Is There a Fiscal Free Lunch in a Liquidity Trap?

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1003r August 2012 Is There a Fiscal Free Lunch in a Liquidity Trap? Christopher J. Erceg Jesper Lindé NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

Is There a Fiscal Free Lunch in a Liquidity Trap? (cid:3) Christopher J. Erceg Jesper LindØ (cid:3)(cid:3) Federal Reserve Board Federal Reserve Board and CEPR First version: April 2009 This version: August 6, 2012 Abstract ThispaperusesaDSGEmodeltoexaminethee⁄ectsofanexpansioningovernmentspending in a liquidity trap. If the liquidity trap is very prolonged, the spending multiplier can be much larger than in normal circumstances, and the budgetary costs minimal. But given this (cid:147)(cid:133)scal freelunch,(cid:148)itisunclearwhypolicymakerswouldwanttolimitthesizeof(cid:133)scalexpansion. Our paperaddressesthisquestioninamodelenvironmentinwhichthedurationoftheliquiditytrap isdeterminedendogenously,anddependsonthesizeofthe(cid:133)scalstimulus. Weshowthatevenif the multiplier is high for small increases in government spending, it may decrease substantially at higher spending levels; thus, it is crucial to distinguish between the marginal and average responses of output and government debt. JEL Classi(cid:133)cation: E52, E58 Keywords: Monetary Policy, Fiscal Policy, Liquidity Trap, Zero Bound Constraint, DSGE Model. (cid:3)Wethank Martin Bodenstein,FabioCanova (theeditor),V.V.Chari,Luca Guerrieri,EricLeeper,RafWouters, and two anonymous referees for very constructive suggestions. We also thank participants at a macroeconomic modelingconferenceattheBankofItalyinJune2009,attheFebruary2010NBEREF&GMeetinginSanFrancisco, at the CEPR 18th ESSIM conference in Tarragona (Spain), at the 2010 SED Meeting in Montreal, and seminar participants at the European Central Bank, Georgetown University, University of Maryland, the Federal Reserve Banks of Cleveland and Kansas City, and the Sveriges Riksbank. Mark Clements, James Hebden, and Ray Zhong provided excellent research assistance. The views expressed in this paper are solely the responsibility ofthe authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. (cid:3)(cid:3) Corresponding Author: Telephone: 202-452-2575. Fax: 202-263-4850 E-mail addresses: christopher.erceg@frb.gov and jesper.l.linde@frb.gov

1. Introduction Keynes argued in favor of aggressive (cid:133)scal expansion during the Great Depression on the grounds that the (cid:133)scal multiplier was likely to be much larger in a liquidity trap than in normal times, and the (cid:133)nancing burden correspondingly smaller.1 Recent analysis using New Keynesian DSGE models (cid:150)including by Eggertsson (2008), Davig and Leeper (2011), and Christiano, Eichenbaum, andRebelo(2011)andWoodford(2011)(cid:150)hascorroboratedKeynes(cid:146)viewbyshowingthatincreases in government spending can indeed have outsized e⁄ects on output when monetary policy allows real interest rates to fall.2 However, these results raise the important question of why policymakers would want to limit the magnitude of (cid:133)scal expansion. Our paper addresses this question by showing that the spending multiplier in a liquidity trap decreases with the level of government spending. The novel feature of our approach is to allow the economy(cid:146)s exit from a liquidity trap (cid:150)and return to conventional monetary policy (cid:150)to be determined endogenously, with the consequence that the multiplier depends on the size of the (cid:133)scal response. Quite intuitively, a large (cid:133)scal response pushes the economy out of a liquidity trap more quickly. Because the multiplier is smaller upon exiting the liquidity trap (cid:150)re(cid:135)ecting that monetary policy reacts by raising real interest rates (cid:150)the marginal impact of a given-sized increase ingovernmentspendingonoutputdecreaseswiththemagnitudeofthespendinghike. Accordingly, it is crucial to understand the marginal e⁄ect of higher government spending on output to make informed choices about the appropriate scale of (cid:133)scal intervention in a liquidity trap. Toward this end, we use a simple New Keynesian model in which policy rates are constrained by the zero bound to derive a government spending multiplier schedule. This schedule shows how the spending multiplier varies with the level of government spending conditional on the state of the economy(whichdetermineshowlongtheliquiditytrapwouldlastintheabsenceof(cid:133)scalstimulus). Akeyresultisthatthespendingmultiplier(cid:150)measuredasthecontemporaneousimpactonoutputof a small increment in government spending (cid:150)is a step function in the level of government spending. If the increment to spending is su¢ ciently small, it does not a⁄ect the economy(cid:146)s exit date from the liquidity trap, and the multiplier is constant at a value that is higher than in a normal situation in which monetary policy would raise real interest rates. However, as spending rises to higher levels, the economy emerges from the liquidity trap more quickly and the multiplier drops (eventually 1A large empirical literature has examined the e⁄ects of government spending shocks, mainly focusing on the post-WWIIperiodinwhichmonetarypolicyhadlatitudetoadjustinterestrates. Thebulkofthisresearchsuggests a government spending multiplier in the range of 0.5 to slightly above unity. See e.g. Hall (2009) and Ramey (2011) and the references therein. 2Some model-based analysis has questioned whether the multiplier is larger in a liquidity trap than in normal times. Cogan et al. (2009) and Drautzburg and Uhlig (2011) conclude that the multiplier is only slightly ampli(cid:133)ed even in liquidity traps lasting 2-3 years. Mertens and Ravn (2010) develop a stylized model which rationalizes a low and possibly negative spending multiplier in a liquidity trap in an environment with multiple equilibria driven by expectationalshocks. Intheirmodel,anincreasein(cid:133)scalspendingcon(cid:133)rmsandreinforcesthepessimisticexpectations of the private sector. 1

leveling o⁄at a value equal to that under normal conditions). Structural factors that a⁄ect in(cid:135)ation expectations, including the slope of the Phillips Curve, play a key role in determining the contour of the government spending multiplier schedule. If prices are fairly responsive to marginal cost (cid:150)as implied by relatively short-lived price contracts (cid:150)the multiplier is extremely high for small increments to government spending, but drops quickly at higher spending levels. By contrast, the multiplier function is much (cid:135)atter if the slope of the Phillips Curve is lower. A second major focus of our analysis is on how the budgetary impact of higher government spending in a liquidity trap di⁄ers from normal times, an important policy question that has receivedrelativelylittleattentionintherecentliterature. Weshow(cid:150)consistentwiththeconjecture of Keynes (1933, 1936) (cid:150)that because the multiplier is higher in a liquidity trap, a given-sized government spending hike can stimulate a much larger response of tax revenues than in normal times, making the (cid:133)scal expansion less costly. Moreover, the multiplier may even be high enough in a deep liquidity trap that the government spending hike becomes self-(cid:133)nancing, re(cid:135)ecting that tax revenue increases enough to pay for the higher spending even at unchanged tax rates. However, while the prospect of such a (cid:147)(cid:133)scal free lunch(cid:148)is clearly attractive, our analysis also underscores the importance of distinguishing average from marginal e⁄ects. Because the multiplier may drop sharply with the level of spending, the marginal impact on government debt may increase rapidly.3 The (cid:135)ipside, which is very relevant in an environment of (cid:133)scal austerity, is that spending cuts (cid:150) at least if perceived as temporary (cid:150)may prolong a recession and boost government debt, with the marginal e⁄ects rising in the size of the cut. Whilethebudgetaryimplicationsofariseingovernmentspendingclearlyaremorefavorablethe largerthespendingmultiplier, ouranalysisshowsthatthecompositionofthetaxbasealsoplaysan important role in determining how a government spending hike a⁄ects the government budget. In particular, the government de(cid:133)cit (and debt stock) rises by less in response to a spending increase if the tax base is more cyclically-sensitive. For example, given that labor income is considerably more procylical than consumption expenditure in the New Keynesian model, the e⁄ects of higher government spending on tax revenue are more favorable to the extent that labor taxes (cid:150)rather than sales taxes (cid:150)comprise a larger fraction of the steady state tax base. In our benchmark model, dynamic tax adjustment occurs exclusively through lump-sum taxes (with distortionary tax rates (cid:133)xed), and government spending is purely exogenous. But we also 3Thereisanimportantdi⁄erencebetweenourframework(cid:150)inwhichhighergovernmentspendingdepressesthereal interestratebecauseofthezerolowerbound(cid:150)andthatofDavigandLeeper(2011). DavigandLeeper(2011)show that the government spending multiplier may be well above unity even after many years under a passive monetary policy regime. Although their model does not impose the zero lower bound, real interest rates remain low because the passive monetary policy stance fails to satisfy the Taylor Principle. As a consequence, tax rates would never have to adjust much in such a regime. By contrast, the outsized multiplier and potential (cid:133)scal free lunch in our model is state contingent, depending on the depth and duration of the liquidity trap; once the liquidity trap ends, the multiplier returns to normal, and there is no (cid:133)scal free lunch. 2

consider the implications of an alternative (cid:133)scal rule whereby the labor income tax rate adjusts to stabilize government debt. If tax rates adjust very gradually to the level of government debt, the multiplier (schedule) associated with a liquidity trap of a given duration is only reduced slightly relative to the lump-sum case, and the e⁄ects of higher spending on the government budget are nearly the same.4 However, we show that a more aggressive tax rule (cid:150)in which the labor tax rate is quite responsive to government debt even in the near-term (cid:150)tends to reduce the multiplier substantially relative to the lump-sum case, especially in a prolonged liquidity trap. At (cid:133)rst glance, this result seemingly contrasts with that of Eggertson (2010) and Christiano, Eichenbaum, and Rebelo (2011), who (cid:133)nd that a purely exogenous rise in a distortionary tax ampli(cid:133)es the multiplier. Theapparentdisparityre(cid:135)ectsthatthein(cid:135)ationresponseisdampedunderanaggressiveendogenous rule, since lower tax rates than in normal times (associated with the higher (cid:133)scal multiplier in a liquidity trap) reduce upward pressure on marginal cost; this e⁄ect is not present under exogenous tax adjustment. We also consider the implications of an endogenous component of government spending that responds to the output gap to proxy for automatic stabilizers. In this environment, the multiplier associated with a given-sized increase in discretionary spending (i.e., the exogenous component) is reduced relative to our benchmark in which all spending is exogenous. Ourpaperconcludesbyexaminingthee⁄ectsofgovernmentspendingshocksonoutputandthe government budget in a more empirically-realistic model. In particular, we utilize a model that is similar to the estimated models of Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2007), but also incorporates (cid:147)Keynesian(cid:148)hand-to-mouth agents and (cid:133)nancial frictions. As argued by Gal(cid:237), L(cid:243)pez-Salido, and VallØs (2007), the inclusion of Keynesian households can help account for the positive response of private consumption to a government spending shock documented in structural VAR studies by e.g., Blanchard and Perotti (2002) and Perotti (2007); more generally, (cid:147)Keynesian(cid:148)hand-to-mouth agents and (cid:133)nancial frictions can increase the multiplier by amplifying the response of the potential real interest rate. We (cid:133)nd that the government spending multiplier exceeds 4 against the backdrop of a deep recession that would generate a 10 quarter liquidity trap in the absence of a (cid:133)scal response, and that small increments to spending generate enough tax revenue to be self-(cid:133)nancing (in fact, the impact on government debt remains persistently negative even at a horizon of (cid:133)ve years). Even so, the government spending multiplier declines fairly abruptly as spending increases. With a spendingincreaseofover3percentofGDP,themarginalmultiplierdeclinesto1.3,andthemarginal impact on government debt is solidly positive. Moreover, while the multiplier is even higher for small increments to spending under calibrations that allow for both prices and wages to adjust more rapidly, the multiplier declines with the level of spending more precipitously. Thus, the multiplier tends to drop sharply in the level of government spending under exactly the conditions (i.e., relatively high in(cid:135)ation responsiveness) that are favorable to a large multiplier. 4Consistent with Uhlig (2010), output is lower in the longer-term under distortionary tax rather than lump-sum tax (cid:133)nancing. 3

Overall, our results corroborate previous analysis suggesting a strong argument in favor of increasing government spending on a temporary basis for an economy facing a deep recession and long-lived liquidity trap. Consistent with the view originally espoused by Keynes, a temporary spending boost can have much larger e⁄ects on output than under usual conditions, and comes at a low cost to the Treasury. But our analysis highlights the importance of taking account of how the multiplier varies with the level of spending at the margin. Insofar as the multiplier can drop quickly with the level of (cid:133)scal spending, larger spending programs may su⁄er from sharply diminishing returns, and hence increase government debt at the margin. Conversely, large-scale (cid:133)scal consolidations under some conditions may run the risk of deepening a recession and boosting government debt. The remainder of this paper is organized as follows. Section 2 derives the multiplier schedule in a stylized New Keynesian model in which all dynamic tax adjustment occurs through lumpsum taxes, and government spending is exogenous; we then consider modi(cid:133)cations that allow for distortionary taxes, and for some component of government spending to be endogenous. Section 3 examines the more empirically-realistic model, and Section 4 concludes. 2. A stylized New Keynesian model As in Eggertsson and Woodford (2003), we use a standard log-linearized version of the New Keynesian model that imposes a zero bound constraint on interest rates. Our framework allows exit from the liquidity trap to be determined endogenously, rather than (cid:133)xed arbitrarily, an innovation that is crucial in showing how the multiplier varies with the level of (cid:133)scal spending. 2.1. The Model The key equations of the model are: pot x = x (cid:27)^(i (cid:25) r ); (1) t t+1t t t+1t t j (cid:0) (cid:0) j (cid:0) (cid:25) = (cid:12)(cid:25) +(cid:20) x ; (2) t t+1t p t j i = max( i;(cid:13) (cid:25) +(cid:13) x ); (3) t (cid:25) t x t (cid:0) 1 pot y = [g g +(1 g )(cid:23) (cid:23) ]; (4) t (cid:30) (cid:27)^ y t (cid:0) y c t mc 1 1 pot r = 1 g (g g )+(1 g )(cid:23) ((cid:23) (cid:23) ) ; (5) t (cid:27)^ (cid:18) (cid:0) (cid:30) mc (cid:27)^ (cid:19) y t (cid:0) t+1 j t (cid:0) y c t (cid:0) t+1 j t (cid:2) (cid:3) 4

where (cid:27)^, (cid:20) , and (cid:30) are composite parameters de(cid:133)ned as: p mc (cid:27)^ = (cid:27)(1 g )(1 (cid:23) ); (6) y c (cid:0) (cid:0) (1 (cid:24) )(1 (cid:12)(cid:24) ) p p (cid:20) = (cid:0) (cid:0) (cid:30) ; (7) p (cid:24) mc p (cid:31) 1 (cid:11) (cid:30) = + + : (8) mc 1 (cid:11) (cid:27)^ 1 (cid:11) (cid:0) (cid:0) All variables are measured as percent or percentage point deviations from their steady state level.5 Equation (1) expresses the (cid:147)New Keynesian(cid:148)IS curve in terms of the output and real interest rate gaps. Thus, the output gap x depends inversely on the deviation of the real interest rate t pot (i (cid:25) )fromitspotentialrater ,aswellasontheexpectedoutputgapinthefollowingperiod. t t+1t t (cid:0) j The parameter (cid:27)^ determines the sensitivity of the output gap to the real interest rate; as indicated by (6), it depends on the household(cid:146)s intertemporal elasticity of substitution in consumption (cid:27), the steady state government spending share of output g , and a (small) adjustment factor (cid:23) which y c scales the consumption taste shock (cid:23) . The price-setting equation (2) speci(cid:133)es current in(cid:135)ation (cid:25) t t todependonexpectedin(cid:135)ationandtheoutputgap,wherethesensitivitytothelatterisdetermined by the composite parameter (cid:20) . Given the Calvo-Yun contract structure, equation (7) implies that p (cid:20) varies directly with the sensitivity of marginal cost to the output gap (cid:30) ; and inversely with p mc the mean contract duration ( 1 ). The marginal cost sensitivity equals the sum of the absolute 1 (cid:24) (cid:0) p valueoftheslopesofthelaborsupplyandlabordemandschedulesthatwouldprevailunder(cid:135)exible prices: accordingly, as seen in (8), (cid:30) varies inversely with the Frisch elasticity of labor supply mc 1, the interest-sensitivity of aggregate demand (cid:27)^, and the labor share in production (1 (cid:11)): The (cid:31) (cid:0) policy rate i follows a Taylor rule subject to the zero lower bound (equation 3). t pot Equation (4) indicates that potential output y varies directly with two exogenous shocks, t includingaconsumptiontasteshock(cid:23) andgovernmentspendingshockg :Bothshocksareassumed t t tofollowanAR(1)processwiththesamepersistenceparameter(1 (cid:26) );e.g.,thetasteshockfollows: (cid:23) (cid:0) (cid:23) = (1 (cid:26) )(cid:23) +" ; (9) t (cid:23) t 1 (cid:23);t (cid:0) (cid:0) where 0 < (cid:26) < 1: Given the front-loaded nature of the shocks, equation (5) indicates that positive (cid:23) realizations of these shocks boosts the potential real interest rate (noting (cid:30) (cid:27)^ > 1); this re(cid:135)ects mc that each shock (cid:150)if positive (cid:150)raises the marginal utility of consumption associated with any given output level. The government does not need to balance its budget each period, and issues nominal debt as needed to (cid:133)nance budget de(cid:133)cits. Under the simplifying assumption that government debt is zero in steady state, the log-linearized government budget constraint is given by: b = (1+r)b +g g (cid:28) s (y +(cid:30) x ) (cid:28) ; (10) G;t G;t 1 y t N N t mc t t (cid:0) (cid:0) (cid:0) 5Weusethenotationy todenotetheconditionalexpectationofavariableyatperiodt+jbasedoninformation t+jt j available at t, i.e., y = E y : The superscript (cid:145)pot(cid:146)denotes the level of a variable that would prevail under t+jt t t+j j completely (cid:135)exible prices, e.g., ypot is potential output. See Appendix A (available online) for the model derivation. t 5

