Board of Governors of the Federal Reserve System. International Finance Discussion Papers. Number July 2010

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1 Board of Governors of he Federal Reserve Sysem Inernaional Finance Discussion Papers Number 3 July 2 Is There a Fiscal Free Lunch in a Liquidiy Trap? Chrisopher J. Erceg and Jesper Lindé NOTE: Inernaional Finance Discussion Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Inernaional Finance Discussion Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors. Recen IFDPs are available on he Web a

2 Is There a Fiscal Free Lunch in a Liquidiy Trap? Chrisopher J. Erceg Federal Reserve Board Jesper Lindé Federal Reserve Board and CEPR Firs version: April 29 This version: July 2 Absrac This paper uses a DSGE model o examine he effecs of an expansion in governmen spending in a liquidiy rap. If he liquidiy rap is very prolonged, he spending muliplier can be much larger han in normal circumsances, and he budgeary coss minimal. Bu given his fiscal free lunch, i is unclear why policymakers would wan o limi he size of fiscal expansion. Our paper addresses his quesion in a model environmen in which he duraion of he liquidiy rap is deermined endogenously, and depends on he size of he fiscal simulus. We show ha even if he muliplier is high for small increases in governmen spending, i may decrease subsanially a higher spending levels; hus, i is crucial o disinguish beween he marginal and average responses of oupu and governmen deb. JEL Classificaion: E52, E58 Keywords: Moneary Policy, Fiscal Policy, Liquidiy Trap, Zero Bound Consrain, DSGE Model. We hank Marin Bodensein, V.V. Chari, Luca Guerrieri, and Raf Wouers for very consrucive suggesions. We also hank paricipans a a macroeconomic modeling conference a he Bank of Ialy in June 29, a he February 2 NBER EF&G Meeing in San Francisco, a he CEPR 8 h ESSIM conference in Tarragona (Spain), a he 2 SED Meeing in Monreal, and seminar paricipans a he European Cenral Bank, Georgeown Universiy, Universiy of Maryland, he Federal Reserve Banks of Cleveland and Kansas Ciy, and he Sveriges Riksbank. Mark Clemens, James Hebden, and Ray Zhong provided excellen research assisance. The views expressed in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Board of Governors of he Federal Reserve Sysem or of any oher person associaed wih he Federal Reserve Sysem. Corresponding Auhor: Telephone: Fax: addresses: and

3 . Inroducion During he pas wo decades, a voluminous empirical lieraure has aemped o gauge he effecs of fiscal policy shocks. This lieraure has been insrumenal in idenifying he channels hrough which fiscal policy affecs he economy, and, in principle, would seem a naural guidepos for policymakers seeking o assess how alernaive fiscal policy acions could miigae business cycle flucuaions. However, i is unclear wheher esimaes of he effecs of fiscal policy from his empirical lieraure which focuses almos exclusively on he poswar period should be regarded as applicable under condiions of a recession-induced liquidiy rap. Keynes (933, 936) argued in suppor of aggressive fiscal expansion during he Grea Depression exacly on he grounds ha he fiscal muliplier was likely o be much larger during a severe economic downurn han in normal imes, and he burden of financing i correspondingly ligher. In his paper, we use a New-Keynesian DSGE modeling framework o examine he implicaions of an increase in governmen spending for oupu and he governmen budge when moneary policy faces a liquidiy rap. A key advanage of he DSGE framework is ha i allows explici consideraion of how he conduc of moneary policy and, in paricular, he zero bound consrain on nominal ineres raes affecs he muliplier. We begin by showing ha he governmen spending muliplier can be amplified subsanially in he presence of a prolonged liquidiy rap. This corroboraes analysis by Eggerson (28) and Davig and Leeper (29), which shows ha governmen spending can have ousized effecs when moneary policy allows real ineres raes o fall, and recen work by Chrisiano, Eichenbaum and Rebelo (29) in a model wih endogenous capial accumulaion. 2 While our workhorse model is a varian of he Chrisiano, Eichenbaum and Evans (25) and Smes-Wouers (27) models, we show ha he spending muliplier is even larger in versions ha embed hand-o-mouh agens (as in The bulk of research suggess a governmen spending muliplier in he range of.5 o slighly above uniy. One srand of he lieraure originaing wih Barro (98, 99) has esimaed he muliplier by examining he response of oupu o changes in miliary spending. This approach has ypically yielded mulipliers in he range of.5-., including in recen work by Hall (29), alhough Ramey (29) esimaed a somewha higher muliplier of.2. As emphasized by Hall, esimaes based on his approach hinge criically on he relaionship beween oupu and spending during WWII and he Korean War, and may be somewha downward-biased due o he "command-economy feaures prevalen in WWII, and because axes were raised markedly during he Korean War. An alernaive approach involves idenifying he governmen spending muliplier using a srucural VAR as in Blanchard and Peroi (22), and Gali, Lopez-Salido, and Valles (27). These sudies repor a governmen spending muliplier of uniy or somewha higher (afer -2 years), hough he cross-couny evidence of Peroi (27) and Mounford and Uhlig (28) is suggesive of a lower muliplier. 2 In conras, Cogan e al. (29) analyze he effecs of governmen spending shocks in he Smes-Wouers model, and conclude ha he muliplier is only slighly amplified under he range of liquidiy rap duraions ha hey consider, which exend beween 4 and 8 quarers. Merens and Ravn (2) develop a sylized model which raionalizes a low and possibly negaive spending muliplier in a liquidiy rap in an environmen wih muliple equilibria driven by expecaional shocks. In heir model, an increase in fiscal spending confirms and reinforces he pessimisic expecaions of he privae secor.

