKHV\PPHULFPRQRSROLVLFFRPSHLLRQHTXLOLEULXP The household s choice is given by: Λ = & 8 [A] P L = Λ 8 [A] r E Λ = Λ [A3] E M Λ =Λ 8 [A4] Opimal capial accumulaion is derived from he firs order condiions of households wih respec o invesmen and capial: q )E ( q 3 K I k P Z P = Λ δ φ [A5] q E K I k 3 P = Λ φ [A6] The firs order condiions of he firm wih respec o he employmen and capial yields he following relaionships: [ ] ) Y (/e L Y ) ( Φ = α [A7] [ ] ) y (/e - K Y Z Φ =α [A8] where H < akes he following expression: P P Y Y P P Y E P P P P Y Y e = ρ ρ φ φ θ [A9] The model also assumes ha every agen has access o a complee and compeiive marke for coningen claims. This is equivalen o say ha firms maximize heir marke value. In such a siuaion, here is a unique real discoun facor saisfying: Λ Λ = ρ ρ [A] Finally, four consrains guaranee ha markes clear. These are given by he producion funcion, capial accumulaion, he governmen budge consrain and he economy wide consrain. T P ) B r (B M M = [A] )K ( I K = δ [A] Φ = K L A Y [A3]

24 3 o obain he economy wide consrain we proceed as follows. Using he definiion of profis Y AC K L Y = = π and imposing he governmen budge consrain in he household budge consrain yields: Y AC Y K I K I C = φ [A4] Price adjusmen coss ake he following quadraic expression: Y P P Y Y AC = φ [A5] e specify he following fiscal policy in erms of he ransfers: B T P = τ [A6] here τ is a posiive consan. Thus, ransfers are deermined o mainain dynamic sabiliy of he model. The equaion for he moneary policy complees he sysem of 7 dynamic equaions wih 7 endogenous variables: 7 prices (3 : = Λ ρ T U ), 5 quaniies (< &, / 7 ), 3 socks ( %. ), he mark-up ((-(/H < )) - ) and he price adjusmen coss ($& < ).

25 $SSHQGL[/LTXLGL\(IIHFDQG/DERU6XSSO\(ODVLFL\ To give some addiional insighs as o why do we need an economy wih low risk aversion and capial we performed he following exercise. The Panel A of Table A presens how changes in he risk aversion ranslae ino: (i) alernaive ineremporal (Frisch) labor supply elasiciies; and (ii) differen impac effecs on nominal ineres raes (r ) and labor (L ). In he economy wihou capial, changes in he ineremporal elasiciy of subsiuion leaves pracically unchanged he iniial impac effec on /, and hus on U. The same exercise in an economy wih capial reveals wo ineresing feaures: firs here is a dramaic increase in he oupu and employmen response; second, and mos imporan, his effec is five imes as large when σ equals as compared wih σ. In he economy wihou capial here is no evidence of liquidiy effec. This resul is robus o alernaive Frisch labor supply elasiciies and seady sae hours worked. In an economy wih capial, low values of risk aversion sill ranslae ino high enough Frisch labor supply elasiciies. Boh circumsances imply ha very small changes in he real wages are associaed wih very imporan impac effecs on labor. 6 Since wih hese preferences boh ineremporal subsiuion and income effecs play a very imporan role in racing ou he impac response of labor, he resul ranslae ino a huge expeced reducion in he marginal uiliy of consumpion and so in an imporan liquidiy effec. 7$%/($/,48,',7<())(&7$'/$%56833/<(/$67,&,7< &'35()(5(&(6 (&<:,787&$3,7$/ σ $/ D Frisch Labor Supply r Impac Effec L Impac Effec %/ D Frisch Labor Supply r Impac Effec L Impac Effec (&<:,7&$3,7$/ σ $/ D Frisch Labor Supply r Impac Effec L Impac Effec %/ D Frisch Labor Supply r Impac Effec Limpac Effec RHÃThe impac effecs correspond o expansionary money shocks. The res of he parameers are aken from Table. The leers r, L corresponds o nominal ineres rae and labor, respecively. 6 In he economy wih capial under and / he impac response of real wages is 4