where b is end-of-period real government debt, (y + (cid:30) x ) equals real labor income, (cid:28) is a G;t t mc t t lump-sum tax, and s is the steady state labor share.6 The government derives tax revenue from N a (cid:133)xed tax on labor income (cid:28) , and from the time-varying lump-sum tax (cid:28) . The tax rate (cid:28) is N t N set so that government spending is (cid:133)nanced exclusively by the distortionary labor tax in the steady state (so (cid:28) s = g ): Lump-sum taxes adjust according to the reaction function: N N y (cid:28) = ’ b : (11) t b G;t 1 (cid:0) Given that agents are Ricardian and that only lump-sum taxes adjust, the (cid:133)scal rule only a⁄ects the evolution of the stock of debt and lump-sum taxes, with no e⁄ect on other macro variables. Our benchmark calibration is fairly standard at a quarterly frequency. We set the discount factor(cid:12) = 0:995;andthesteadystatenetin(cid:135)ationrate(cid:25) = :005; thisimpliesasteadystateinterest rate of i = :01 (i.e., four percent at an annualized rate). We set the intertemporal substitution elasticity (cid:27) = 1 (log utility); the capital share parameter (cid:11) = 0:3; the Frisch elasticity of labor supply 1 = 0:4; the government share of steady state output g = 0:2; and the scale parameter (cid:31) y on the consumption taste shock (cid:23) = 0:01: We examine a range of values of the price contract c duration parameter (cid:24) to highlight the sensitivity of the (cid:133)scal multiplier to the Phillips Curve slope p (cid:20) : We assume that monetary policy would completely stabilize in(cid:135)ation and the output gap in p the absence of a zero bound constraint, which can be regarded as a limiting case in which the coe¢ cients on in(cid:135)ation, (cid:13) , and the output gap, (cid:13) , in the interest rate reaction function become (cid:25) x arbitrarily large. The tax rule parameter ’ is set equal to :01, which implies that the contribution b of lump-sum taxes to the response of government debt is extremely small in the (cid:133)rst couple of years following a shock (so that almost all variation in tax revenue comes from (cid:135)uctuations in labor tax revenues). Finally, the preference and government spending shocks are assumed to follow an AR(1) process with persistence of 0:9, so that (cid:26) = 0:1 in equation (9): (cid:23) 2.2. Impulse Responses to a Rise in Government Spending The e⁄ects of (cid:133)scal policy in a liquidity trap depend crucially on agents(cid:146)perceptions about the likely duration of the liquidity trap. The liquidity trap duration in turn generally depends on a number of factors, including the parameters of the monetary policy rule, the type of shocks causing the liquidity trap, and the (cid:133)scal response. For simplicity, we follow the recent literature (cid:150)including Eggertson and Woodford (2003), and Eggertson (2009), and Adam and (2008) (cid:150)by assuming that the liquidity trap is caused by an adverse taste shock (cid:23) that sharply depresses t 6In(10),realgovernmentdebtb andrealtransfers(cid:28) arede(cid:133)nedasashareofsteadystateGDPandexpressed G;t t as percentage point deviations from their steady state values. That is, b = BG;t b , where B is nominal G;t PtY (cid:0) G G;t government debt, P is the price level, and Y is real steady state output; and sim(cid:16)ilarly(cid:17), (cid:28) = Tt (cid:28) :Because of t t PtY (cid:0) our simplifying assumption that the steady state government debt b G =0; a time-varying rea(cid:16)l inte(cid:17)rest rate does not enter in eq. (10). In the full model analyzed in Section 3, we allow for positive steady state government debt, and hence a role for time-varying debt service costs. 6

pot the potential real interest rate r : Because this type of shock does not imply a tradeo⁄between t stabilizing in(cid:135)ation and the output gap, our assumed highly aggressive monetary policy reaction function would keep both in(cid:135)ation at target and output at potential if unconstrained by the zero lower bound. Thus, if the zero bound is not binding, equation (1) implies that the nominal interest pot pot rate i simply tracks r (i.e., i = r , recalling that both variables are measured as percentage t t t t point deviations from baseline).7 A key implication is that the duration of the liquidity trap pot pot depends solely on how long r remains below i: Given that r is a simple function of both the t t (cid:0) taste and government spending shocks by equation (5), this setting makes it very tractable to show how di⁄erent government spending choices a⁄ect the duration of the liquidity trap and hence the spending multiplier. Figure 1.a illustrates how the liquidity trap duration and the path of the policy rate is deterpot mined. The solid line shows the impact of an adverse taste shock (cid:23) on r : The liquidity trap t t pot lasts as long as r < i; over which period i = i = 1 percent (the (cid:133)gure shows the annualized t t (cid:0) (cid:0) (cid:0) pot interestrate, so 4percent). BeginninginperiodT , whichisthe(cid:133)rstperiodinwhichr exceeds n t (cid:0) pot i, the policy rate i simply rises with r (the taste shock is scaled so that the liquidity trap lasts t t (cid:0) for T = 8 quarters). A government spending shock equal to one percentage point of steady state n pot GDP(cid:150)shownbythedashedline(cid:150)simplyo⁄setssomeofthedeclineinr inducedbythenegative t pot taste shock, shifting up the path of r in a proportional manner (recalling the shocks are equally t persistent). Becausethisgovernmentspendinghiketurnsouttobetoosmalltoa⁄ecttheduration of the liquidity trap, monetary policy continues to hold the nominal interest rate unchanged for T = 8 quarters. n The solid lines in Figure 2 show the dynamic e⁄ects of the taste shock (cid:23) on the real interest t rate, output gap, in(cid:135)ation, and government debt as a share of GDP. To highlight the role of expected in(cid:135)ation in amplifying the e⁄ects of the shock, it is useful to begin with a limiting case in which in(cid:135)ation is (essentially) constant, which is achieved by setting (cid:20) in equation (2) arbitrarily p close to zero. The left column of Figure 2 shows this limiting case. Because the taste shock pot causes the potential interest rate r to decline persistently below the policy rate i; output falls t (cid:0) below potential (noting equation (1), and that the real rate tracks the policy rate). The dashdotted lines show the partial e⁄ect of the one percentage point government spending rise (i.e., the di⁄erence between the response to both shocks and the taste shock alone). Because the pot increase in government spending boosts r while leaving the real interest rate una⁄ected, the t higher spending has a positive e⁄ect on the output gap while the economy remains in a liquidity 7The result that the monetary policy rule in equation (3), if unconstrained, can stabilize in(cid:135)ation and the output gap at target (implying the the policy rate tracks rpot) is shown in Woodford (2003), and reproduced in Appendix t A: Moreover, given the absence of a policy tradeo⁄, the relative weight on in(cid:135)ation stabilization and output gap stabilization in the rule is immaterial so long as one policy rule coe¢ cient is su¢ ciently large. Other shocks that induced the same path of rpot as the taste shock (cid:150)including to tax rates or to productivity (cid:150)would have the same t implications for the spending multiplier in this model (provided that the shocks did not induce a policy tradeo⁄, as discussed below). 7

trap. The government spending multiplier, 1 dy t m = ; (12) t g dg y t which is equal to the sum of the output gap response of 0.5 shown in the (cid:133)gure and the potential output response of 0.2, (not shown), is considerably larger than in a normal situation in which the output gap would be una⁄ected. The multiplier is ampli(cid:133)ed substantially more when expected in(cid:135)ation responds to shocks, as illustrated in the right column of Figure 2 for a calibration implying a mean duration of price contracts of 5 quarters (i.e. (cid:24) = 0:8). In this case, the negative taste shock causes expected p in(cid:135)ation to fall persistently, which raises the real interest rate and augments the output decline relative to the no-in(cid:135)ation response case. Higher government spending partially reverses these pot e⁄ects by boosting r and expected in(cid:135)ation. The impact spending multiplier, m is about 2. t 0 Because the (cid:133)scal multiplier is relatively high in a liquidity trap, the e⁄ects of a rise in government spending on government debt are smaller than in normal times in which policy rates adjust. For the case of 5 quarter contracts, government debt falls for several quarters (lower right panel), a sharp contrast from the progressive rise in debt that occurs in normal times. These debt dynamics largely re(cid:135)ect that labor income tax revenue, and hence the primary balance p , varies sharply Gt with the spending multiplier: 1 dp Gt pot = g m +(cid:30) (m m ) 1: (13) g dg y f t mc t (cid:0) t g(cid:0) y t where the bracketed term premultiplied by g re(cid:135)ects the response of labor income to higher govy ernment spending. Thus, holding lump-sum taxes unchanged (as assumed in equation 13), the primary balance p (expressed relative to GDP) actually improves in response to higher spend- Gt ing if m > 1 ( 1 +(cid:30) m pot ); which corresponds to a multiplier greater than unity under our t 1+(cid:30) mc gy mc t pot benchmark calibration. By contrast, in normal times the multiplier of m implies that labor tax t pot revenue rises by g m ; or only about 0.05 under our calibration, so that a 1 percentage point rise y t in spending causes the primary de(cid:133)cit to rise by 0.95 percent. Accordingly, conditions that imply a larger and more persistent boost in the multiplier than under the 5 quarter contracts calibration cause debt to fall even more than shown in the (cid:133)gure. These include a higher Phillips Curve slope (as suggested by comparing the lower two panels), or a longer-lived liquidity trap. For example, in the case of four quarter price contracts (cid:150)which imply an impact multiplier of 3 (cid:150)debt falls progressively. Debt eventually converges to baseline through a reduction in lump-sum taxes, which may be regarded as tantamount to a (cid:147)(cid:133)scal free lunch.(cid:148) 2.3. The Multiplier and the Size of Fiscal Spending In the log-linearized model that ignores the zero bound constraint, the government spending multiplier is invariant to the size of the change in spending. By contrast (cid:150)as we next proceed to show 8

(cid:150)the multiplier in a liquidity trap declines in the level of government spending. Intuitively, this behavior re(cid:135)ects that the multiplier varies positively with the duration of the liquidity trap, and that the duration shortens as the level of spending rises. pot Because government spending and taste shocks have the same linear dynamic e⁄ects on r t pot under our assumption that the shocks are equally persistent, and only the path of r matters for t pot the output gap and in(cid:135)ation response, we can simply focus on how r a⁄ects the output gap and t in(cid:135)ation in a liquidity trap. Solving the IS curve forward yields: Tn 1 Tn (cid:0) pot x = (cid:27)^ ( i r )+(cid:27)^ (cid:25) +x ; (14) t (cid:0) (cid:0) (cid:0) t+j j t t+j j t t+Tn j t j=0 j=1 X X where T is the duration of the liquidity trap. Hence, the output gap x in a liquidity trap depends n t on four terms. First, it depends on the cumulative gap between the nominal interest rate i (cid:0) and the potential real interest rate over the interval in which the economy remains in a liquidity trap. This cumulative interest rate gap Tn 1( i r pot ) can be interpreted as indicating j=(cid:0)0 (cid:0) (cid:0) t+jt j how shocks to the potential real interest rate would a⁄ect the output gap if expected in(cid:135)ation P remained constant. Second, the output gap depends on cumulative expected in(cid:135)ation over the liquidity trap (or equivalently, the log change in the price level log(P ) log(P )); as indicated t+Tn t (cid:0) above, the e⁄ects of shocks to the potential real rate on the output gap are ampli(cid:133)ed through changes in expected in(cid:135)ation. Third, the current output gap also depends on the expected output gap x when the economy exits the liquidity trap, though both the terminal output gap and t+Tnt j in(cid:135)ation terms drop under the assumption that monetary policy completely stabilizes the economy (x = (cid:25) = 0). Finally, t+T can be interpreted as the exit date of the liquidity trap and t+Tnt t+Tnt n j j is determined endogenously as the (cid:133)rst period in which the expected potential real interest rate exceeds i: Thus: (cid:0) pot T = min(r > i); (15) n j t+j j t (cid:0) where j = 0;1;2;::: pot T depends both on the size and persistence of the shocks to r : The relation between T n t n pot pot and r under our baseline calibration is shown in Figure 1.b. Because T is only a⁄ected as r t n t pot pot exceeds certain threshold values, it is a step function in the level of r (rising as r assumes t t pot more negative values): Thus, a slightly larger adverse taste shock that caused r to drop more t than shown in Figure 1.a would leave the duration of the liquidity trap unchanged at 8 quarters; but a large enough adverse shock would extend the duration of the trap, and a su¢ ciently smaller shock would shorten it. In the limiting case in which expected in(cid:135)ation remains constant, we can derive a simple closed pot form solution for the multiplier. Because r follows an AR(1) with persistence parameter 1-(cid:26) ; t v equation (14) implies that the output gap x equals: t x = (cid:27)^ Tn (cid:0) 1 ( i (1 (cid:26) )jr pot ) = (cid:27)^iT +(cid:27)^r pot 1 (cid:0) (1 (cid:0) (cid:26) v )Tn < 0: (16) t (cid:0) (cid:0) (cid:0) (cid:0) v t (cid:0) n t (cid:26) j=0 v X 9

For changes in government spending that are small enough to keep the liquidity trap duration unchangedatT periods,themultiplier 1 dyt isderivedbydi⁄erentiatingequation(16)withrespect n gy dgt dypot to g ; and adding the e⁄ect on potential output ( t ) : t dgt 1 dy 1 dx dy pot 1 (1 (cid:26) )Tn 1 dr pot 1 dy pot t = ( t + t ) = (cid:27)^ (cid:0) (cid:0) v t + t : (17) g dg g dg dg (cid:26) g dg g dg y t y t t v y t y t The (cid:133)rst term (cid:150)the output gap component (cid:150)is positive. It varies directly with the duration of the underlying liquidity trap T , re(cid:135)ecting that (cid:133)scal policy can only a⁄ect the output gap over the n periodinwhichtheeconomyremainsinthetrap. Thesecondterm 1 dy t pot isequaltothespending gy dgt multiplierinthe(cid:135)exiblepriceequilibrium,aswellasduringnormaltimesgivenourassumptionthat monetary policy, if unconstrained, keeps output at potential. From equations (4); (6); and (8), thelattermaybeexpressedas 1 dy t pot = 1 whichislessthanunitysince (cid:30) (cid:27)^ = 1+ ((cid:11)+(cid:31))(cid:27)^ > 1: Substituting 1 dr t pot = 1(1 gy d 1 gt )(cid:26) (cid:30) i m nt c (cid:27) o ^ equation (17), the multiplier ca m n c be expres 1 s (cid:0) ed (cid:11) in the gy dgt (cid:27)^ (cid:0) (cid:30) mc (cid:27)^ v simple form: 1 dy 1 t = 1 (1 )(1 (cid:26) )Tn: (18) g dg (cid:0) (cid:0) (cid:30) (cid:27)^ (cid:0) v y t mc The solid lines in the upper left panel of Figure 3 show how the marginal multiplier varies with the duration of the liquidity trap, where the latter is indicated by the tick marks along the upper axis. The multiplier associated with a tiny increment to government spending in an 8 quarter liquidity trapisabout0.7, butrisestoabout0.8againstthebackdropofan12quarterliquiditytrap(caused by a larger contractionary taste shock than in Figure 1.a). The multiplier increases monotonically with the duration of the trap, but in a concave manner; and importantly, the multiplier remains less than or equal to unity provided that the liquidity trap is of (cid:133)nite duration, however long. These results provide a key stepping stone for understanding how the multiplier varies with the size of the increase in government spending. While the foregoing analysis examined the e⁄ects of tiny increments to government spending against the backdrop of di⁄erent initial conditions (i.e., associated with liquidity traps of varying length), we now take (cid:147)initial conditions(cid:148)(cid:150)summarized by a given-sized taste shock (cid:150)as (cid:133)xed, and assess how progressively larger increases in government spending a⁄ect the multiplier by reducing the duration of the liquidity trap. In this vein, the upper left panel of Figure 3 can be reinterpeted as showing how the multiplier varies with alternative levels of government spending (cid:150)that is, a government spending multiplier schedule. For concreteness, we assume that the liquidity trap is generated by the same adverse taste shock shown in Figure 1.a, so that the (cid:147)0(cid:148)government spending level on the lower horizontal axis implies an 8 quarter liquidity trap (shown by the tick mark on the upper horizontal axis). For a government spending hike of less than a threshold value of 1.2 percent of GDP, the duration of the liquidity trap remains unchanged at T = 8 quarters, and the multiplier equals about 0.7.8 n As government spending exceeds this threshold, the potential real rate is boosted enough that the 8As seen in the (cid:133)gure, cuts in government spending exert a progressively more negative marginal impact as they become large enough to extend the duration of the liquidity trap. 10