4 Galí, López-Salido, and Vallés 27) and financial fricions (as in Bernanke, Gerler, and Gilchris 999, and Chrisiano, Moo, and Rosagno 27). Moreover, an increase in governmen spending agains he backdrop of a deep liquidiy rap pus less upward pressure on public deb han under normal circumsances, reflecing ha he larger oupu response ranslaes ino much higher ax revenues. A firs blush, hese resuls seem highly supporive of Keynes argumen for fiscal expansion in response o a recession-induced liquidiy rap he benefis are exremely high, and he budgeary expense o achieve i very low. Bu his raises he imporan quesion of why policymakers would wan o limi he magniude of fiscal expansion, and hus pass up on wha appears o be a fiscal free lunch. Our paper addresses his quesion by showing ha he spending muliplier in a liquidiy rap decreases wih he level of governmen spending. The novel feaure of our approach is o allow he economy s exi from a liquidiy rap and reurn o convenional moneary policy o be deermined endogenously, wih he consequence ha he muliplier depends on he size of he fiscal response. rap more quickly. Quie inuiively, a large fiscal response pushes he economy ou of a liquidiy Because he muliplier is smaller upon exiing he liquidiy rap reflecing ha moneary policy reacs by raising real ineres raes he marginal impac of a given-sized increase in governmen spending on oupu decreases wih he magniude of he spending hike. This dependence of he governmen spending muliplier on he scale of fiscal expansion evidenly conrass wih a sandard linear framework in which he muliplier is invarian o he size of he spending shock. 3 The implicaion ha he muliplier declines in he level of spending provides a poenially imporan raionale for limiing he size of fiscal spending packages in a liquidiy rap. If so, i becomes crucial o characerize he marginal response of oupu and public deb o higher governmen spending o make informed choices abou he appropriae scale of fiscal inervenion in a liquidiy rap. A major focus of our paper consiss of providing such a quaniaive characerizaion in an array of nesed DSGE models. Secion 2 analyzes he effecs of governmen spending shocks in a simple hree equaion New Keynesian model in which policy raes are consrained by he zero lower bound. Similar o previous research (e.g., Eggerson 28), he liquidiy rap is generaed by an adverse ase shock 3 Bodensein, Erceg, and Guerrieri (29) show in he conex of simulaions of a large-scale open economy model ha he conracionary effec of foreign shocks on he domesic economy increases nonlinearly in he size of he shock when he domesic economy is consrained by he zero bound. 2