26 5HIHUHQFHV Basu, S. and J.G. Fernald (997): "Reurns o Scale in US Producion: Esimaes and Implicaions", -RXUQDORI3ROLLFDO(FRQRP\, 5, Blanchard O.J. and C.M. Khan (98): "The soluion of Linear Difference Models under Raional Expecaions", (FRQRPHULFD 48, Chari, V.V., P.J. Kehoe and E.R. McGraan (996): "Sicky Price Models of he Business Cycle: Can he Conrac Muliplier Solve he Persisence Problem?, NBER P 589. Chari, V.V., P.J. Kehoe and E.R. McGraan (998): "Moneary shocks ad real ineres raes in sicky price models of inernaional business cycle, Federal Reserve Bank of Minneapolis, Research Dp. Saff Repor 33. Chrisiano, L., M. Eichenbaum and C. Evans (997): "Sicky Price and Limied Paricipaion Models: A Comparison", (XURSHDQ(FRQRPLF5HYLHZ, 4 (6), -49. Chrisiano, L., M. Eichenbaum and C. Evans (998): "Moneary Policy Shocks: ha Have we Learned and o ha End?" NBER P 64. Cooley, T. and G. Hansen (995): "Money and he Business Cycle", in T. Cooley (edior) )URQLHUVRI%XVLQHVV&\FOH5HVHDUFK, Princeon Universiy Press, Princeon, New Yersey, Fuers,T.(99):"Liquidiy, Loanable Funds and Real Aciviy",-RXUQDORIRQHDU\(FRQRPLFV 9, 3-4. Hairaul, J. and F. Porier (993): "Money, New-Keynesian Macroeconomics and he Business Cycle",(XURSHDQ(FRQRPLF5HYLHZ 37, Jeanne, O, (994): "Nominal Rigidiies and he Liquidiy Effec", mimeo ENCP-CERAS. Goldfeld, S. and D. Sichel (99): "The Demand for Money", in B. Friedman and F. Hahn (eds.), DQGERRNRIRQHDU\(FRQRPLFV vol. I Amserdam: Norh Holland,

27 Greenwood, J.; Zvi Hercowiz and G. Huffman (988): "Invesmen, Capaciy of Uilizaion and he Real Business Cycle", $PHULFDQ(FRQRPLF5HYLHZ, 78, Kim, J. (998): "Moneary Policy in a Sochasic Equilibrium Model wih Real and Nominal Rigidiies". Divisions of Research & Saisics and Moneary Affairs. Federal Reserve Board, IRUKFRPLQJ-RXUQDORIRQHDU\(FRQRPLFV King, R.G.; Plosser, G. and S. Rebelo (988): "Producion, Growh and Business Cycles I: The Basic Neoclassical Model", -RXUQDORIRQHDU\(FRQRPLFV,, King, G.R. and ason, M. (996): Money, Prices, Ineres Raes and he Business Cycle 5HYLHZRI(FRQRPLFVDQG6DLVLFV, 78, Leeper, E. (99): " Equilibria under "Acive" and "Passive" Moneary and Fiscal Policies", -RXUQDORIRQHDU\(FRQRPLFV 7, Lucas, R.E. (98): "Mehods and Problems in Business Cycle Theory", -RXUQDORIRQH\&UHGL DQG%DQNLQJ, (4), Lucas, R.E: (988):"Money Demand in he Unied Saes: A Quaniaive Review &DUQHJLH 5RFKHVHU&RQIHUHQFH6HULHVRQ3XEOLF3ROLF\, 9, Roemberg, J.(98): "Sicky Prices in he Unied Saes", -3( 9, 87-. Sbordone, A.(998):"Price and Uni Labor Coss: A New Tes of Price Sickiness",IIES.P 653. Sims, C. (995): "Solving Linear Raional Expecaions Models", mimeo, Yale Universiy. hied, T. (99): "Deb, Liquidiy Consrains and Corporae Invesmen: Evidence from Panel Daa", 7KH-RXUQDORI)LQDQFH 47, oodford, M. (996): "Conrol of he Public Deb: A Requiremen for Price Sabiliy", NBER P