liquidity trap is shortened by one period (as determined by equation 15), and the multiplier falls discontinuously (to the value implied by equation (18) with T = 7). The multiplier continues to n decline in a step-wise fashion (cid:150)with equation (15) determining the threshold levels of spending at which the multiplier drops (cid:150)until leveling o⁄at a constant value of 1 dy t pot corresponding to a gy dgt spending level high enough to keep the economy from entering a liquidity trap. Given that the multiplier declines with spending, the average change in output per unit increase in government spending, 1 (cid:1)yt,lieswellabovethemarginalresponse 1 dyt;todi⁄erentiatebetweentheseconcepts gy (cid:1)gt gy dgt inthe(cid:133)gure,theformerislabeledthe(cid:147)averagemultiplier,(cid:148)andthelatterthe(cid:147)marginalmultiplier(cid:148) (in a slight abuse of terminology, since the multiplier is inherently a marginal concept). The relationship between the multiplier and the size of government spending can be given an alternative graphical interpretation using Figure 1.a. Recall that the e⁄ect of the adverse taste shock alone on the potential real interest rate is shown by the solid line. This shock(cid:146)s e⁄ect on the output gap is proportional to Tn 1( i r pot ); the sum of the bold vertical line segments j=(cid:0)0 (cid:0) (cid:0) t+jt pot j between i and the path of r (the (cid:147)interest rate gaps(cid:148)) implied by the taste shock through (cid:0) t+j P period T 1: The 1 percent of GDP rise in government spending shown by the dashed line leaves n (cid:0) the liquidity trap duration unchanged, implying that the higher government spending narrows the pot gap between between i and r over a full T = 8 periods. But for a spending program equal to (cid:0) t+j n pot 2 percent of GDP (the dash-dotted line), r exceeds i: Thus, further increments to spending t+Tn 1 (cid:0) (cid:0) havenoe⁄ectontheinterestrategapatT 1;asthee⁄ectonthepotentialrealrateinthisperiod n (cid:0) is completely o⁄set by monetary policy. Because additional spending only shrinks the interest rate gap for 7 quarters, the multiplier is lower. Thevariationinthemultiplierwithspendingbecomesmorepronouncedwithanupward-sloping Phillips Curve. When expected in(cid:135)ation responds, movements in the potential real interest rate pot r have larger e⁄ects on the output gap than implied by equation (16), so that the same taste t shock has a larger contractionary e⁄ect, and higher government spending has a more stimulative pot e⁄ect. To see how the e⁄ects of variation in r are magni(cid:133)ed, equations (1) and (2) can be solved t forward (imposing the zero bound constraint that i = i) to express in(cid:135)ation in terms of current t (cid:0) and future interest rate gaps: Tn 1 (cid:0) pot (cid:25) = (cid:27)^(cid:20) (j)( i r ); (19) t (cid:0) p (cid:0) (cid:0) t+jt j j=0 X where the weighting function (j) is given by j (j) = (cid:21) (j 1)+(cid:21) ; (20) 1 (cid:0) 2 with the initial condition (0) = 1, and where (cid:21) and (cid:21) are determined by: 1 2 (cid:21) +(cid:21) = 1+(cid:12)+(cid:27)^(cid:20) ; (cid:21) (cid:21) = (cid:12): (21) 1 2 p 1 2 Given that (cid:20) > 0, the coe¢ cients (j) premultiplying the interest rate gap grow exponentially p with the duration of the liquidity trap T . Moreover, the contour is extremely sensitive to (cid:20) , as n p 11

illustratedinFigure1.cforseveralvaluesof(cid:20) associatedwithpricecontractdurationsrangingfrom p four to ten quarters.9 The convex pattern of weights re(cid:135)ects that de(cid:135)ationary pressure associated pot with any given-sized interest rate gap ( i r ) is compounded as the liquidity trap lengthens (cid:0) (cid:0) t+jt j by the interaction between the response of the output gap and expected in(cid:135)ation. ConsistentwiththeanalysisofEggertson(2009and2010),Christiano,Eichenbaum,andRebelo (2011), and Woodford (2011), the multiplier can be ampli(cid:133)ed substantially relative to normal circumstances in a long-lived liquidity trap, especially if the Phillips Curve slope is relatively high. This is illustrated in the lower left panel of Figure 3, which plots the impact government spending multiplier under alternative speci(cid:133)cations of the parameter (cid:24) that imply price contracts with a p mean duration between 4 and 10 quarters. With short enough price contracts, the multiplier increases in a sharply convex manner with the duration of the liquidity trap, in contrast to the concave relation that obtains when expected in(cid:135)ation is less responsive. For example, in a liquidity trap lasting 12 quarters (see the tick marks on the upper axis), the multiplier is about 7 with (cid:133)ve quarter contracts, and 25 with four quarter contracts. Clearly, (cid:133)scal policy can be very e⁄ective in providing economic stimulus. Even so, under exactly the same conditions in which the government spending multiplier is very large (cid:150)a long-lived trap, and shorter-lived price contracts (cid:150)the multiplier drops substantially as government spending increases. Intuitively, the multiplier is large under these conditions because (cid:133)scal stimulus helps reverse the strong de(cid:135)ationary pressure arising from the adverse taste shock. Butinsofarasthehigherspendingisverye¢ caciousandshortenstheliquiditytrap,thede(cid:135)ationary pressureabatesandthebene(cid:133)tsofadditionalstimulusdiminishsubstantially. Inthisvein,thelower left panel of Figure 3 can be interpreted as showing how the impact multiplier varies with the level of government spending assuming that the taste shock induces an eight quarter liquidity trap.10 The multiplier associated with four quarter price contracts drops from about 4 for a spending level of 1 percent of GDP to about 1:5 for spending increments above 3:5 percent of GDP. The marginal multiplierschedulesforthecasesof5and10quarterpricecontractsareconsiderably(cid:135)atter,though the multipliers clearly decrease in the level of spending.11 Our assumptions that the timing of (cid:133)scal stimulus coincides exactly with the arrival of the shock causing the liquidity trap, and that the spending shock is equally persistent, are useful for expositional clarity in showing how the multiplier varies with spending. But while the downward- 9Asisclearfrom equations(19)and(21);theaggregatedemandelasticity(cid:27)^ alsoin(cid:135)uencesthein(cid:135)ationresponse, re(cid:135)ecting that the contour is determined by the product of (cid:27)^ and (cid:20) : p 10Asin the upperleftpanel,the (cid:147)0(cid:148)spending levelon the lowerhorizontalaxisimpliesan 8 quarterliquidity trap (denoted by the tick marks on the upper horizontal axis). 11The two-state Markov framework adopted by Eggertson (2009 and 2010), Christiano, Eichenbaum, and Rebelo (2011), and Woodford (2011) provides a great deal of clarity in identifying factors that can potentially account for a high multiplier, which is the focus of their analysis. But given that the depth of the recession (cid:150)and associated fall inthepotentialrealinterestrate(cid:150)isassumedtobeconstantintheliquiditytrapstate,themultiplieralsoturnsout to be constant in a liquidity trap irrespective of the level of spending (i.e., until spending rises enough to snap the economy out of the liquidity trap entirely). 12

sloping multiplier schedule is a robust implication, the exact contour of the government spending multiplier schedule depends on a range of factors, including both the timing and persistence of government spending shocks, as well as on the characteristics of the adverse shocks causing the liquidity trap. Christiano, Eichenbaum, and Rebelo (2011) show that the spending multiplier tends to be larger if the spending is timed to coincide with the period in which monetary policy is constrained by the zero lower bound. Erceg and Linde (2010) emphasize that even the pro(cid:133)le of (cid:133)scal stimulus over the period in which monetary policy is constrained can markedly a⁄ect the multiplier: in particular, signi(cid:133)cant lags in the implementation of the spending hike (cid:150)through damping the impact on the potential real rate (cid:150)can reduce the multiplier substantially.12 The multiplier schedule also depends on the conduct of monetary policy. In general, the coe¢ cients of the monetary rule in(cid:135)uence the size of the government spending multiplier both through a⁄ecting the duration of the liquidity trap, and through their e⁄ect on the (cid:147)normal times(cid:148) spending multiplier that applies when monetary policy is no longer constrained. The relative weight attached to in(cid:135)ation compared with output gap stabilization in the monetary rule can have particularly important implications for the (cid:133)scal multiplier in the event that the underlying shocks induceapolicytradeo⁄betweenstabilizingin(cid:135)ationandtheoutputgap. Asanillustration,suppose that the adverse demand (i.e., taste) shock in our baseline were accompanied by a markup shock that boosted in(cid:135)ation. Under some conditions, a monetary policy rule that put a large weight on in(cid:135)ation stabilization would imply a shorter-lived liquidity trap than a policy rule more tilted towards stabilizing the output gap, implying a smaller (cid:133)scal multiplier in the former case.13 Our benchmark monetary policy rule assumes that policy rates, if constrained, respond only to contemporaneous values of in(cid:135)ation and the output gap. The government spending multiplier would tend to be lower under an inertial, or (cid:147)history dependent(cid:148)policy rule that more closely approximated the optimal policy under commitment (Woodford 2003). In this case, monetary policy is more e⁄ective in cushioning the economy from the e⁄ects of adverse demand (or (cid:133)nancial) shocks, which reduces the bene(cid:133)ts of discretionary (cid:133)scal actions.14 2.4. Marginal Impact on Government Budget Wenextconsiderthebudgetaryimpactofgovernmentspendinginourbenchmarkmodel. Theright panels of Figure 3 show how the response of the government debt/GDP ratio after four quarters 12Interestingly, committing to a constant increase in government spending through date T yields a somewhat N 1 (cid:0) larger marginal multiplier than shown in Figure 3. This policy boosts the potential interest rate sharply at T ; N 1 (cid:0) whichisveryusefulinlimitingsomeofthedownwardpressureonin(cid:135)ationassociatedwithalong-livedliquiditytrap (noting that equation 19 implies that the response of expected in(cid:135)ation is highly sensitive to the real interest rate gapatmoredistanthorizons). Evenso,themultiplierdropsevenmoreprecipitouslywiththelevelofspendingthan under our benchmark calibration (re(cid:135)ecting that spending remains at its initial level for "too long" as the liquidity trap duration shortens). 13This case is illustrated in Figure A.1 of Appendix A. 14This case is illustrated in Figure A.2 of Appendix A. 13

varies with the level of government spending assuming that the baseline taste shock generates an eight quarter liquidity trap. Consistent with the impulse responses discussed earlier, a 1 percent of GDP rise in government spending reduces government debt by 1/2 percentage point under our benchmarkcalibrationwith(cid:133)vequarterpricecontracts(lowerrightpanel), andover1:5percentage points under four quarter price contracts; as noted, the government debt response in the latter case remains well below baseline even for several years. With longer price contracts of 10 quarters, the lower spending multiplier implies a rise in government debt; hence, taxes must increase, though by much less than in normal times. Under all of these calibrations, the implications for the debt/GDP ratio would be even more favorable if we allowed for a positive steady state debt/GDP ratio, re(cid:135)ecting that higher in(cid:135)ation (cid:150)and a consequently higher price level (cid:150)would reduce the real value of the outstanding stock government debt. While the possibility of a (cid:133)scal free lunch (cid:150)or even a cheap lunch (cid:150)is intriguing, our analysis highlights that it is vital for policymakers to distinguish between the marginal and average e⁄ects on government debt (and hence taxes) of di⁄erent-sized spending programs. Although government debtmayfallinresponsetosmallspendingincrementsunderconditionsthatimplyahighspending multiplier (cid:150)i.e., a substantial responsiveness of expected in(cid:135)ation (cid:150)larger increases in spending may put sizeable upward pressure on government debt and tax rates, as is particularly apparent in the case of four quarter contracts in the lower right panel. Our analysis also has implications for how (cid:133)scal consolidation may a⁄ect the economy when monetary policy is constrained by the zero bound. Figure 3 indicates that a persistent reduction in the level of government spending (cid:150)if large enough (cid:150)lengthens the duration of the liquidity trap, and can have very contractionary e⁄ects if the multiplier schedule is su¢ ciently convex. In this case, a reduction in government spending may boost government debt persistently. For example, a 5 percent of GDP spending cut under our benchmark causes the debt/GDP ratio to rise by almost 2 percentage points after 4 quarters, and the debt/GDP ratio remains about 1.3 percentage point abovebaselineevenafter3years.15 Thisanalysisofatemporaryspendingcutmaywelloverstatethe negative impact of (cid:133)scal consolidation to the extent that the latter is perceived as more enduring (in which case crowding in e⁄ects could be substantially larger). Even so, it provides a strong caution that (cid:133)scal consolidation that is perceived as temporary (cid:150)perhaps due to low credibility (cid:150) can generate a deep output contraction and cause government debt to rise for a prolonged period. Insofar as larger spending cuts may prolong a liquidity trap signi(cid:133)cantly and increase the marginal impact on output and government debt, it is clearly important for policymakers to understand the full contour of the multiplier schedule. Inadditiontotheoutputmultiplier,thegovernmentdebtresponsealsodependscruciallyonthe composition of the tax base, and the cyclical responsiveness of its key components. If the tax base is less (more) cyclically-sensitive, the consequences of higher government spending for government 15For visual clarity, Figure 3 does not show the e⁄ects of cuts in spending on the debt/GDP ratio (since a cut implies a rise in debt). 14

debt and taxes may appear less (more) benign than in Figure 3, even if the spending multiplier is unchanged. To illustrate this, it is helpful to amend the benchmark model slightly by assuming that government spending in the steady state is (cid:133)nanced by both a labor tax (cid:28) and a sales tax N (cid:28) that is levied on private consumption. Under the assumption that government debt is zero in C steady state, the government budget constraint implies that tax rates satisfy the relation that g = y (cid:28) s + (cid:28) (1 g ), where s is the steady state labor income share. Retaining the assumption N N C y N (cid:0) that only lump-sum taxes adjust dynamically, so that the dynamics of the government spending multiplier are una⁄ected by the form of tax (cid:133)nancing, government debt evolves according to: b = (1+r)b +g g (cid:28) s (y +(cid:30) x ) (cid:28) (1 g )c (cid:28) : (22) G;t G;t 1 y t N N t mc t C y t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) Hence, the impact of a one percent of GDP rise in government spending on the government debt/GDP ratio depends on the response of labor income y + (cid:30) x ; of private consumption t mc t c (the base for the sales tax, which equals 1 (y g g )), and on the share of government spendt 1 gy t (cid:0) y t (cid:0) ing (cid:133)nanced by each type of tax. The response of labor income to a one percent of GDP rise in government spending may be expressed as 1 dyt +(cid:30) 1 dxt = m +(cid:30) (m m pot ), where m is gy dgt mcgy dgt t mc t (cid:0) t t the spending multiplier. The term involving (cid:30) is nonnegative, and increases sharply with the mc multiplier if marginal cost is relatively sensitive to the output gap (i.e., if (cid:30) is relatively high). mc The response of consumption to the same spending hike is given by 1 dct = 1 (m 1), which gy dgt 1 gy t (cid:0) (cid:0) implies much less variation with the multiplier. For example, with a multiplier of unity, labor tax revenue would rise 5 percent in response to the spending increase, while no revenue would accrue from a sales tax (since private consumption would remain unchanged). As noted earlier, the primary balance would improve (cid:150)and hence government debt decline (cid:150)under a pure labor income tax if the multiplier exceeded unity, while the multiplier would have to exceed 5 for government debt to decline under a pure sales tax. Thus, the cyclical responsiveness of the tax base can play a major role in determining the budgetary implications of higher government spending. From a policy perspective, the budgetary implications of (cid:133)scal stimulus may diverge markedly across countries with di⁄erent tax bases, such as between the United States (where labor taxes, inclusive of payroll taxes, comprise a large share of revenue at the federal level) and European countries in which sales taxes are more important. From the perspective of the literature, our model(cid:146)s implication that a (cid:133)scal expansion may induce government debt to contract has an important parallel in the analysis of Davig and Leeper (2011). Within the context of a New Keynesian sticky price model, these authors show that a spending expansion both has large e⁄ects on output and can cause government debt to decline if monetarypolicybehavespassively(allowingrealinterestratestofall),andif(cid:133)scalpolicyis(cid:147)active(cid:148) (so that the tax rule implies a small response of taxes to debt). In this environment, because tax policy is not expected to be aggressive enough to generate eventual primary surpluses, the price level must rise, reducing government debt enough to satisfy the government(cid:146)s intertemporal budget constraint. Importantly, the implications for the (cid:133)scal multiplier and debt re(cid:135)ect the imprint of 15

the (cid:147)passive monetary, active (cid:133)scal(cid:148)regime, so that taxes or tax rates would never have to adjust muchinsucharegime(andthespendingmultiplierwouldalwaysbehigh). Bycontrast, ourmodel assumes that monetary policy is active once the zero bound constraint is no longer binding, and that (cid:133)scal policy is passive, so that the rule is committed to adjusting taxes aggressively enough to stabilize debt. Thus, both the multiplier and budgetary implications of alternative spending choices are highly state-dependent, rather than a characteristic of the regime. 2.5. Distortionary vs. Lump-Sum Taxes We next compare our benchmark speci(cid:133)cation with lump-sum taxes to an alternative in which the labor income tax rate (cid:28) adjusts according to the rule: N;t (cid:28) = b + g g (cid:28) s (y +(cid:30) x ) : (23) N;t 1 G;t 1 2 y t N N t mc t (cid:0) f (cid:0) g According to (23); the tax rate on labor income (cid:28) reacts endogenously to the lagged stock of N;t government debt b ; and to the primary budget de(cid:133)cit that would accrue if the labor income G;t 1 (cid:0) tax rate remained (cid:133)xed at its steady state value of (cid:28) (the term in parentheses that postmultiplies N ): The latter term is convenient for analyzing tax rules designed to stabilize debt aggressively, 2 with a balanced budget rule a special case in which = 0 and = 1 : 1 2 sN Allowing the labor tax rate to be determined endogenously has potentially important implicationsforthegovernmentspendingmultiplierthatarisethroughe⁄ectsonin(cid:135)ation,potentialoutput, and the potential real rate. With regard to in(cid:135)ation, the price-setting equation (2) becomes: 1 pot (cid:25) = (cid:12)(cid:25) +(cid:20) (cid:30) x + ((cid:28) (cid:28) ) : (24) t t+1 j t p f mc t (1 (cid:28) N ) N;t (cid:0) N;t g (cid:0) The salient change relative to the standard price-setting equation is that real marginal cost (cid:150)the pot expressioninparentheses(cid:150)dependsonthe(cid:147)taxgap(cid:148)(cid:28) (cid:28) inadditiontothestandardoutput N;t (cid:0) N;t gap term. The tax gap enters because the endogenous labor tax rule (23) implies that the tax pot rate (cid:28) may di⁄er from its level under (cid:135)exible prices of (cid:28) : As suggested by equation (23), if a N;t N;t shock to government spending boosts labor income by more than would occur under (cid:135)exible prices (because the multiplier is large), the tax reaction function implies a lower labor income tax rate pot ((cid:28) ) than would occur under (cid:135)exible prices ((cid:28) ): This puts downward pressure on marginal cost N;t N;t and in(cid:135)ation, which reduces the multiplier.16 16The tax gap is inversely related to the current output gap x and directly related to the lag of the debt gap t (b bpot ): G;t (cid:0) 1 (cid:0) G;t (cid:0) 1 1 ((cid:28) (cid:28)pot)= (cid:28) s (1+(cid:30) )x + (b bpot ): (25) N;t (cid:0) N;t (cid:0) 2 N N mc t 1(1+ 2(1 (cid:0) sN (cid:28)N) ) G;t (cid:0) 1 (cid:0) G;t (cid:0) 1 This equation follows from equation (23) and the government budget constraint (equation 11, after making the appropriate substitution for the distortionary tax rate in place of the the lump-sum tax). Because the debt gap variesinverselywithpastoutputgaps(i.e.,positiveoutputgapsreducedebtrelativetothelevelimpliedunder(cid:135)exible prices), the tax gap varies inversely with current and past output gaps. 16