5 ha sharply depresses he poenial real ineres rae. A key resul of our analysis is ha he governmen spending muliplier measured as he conemporaneous impac on oupu of a very small incremen in governmen spending is a sep funcion in he level of governmen spending. If he level of spending is suffi cienly small, higher governmen spending does no affec he economy s exi dae from he liquidiy rap, and he muliplier is consan a a value ha is higher han in a normal siuaion in which moneary policy would raise real ineres raes. However, as spending rises o higher levels, he economy emerges from he liquidiy rap more quickly, and he muliplier drops. The muliplier coninues o drop discreely as governmen spending rises furher reflecing a progressive shorening of he liquidiy rap unil spending is high enough o keep he economy from falling ino a liquidiy rap. Beyond his level of spending, he muliplier levels ou a a value equal o ha under normal condiions in which policy raes are unconsrained. The simple New Keynesian model is a convenien ool for illusraing he salien role of inflaion expecaions in deermining how he muliplier varies wih he level of governmen spending. If prices are fairly responsive o marginal cos as implied by relaively shor-lived price conracs he muliplier is exremely high for small incremens o governmen spending, bu drops quickly a higher spending levels. Thus, he large mulipliers ha apply o small fiscal expansion should no be inferred o carry over o much larger fiscal expansions, and i is paricularly imporan o ake accoun of he endogeneiy of he muliplier under such condiions. By conras, he muliplier funcion is much flaer if he slope of he Phillips Curve is lower, and even a low spending levels i isn dramaically differen han in normal imes. The simple model is also convenien ool for assessing oher empirically relevan facors ha may affec he muliplier, including implemenaion lags in spending. Implemenaion lags may dampen he muliplier significanly, and even cause he muliplier o be negaive if he lags are suffi cienly long. Thus, echoing Friedman (953), he effi cacy of fiscal policy in macroeconomic sabilizaion even in a liquidiy rap can be hampered by long and variable lags. The implicaions of he sylized model prove useful in inerpreing he behavior of he governmen spending muliplier in more empirically-realisic models. In Secion 3, we analyze a workhorse model ha is very similar o he esimaed models of Chrisiano, Eichenbaum and Evans (25) and Smes and Wouers (27). Secion 4 exends he workhorse model by including boh Keynesian hand-o-mouh agens and financial fricions. These addiional feaures boos he muliplier by amplifying he effecs of governmen spending shocks on he poenial real ineres rae; moreover, as argued by Galí, López-Salido, and Vallés (27), he inclusion of Keynesian households 3

6 can help accoun for he posiive response of privae consumpion o a governmen spending shock documened in srucural VAR sudies by e.g., Blanchard and Peroi (22) and Peroi (27). 4 Given he relevance of iniial condiions ha deermine he duraion and deph of he liquidiy rap for he spending muliplier, we analyze he muliplier agains he backdrop of a severe recession scenario ha aemps o capure some of he feaures of he U.S. experience during he recen financial crisis. For each model varian, such a scenario is consruced by a sequence of adverse consumpion demand shocks ha depress oupu by abou 8 percen relaive o seady sae, and ha generae a liquidiy rap lasing eigh quarers. In our workhorse model, he muliplier implied by a sandard-sized percen of GDP increase in governmen spending is. in he firs four quarers following he shock The muliplier is.6 in he augmened model when he share of Keynesian households hose ha consume heir enire aferax income equals 5 percen, which would seem a he upper end of he plausible range. Under eiher model varian, he muliplier is considerably larger and more persisen ha under normal condiions in which moneary policy would raes ineres raes immediaely, and he ousized oupu effecs imply a smaller rise in governmen deb. Even so, he muliplier declines noiceably as he fiscal spending package exceeds 2-3 percen of GDP, reflecing ha simulus of ha magniude is suffi cien o reduce he duraion of he liquidiy rap by a couple of quarers; for example, he muliplier drops from. o.7 in he workhorse model. Moreover, implemenaion lags can markedly reduce he muliplier. Thus, agains he backdrop of an eigh quarer liquidiy rap, here may be subsanial benefis of increasing some forms of spending ha have shor implemenaion lags; bu argumens favoring such programs based on an ousized muliplier would seem o apply only for spending packages fairly modes in scale. The governmen spending muliplier under our benchmark calibraion increases markedly as he liquidiy rap duraion exends much beyond wo years, mainly reflecing ha expeced inflaion becomes much more responsive o shocks. Wih he high muliplier, governmen deb falls sharply, so ha he fiscal expansion more han pays for iself. However, while such condiions of deep recession provide a srong raionale for fiscal acivism, our analysis shows ha he muliplier under hese condiions ends o drop sharply in he level of governmen spending. For example, in he workhorse model wih an quarer liquidiy rap, he muliplier is 3.3 for small increases in 4 As discussed in recen papers by Leeper, Walker and Yang (29) and Ramey (29), idenified VARs can produce misleading resuls if some of he fiscal expansion is anicipaed. Accordingly, Fisher and Peers (29) idenify governmen spending shocks wih saisical innovaions o he accumulaed excess reurns of large U.S. miliary conracors, and find ha posiive spending shocks are associaed wih an oupu muliplier above uniy and increases in hours and consumpion. 4