28 ),*85( 3$(/$,38/6(5(636(7$3(56,67(7(<*5:76&. 6(3$5$%/(35()(5(&(6:,787&$3,7$/.7 Real Balances.8 Nominal Ineres Rae Markup.7 Consumpion Quarers afer he shock Quarers afer he shock RH3DUDPHHU9DOXHV φ Y DQG c =. 3$(/%5%867(667&$*(6,7(5,6.$9(56, Nominal Ineres Rae.8 Real Balances.5 ε c =.7 ε c =..6 Impac Effec.5 Impac Effec risk aversion(σ) RH3DUDPHHU9DOXHV φ Y DQG = risk aversion(σ)

29 ),*85( 3$(/$,38/6(5(636(7$3(56,67(7(<*5:76&. 6(3$5$%/(35()(5(&(6&'*:,787&$3,7$/.8 Real Balances.5 Nominal Ineres Rae Oupu.8 Consumpion.6 GHH.6. CD Quarers afer he shock Quarers afer he shock RH3DUDPHHU9DOXHV D Y φ Y DQG c =. 3$(/%5%867(667&$*(6,7(5,6.$9(56,.5 Nominal Ineres Rae (CD).6 Real Balances (CD) Impac Effec Impac Effec Nominal Ineres Rae (GHH).6 Real Balances (GHH).9 Impac Effec ε c = ε c =. Impac Effec risk aversion(σ). 5 5 risk aversion(σ) RH3DUDPHHU9DOXHV D Y φ Y DQG =.5. 8

30 ),*85( 3$(/$,38/6(5(636(7$3(56,67(7(<*5:76&. 6(3$5$%/(35()(5(&(6&':,7&$3,7$/.7 Real Balances.6 Nominal Ineres Rae σ= σ= Oupu.7 Consumpion Quarers afer he shock Quarers afer he shock RH3DUDPHHU9DOXHV D φ Y =85, φ k DQG c =. 3$(/%5%867(667&$*(6,7(5,6.$9(56, &3$5,*&':,7*35()(5(&(6.8 Nominal Ineres Rae (CD).7 Real Balances (CD) Impac Effec.6. Impac Effec ε c = ε c = Nominal Ineres Rae (GHH).65 Real Balances (GHH).55.6 Impac Effec.5 5 Impac Effec risk aversion(σ) risk aversion(σ) RH3DUDPHHU9DOXHV D Y φ Y =85, φ k DQG =.5. 9

31 ),*85( 6(6,7,9,7<$$/<6,6 6(3$5$%/(35()(5(&(6&':,7&$3,7$/ Impac Effec Nominal Ineres Rae φ k Impac Effec Real Balances φ k Impac Effec Impac Effec ρ µ ρ µ Impac Effec Impac Effec φ y φ y Noes: (L&RQLQXRXVOLQH DQGKHGRHGOLQH =.75. (ii) In he firs panel (Robusness o φ k ZHVH D φ y DQG c =.5. (iii) In he second panel (Robusness o ZHVH D φ y =85, φ k DQG c =. (iv) In he hird panel (Robusness o φ y ZHVH D φ k c DQG =.5. 3

32 Descripion 7$%/( $%$6(/,(9$/8(6)5&$/,%5$7,3$5$(7(56 Parameer Value Discoun Facor β (.97) /4 3UHIHUHQFHV D6HSDUDEOHV /DERUVXSSO\HODVLFL\. Risk aversion σ. ERQ6HSDUDEOHV&REE'RXJODV&' Risk aversion σ. FRQ6HSDUDEOHV*UHHQZRRGHUFRZL]XIIPDQ* Labor supply elasiciy. RQH\'HPDQG3URSHULHV Consumpion elasiciy ( c ) (-σ. Ineres rae semi-elasiciy ( r ) U. 7HFKQRORJ\DQG&DSLDO$FFXPXODLRQ Capial income share α.33 Depreciaion rae δ (.) /4 Capial adjusmen cos parameer φ k 3ULFH6HLQJ 6HDG\6DHDUNXS. 3ULFH$GMXVPHQFRVSDUDPHHU φ y 85 RQHDU\3ROLF\ Auocorrelaion of Money Growh Shocks ρ µ.5 %67($'<67$7( Values Descripion (in annual erms) µ.5 R.8 M/B.33 L.3 C/Y.74 K/Y.5 I/Y.5 Φ/Y. Price Adjus.Cos/Y (%).7 Capial Adjus.Coss/I (%).74 3