The second key channel through which distortionary taxes in(cid:135)uences the multiplier is through potential output, which may be expressed: g (cid:27)^g (1 g ) y pot = y g 2 y g 1 b pot + (cid:0) y (cid:23) (cid:23) ; (26) t (cid:30) (cid:27)^(cid:14) t (cid:0) (cid:30) (cid:27)^(cid:14) (1 (cid:28) )(cid:14) t (cid:0) (cid:30) (1 (cid:28) )(cid:14) G;t 1 (cid:30) (cid:27)^(cid:14) c t mc 1 mc 1 N 1 mc L 1 (cid:0) mc 1 (cid:0) (cid:0) where 0< (cid:14) 1 5 1.17 The standard wealth e⁄ect of a government spending shock is captured by the (cid:133)rst term: as in the benchmark model, higher government spending reduces consumption, expands pot potential labor supply, and boosts y : The second term is negative provided that > 0, and t 2 pot re(cid:135)ectsthaty isreducedtotheextentthatthelabortaxratereactscontemporaneouslytohigher t government spending. The third term re(cid:135)ects how past increments to government spending (cid:150)by pot raising potential government debt b and hence the tax labor rate by equation (23) (cid:150)reduce G;t 1 pot (cid:0) potentialoutputy :Thus, labortax(cid:133)nancingtendstoreducethespendingmultiplierbyreducing t the response of potential output relative to lump-sum (cid:133)nancing. pot The third channel is through the potential real interest rate r . Both the form of the IS curve t pot given by equation (1) and the equation determining r are the same as under lump-sum taxes; t the latter can be written: pot pot pot r = y y +g (g g )+v ((cid:23) v ): (27) t t+1 j t(cid:0) t y t (cid:0) t+1 j t c t (cid:0) t+1 j t pot The e⁄ect of higher spending on r under the alternative modes of (cid:133)nancing (cid:150)and thus the t stimulus to aggregate demand (cid:150)depends on the response of potential output growth. Although distortionary taxes depress the response of the level of potential output to a spending hike relative to lump-sum taxes, the growth rate of potential output may be higher or lower depending on the parameters of the tax rule (with a larger response of potential output growth possible under a rule that responds aggressively to the de(cid:133)cit, as we will illustrate). In the case in which the tax rate reaction function reaction reacts only to the stock of debt ( > 0 and = 0), the three channels discussed above operate in the same direction to reduce 1 2 the government spending multiplier. This is illustrated in Figure 4, which compares the e⁄ects of a 1 percent of GDP rise in government spending under our benchmark with lump-sum (cid:133)nancing to an alternative ((cid:147)simple debt labor tax rule(cid:148)) which sets = 0:01 in the tax reaction function 1 (23) (this calibration implies that a 10 percentage point increase in the annualized debt/GDP ratio boosts the labor tax rate by 0.4 percentage points). The peak multiplier in the latter case is only about 3/4 as large as under lump-sum taxes. The aggregate demand stimulus is smaller under the pot pot labor tax because r rises by less (potential output growth is relatively lower because (cid:28) rises t N;t gradually as potential government debt increases), and because in(cid:135)ation rises less as the negative tax gap restrains pressure on marginal cost. On the supply side, the response of potential output is also smaller. The smaller multiplier translates into a more substantial increase in government debt. Given that in(cid:135)ation is less responsive under labor tax (cid:133)nancing, the disparity between the 17The parameter (cid:14) =1 2 sN : 1 (cid:0) (cid:30)mc(1 (cid:0) (cid:28)N) 17

spending multiplier under these alternative taxation schemes becomes considerably larger as the durationoftheliquiditytraplengthensbeyondtheeightquartersshowninthe(cid:133)gure. Forexample, the multiplier exceeds 7 for a 12 quarter liquidity trap under lump-sum taxes, but is only slightly above 2 under labor tax (cid:133)nancing.18 Conversely, the di⁄erences across taxation schemes become smaller to the extent that the tax rule responds more gradually to government debt. For example, under an inertial form of the same distortionary tax rule considered in Figure 4 (cid:150)of the form (cid:28) = 0:98(cid:28) +(1 0:98) b N;t N;t 1 1 G;t 1 (cid:0) (cid:0) (cid:0) (cid:150)the spending multiplier and government debt responses are virtually identical to the lump-sum tax case (and hence are omitted for visual clarity). Under this more inertial tax rule, the tax gap in equation (24) is smaller, so in(cid:135)ation determination isn(cid:146)t much a⁄ected relative to lump-sum tax case.19 As the liquidity trap duration extends beyond the eight quarters shown in the (cid:133)gure, the multiplier rises only a bit faster with duration under lump-sum taxes than labor tax (cid:133)nancing.20 Given that previous research, including by Eggertson (2010) and Christiano, Eichenbaum, and Rebelo (2011), has shown that the government spending multiplier is ampli(cid:133)ed in a liquidity trap if accompanied by an exogenous rise in the labor tax rate, our result that the multiplier is reduced might appear to hinge on the gradual adjustment of tax rates to debt implied by setting = 0 in 2 the tax reaction function (23). In particular, a key channel highlighted by this previous literature in accounting for greater stimulus is that a higher distortionary tax raises the potential real rate pot r relative to the lump-sum case, which boosts expected in(cid:135)ation and hence the multiplier (given t that monetary policy does not react).21 However, the di⁄erent implications turn out to mainly re(cid:135)ect that we specify an endogenous tax reaction function, rather than assume that the labor tax adjusts exogenously. In our framework, the e⁄ects of endogenous tax changes on marginal cost and in(cid:135)ation implied by equation (24), which lower the responsiveness of in(cid:135)ation, can dominate the behavior of the multiplier. Consequently, even a tax reaction function that reacts aggressively to the current de(cid:133)cit in order to stabilize government debt and that generates a much larger rise in pot r can imply a smaller multiplier than in the lump-sum case. t To show how the multiplier may be reduced (relative to the lump-sum case) even under an aggressive tax rule, it is useful to consider the special case of a balanced budget rule (which sets = 0 and = 1 in equation (23). Figure 4 shows the e⁄ects of the 1 percent of GDP increase 1 2 sN pot in government spending under the balanced budget rule. Because y declines in a front-loaded t 18The upper panels of Figure A.1 in Appendix A shows how the (marginal) government spending multiplier and government debt response vary with the liquidity trap duration under both lump-sum taxes and the distortionary tax rate rule; the (cid:133)gure also includes an inertial version of the latter, and a balanced budget rule. 19Moreover, because (cid:28)pot adjust even more slowly to potential debt, the evolution of ypot and rpot is also very N;t t t similar to the case of lump-sum taxes. 20See Figure A.3 in Appendix A. 21Because the tax hike is assumed to be front-loaded, potential output falls immediately, but potential output growth(andhencerpot)increases. Thus,thetaxhikestimulatesaggregatedemandifinterestratesareleftunchanged. t 18

manner under the balanced budget rule, r pot rises more sharply than in the lump-sum tax case.22 t pot Whilethepartiale⁄ectofanincreaseinr istoboostthemultiplierrelativetothelump-sumcase, t the overall impact hinges on the extent to which in(cid:135)ation behavior is in(cid:135)uenced by the endogenous pot reaction of the labor tax. In the case of a balanced budget rule, the tax gap (cid:28) (cid:28) is simply N;t (cid:0) N;t proportional to the current output gap, so that endogenous tax adjustment in e⁄ect reduces the Phillips Curve slope by a factor that depends on the steady state level of the distortionary tax, i.e., equation (24) becomes: (cid:28) N (cid:25) = (cid:12)(cid:25) +((cid:20) s )x : (28) t t+1 j t p (cid:0) 2(1 (cid:28) N ) N t (cid:0) Under our benchmark calibration with a government spending share of 20 percent ((cid:28) = 0:27); N Figure 4 shows that the in(cid:135)ation response is noticeably smaller under the balanced budget rule than under lump-sum taxes, which explains the smaller output response. The lower Phillips Curve slope also means that the gap between the spending multipliers rises as the liquidity trap duration becomes more prolonged. But interestingly, because the e⁄ect of the tax gap on the Phillips Curve slope declines as the tax rate becomes very low, our model can also yield results more in line with the exogenous tax speci(cid:133)cations considered in the literature. For example, a calibration with a steady state government spending share of 5 percent generates a larger spending multiplier under the balanced budget rule with distortionary taxes than under lump-sum taxes if the liquidity trap pot extends beyond six quarters (since the e⁄ect of higher r dominates the e⁄ect of a lower Phillips t Curve slope under this calibration).23 Overall, our analysis suggests that the response of output and government debt to government spending under distortionary taxes is probably only modestly lower than under lump-sum taxes provided that tax rates adjust fairly inertially to government debt. This seems important from a practical perspective, as tax rates typically exhibit considerable inertia. Even so, our analysis underscoresthatthenatureoftaxadjustmentcanpotentiallyhavesubstantialconsequencesforthe multiplier if taxes adjust rapidly to debt or current de(cid:133)cits, an environment that may become more relevant going forward in the wake of mounting concerns about (cid:133)scal sustainability. In terms of the recent literature on (cid:133)scal spending multipliers, our results help explain why Drautzburg and Uhlig (2011) - who uses an aggressive labor income tax rule to stabilize debt - obtain only a moderately higherspendingmultiplierinaliquiditytrapthaninnormaltimes, whileotherauthors(cid:150) including Christiano, Eichenbaum and Rebelo (2011), Eggertsson (2010) and Woodford (2011) (cid:150)(cid:133)nd much higher multipliers under the assumption of lump-sum (cid:133)nancing. 22Referring to equation (26), the front-loaded decline in potential output re(cid:135)ects that the wealth e⁄ect of higher government spending (cid:150)the (cid:133)rst term (cid:150)is more than o⁄set by the e⁄ect of immediately higher labor tax rates (cid:150)the second term. 23FigureA.3inAppendixAshowsmarginalmultipliersunderthebalancedbudgetruleforalternativegovernment spending shares. 19

2.6. Endogenous Government Spending The (cid:133)scal multiplier may also be a⁄ected by the presence of automatic stabilizers.24 In particular, while some component of government spending gexo may be exogenous (and follow the same t autoregression as in our benchmark), another component gendo may be endogenous and vary with t cylical conditions. To illustrate some key channels through which endogenous spending operates, it is instructive to consider the simple speci(cid:133)cation gendo = (cid:22)x , where (cid:22) > 0; so that a 1 percent t t (cid:0) rise in the output gap induces a contraction of (cid:22) percent in the endogenous component of spending (with gendo equal to zero in the steady state). In this case, the New Keynesian IS curve given by t equation(1)isunchanged, exceptthattheinterest-sensitivityofdemandisreducedfrom(cid:27)^ to(cid:27)^A = (cid:27)^ : Intuitively, because the endogenous spending (cid:147)leans against the wind(cid:148)by contracting spend- 1+(cid:22) ing when output rises above potential, the response of the output gap to a given-sized interest rate gap is reduced. The endogenous spending response (cid:150)as in the case of the distortionary tax rule (cid:150)also a⁄ects the slope of the Phillips Curve. Because government spending falls when the output gap is positive, labor supply shifts inward (due to a positive wealth e⁄ect), which translates into a heightenedsensitivityofrealmarginalcosttotheoutputgap. Again, theformofthePhillipsCurve is unchanged, but the marginal cost sensitivity to output increases from (cid:30) to (cid:30)A = (cid:11)+(cid:31) + 1+(cid:22): mc mc 1 (cid:11) (cid:27)^ (cid:0) Given that the New Keynesian model equations (1)-(5) are unchanged aside from these parameter adjustments (cid:150)and importantly, the duration of the liquidity trap is una⁄ected (cid:150)we can easily analyze the implications of alternative response coe¢ cients (cid:22). Thus, in the special case of a (cid:135)at Phillips Curve, the spending multiplier associated with a given-sized increase in the exogenous component of spending gexo is smaller if (cid:22) > 0, since automatic stabilizers reduce the interestt sensitivity of demand; the quantitative e⁄ect on the multiplier is given by equation (18). While it is natural to conjecture that the higher Phillips Curve slope under automatic stabilizers might raise the multiplier under certain conditions, the in(cid:135)ation responsiveness depends on the product of (cid:30)A and (cid:27)^ (from equations 19 and and 7), which declines in (cid:22): Thus, automatic stabilizers mc unambiguously reduce the multiplier. Moreover, the multiplier schedule relating the multiplier to the level of spending becomes less convex as (cid:22) increases.25 The responsiveness of automatic stabilizers to cyclical conditions may have important implications for the expected bene(cid:133)ts of discretionary (cid:133)scal stimulus in the presence of uncertainty. In particular, decisions about the size of a discretionary (cid:133)scal spending program must often be made against the backdrop of considerable uncertainty about how long the liquidity trap would be likely to last in the absence of (cid:133)scal stimulus. To the extent that the multiplier is convex in the liquidity trap duration (cid:150)and the duration is uncertain (cid:150)the expected spending multiplier will exceed the 24We thank Eric Leeper for suggesting an analysis of automatic stabilizers and uncertainty along the lines of this subsection. 25Figure A.4 in Appendix A compares the e⁄ects of a one percent of baseline GDP increase in the exogenous componentofgovernmentspendingunderourbenchmark with (cid:22)=0toan alternativespeci(cid:133)cation in which (cid:22)isset to unity. 20

multiplier under perfect foresight (i.e., the multiplier associated with the mean expected duration of the liquidity trap). Quite intuitively, this wedge between the multiplier under uncertainty and perfectforesightre(cid:135)ectsthatthepayo⁄to(cid:133)scalexpansioninbadstatesisespeciallyhigh. Because automatic stabilizers reduce the convexity of the multiplier schedule, the di⁄erence between the expected multiplier and multiplier under perfect foresight is comparatively smaller. Accordingly, the desirability of committing to a large (cid:133)scal expansion to hedge against adverse tail risks would appear stronger to the extent that automatic stabilizers are relatively weak.26 3. Fiscal Stimulus in a New-Keynesian Model with Keynesian Households and Financial Frictions In this section, we examine how the spending multiplier and government debt responses depend on the level of spending in a more empirically realistic framework with endogenous capital accumulation. The core of our model is a close variant of the models developed and estimated by Christiano, Eichenbaum and Evans (2005), CEE henceforth, and Smets and Wouters (2003, 2007), SW henceforth. CEE show that their model can account well for the dynamic e⁄ects of a monetary policyinnovationduringthepost-warperiod. SWconsideramuchbroadersetofshocks, andargue that their model (cid:150)which is estimated by Bayesian methods (cid:150)is able to (cid:133)t many key features of U.S. and euro area-business cycles. However, we depart from the CEE/SW environment in two substantive ways. First, we assume that a fraction of the households are (cid:147)Keynesian(cid:148), and simply consume their current after-tax income. Gal(cid:237), L(cid:243)pez-Salido and VallØs (2007) show that the inclusion of non-Ricardian households helps account for structural VAR evidence indicating that private consumption rises in response to higher government spending, and also allows their model to generate a higher spending multiplier. Second, to capture (cid:133)nancial channels omitted from the CEE/SW framework, we incorporate a (cid:133)nancial accelerator following the basic approach of Bernanke, Gertler and Gilchrist (1999). In this framework, the corporate (cid:133)nance premium varies with the degree of leverage of the economy due to an agency problem in private lending markets.27 We set the share of Keynesian households to optimizing households to 0.5, implying that the former comprise about 1/3 of aggregate consumption in the steady state, and calibrate the parameters a⁄ecting the (cid:133)nancial accelerator as in BGG (1999). However, we also report some results from a CEE/SW-type speci(cid:133)cation to help gauge the sensitivity to these factors. Given space limitations, we relegate most of the remaining details about the model, solution 26While a complete treatment of the interaction between automatic stabilizers and uncertainty is well beyond the scope of this paper, Figure A.4 of Appendix A illustrates how automatic stabilizers a⁄ect the expected spending multiplier in a simple framework in which the government spending choice must be made before the size of the adverse shock(s) is revealed (see panels B and C and the associated discussion). 27Following Christiano, Motto and Rostagno (2008), we assume that the debt contract between the entrepreneurs and lenders (households) is written in nominal terms (rather than real terms as in BGG 1999). 21