7 governmen spending, bu declines o.2 as spending rises above 2 percen of GDP. Inuiively, jus as small adverse shocks can exer a large conracionary impac in a long-lived liquidiy rap and deep recession, small fiscal expansions may be highly effecive in miigaing he recession; bu wih a shallower recession, he benefis of addiional simulus drop subsanially. We conduc exensive sensiiviy analysis o assess how he relaionship beween he muliplier and level of governmen spending is affeced by he slopes of he price- and wage-seing schedules. Our benchmark calibraion implies a fairly sluggish price and wage adjusmen, wih he slopes of he price- and wage-seing schedules a he lower end of empirical esimaes: for example, price conracs have an effecive duraion of en quarers. 5 If boh prices and wages were more responsive, he muliplier could be very high even for a liquidiy rap lasing under wo years; under such condiions, fiscal policy would be a very poen ool for reversing he sizeable deflaionary pressures ha would occur in he wake of recession. 6 However, he muliplier also drops abruply wih he size of he fiscal simulus, as greaer simulus shorens he duraion of he liquidiy rap. Thus, wih four quarer price and wage conracs, he muliplier exceeds in our workhorse model for a very small increase in spending, bu drops o uniy when governmen spending is boosed more han percen of GDP. Given he resilence of shor-run expeced inflaion during he pas recession, we are somewha skepical of calibraions ha imply such exreme variaions in he muliplier across spending levels. Bu a he leas, our analysis underscores ha he muliplier ends o decline sharply in he level of governmen spending under exacly he same condiions ha are favorable o a high muliplier, i.e., when expeced inflaion is highly responsive. Taken ogeher, our resuls sugges a somewha nuanced view of he role of fiscal policy in a liquidiy rap. For an economy facing a deep recession ha appears likely o keep moneary policy consrained by he zero bound for well over wo years, here is a srong argumen for increasing governmen spending on a emporary basis. Consisen wih he views originally espoused by Keynes, his emporary boos can have much larger effecs han under usual condiions, and comes a a low cos o he Treasury. Bu as he muliplier can drop quickly wih he level of fiscal spending, larger spending programs may suffer from sharply diminishing reurns, and may increase governmen deb a he margin. Agains he backdrop of a shorer-lived liquidiy rap of less han wo years, he muliplier is probably only slighly above uniy even in he ideal siuaion in 5 We sress ha he implied slope of he Phillips Curve is consisen wih empirical esimaes (albei a he lower end). In his paper, we find i convenien o simply map a given slope of he Phillips curve ino an effecive conrac duraion under he assumpion ha marginal coss are idenical across firms. However, he same slope of he Phillips curve can be consisen wih a much shorer price conrac duraion if capial is firm-specific, as shown by e.g. Galí, Gerler, and López-Salido (2) and Alig, Chrisiano, Eichenbaum and Lindé (2). 6 As shown below, he muliplier isn nearly as large if only price conracs are shorer. 5

8 which fiscal simulus can be implemened immediaely. Such an environmen clearly presens more risks ha a fiscal spending program will fall shor of he objecive of producing a large oupu gain a minimal public cos, especially if he simulus plan is large in scale and has subsanial implemenaion lags. 2. A sylized New Keynesian model As in Eggersson and Woodford (23), we use a sandard log-linearized version of he New Keynesian model ha imposes a zero bound consrain on ineres raes. Our framework allows exi from he liquidiy rap o be deermined endogenously, raher han fixed arbirarily, an innovaion ha is crucial in showing how he muliplier varies wih he level of fiscal spending. 2.. The Model The key equaions of he model are: x = x + ˆσ(i π + r po ), () π = βπ + + κ p x, (2) i = max ( i, γ π π + γ x x ), (3) r po = ˆσ ( ) [gy (g g φ + ) + ( g y )ν c (ν ν ] + ) (4) mcˆσ where ˆσ, κ p, and φ mc are composie parameers defined as: ˆσ = σ( g y )( ν c ) (5) κ p = ( ξ p)( βξ p ) ξ p φ mc (6) φ mc = χ α + ˆσ + α α and where x is he oupu gap, π is he inflaion rae, i is he shor-erm nominal ineres rae, and r po is he poenial (or naural ) real ineres rae. All variables are measured as percen or percenage poin deviaions from heir seady sae level. 7 7 We use he noaion y +j o denoe he condiional expecaion of a variable y a period + j based on informaion available a, i.e., y +j = E y +j. The superscrip po denoes he level of a variable ha would prevail under compleely flexible prices, e.g., y po is poenial oupu. 6 (7)