method, and calibration to Appendix B.28 Even so, it is important to highlight two features. First, in the model(cid:146)s (cid:133)scal block, government revenue is assumed to be derived from taxes on labor and capital.29 While the tax rate on capital income is (cid:133)xed, the distortionary tax on labor income reacts to annualized government debt according to the calibrated rule: (cid:28) = 0:92(cid:28) +(1 :92)0:1~b : (29) N;t N;t 1 G;t (cid:0) (cid:0) Because this tax rule has substantial inertia (cid:150)and is not very aggressive even in the long-run (cid:150)the consequences for the multiplier are very similar to lump-sum taxes.30 Second, our calibration of both the parameters of the monetary policy rule and the Calvo price and wage contract duration parameters (cid:150)while within the range of empirical estimates (cid:150)tilt in the direction of reducing the sensitivity of in(cid:135)ation to shocks. In particular, the monetary rule that is followed when policy is unconstrained is a Taylor rule with a fairly aggressive long-run coe¢ cient of 3 on in(cid:135)ation, of unity on the output gap, and 0.7 on the lagged interest rate. Our choice of a price contract duration parameter of (cid:24) = :90 implies a Phillips Curve slope of about P .007, which is on the low side of the median estimates reported in the empirical literature, even if well within reported con(cid:133)dence intervals; and wages exhibit a commensurate degree of stickiness.31 These parameter choices are aimed at capturing the resilience of core in(cid:135)ation, and measures of expected in(cid:135)ation, during the global recession. 3.1. Dynamic E⁄ects of Government Spending Figure 5 shows the e⁄ects of a front-loaded increase in government expenditures equal to 1 percent ofsteadystateoutputunderourbenchmarkcalibration. Thegovernmentspendingshockfollowsan AR(1)withapersistenceof0.9. Thespendinghikeoccursagainstthebackdropofinitialconditions consistent with a deep recession and liquidity trap expected to last eight quarters (these initial conditions, which are generated by a sequence of adverse taste shocks, are described in Appendix B). The (cid:133)gure shows results both for the benchmark model (labeled ZLB Full Model), and for 28In the models used in this paper, we have worked with log-linearized equations, aside from imposing the zero lower bound on policy rates. Given that we examine model dynamics well away from the steady state, a useful extension of our work would be to solve all model equations using nonlinear methods. 29Given a steady state government spending share of 20 percent and debt/GDP ratio of 50 percent, the steady state tax rate on labor income is 27 percent, and capital income 20 percent. 30Thecoe¢ cientsintherulearetakenfrom Traum andYang(2011),whoestimateaDSGE modelusingBayesian methods; but even simple regression analysis suggests a high degree of tax smoothing. The multiplier would be reduced substantially with a more aggressive response to debt (both for reasons noted in Section 2.5, and because it reduces the disposable income of Keynesian households). 31The median estimates of the Phillips Curve slope in recent empirical studies by e.g. Adolfson et al (2005), Altig etal. (2011),Gal(cid:237)and Gertler(1999),Gal(cid:237),Gertler,and L(cid:243)pez-Salido (2001),LindØ(2005),and Smetsand Wouters (2003; 2007) are in the range of 0:009 :014: Given our speci(cid:133)cation of the steady-state wage markup and a wage (cid:0) contract duration parameter of (cid:24) =0:85 along with a wage indexation parameter of (cid:19) =0:9 wage in(cid:135)ation is w (cid:0) w (cid:0) about as responsive to the wage markup as price in(cid:135)ation is to the price markup. 22

a variant (labeled ZLB CEE/SW) which excludes (cid:133)nancial frictions and Keynesian households. Results for each model variant are also presented for normal times in which monetary policy is unconstrained by the zero lower bound. Under either model variant, the (cid:133)scal policy expansion implies larger e⁄ects on output in a liquidity trap than in a normal situation in which policy is unconstrained. As in the stylized model in Section 2, this re(cid:135)ects that higher government spending in a liquidity trap boosts the potential real interest rate while causing real interest rates to fall (as seen in the (cid:133)gure, the nominal interest rate does not respond for some time and expected in(cid:135)ation rises). Because the rise in the potential interest rate is ampli(cid:133)ed by Keynesian households and (cid:133)nancial frictions, the output response is considerably larger in the full model than in the CEE/SW alternative when the economy is constrained by the zero bound; by contrast, the disparity in the output responses across models is much smaller in normal conditions, re(cid:135)ecting that monetary policy would raise interest rates more to o⁄set a bigger rise in the potential real rate. The lower right panel shows the present value government spending multiplier as in Uhlig (2010), which at horizon K is de(cid:133)ned as 1 K(cid:12)K(cid:1)y m = 0 t+K : (30) K g y P K 0 (cid:12)K(cid:1)g t+K Thus, the impact multiplier m is simply given P by 1 (cid:1)yt:The implied government spending multi- 0 gy (cid:1)gt plier exceeds 1.5 in the full model for well over a year after the spending shock, but is only around unity in the CEE/SW model. Against the backdrop of the eight quarter liquidity trap, the response of the government debt/GDP ratio is considerably smaller than under normal conditions under either model variant. In the full model, the government debt/GDP ratio rises only about half as much at a medium-run horizon of 3-5 years as under normal conditions, implying a much smaller rise in the labor tax rate thaninnormaltimes.32 Thesmallerdebt/GDPresponseisattributabletothreefactors. First, the higher multiplier in the liquidity trap boosts labor tax receipts substantially (i.e., holding the labor tax rate constant). Second, the fall in real interest rates lowers the cost of debt service relative to a normal situation in which real interest rates rise. Finally, capital income and hence revenue from the capital income tax rises by more. 3.2. Marginal vs. Average Responses The solid line in the upper left panel of Figure 6 shows the multiplier as function of government spendingforthefullmodelwithKeynesianhouseholdsand(cid:133)nancialfrictions.Althoughbenchmark parametersareunchanged,weconsiderasomewhatdeeperliquiditytrapthatwouldlast10quarters absent (cid:133)scal intervention (generated by larger taste shocks) to illustrate conditions su¢ cient to 32Note that while the tax-rule (29) responds to government debt as a ratio of annualized trend nominal output BGt ; the (cid:133)gure reports government debt relative to actual output BGt : The di⁄erence between the responses 4PtY 4PtYt becomes negligible after about 10 quarters as the e⁄ect of the spending shock on output dissipates. 23

produce a (cid:147)(cid:133)scal free lunch.(cid:148) The average multiplier reported in the (cid:133)gure is for a four quarter horizon (i.e. m in eq. 30); similarly, the marginal multiplier is derived from m , albeit for 3 3 in(cid:133)nitesmal increments to spending that keep the liquidity trap duration unchanged. As in the stylized model in Section 2, the marginal multiplier follows a step function. The marginal multiplier is nearly 5 for small additions to spending of less than 0:5 percent of GDP, but drops to 2.2 for a somewhat larger increments; thus, the average multiplier of 3:5 associated with a one percent of GDP spending hike lies well above the marginal.33 The rapid fallo⁄ in the multiplier re(cid:135)ects that (cid:133)scal stimulus is very e⁄ective in mitigating the e⁄ects of recession when monetary policy is constrained for a prolonged period; but with a shallower recession, the bene(cid:133)ts of additional stimulus decline substantially.34 Thesolidlineintherightupperpanelplotsthemarginalresponseofthegovernmentdebt/GDP ratio for various levels of government spending. The debt responses shown are at a (cid:147)medium-run(cid:148) horizon of 12 quarters, and indicate the direction in which labor tax rates must eventually move. For a long-lived 10 quarter liquidity trap, the marginal multiplier is large enough that government debt falls substantially below baseline after several years, consistent with an enduring reduction in labor tax rates and a (cid:133)scal free lunch. A large rise in labor tax revenue (notwithstanding a slight declineinthelabortaxrate)playsthebiggestroleinallowinggovernmentdebttofallatthemargin, while lower debt servicing costs also make a major contribution (as higher nominal GDP reduces the ratio of existing nominal debt to GDP).35 As additional spending shrinks the duration of the liquidity trap below 10 quarters (see the upper tick marks), the marginal e⁄ect on government debt turns positive, even though the average response (cid:150)shown by the dotted line (cid:150)remains negative. These results underscore how marginal increments to spending can put signi(cid:133)cant upward pressure on government debt and tax rates even when the average e⁄ects appear small. The contour of the multiplier schedule can be highly sensitive to parameters determining the responsivenessofin(cid:135)ation. ThelowerleftpanelofFigure6showsthemultiplierscheduleunderthe benchmarkcalibrationandseveralalternativesbasedoninitialconditionsthatimplyaliquiditytrap of 8 quarters (as in Figure 5). While the multiplier is below 2 under our benchmark calibration at all spending levels, the multiplier can be much higher under calibrations that imply a larger response of expected in(cid:135)ation. The ampli(cid:133)cation of the multiplier is especially dramatic under a calibration in which both price and wage contracts are relatively short-lived ((cid:147)more (cid:135)exible prices and wages(cid:148)), with the contract duration speci(cid:133)ed at four and (cid:133)ve quarters, respectively. However, although the multiplier exceeds 9 for low spending levels below 0:3 percent of GDP, the multiplier 33The tick marks in along the upper horizontal axis indicate how the duration of the liquidity trap varies with the level of government spending. 34Underlump-sumtaxadjustment,themultiplierisslightlyhigherforadeepliquiditytrapof10quarters,butnearly identical for short-lived liquidity traps, re(cid:135)ecting that the tax rule responds gradually (cid:150)and not very aggressively (cid:150) to government debt. 35Figure B.2 in Appendix B, and the associated discussion, provides a more detailed analysis of the channels through which higher government spending generates a (cid:133)scal free lunch in a 10 quarter liquidity trap. 24

drops precipitously as spending increases and the liquidity trap duration shortens. This contour of the multiplier schedule re(cid:135)ects that even relatively small adverse shocks can cause a deep recession whenin(cid:135)ationisfairlyresponsive; hence, onlyasmalldoseof(cid:133)scalexpansioniscalledfortoreverse most of these adverse e⁄ects. A second alternative examines a calibration in which prices are less sticky than under our baseline (i.e. the contract duration parameter (cid:24) is lowered from 0:90 to p 0:75), but wage-setting remains unaltered. The multiplier schedule in this case (the dotted line) is only slightly higher than under our benchmark, re(cid:135)ecting that the sluggish behavior of wages keeps price in(cid:135)ation from moving as much as under the previous alternative. This calibration underscores that the much higher multiplier under the (cid:147)more (cid:135)exible price and wage(cid:148)calibration hinges on both prices and wages being much more (cid:135)exible than under our benchmark. Finally, the marginal multiplier is also larger under the standard Taylor rule ((cid:147)looser policy rule(cid:148)) than the benchmark for small spending increments, but the quantitative disparity appears small. 4. Conclusions For an economy facing a deep recession and prolonged liquidity trap, there is a strong argument for increasing government spending on a temporary basis. But our analysis highlights the importance ofrecognizingthatthemarginalbene(cid:133)tsof(cid:133)scalstimulusmaydropsubstantiallyasspendingrises, so that there is some risk that larger spending programs may have a low marginal payo⁄, and put substantial pressure on government budgets. Governments and central banks have an array of options in addition to stimulative (cid:133)scal policy for mitigating the e⁄ects of a liquidity trap. For example, many central banks have used the asset side of their balance sheet to support credit markets by providing liquidity and purchasing longterm securities. Although the models we have examined are not designed to assess the e⁄ectiveness ofsuchactions,ouranalysishighlightstheimportanceofanalyzingthee⁄ectsofsuchactionsjointly with the (cid:133)scal stimulus packages in order to properly assess their marginal impact. As emphasized by Canova and Pappa (2011), a major issue for future research is to assess whether conditions that have been identi(cid:133)ed as likely to make (cid:133)scal policy highly e⁄ective hold empirically.36 Many recent papers, including our own, have used calibrated models with a binding zero lower bound constraint to show that a sizeable response of in(cid:135)ation plays a crucial role in generating a large spending multiplier well above unity; this is also true in models in models in which the monetary policy regime is passive at least for some time.37 However, the resilience of in(cid:135)ation in the aftermath of the global (cid:133)nancial crisis gives reason to question whether in(cid:135)ation is 36These authors provide empirical evidence suggesting that the conditions for a high multiplier did not appear to be satis(cid:133)ed during the global recession. 37Forexample,DavigandLeeper(2011)showinaregime-switchingmodelthatthegovernmentspendingmultiplier under a passive monetary policy regime is around 1-1/2 after 10 quarters, roughly twice as high as under an active policyregime. Thedisparitymainlyre(cid:135)ectsamuchlargerandmorepersistentresponseofin(cid:135)ationunderthepassive regime. 25

as responsive to (cid:133)scal policy, and to macro shocks generally, as implied by existing models that are calibrated based on estimates derived from pre-crisis data. Even more directly, recent analysis by Canova and Pappa (2011) (cid:150)using a structural VAR with sign restrictions (cid:150)found that stimulative government spending shocks induce only a transient increase in in(cid:135)ation, rather than the persistent in(cid:135)ation rise required for a big spending multiplier. In future research, it will be important to draw on evidence from the global recession to further re(cid:133)ne our empirical understanding of the role of di⁄erent factors and policies in in(cid:135)uencing the response of in(cid:135)ation to (cid:133)scal policy, including the characteristics of the monetary and (cid:133)scal policy regimes, the parameters of the price and wage Phillips Curve, and the nature of the shocks driving the economy into a liquidity trap. There are also open questions about whether the traditional channels through which (cid:133)scal policy a⁄ects aggregate demand remain operative in a severe recession. The potency of the interest rate channel might be impaired to the extent that tight credit and heavy debt burdens reduce the interest-sensitivityofhouseholdsand(cid:133)rms. AsarguedbyMertenandRavn(2010), thestimulative e⁄ects of government spending may also be muted if the source of recession is a self-fu(cid:133)lling loss in con(cid:133)dence, re(cid:135)ecting that the higher spending is perceived as a negative signal about the state of the economy. Conversely, various types of (cid:133)scal interventions could have a heightened impact through easing collateral constraints on borrowers, reducing precautionary savings, or by a⁄ecting (cid:133)nancialmarketriskpremia. Fromamodelingperspective, addressingsomeofthesequestionswill requireanon-linearstochasticframeworktocapturekeychannelsthroughwhich(cid:133)scalinterventions mayoperateinthepresenceofuncertaintysuchasinrecentworkbyBi,Leeper,andLeith(2011).38 38These authors examine how the e⁄ects of (cid:133)scal consolidation vary with the state of the economy, including the level of government debt. 26

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Figure 1.a: Negative Taste Shock and Fiscal Response 0 −i Tn −5 Potential real rate (taste shock only) Nominal interest rate (taste shock only) Pot real rate − 1% g(t) increase Pot real rate − 2% g(t) increase −10 0 4 8 Quarters Figure 1.b: Liquidity Trap Duration and Potential Real Rate 15 10 5 0 −14 −12 −10 −8 −6 −4 −2 0 Potential Real Interest Rate noitarud part ytidiuqiL 2000 1500 1000 500 0 0 2 4 6 8 10 12 )j( ψ Figure 1.c: Weights on Leads of the Interest Rate Gap in Inflation Equation Benchmark (5 qtr) 10 qtr contracts 4 qtr contracts j=0,1,... periods ahead

Figure 2: Immediate Rise in Government Spending No Inflation Response 5 Quarter Price Contracts Real Interest Rate Real Interest Rate 2 10 0 5 −2 0 −4 −6 −5 0 4 8 12 16 0 5 10 15 20 Quarters Quarters Output Gap Output Gap 2 5 0 0 −2 −5 −4 −10 0 4 8 12 16 0 4 8 12 16 Quarters Quarters Inflation Inflation 0 0 −10 −10 Taste shock only Taste shock only Both shocks Both shocks −20 Government only −20 Government only 0 4 8 12 16 0 4 8 12 16 Quarters Quarters Government Debt/GDP Government Debt/GDP 10 10 5 5 0 0 0 4 8 12 16 0 4 8 12 16 Quarters Quarters

Figure 3: Spending Multipliers and Government Debt Responses in Simple New−Keynesian Model 0.8 0.6 0.4 0.2 0 −10 −5 0 5 10 15 % Change in Govt Spend (Share of GDP) reilpitluM gnidnepS tnemnrevoG Zero Lower Bound Duration 12 11 10 9 8 7 6 5 4 3 2 1 1 0.8 0.6 0.4 0.2 Marginal multiplier Average multiplier 0 0 5 10 15 PDG lautcA ot tbeD tnemnrevoG No Inflation Response Zero Lower Bound Duration 8 7 6 5 4 3 2 1 % Change in Govt Spend (Share of GDP) 30 25 20 15 10 5 0 −10 −5 0 5 10 15 % Change in Govt Spend (Share of GDP) reilpitluM gnidnepS tnemnrevoG Zero Lower Bound Duration 12 11 10 9 8 7 6 5 4 3 2 1 1 5 qtr contracts 10 qtr contracts 0.5 4 qtr contracts 0 −0.5 −1 −1.5 0 5 10 15 % Change in Govt Spend (Share of GDP) PDG lautcA ot tbeD tnemnrevoG With Inflation Response − Alternative Contract Durations Zero Lower Bound Duration 8 7 6 5 4 3 2 1 5 qtr contracts 10 qtr contracts 4 qtr contracts

Figure 4: Immediate Government Spending Rise Under Alternative Financing Assumptions: Lump−Sum Vs. Labor−Income Taxes 2 1.5 1 0.5 0 0 4 8 12 16 Quarter tnecreP Output Potential Output Lump−Sum Taxes (benchmark) 0.2 Simple−Debt Labor−Tax Rule Balanced−Budget Labor−Tax Rule 0.1 0 −0.1 −0.2 0 4 8 12 16 tnecreP Quarter Inflation (APR) 4 3 2 1 0 0 4 8 12 16 tnecreP Nominal Interest Rate (APR) 0.2 0.1 0 0 4 8 12 16 Quarter tnecreP Quarter Potential Real Interest Rate (APR) 0.6 0.4 0.2 0 4 8 12 16 tnecreP Government debt (trend GDP share) 1 0.5 0 0 4 8 12 16 Quarter tnecreP Quarter Actual Labor Tax Rate 0.5 0 −0.5 0 4 8 12 16 tnecreP Actual minus Potential Labor Tax Rate 1 0 −1 −2 −3 0 4 8 12 16 Quarter tnecreP Quarter

Figure 5: Spending Hike in Normal Times and a Liquidity Trap in Full Model With Keynesian Agents and Financial Frictions and in CEE−SW Model Inflation (APR) 0.6 0.4 0.2 0 −0.2 0 4 8 12 16 tnecreP Nominal Interest Rate (APR) 0.4 0.3 0.2 0.1 0 −0.1 0 4 8 12 16 Quarter tnecreP Quarter Real Interest Rate (APR) 0.4 0.2 0 −0.2 −0.4 0 4 8 12 16 tnecreP 6 4 2 0 −2 0 4 8 12 16 Quarter Quarter tnecreP Potential Real Interest Rate (APR) ZLB Full Model Normal Full Model ZLB CEE/SW Normal CEE/SW Output 2 1.5 1 0.5 0 −0.5 0 4 8 12 16 tnecreP Govt Spending (trend GDP share) 1 0.8 0.6 0.4 0.2 0 0 4 8 12 16 Quarter tnecreP Quarter Government Debt to Actual Output 3 2 1 0 −1 0 4 8 12 16 tnecreP Government Spending Multiplier 2 1.5 1 0.5 0 0 4 8 12 16 Quarter reilpitluM Quarter

5 4 3 2 1 0 0 2 4 6 8 % Change in Govt Spend (Share of GDP) reilpitluM gnidnepS tnemnrevoG Zero Lower Bound Duration 10 9 8 7 6 5 4 3 2 1 2 Marginal multiplier Average multiplier 1 0 −1 −2 −3 −4 0 2 4 6 8 % Change in Govt Spend (Share of GDP) PDG lautcA ot tbeD tnemnrevoG Figure 6: Marginal Output Multipliers and Government Debt Responses in Full Model with Keynesian Agents and Financial Frictions Benchmark Calibration (10 quarter liquidity trap absent stimulus) Zero Lower Bound Duration 10 9 8 7 6 5 4 3 2 1 Marginal response Average response 10 8 6 4 2 0 0 1 2 3 4 5 6 % Change in Govt Spend (Share of GDP) reilpitluM gnidnepS tnemnrevoG Benchmark and Alternative Calibrations (8 quarter liquidity trap absent stimulus) Zero Lower Bound Duration (more flexible p and w) 8 7 6 5 4 3 2 1 Benchmark calibration 2 More flexible p and w 0 More flexible prices Looser policy rule −2 −4 −6 −8 0 1 2 3 4 5 6 % Change in Govt Spend (Share of GDP) PDG lautcA ot tbeD tnemnrevoG Zero Lower Bound Duration (more flexible p and w) 8 7 6 5 4 3 2 1 Benchmark calibration More flexible p and w More flexible prices Looser policy rule

Appendix A. The Simple New-Keynesian Model Appendix A contains two parts. Section A.1 describes and derives the model used in Section 2, including both the benchmark model with lump-sum taxes and the variant with distortionary labor income taxes.A.1 In Section A.2, we discuss some additional results referred to in Section 2 of the main text (including several supplementary (cid:133)gures). A.1. The Model A.1.1. Households The utility functional for the representative household is Et 1 (cid:12)j 1 1 1 (C t+j (cid:0) C(cid:23) t+j )1 (cid:0)(cid:27) 1 (cid:0) N 1+ t 1 + + j (cid:31) (cid:31) +(cid:22) 0 F MB t P +j+1 (h) (A.1) j=0 ( (cid:0) (cid:27) (cid:18) t+j (cid:19) ) X where the discount factor (cid:12) satis(cid:133)es 0 < (cid:12) < 1: The period utility function depends on the household(cid:146)s current consumption C as deviation from a (cid:147)reference level(cid:148)C(cid:23) , where a positive t t+j taste shock (cid:23) raises this reference level and thus the marginal utility of consumption associated t with any given consumption level. The period utility function also depends inversely on hours worked N : Following Eggertsson and Woodford (2003), the subutility function over real balances, t F MBt+j+1(h) ,isassumedtohaveasatiationpointforMB=P. Hence,inclusionofmoney-which Pt+j is (cid:16)a zero nomin(cid:17)al interest asset - provides a rationale for the zero lower bound on nominal interest rates. However, we maintain the assumptions that money is additive and that (cid:22) is arbitrarily 0 small so that changes in real money balances have negligible implications for seignorage. Together, these assumptions imply that we can disregard the implications of money for government debt and output. The household(cid:146)s budget constraint in period t states that its expenditure on goods and net purchases of (zero-coupon) government bonds B must equal its disposable income: G;t P (1+(cid:28) )C +B +MB = (1 (cid:28) )W N +(1+i )B +MB T +(cid:0) (A.2) t C;t t G;t t+1 N;t t t t 1 G;t 1 t t t (cid:0) (cid:0) (cid:0) (cid:0) Thus, the household purchases the (cid:133)nal consumption good (at a price of P ) and subject to a sales t tax (cid:28) . Each household earns after-tax labor income (1 (cid:28) )W N ((cid:28) denotes the tax rate), C;t N;t t t N;t (cid:0) pays a lump-sum tax T (this may be regarded as net of any transfers), and receives a proportional t share of the pro(cid:133)ts (cid:0) of all intermediate (cid:133)rms. t In every period t, the household maximizes the utility functional (B.8) with respect to its consumption, labor supply and bond holdings. Forming the Lagrangian and computing the (cid:133)rst- A.1Inderivingthemodel,wealsoincludeasalestaxtocomplementthediscussioninSection2.4onthecomposition of the tax base. 36

order conditions w.r.t. [ C N B ], we obtain t t G;t 1 (C t C(cid:23) t ) (cid:0)(cid:27) (cid:21) t P t (1+(cid:28) C;t ) = 0; (cid:0) (cid:0) (cid:31) N +(cid:21) (1 (cid:28) )W = 0; t t N;t t (cid:0) (cid:0) (cid:21) +(cid:12)(1+i )E (cid:21) = 0; t t t t+1 (cid:0) and by de(cid:133)ning (cid:3) (cid:21) P as the pre-tax cost of consumption in utility units, we can rewrite the t t t (cid:17) (cid:133)rst-order conditions as 1 (C t C(cid:23) t ) (cid:0)(cid:27) (cid:3) = (cid:0) ; t (1+(cid:28) ) C;t W (cid:31) t N = (cid:3) (1 (cid:28) ) ; t t (cid:0) N;t P t (1+i ) t (cid:3) = (cid:12)E (cid:3) ; t t t+1 1+(cid:25) t+1 where we have introduced the notation 1+(cid:25) = P =P . t+1 t+1 t By substituting out for (cid:3) , we derive the consumption Euler equation t 1 1 (C t C(cid:23) t ) (cid:0)(cid:27) (1+i t ) (C t+1 C(cid:23) t+1 ) (cid:0)(cid:27) (cid:0) = (cid:12)E (cid:0) ; (A.3) t (1+(cid:28) ) 1+(cid:25) (1+(cid:28) ) C;t t+1 C;t+1 and the following labor supply schedule N (cid:31) (1 (cid:28) )W mrs t = (cid:0) N;t t : (A.4) t (cid:17) (C t C(cid:23) t ) (cid:0)(cid:27) 1 (1+(cid:28) C;t ) P t (cid:0) (A.3) and (A.4) are the key equations for the household side of the model. A.1.2. Firms We assume a familiar setting with a continuum of monopolistically competitive (cid:133)rms to rationalize Calvo-style price stickiness. The framework in the stylized model is identical to that described below in the full model with capital (Appendix B.1.1), with two important exceptions. First, aggregate capital is assumed to be (cid:133)xed, so that aggregate production is given by Y = K(cid:11)N1 (cid:11): (A.5) t t(cid:0) Despite the (cid:133)xed aggregate stock, shares of the aggregate capital stock can be freely allocated across the f (cid:133)rms, implying that real marginal cost, MC (f)=P is identical across (cid:133)rms and equal t t to MC W =P W =P t t t t t = = : (A.6) P MPL (1 (cid:11))K(cid:11)N (cid:11) t t t(cid:0) (cid:0) The second notable di⁄erence relative to the setup in the full model with capital is that here we do not allow for dynamic indexation to lagged in(cid:135)ation. Instead, all (cid:133)rms which are not allowed 37

to reoptimize their prices in period t (which is the case with probability (cid:24) ), update their prices p according to the following formula P~ = (1+(cid:25))P ; (A.7) t t 1 (cid:0) where (cid:25) is the steady-state (net) in(cid:135)ation rate and P~ is the updated price. t opt GivenCalvo-stylepricingfrictions, (cid:133)rmf thatisallowedtoreoptimizeitsprice(P (f))solves t the following problem max E 1 (cid:24)j (1+(cid:25))jP opt (f) MC Y (f) t p t;t+j t t+j t+j Popt(f) (cid:0) t X j=0 h i where is the stochastic discount factor (the conditional value of future pro(cid:133)ts in utility units, t;t+j i.e. (cid:12)jE (cid:21)t+j, recalling that the household is the owner of the (cid:133)rms), (cid:18) the net markup and the t (cid:21)t p (1+(cid:18)p) demand for (cid:133)rm f is given by Y (f) = P t(cid:3) (f) (cid:0) (cid:18)p Y . The (cid:133)rst-order condition is given by t+j Pt t h i E 1 (cid:24)j (1+(cid:25))j (cid:0) 1 (cid:0) (1+(cid:18) p ) 1 MC Y (f) = 0; t j=0 p t;t+j " (cid:18) p (cid:0) (cid:18) p P t opt (f) t+j # t+j X which after multiplying through by - (cid:18)p P opt (f) can be rewritten as 1+(cid:18)p t 1 (1+(cid:25))jP opt (f) E (cid:24)j t MC Y (f) = 0 (A.8) t p t;t+j 1+(cid:18) (cid:0) t+j t+j " p # j=0 X By implication of equations (B.2) and (A.7), the evolution of the (cid:133)nal goods price is given by 1 (cid:18)p opt (cid:0)(cid:18)p (cid:0)(cid:18)p 1 (cid:0) P = 1 (cid:24) P +(cid:24) ((1+(cid:25))P ) (A.9) t 2 (cid:0) p t p t (cid:0) 1 3 (cid:16) (cid:17) (cid:0) (cid:1) 4 5 where we have used the fact that all (cid:133)rms that reoptimize will set the same price (because they face the same costs for labor and capital), and that the updating price for the non-optimizing (cid:133)rms equals the past aggregate price (as we consider a continuum of (cid:133)rms which does not re-optimize). A.1.3. Government The evolution of nominal government debt is determined by the following equation B = (1+i )B +P G (cid:28) P C (cid:28) W N T MB +MB (A.10) G;t t 1 G;t 1 t t C;t t t N;t t t t t+1 t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) where G denotes real government expenditures on the (cid:133)nal good Y : Scaling with 1=(P Y); we t t t obtain B (1+i )B G C W N T MB MB G;t t 1 G;t 1 t t t t t t+1 t = (cid:0) (cid:0) + (cid:28) C;t (cid:28) N;t + : (A.11) P Y (1+(cid:25) ) P Y Y (cid:0) Y (cid:0) P Y (cid:0) P Y (cid:0) P Y P Y t t t 1 t t t t (cid:0) 38

In the benchmark model where lump-sum taxes stabilize the evolution of government debt (as shareofnominaltrendGDP,b BG;t),weassumethatlump-sumtaxesasshareofnominaltrend G;t (cid:17) PtY GDP, (cid:28) Tt , follows the rule (11). In the variant of the model with distortionary labor-income t (cid:17) PtY taxes, the lump-sum tax rule is replaced by the rule (23). Turning to the central bank, it is assumed to adhere to the non-linear Taylor-type policy rule (in log-linearized form) in equation (3), where i denotes the steady-state (net) nominal interest rate, which is given by r+(cid:25) where r 1=(cid:12) 1. (cid:17) (cid:0) A.1.4. The Aggregate Resource Constraint We now turn to discuss the derivation of the aggregate resource constraint. Let Y denote the t(cid:3) unweighted average (sum) of output for each (cid:133)rm f, i.e. 1 Y = Y (f)df t(cid:3) t Z0 (1+(cid:18)p) Recalling that Y (f) = P t(cid:3) (f) (cid:0) (cid:18)p Y , it follows that t+j Pt t h i (1+(cid:18)p) 1 1 P t (f) (cid:0) (cid:18)p Y = Y (f)df = Y df t(cid:3) t P t Z0 Z0 (cid:20) t (cid:21) 1 (cid:0) (1 (cid:18) + p (cid:18)p) 1 (cid:0) (1+(cid:18)p) (1 (cid:0) + (cid:18) (cid:18) p p) (cid:0) (1 (cid:18) + p (cid:18)p) = P t (f) (cid:18)p df Y t P 2 3 (cid:18) t (cid:19) (cid:18)Z0 (cid:19) (1+(cid:18)p)4 5 = P t(cid:3) (cid:0) (cid:18)p Y ; t P t (cid:18) (cid:19) where Y is aggregate output of the (cid:133)nal good sector, as de(cid:133)ned above, and P is the indicated t t(cid:3) weighted average of the individual prices, de(cid:133)ned as (cid:18)p 1 (cid:0) (1+(cid:18)p) (1 (cid:0) +(cid:18)p) P t(cid:3) P t (f) (cid:18)p df : (A.12) (cid:17) (cid:18)Z0 (cid:19) Notice how the weights for P di⁄er from what they are for the aggregate price level P (see eq. t(cid:3) t B.2): Now, actual output is Y ; and this is what is available to be divided into private consumption t and government spending: Y = C +G : (A.13) t t t Using the de(cid:133)nition of the production function (A.5), we can write the resource constraint in real terms as follows: (1+(cid:18)p) C +G P t(cid:3) (cid:18)p K(cid:11)N 1 (cid:11): (A.14) t t t (cid:0) (cid:20) P t (cid:18) (cid:19) (cid:17) Yt (cid:17) Y t(cid:3) The sticky price distortion clearly introduces a wedge between input use and the output available | {z } | {z } for consumption (including by the government). Even so, this term vanishes in the log-linearized version of the model. 39

A.1.5. Equilibrium We now collect the equilibrium relationships in the model and derive a log-linear approximation of the model. Collecting the equations First, we may regard the households equations (A.3) and (A.4) as determining C and N , and marginal cost relation equation (A.6) as determining MC =P ; and t t t t the aggregate resource constraint (A.14) as determining the real wage W =P . The Taylor-type t t policy rule determines the nominal interest rate i , and the (cid:133)rms pricing equations (A.8) and (A.9) t determines the evolution of the aggregate price level P , whereas the (shadow) gross real interest t rate 1+r is determined by the Fisher relationship t (1+i ) t 1+r = E (A.15) t t (1+(cid:25) ) t+1 Finally, the (cid:133)scal budget constraint (A.11) determines the evolution of government debt B , and G;t the (cid:133)nal goods resource constraint (A.13) relate consumption and government spending to (cid:133)nal output Y . The other (cid:133)scal variables, G ;(cid:28) ;(cid:28) and (cid:28) , are exogenous or determined by policy t t C;t N;t t rules: Log-linear Approximation of Model We will now derive the equations in Section 2 in turn. We start with the sticky price equilibrium conditions, and then discuss the (cid:135)ex-price equilibrium. In general, a log-linearized variable is denoted with lower case letters, and derived as dX t x = ; (A.16) t X except in the special case X = 0 when the log-linearized variable is simply given by dX (e..g t government debt as share of nominal trend GDP, and the lump-sum tax rate). Moreover, for in(cid:135)ation and interest rates, we use the approximation that d(1+x ) x because x is small. t t t (cid:25) Finally, notice that for distortionary tax rates, we use d(cid:28) (cid:28) (thus, rather than introducing X;t X;t (cid:17) new notation, the tax rates are henceforth understood to be in deviations from their steady state level; this is also the case for the preference shock (cid:23) ). t Totally di⁄erentating the government debt evolution equation (A.11);we obtain (dropping the seignorage term which is assumed to be arbitrarily small) 1 (cid:11) b = (1+r)b +g g c ((cid:28) +(cid:28) c ) (cid:0) ((cid:28) +(cid:28) (cid:16) +(cid:28) n ) (cid:28) +b (1+r)(i (cid:25) ); G;t G;t (cid:0) 1 y t (cid:0) y C;t C t (cid:0)1+(cid:18) p N;t N t N t (cid:0) t G t (cid:0) 1 (cid:0) t (A.17) where we have introduced the notation that (cid:16) represents the real wage (as percent deviation from t steady state, i.e. d(W =P )=(W=P)), de(cid:133)ned g G=Y, and used that WN = 1 (cid:11) s and our t t y (cid:17) PY 1+(cid:0)(cid:18)p (cid:17) N simplifying assumption that b = 0. Assuming that the labor income tax is the only tax which G balances the budget in steady state, it then follows that: 1 (cid:11) g = (cid:0) (cid:28) ; (A.18) y N 1+(cid:18) p 40