9 Equaion () expresses he New Keynesian IS curve in erms of he oupu and real ineres rae gaps. Thus, he oupu gap x depends inversely on he deviaion of he real ineres rae (i π + ) from is poenial rae r po, as well as direcly on he expeced oupu gap in he following period. The parameer ˆσ deermines he sensiiviy of he oupu gap o he real ineres rae; as indicaed by (5), i depends on he household s ineremporal elasiciy of subsiuion in consumpion σ, he seady sae governmen spending share of oupu g y, and a (small) adjusmen facor ν c which scales he consumpion ase shock ν. The price-seing equaion (2) specifies curren inflaion o depend on expeced inflaion and he oupu gap, where he sensiiviy o he laer is deermined by he composie parameer κ p. Given he Calvo-Yun conrac srucure, equaion (6) implies ha κ p varies direcly wih he sensiiviy of marginal cos o he oupu gap φ mc, and inversely wih he mean conrac duraion ( ξ p ). The marginal cos sensiiviy equals he sum of he absolue value of he slopes of he labor supply and labor demand schedules ha would prevail under flexible prices: accordingly, as seen in equaion (7), φ mc varies inversely wih he Frisch elasiciy of labor supply χ, he composie parameer ˆσ deermining he ineressensiiviy of aggregae demand, and he labor share in producion ( α). Equaion (4) indicaes ha he poenial real ineres rae is driven by wo exogenous shocks, including a consumpion ase shock ν and governmen spending shock g. Each shock, if posiive, raises he marginal uiliy of consumpion associaed wih any given oupu level, which pus upward pressure on he real ineres rae if he shock is fron-loaded. 8 The consumpion ase shock and governmen spending shock are assumed o follow an AR() process wih he same persisence parameer ( ρ ν ), e.g., he ase shock follows: ν = ( ρ ν )ν + ε ν,, (9) Given he same sochasic srucure for he shocks, i is eviden from equaion (4) ha hese shocks affec he poenial real ineres rae in an idenical way. The log-linearized equaion for he sock of governmen deb assuming for simpliciy ha he seady sae governmen deb sock is zero is given by: b = ( + r)b + g y (g l ζ ) τ, () 8 The effec of each shock on he marginal uiliy of consumpion λ c can be expressed: νc( gy) λ c, = c + ν = ˆσ ˆσ [ ] (gyg y ) + ν c( g y)ν ˆσ g y (8) where c is consumpion, y oupu, and g governmen spending. 7

10 where l is labor hours, ζ is he real wage, b is end-of-period governmen deb, and τ is a lumpsum ax (wih boh b and τ expressed as a share of seady sae GDP). The governmen derives ax revenue from a fixed ax on labor income τ L, and from he ime-varying lump-sum ax τ. The ax rae τ L is se so ha governmen spending is financed exclusively by he disorionary labor ax in he seady sae. Lump-sum axes adjus according o he reacion funcion: τ = ϕ τ τ + ϕ b b, () Given ha agens are Ricardian and ha only lump-sum axes adjus dynamically, he fiscal rule only affecs he evoluion of he sock of deb and lump-sum axes, wih no effec on oher macro variables. In Secion 3, we consider he implicaions of rules in which disorionary axes adjus dynamically. Our benchmark calibraion is fairly sandard. The model is calibraed a a quarerly frequency. We se he discoun facor β =.995, and he seady sae ne inflaion π =.5; his implies a seady sae ineres rae of i =. (one percen a a quarerly rae, or four percen a an annualized rae). We se he ineremporal subsiuion elasiciy σ = (i.e., logarihmic period uiliy), he capial share parameer α =.3, he Frisch elasiciy of labor supply χ =.4, he governmen share of seady sae oupu g y =.2, and he scale parameer on he consumpion ase shock ν c =.. We examine a range of values of he price conrac duraion parameer ξ p o highligh he sensiviy of he fiscal muliplier o he Phillips Curve slope κ p. I is convenien o assume ha moneary policy would compleely sabilize oupu and inflaion in he absence of a zero bound consrain, which can be regarded as a limiing case in which he coeffi cien on inflaion γ π in he ineres rae reacion funcion becomes arbirarily large. The parameers of he ax rule are se so ha ϕ b =., and ϕ τ =.98. Imporanly, he very small value of he coeffi cien ϕ b implies ha he conribuion of lump-sum axes o he response of governmen deb is exremely small in he firs couple of years following a shock (so ha almos all variaion in ax revenue comes from flucuaions in labor ax revenues). Finally, he preference and governmen spending shocks are assumed o follow he AR() process in equaion (9) wih persisence of.9, so ha ρ ν = Impulse Responses o a Fron-Loaded Rise in Governmen Spending The effecs of fiscal policy in a liquidiy rap depend on agens percepions abou how long he liquidiy rap would las in he absence of fiscal simulus, as well as he severiy of he associaed 8