implyingthatthelog-linearizedbudgetconstraintinthebenchmarkmodelwithlump-sumtaxescan be written as (10) in Section 2. However, in the model in Section 2.5 where dynamic adjustments in (cid:28) stabilizes government debt, (A.17) is the budget constraint when setting (cid:28) = (cid:28) = 0 for N;t C;t t all t and (cid:28) = 0. C To derive a log-linearized representation for real marginal cost, we work from the equation (A.6), which implies (cid:11) mc = (cid:16) y +n = (cid:16) + y ; t t (cid:0) t t t 1 (cid:11) t (cid:0) where the second equality follows from (A.5): By noting that real marginal cost is constant in the (cid:135)ex-price equilibrium, we have (cid:11) pot pot pot pot pot (cid:16) y +n = (cid:16) + y = 0: (A.19) t (cid:0) t t t 1 (cid:11) t (cid:0) Accordingly, we can write (log-linearized) real marginal cost as (cid:11) pot pot mc = (cid:16) (cid:16) + y y : (A.20) t t (cid:0) t 1 (cid:11) t (cid:0) t (cid:16) (cid:17) (cid:0) (cid:16) (cid:17) In order to write this equation solely in terms of the output gap, pot x y y ; (A.21) t t t (cid:17) (cid:0) we need to derive a log-linearized equation for the real wage. To obtain such a measure, we loglinearize equation (A.4) to obtain 1 (cid:28) (cid:28) N;t C;t (cid:31)n + (c (cid:23)(cid:23) ) = (cid:16) ; t (cid:27)(1 (cid:23)) t (cid:0) t t (cid:0) 1 (cid:28) (cid:0) 1+(cid:28) N C (cid:0) (cid:0) again recalling that (cid:28) for j = [N;C] and (cid:23) are to be interpreted as percentage point deviations. j;t t By log-linearizing and substituting the aggregate resource constraint in (A.13) into this expression, we obtain 1 1 (cid:28) (cid:28) N;t C;t (cid:16) = (cid:31)n + (y g g ) (cid:23)(cid:23) + + ; t t (cid:27)(1 (cid:23)) 1 g t (cid:0) y t (cid:0) t 1 (cid:28) 1+(cid:28) y N C (cid:0) (cid:18) (cid:0) (cid:19) (cid:0) andusing(A.5),i.e. thatn = 1 y ;we(cid:133)nallyderivethefollowingexpressionforthelog-linearized t 1 (cid:11) t (cid:0) real wage: (cid:31) 1 g (cid:23) 1 1 y (cid:16) = + y g (cid:23) + (cid:28) + (cid:28) : t 1 (cid:11) (cid:27)(1 (cid:23))(1 g ) t (cid:0)(cid:27)(1 (cid:23))(1 g ) t (cid:0)(cid:27)(1 (cid:23)) t 1 (cid:28) N;t 1+(cid:28) C;t y y N C (cid:18) (cid:0) (cid:0) (cid:0) (cid:19) (cid:0) (cid:0) (cid:0) (cid:0) (A.22) Next, we log-linearize the consumption Euler equation, (A.3), to get c (cid:23)v 1 c (cid:23)(cid:23) t t t+1 t+1 (cid:0) = E i (cid:25) (cid:1)(cid:28) (cid:0) ; t t t+1 C;t+1 (cid:0)(cid:27)(1 (cid:23)) (cid:0) (cid:0) 1+(cid:28) (cid:0) (cid:27)(1 (cid:23)) c (cid:0) (cid:20) (cid:0) (cid:21) where we have used that 1+i 1 = (cid:12) = (cid:12)(1+r): 1+(cid:25) 41

By substituting the log-linearized aggregate resource constraint (A.13) into this expression, and de(cid:133)ning: (cid:27)^ (cid:27)(1 (cid:23))(1 g ): (A.23) y (cid:17) (cid:0) (cid:0) we obtain after some re-arranging: (cid:27)^ y = E y (cid:27)^(i E (cid:25) ) g E (cid:1)g (1 g )(cid:23)E (cid:1)(cid:23) + E (cid:1)(cid:28) ; (A.24) t t t+1 t t t+1 y t t+1 y t t+1 t C;t+1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 1+(cid:28) c which is the log-linearized IS curve equation. Using the labor supply equation (A.22) and labor demand equation (A.19) under (cid:135)exible prices, we get (cid:31) 1 (cid:11) g (cid:23) 1 1 pot y pot + + y = g + (cid:23) (cid:28) (cid:28) ; 1 (cid:11) (cid:27)^ 1 (cid:11) t (cid:27)^ t (cid:27)(1 (cid:23)) t (cid:0) 1 (cid:28) N;t(cid:0) 1+(cid:28) C;t N C (cid:18) (cid:0) (cid:0) (cid:19) (cid:20) (cid:0) (cid:0) (cid:21) pot where we use the notation z for endogenous variables, and simply z for exogenous variables. t t pot Notice that (cid:28) for the moment is treated as an endogenous variable as it potentially depends on N;t other endogenous variables via (23). Using the notation (cid:31) 1 (cid:11) (cid:30) + + ; (A.25) mc (cid:17) 1 (cid:11) (cid:27)^ 1 (cid:11) (cid:0) (cid:0) the solution for potential output can be written 1 (cid:27)^ (cid:27)^ pot pot y = g g +(1 g )(cid:23)(cid:23) (cid:28) (cid:28) : (A.26) t (cid:30) (cid:27)^ y t (cid:0) y t (cid:0) 1 (cid:28) N;t(cid:0) 1+(cid:28) C;t mc (cid:20) (cid:0) N C (cid:21) To get a tractable solution for the potential real interest rate, we use the de(cid:133)nition in (A.23) to rearrange (A.24) as: 1 g 1 g 1 pot pot y y r = E (cid:1)y E (cid:1)g (cid:0) (cid:23)E (cid:1)(cid:23) + E (cid:1)(cid:28) ; t (cid:27)^ t t+1(cid:0) (cid:27)^ t t+1 (cid:0) (cid:27)^ t t+1 1+(cid:28) t C;t+1 c pot and by substituting the expression for y in (A.26) into this equation, we obtain t r t pot = (cid:27)^(cid:30) 1 mc E t " 1 1 g (cid:28) (cid:27)^ y N (cid:1) (cid:1) g (cid:28) t+ p N o 1 ; t t + +1 1 (cid:0) (cid:0) (cid:27)^ g 1 y + (cid:23) 1 (cid:28) (cid:1) C (cid:23) (cid:1) t+ (cid:28) 1 C (cid:0) ;t+1 # (cid:0) g (cid:27)^ y E t (cid:1)g t+1 (cid:0) 1 (cid:0) (cid:27)^ g y (cid:23)E t (cid:1)(cid:23) t+1 + 1+ 1 (cid:28) c E t (cid:1)(cid:28) C;t+1 ; (cid:0) which can be rearranged as r pot = 1 1 1 E [ g (cid:1)g (1 g )(cid:23)(cid:1)(cid:23) ] 1 E (cid:1)(cid:28) pot + 1 1 1 E (cid:1)(cid:28) ; t (cid:27)^ (cid:0) (cid:27)^(cid:30) mc t (cid:0) y t+1 (cid:0) (cid:0) y t+1 (cid:0)(cid:27)^(cid:30) mc (1 (cid:0) (cid:28)N) t N;t+1 (cid:0) (cid:27)^(cid:30) mc 1+(cid:28)c t C;t+1 (cid:16) (cid:17) (cid:16) (cid:17) (A.27) which is the general solution for the potential real interest rate. The Benchmark Model With Lump-sum Taxes From the equations above, it is an easy task to derive the benchmark model with lump-sum taxes. In this version of the model, 42

(cid:28) = (cid:28) = 0 for all t. Accordingly, equation (5) follows from (A.27), and (4) follows from N;t C;t (A.26). The IS-curve (1) obtains from (A.24) which holds for actual and potential output, so that: (E y (cid:27)^(i E (cid:25) ) g E (cid:1)g (1 g )(cid:23)E (cid:1)(cid:23) ) t t+1 t t t+1 y t t+1 y t t+1 y t (cid:0) y t pot = (cid:0) E t (cid:0) y t p + ot 1(cid:0) (cid:0) (cid:27)^r t pot (cid:0) g y (cid:0) E t (cid:1)g t+1 (cid:0) (1 (cid:0) (cid:0) g y (cid:0) )(cid:23)E t (cid:1)(cid:23) t+1 ; (cid:16) (cid:17) which can be written as equation (1) by using the de(cid:133)nitions (A.21) and (A.23): As is well-known, log-linearization of (A.8) and (A.9) around the in(cid:135)ation target (cid:25) results in the following Phillips curve 1 (cid:24) 1 (cid:12)(cid:24) p p (cid:25) = (cid:12)E (cid:25) = (cid:0) (cid:0) mc : (A.28) t t t+1 t (cid:24) (cid:0) (cid:1)(cid:0)p (cid:1) To write the model in terms of the output gap x instead of mc as in the text, we use (A.20) and t t (A.22), which in the model with time-varying lump-sum taxes simpli(cid:133)es to (cid:11) pot pot mc = (cid:16) (cid:16) + y y t t (cid:0) t 1 (cid:11) t (cid:0) t (cid:16) (cid:31) (cid:17) (cid:0) 1(cid:16) (cid:17) (cid:11) pot pot = + y y + y y 1 (cid:11) (cid:27)(1 (cid:23))(1 g ) t (cid:0) t 1 (cid:11) t (cid:0) t y (cid:18) (cid:0) (cid:0) (cid:0) (cid:19) (cid:16) (cid:17) (cid:0) (cid:16) (cid:17) = (cid:30) x ; mc t where x is de(cid:133)ned accordingly with (A.21) and (cid:30) is de(cid:133)ned as in (A.25). Using this in (A.28), t mc we obtain (2) with (cid:20) de(cid:133)ned as in (7). p As mentioned previously, (10) obtains from (A.17) by using (cid:28) = 0 and (A.18). Apart from C pot the equations stated in the main text, we use (A.26) to compute y , which enables us to compute t pot actual output as y = x +y : To get hours worked and real wage in (10), we use (A.22) and t t t n = 1 y . t 1 (cid:11) t (cid:0) The Model With Exogenous Distortionary Labor-Income Taxes In this variant of the model we have that (cid:28) varies exogenously, but we still assume that (cid:28) = (cid:28) = 0 for all t. N;t C;t C Since labor income taxes are exogenous, all the model equations are identical, except that the labor pot pot income tax a⁄ects r according to (A.27) and y according to (A.26). In this version of the t t model, the IS-curve (1) is identical to the benchmark model. Finally, (cid:28) enters the government N;t budget constraint (A.17). The Model With Endogenous Distortionary Labor-Income Taxes In this variant of the model (cid:28) varies endogenously according to the rule given by equation (23 in Section 2.5, but N;t westillassume(cid:28) = (cid:28) = 0forallt. Alsointhisversionofthemodel, theIS-curve(1)isidentical C;t C to the benchmark model. However, the expression for marginal costs changes, because it follows from (A.22) that the following wedge between the actual and potential labor tax-rate will enter 43

into marginal costs: (cid:11) pot pot mc = (cid:16) (cid:16) + y y t t (cid:0) t 1 (cid:11) t (cid:0) t (cid:16) (cid:31) (cid:17) (cid:0) 1(cid:16) (cid:17) 1 (cid:11) pot pot pot = + y y + (cid:28) (cid:28) + y y 1 (cid:11) (cid:27)(1 (cid:23))(1 g ) t (cid:0) t 1 (cid:28) N;t (cid:0) N;t 1 (cid:11) t (cid:0) t y N (cid:18) (cid:0) (cid:0) (cid:0) (cid:19) (cid:16) (cid:17) (cid:0) (cid:16) (cid:17) (cid:0) (cid:16) (cid:17) 1 pot = (cid:30) x + (cid:28) (cid:28) ; mc t 1 (cid:28) N;t (cid:0) N;t N (cid:0) (cid:16) (cid:17) implying that a negative gap between the actual and potential labor income tax rate will put downward pressure on marginal costs and hence in(cid:135)ation. In all other aspects, this variant is identical to the model with exogenous (cid:28) , with the exception that the exogenous process for (cid:28) N;t N;t is replaced with the rule (23) and that (cid:28) = 0 for all t. t A.2. Additional Results in Simple Benchmark Model Below, we report and discuss brie(cid:135)y some additional results referred to in Section 2 of the main text. A.2.1. Complete Stabilization when Policy Unconstrained (Section 2.2) As indicated in Section 2.2, Woodford (2003) shows that the monetary policy rule in equation (3), if unconstrained, can fully stabilize in(cid:135)ation at target and output at potential if the coe¢ cients on in(cid:135)ation and/or the output gap are su¢ ciently large (i.e., in(cid:135)ation and the output gap become arbitrarily close to zero). Here we reproduce this result in our augmented model that includes exogenous shocks to the labor and sales tax rates to illustrate that other shocks that generate pot the same path of r as the taste shock would have identical implications for the duration of the t liquidity trap, the path of policy rates, and the multiplier. To show the full stabilization result, substitute the unconstrained monetary policy rule into the IS curve equation (1)to get: pot x = E x (cid:27)^((cid:13) (cid:25) +(cid:13) x E (cid:25) r ); (A.29) t t t+1 (cid:25) t x t t t+1 t (cid:0) (cid:0) (cid:0) which can be rearranged as: pot (1+(cid:27)^(cid:13) )x = E x (cid:27)^(cid:13) (cid:25) +(cid:27)^E (cid:25) +(cid:27)^r : (A.30) x t t t+1 (cid:25) t t t+1 t (cid:0) Taking the lead of this equation, multiplying by (cid:12); and subtracting the result from equation (A.30) yields: pot pot (1+(cid:27)^(cid:13) )[x (cid:12)E x ] = [x (cid:12)E x ] (cid:27)^(cid:13) [(cid:25) (cid:12)E (cid:25) ]+(cid:27)^[(cid:25) (cid:12)E (cid:25) ]+(cid:27)^[r (cid:12)E r ]: x t (cid:0) t t+1 t+1 (cid:0) t t+2 (cid:0) (cid:25) t (cid:0) t t+1 t+1 (cid:0) t t+2 t (cid:0) t t+1 (A.31) Grouping terms in x and its leads on the left hand side of the equality, using the Phillips curve t (2), and moving the exogenous shocks to the right, yields: 44

1+(cid:12)(1+(cid:27)^(cid:13) )+(cid:27)^(cid:20) 1+(cid:27)^(cid:13) +(cid:27)^(cid:20) (cid:13) (cid:27)^ E x x p E x + x p (cid:25)x = r pot (cid:27)^E r pot ; (A.32) t t+2 (cid:0) (cid:12) t t+1 (cid:12) t (cid:12) t (cid:0) t t+1 This equation can be written in the alternative form in the lead operator: (cid:27)^ E (cid:9)(L 1)x = (cid:21) = r pot (cid:27)^E r pot ; (A.33) t (cid:0) t x;t (cid:12) t (cid:0) t t+1 where 1+(cid:12)(1+(cid:27)^(cid:13) )+(cid:27)^(cid:20) 1+(cid:27)^(cid:13) +(cid:27)^(cid:20) (cid:13) (cid:9)(L 1) = L 2 x p L 1+ x p (cid:25) = (L 1 v )(L 1 v ); (A.34) (cid:0) (cid:0) (cid:0) (cid:0) 1 (cid:0) 2 (cid:0) (cid:12) (cid:12) (cid:0) (cid:0) and (cid:21) in equation (A.33) depends only on the exogenous shocks, and the parameters v and v x;t 1 2 in equation (A.34) have the following relation to the structural parameters: 1+(cid:12)(1+(cid:27)^(cid:13) )+(cid:27)^(cid:20) 1+(cid:27)^(cid:13) +(cid:27)^(cid:20) (cid:13) v +v = x p ; v v = x p (cid:25) (A.35) 1 2 1 2 (cid:12) (cid:12) The exogenous shocks g ; (cid:23) ; (cid:28) and (cid:28) are assumed to follow (cid:133)rst order autoregressions t t N;t C;t with persistence coe¢ cients (1 (cid:26) ); (1 (cid:26) ); (1 (cid:26) ); and (1 (cid:26) ); respectively (as a slight G v n c (cid:0) (cid:0) (cid:0) (cid:0) generalization of the model in the main text, which imposes (cid:26) = (cid:26) ): Accordingly, equation G v (A.27) for the potential real rate may be expressed (noting E (cid:1)g = (cid:26) g , and similarly for t t+1 G t (cid:0) the other shocks): r pot = 1 1 1 [g (cid:26) g +(1 g )(cid:23)(cid:26) (cid:23) + 1 (cid:26) (cid:28) + 1 1 1 (cid:26) (cid:28) : (A.36) t (cid:27)^ (cid:0) (cid:27)^(cid:30) mc y G t (cid:0) y v t (cid:27)^(cid:30) mc (1 (cid:0) (cid:28)N) n N;t (cid:0) (cid:27)^(cid:30) mc 1+(cid:28)c c C;t (cid:16) (cid:17) (cid:16) (cid:17) Using equation (A.36);(cid:21) may be expressed in terms of the exogenous shocks as: xt 1 (cid:21) = 1 1 [g (cid:26) (1 (cid:12)+(cid:12)(cid:26) )g +(1 g )(cid:23)(cid:26) (1 (cid:12)+(cid:12)(cid:26) )(cid:23) + (cid:27)^ (cid:26) (cid:28) ]+ 1 (cid:26) (cid:28) x;t (cid:12) (cid:0) (cid:27)^(cid:30) mc y G (cid:0) G t (cid:0) y v (cid:0) v t 1+(cid:28)c c C;t (cid:12)(cid:30) mc (1 (cid:0) (cid:28)N) n N;t (cid:16) (cid:17) (A.37) or equivalently: (cid:21) = (cid:30) g +(cid:30) (cid:23) +(cid:30) (cid:28) +(cid:30) (cid:28) : (A.38) x;t xg t xv t xc C;t xn N;t We next premultiply equation (A.33) by 1 to yield: v1v2 1 1 1 1 E (cid:9)(L 1)x = E ( L 1 1)( L 1 1)x = (cid:21) : (A.39) t (cid:0) t t (cid:0) (cid:0) t x;t v v v (cid:0) v (cid:0) v v 1 2 1 2 1 2 As shown in Woodford (2003) Chapter 4 (and proved in his Appendix C.2), the roots v and v of 1 2 thecharacteristicequation(A.35)lieoutsidetheunitcircle, andhencehaveadeterminatesolution, provided that (cid:13) + 1 (cid:12)(cid:13) > 1 (the output gap is not annualized here, and hence not divided by (cid:25) (cid:20)(cid:0) p x 4 as in Woodford(cid:146)s equation C.18). Solving equation (A.39) forward, the output gap may be expressed in the form: 45