11 recession. 9 Thus, we mus firs specify a shock(s) ha pushes he economy ino a liquidiy rap, and hen consider how fiscal policy operaes agains he backdrop of hese iniial condiions. Clearly, a number of differen shocks could cause he noional ineres rae he level of he ineres rae if moneary policy were unconsrained o fall persisenly below zero, and consequenly generae a liquidiy rap. For simpliciy, we follow he recen lieraure including Eggerson and Woodford (23), and Eggerson (29), and Adam and Billi (28) by assuming ha he liquidiy rap is caused by an adverse ase shock ν ha sharply depresses he poenial real ineres rae r po. The response of r po o he adverse ase shock is depiced by he solid line in Figure a. Given he assumpion ha moneary policy would fully sabilize inflaion and he oupu gap if feasible, he nominal ineres rae i simply racks r po provided ha he implied nominal rae is non-negaive (i.e., i = r po, recalling ha boh variables are measured as percenage poin deviaions from baseline). The concurrence of he nominal and poenial real ineres rae is apparen in Figure a beginning in period T n, which is he firs period in which r po exceeds i = percen (he figure shows he annualized ineres rae, so -4 percen). However, because r po < i prior o T n, equaions ()-(3) imply ha he nominal ineres rae mus equal is lower bound of i. so ha he liquidiy rap lass for T n = 8 quarers. The ase shock is scaled The solid lines in Figure 2 shows he effecs of he ase shock on he oupu gap, inflaion, he real ineres rae, and governmen deb (relaive o baseline GDP). To highligh he role of expeced inflaion in amplifying he effecs of he shock, i is useful o begin by illusraing he effecs in a limiing case in which inflaion (and hence expeced inflaion) is consan. This is achieved by assuming ha he average duraion of price conracs is arbirarily long, so ha κ p (in equaion 2) is close o zero. The lef column of Figure 2 shows his limiing case. The real ineres rae declines in locksep wih he nominal ineres rae (i.e., by i = 4 percen). However, because he poenial ineres rae declines by more, and remains persisenly below i, oupu falls persisenly below poenial. The governmen deb/gdp raio increases subsanially, mainly because revenue from he labor income ax declines in response o lower labor demand and falling real wages. We nex consider he effec of a one percenage poin of (seady sae) GDP rise in governmen 9 In he more empirically realisic models considered in Secions 3 and 4, he underlying shock process is chosen o roughly mach he decline in U.S. GDP during he recen recession. However, in his secion we simply scale he ase shock o generae a liquidiy rap of he same duraion (8 quarers) as in he models considered subsequenly. The parameer ξ p is se equal o.9995, implying a mean duraion of price conracs of 2 quarers, and a value of κ p of.28. 9

12 spending agains his backdrop. offses some of he decline in r po shifs upward in a proporional manner. As seen in Figure a, he higher governmen spending simply induced by he negaive ase shock, so ha he pah of r po Because he governmen spending hike is oo small o affec he duraion of he liquidiy rap, moneary policy coninues o hold he nominal ineres rae unchanged for T n = 8 quarers. As seen in he lef column of Figure 2, his invariance of he nominal rae implies ha he higher governmen spending has no effec on he pah of he real ineres rae (dashed lines) over his period. Accordingly, he oupu gap is less negaive in response o he combined shocks, since he real ineres rae remains unchanged even hough he poenial real ineres rae pah is higher. The (parial) effec of he governmen spending rise he difference beween he response o he combined shocks and ase shock alone is depiced by he dash-doed line(s) in he figure. Recalling ha governmen spending has no effec on he oupu gap ouside of he liquidiy rap, and only affecs poenial oupu, he spending muliplier is clearly larger in a liquidiy rap. The higher governmen spending has virually no effec on he pah of governmen deb: wih he ousized muliplier, ax revenues rise enough o finance he fiscal expansion. The governmen spending muliplier g y dy dg, i.e., he percenage increase in oupu in response o a one percen of baseline GDP rise in governmen spending, is abou.7 in his case. The muliplier is he sum of he oupu gap response of nearly.5 shown in he figure, plus he effec on poenial oupu (no shown). The muliplier is amplified subsanially more when expeced inflaion responds o shocks, as illusraed in he righ column of Figure 2 for a calibraion implying a mean duraion of price conracs of 5 quarers ( ξ p =.8). In his case, he negaive oupu gap due o he ase shock causes inflaion o fall persisenly (solid lines). Wih expeced inflaion falling more han he nominal ineres rae, he real ineres rae rises, which reinforces he conracion in oupu. Thus, he same-sized fall in r po has much larger adverse effecs on oupu when expeced inflaion reacs. Conversely, in addiion o boosing r po, higher governmen spending causes expeced inflaion o rise, and hence exers a more simulaive effec on oupu han when expeced inflaion remains consan. The peak oupu gap response of.8 seen in he figure implies a spending muliplier of 2., and ranslaes ino a subsanial and persisen reducion in governmen deb The Muliplier and he Size of Fiscal Spending In he log-linearized model ha ignores he zero bound consrain, he governmen spending muliplier is invarian o he size of he change in spending. By conras as we nex proceed o show

13 he muliplier in a liquidiy rap declines in he level of governmen spending. Inuiively, his behavior reflecs ha he muliplier varies posiively wih he duraion of he liquidiy rap, and ha he duraion shorens as he level of spending rises. r po Because governmen spending and ase shocks have he same linear effecs on r po, and only maers for he oupu gap and inflaion response, i is convenien o simply analyze how r po affecs hose variables in a liquidiy rap. j= Solving he IS curve forward yields: T n Tn x = ˆσ ( i r po +j ) + ˆσ π +j + x T (2) The oupu gap x in he curren period depends on four erms. Firs, i depends on he cumulaive gap beween he nominal ineres rae i and he poenial real ineres rae over he inerval in which he economy remains in a liquidiy rap. j= This cumulaive ineres rae gap T n j= ( i rpo +j ) can be inerpreed as indicaing how shocks o he poenial real ineres rae would affec he oupu gap if expeced inflaion remained consan. Second, he oupu gap depends on cumulaive expeced inflaion over he liquidiy rap (or equivalenly, he log change in he price level log(p T n ) log(p )); as indicaed above, he effecs of shocks o he poenial real rae on he oupu gap can be amplified hrough changes in expeced inflaion. Third, he curren oupu gap also depends on he expeced oupu gap x Tn when he economy exis he liquidiy rap, hough boh he erminal oupu gap and inflaion erms drop under he assumpion ha moneary policy compleely sabilizes he economy (x Tn = π T n = ). Finally, he exi dae T n is deermined endogenously as he firs period in which he expeced poenial real ineres rae exceeds i. Thus: T n = min(r po j +j > i) (3) In general, his exi dae depends boh on he size and persisence of he shocks o r po. The relaion beween he exi dae and r po under our baseline calibraion in which boh he ase and governmen spending shocks follow an AR() wih persisence equal o ( ρ ν ) =.9 is shown in Figure b. Because he exi dae is only affeced as r po a sep funcion in he level of r po (rising as r po larger adverse ase shock ha caused r po exceeds cerain hreshold values, i is assumes more negaive values). Thus, a slighly o drop more han shown in Figure a would leave he duraion of he liquidiy rap unchanged a 8 quarers; bu a large enough adverse shock would exend he duraion of he rap, and a suffi cienly smaller shock would shoren i. In he limiing case in which expeced inflaion remains consan, we can derive a simple closed form soluion for he muliplier. Because r po follows an AR() wih persisence parameer -ρ v,

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