(cid:30) (cid:30) (cid:30) (cid:30) x = xg g + xv(cid:23) + xc(cid:28) + xn(cid:28) ; (A.40) t t t C;t N;t m m m m g v c n where the coe¢ cients m is given by: g 1 (cid:27)^(cid:13) (cid:27)^(cid:20) m = v v (1 (cid:26) )(v +v )(1 (cid:26) )2 = (cid:26) ( (1 (cid:26) ))+ x(1 (1 (cid:26) )(cid:12))+ p ((cid:13) (1 (cid:26) )) > 0 g 1 2 (cid:0) (cid:0) G 1 2 (cid:0) G G (cid:12)(cid:0) (cid:0) G (cid:12) (cid:0) (cid:0) G (cid:12) (cid:25) (cid:0) (cid:0) G (A.41) The coe¢ cients m , m ; and m are identical in form, but with di⁄erent persistence parameters v c n (cid:26) ;(cid:26) , and (cid:26) ; respectively. Because m depends linearly on the monetary rule coe¢ cients (cid:13) and v c n g (cid:25) (cid:13) , whiletheparameters(cid:30) ;(cid:30) ;(cid:30) ;(cid:30) donot, equation(A.40)impliesthattheoutputgapmay x xg xv xc xn be kept arbitrarily close to zero for a rule that reacts aggressively enough either to in(cid:135)ation, the output gap, or both. Moreover, given the form of the price-setting equation (2), in(cid:135)ation can also be fully stabilized. With in(cid:135)ation and the output gap at target, the IS curve given by equation pot (1) implies that the policy rate i simply tracks r : t t As seen from equation (A.36), shocks to the labor tax rate (cid:28) or sales tax rate (cid:28) can be N;t C;t pot scaledtohavethesamee⁄ectonthepath ofr (providedthepersistenceparametersareidentical). t pot This would also be true for productivity shocks (in a slight generalization of the model). Since r t is the only shock in the model in a reduced form sense, an alternative con(cid:133)guration of structural pot shocks (to taste, tax rates, or productivity) that produced the same path for r would have the t same implications for the duration of the liquidity trap and for the (cid:133)scal multiplier. A.2.2. Allowing for a Price Markup Shock (Section 2.3) Complementing our discussion in Section 2.3, Figure A.1 shows the e⁄ects of an equally-sized government spending hike under a baseline that incorporates a price markup shock in addition to the taste shock. The price markup shocks induces a trade-o⁄ between in(cid:135)ation and output gap stabilization. Speci(cid:133)cally, the aggregate supply curve (2) now incorporates a price markup shock (cid:18) : p;t (cid:25) = (cid:12)(cid:25) +(cid:20) (x +(cid:18) ) t t+1t p t p;t j where (cid:18) = 0:5(cid:18) +" : In all other respects, the model is unchanged. p;t p;t 1 p;t (cid:0) We consider two alternative monetary policy rules that di⁄er in the relative weight that they put on stabilizing the output gap compared to in(cid:135)ation. At one extreme, we assume that the Taylor rule is very aggressive in stabilizing the output gap, setting (cid:13) = 1000 and (cid:13) = 1: In x (cid:25) this case, the path of the policy rate is essentially identical to that in the baseline with only taste shocks,re(cid:135)ectingthatthemarkupshockhasverysmalle⁄ectsonin(cid:135)ationtwoormoreyearsahead. Accordingly, a temporary rise in government spending has the same impact as in our benchmark case which excludes markup shocks. Alternatively, we assume that the Taylor rule coe¢ cients impose (cid:13) = 1000 and (cid:13) = 0; so that the rule focuses exclusively on stabilizing in(cid:135)ation. In (cid:25) x 46

this case, the upward pressure on in(cid:135)ation due to the markup shock shortens the duration of the liquidity trap by several quarters (even assuming no (cid:133)scal shock), re(cid:135)ecting that policy only cares about keeping in(cid:135)ation at target under this rule. With a shorter liquidity trap, (cid:133)scal stimulus causes a larger and faster adjustment of policy rates as seen in the (cid:133)gure. As a consequence, the multiplier is considerably smaller in this case than under strict output gap targeting. A.2.3. History-Dependent Policy Rules (Section 2.3) Following the discussion at the end of Section 2.3, we consider the implications of rules that allow for history dependence in the conduct of monetary policy. Speci(cid:133)cally, we add a price level gap term so that the policy rule assumes the form: i = max i;(cid:13) (cid:25) +(cid:13) x +(cid:13) p ; t (cid:25) t x t p t (cid:0) (cid:0) (cid:1) where the price level gap variable p is de(cid:133)ned as t p = (cid:25) +p : t t t 1 (cid:0) Inthebenchmarkformulationofourrule,weassume(cid:13) = (cid:13) = 1000and(cid:13) = 0. Inthealternative (cid:25) x p monetary rule that includes a response to p , we assume that (cid:13) = 10. In addition, we study the t p implications of a more standard Taylor rule which sets (cid:13) = 1:5 and (cid:13) = :125, and then add p by (cid:25) x t setting (cid:13) = :125: Under each of the rules, we adjust the size of the negative taste shock to imply p aneightquarterliquiditytrapasinthebaselinescenario(thisistheonlytypeofshockconsidered). Thee⁄ectsofspendinghikeunderthefouralternativerulesareshowninFigureA.2. Underour benchmark policy rule, the spending multiplier is damped modestly with the inclusion of a weight on the price level gap. However, the di⁄erences are considerably larger under the standard Taylor rule. Because the standard Taylor rule is not very e⁄ective in keeping output and in(cid:135)ation near baseline in our model, there is a very large bene(cid:133)t to (cid:133)scal stimulus (cid:150)the (cid:133)scal multiplier exceeds 5. But when the Taylor rule is modi(cid:133)ed to include a price level response, the adverse e⁄ects of the taste shock are diminished, and the bene(cid:133)ts to (cid:133)scal stimulus substantially reduced. Clearly, history dependent rules that imply a credible commitment to stabilizing the price level can have important implications for the spending multiplier. A.2.4. Marginal Multiplier Schedule: Distortionary vs. Lump-Sum Taxes (Section 2.5) ThisdiscussioncomplementsthatinSection2.5ofthemaintextbypresenting(marginal)multiplier and government debt schedules under di⁄erent rules for the tax reaction function. In particular, the upper panels of Figure A.3 shows how the government spending multiplier and government debt response vary with the duration of the liquidity trap both under our benchmark of lump-sum taxes (the solid blue lines) and several alternatives in which the distortionary tax rate reacts to the lagged stock of debt ((cid:147)Simple Debt Labor-Tax Rule(cid:148)), the inertial version of the same rule 47

with a lag of the tax rate ((cid:147)Labor Tax Rule with Smoothing(cid:148)), and the balanced budget rule. As discussed in Section 2.5, it is apparent from the upper left panel that the disparity between the multiplier under lump-sum taxes and the simple debt rule (the green dashed line) is quite small for a liquidity trap duration of less than eight quarters; however, as the liquidity trap duration is more prolonged, the disparity increases markedly, re(cid:135)ecting mainly that in(cid:135)ation is much less responsive under distortionary taxes. While the multiplier exceeds 7 in a 12 quarter liquidity trap under lump-sum tax adjustment, the multiplier is only slightly above 2 under labor tax (cid:133)nancing. With the highly inertial form of the debt targeting rule (the dashed black lines), the multiplier and government spending debt response are nearly identical to that under lump-sum taxes for an eight quarter liquidity trap (hence, as noted in the text, this case was omitted from Figure 4). For evenlonger-livedtraps, themultipliergrowssomewhatmoreslowlywiththeliquiditytrapduration under the labor tax rule with smoothing, but the multiplier still remains pretty close to its level under lump-sum taxes. Finally, Figure A.3 also considers a balanced budget rule. As seen in the upper panel, the multiplier under the balanced budget rule lies uniformly below the lump-sum tax multiplier (and in particular, increases a bit more gradually as the liquidity trap duration lengthens). However, as seen in the bottom panel (cid:150)and as discussed in the main text (cid:150)the balanced budget multiplier actually grows more quickly than the lump-sum tax multiplier if the government spending share is relatively low (since the sensitivity of in(cid:135)ation to the output gap is nearly as large as under lump-sum taxes in this case). Accordingly, the multiplier under the balanced budget rule rises above the lump-sum tax multiplier if the liquidity trap duration exceeds six quarters. A.2.5. Endogenous Government Spending (Section 2.6) ComplementingthediscussioninSection2.6,PanelAofFigureA.4showshowautomaticstabilizers reduce the government spending multiplier by comparing the e⁄ects of a one percent of baseline GDPincreaseintheexogenouscomponentofgovernmentspendingunderourbenchmarkwith(cid:22) = 0 toanalternativespeci(cid:133)cationinwhich(cid:22)issettounity. Theimprovementinthedebt/GDPratiois also somewhat smaller under automatic stabilizers. As discussed in the text, the dampening of the multiplier and government debt response would be even more pronounced if automatic stabilizers were stronger, i.e., for higher values of (cid:22): To illustrate the interaction between automatic stabilizers and uncertainty about the path of shocks, it is insightful to consider a simple framework in which (cid:133)scal spending decisions must be made before the government and public become fully informed about the severity of the adverse shock(s) a⁄ecting the economy. In this vein, the lower panels of Figure A.4 compare the e⁄ects of a 1 percent of GDP rise in government spending under perfect foresight with the expected responses to the same spending hike when information about the taste shock becomes revealed only after the spending decision has been made; in the latter case, agents assign a 50 percent probability to a taste shock that would generate an 8 quarter liquidity trap absent (cid:133)scal stimulus (as in the 48

benchmark), a 25 percent probability to a more severe shock that would produce a 12 quarter liquidity trap, and a 25 percent probability to a less severe shock that would generate a 4 quarter liquidity trap. Panel B shows the case for an economy without automatic stabilizers, while panel C shows a corresponding case with stabilizers (with (cid:22) = 1).A.2 Without automatic stabilizers, the expected multiplier lies substantially above the multiplier under perfect foresight, and there is a much larger expected decline in government debt. These implications re(cid:135)ect the relatively high degree of convexity of the multiplier schedule in the absence of automatic stabilizers, which makes the payo⁄ to (cid:133)scal expansion if the bad state materializes especially high. By contrast, with automatic stabilizers as in the lower panel, the di⁄erence between the expected multiplier and multiplier under perfect foresight is comparatively smaller; the gap would narrow further if the stabilizers were even more responsive (i.e., for higher (cid:22)): A.2The size of the taste shocks in each of the three states is the same with and without the automatic stabilizers. 49

Figure A.1: Effects of Spending Hike in Simple Model For Alternative Policy Rules Under Baseline Generated by Both Taste and Markup Shocks. 2 1.5 1 0.5 0 4 8 12 16 Quarter tnecreP Output Potential Output Benchmark Rule−Taste Shock Only Strict Output Targeting 0.2 Strict Inflation Targeting 0.15 0.1 0.05 0 4 8 12 16 tnecreP Quarter Inflation (APR) 4 3 2 1 0 4 8 12 16 tnecreP Price Level 2.5 2 1.5 1 0 4 8 12 16 Quarter tnecreP Quarter Nominal Interest Rate (APR) 0.2 0.15 0.1 0.05 0 0 4 8 12 16 tnecreP Real Interest Rate (APR) 0 −1 −2 0 4 8 12 16 Quarter tnecreP Quarter Government debt (trend GDP share) 1 0.5 0 0 4 8 12 16 tnecreP Government Spending (Trend GDP share) 1 0.8 0.6 0.4 0.2 0 4 8 12 16 Quarter tnecreP Quarter

Figure A.2: Immediate Government Spending Rise in Simple Model Under Monetary Policy Rules With and Without History Dependence. 5 4 3 2 1 0 4 8 12 16 Quarter tnecreP Output Potential Output Benchmark Rule Benchmark with PL term 0.2 Taylor Rule Taylor Rule with PL term 0.15 0.1 0.05 0 4 8 12 16 tnecreP Quarter Inflation (APR) 15 10 5 0 0 4 8 12 16 tnecreP Price Level 10 8 6 4 2 0 4 8 12 16 Quarter tnecreP Quarter Nominal Interest Rate (APR) 0.15 0.1 0.05 0 0 4 8 12 16 tnecreP Real Interest Rate (APR) 0 −2 −4 −6 −8 0 4 8 12 16 Quarter tnecreP Quarter Government debt (trend GDP share) 1 0 −1 −2 −3 0 4 8 12 16 tnecreP Government Spending (Trend GDP share) 1 0.8 0.6 0.4 0.2 0 4 8 12 16 Quarter tnecreP Quarter

Figure A.3: Marginal Output and Government Debt Multipliers Under Alternative Financing Assumptions: Lump−Sum Vs. Distortionary Labor−Income Taxes 8 6 4 2 0 12 11 10 9 8 7 6 5 4 3 2 1 0 Liquidity Trap Duration reilpitluM gnidnepS tnemnrevoG 1 Lum−Sum Tax Rule Simple−Debt Labor−Tax Rule 0 Balanced−Budget Labor−Tax Rule Labor−Tax Rule with Smoothing −1 −2 −3 −4 −5 12 11 10 9 8 7 6 5 4 3 2 1 0 PDG ot tbeD tnemnrevoG Benchmark Parameterization (Steady State Government Spending Share 0.2) Liquidity Trap Duration 12 10 8 6 4 2 0 12 11 10 9 8 7 6 5 4 3 2 1 0 reilpitluM gnidnepS tnemnrevoG 1 0.5 0 −0.5 −1 12 11 10 9 8 7 6 5 4 3 2 1 0 Liquidity Trap Duration PDG ot tbeD tnemnrevoG Parameterization With Lower Steady State Government Spending Share (G/Y = 0.05) Liquidity Trap Duration

Figure A.4: Immediate Government Spending Rise: Assessing the Impact of Automatic Stabilizers and Shock Uncertainty Panel A: Automatic Stabilizers Output Gap Government Debt/GDP 3 1 Benchmark, No Auto Stab 2.5 Automatic Stabilizers 0.5 2 0 1.5 −0.5 1 −1 0.5 0 −1.5 0 4 8 12 16 0 4 8 12 16 Panel B: Shock Uncertainty − No Automatic Stabilizers Output Gap Government Debt/GDP 3 1 Benchmark, Perf Foresight 2.5 With Uncertainty 0.5 2 0 1.5 −0.5 1 −1 0.5 0 −1.5 0 4 8 12 16 0 4 8 12 16 Panel C: Shock Uncertainty − With Automatic Stabilizers Output Gap Government Debt/GDP 3 1 Perfect Foresight 2.5 Uncertainty 0.5 2 0 1.5 −0.5 1 −1 0.5 0 −1.5 0 4 8 12 16 0 4 8 12 16 Quarters Quarters

Appendix B. The New-Keynesian Model with Keynesian Agents and Financial Frictions This appendix contains two parts. Section B.1 describes the model used in Section 3. Section B.2 discusses some additional results referred to in the main text, including the construction of the baseline for our simulations, and a decomposition of the sources of improvement in the debt/GDP ratio that underlie a "(cid:133)scal free lunch." B.1. The Model The model is essentially a variant of the CEE/SW model augmented with (cid:147)Keynesian(cid:148)households, as in Erceg, Guerrieri and Gust (2006), and (cid:133)nancial frictions, following Bernanke, Gertler and Gilchrist (1999). As such, our model incorporates nominal rigidities by assuming that labor and product markets exhibit monopolistic competition, and that wages and prices are determined by staggered nominal contracts of random duration (following Calvo (1983) and Yun (1996)). In addition, the model includes an array of real rigidities, including habit persistence in consumption, and costs of changing the rate of investment. Monetary policy follows a Taylor rule, and (cid:133)scal policy speci(cid:133)es that taxes respond to government debt. B.1.1. Firms and Price Setting FinalGoodsProduction Weassumethatasingle(cid:133)naloutputgoodY isproducedusingacontinuum t of di⁄erentiated intermediate goods Y (f). The technology for transforming these intermediate t goods into the (cid:133)nal output good is constant returns to scale, and is of the Dixit-Stiglitz form: 1 1 1+(cid:18)p Y t = Y t (f)1+(cid:18)p df (B.1) (cid:20)Z0 (cid:21) where (cid:18) > 0. p Firms that produce the (cid:133)nal output good are perfectly competitive in both product and factor markets. Thus, (cid:133)nalgoodsproducersminimizethecostofproducingagivenquantityoftheoutput index Y , taking as given the price P (f) of each intermediate good Y (f). Moreover, (cid:133)nal goods t t t producers sell units of the (cid:133)nal output good at a price P that can be interpreted as the aggregate t price index: 1 1 (cid:0) (cid:18)p P t = P t (f)(cid:0)(cid:18)p df (B.2) (cid:20)Z0 (cid:21) Intermediate Goods Production A continuum of intermediate goods Y (f) for f [0;1] is produced t 2 by monopolistically competitive (cid:133)rms, each of which produces a single di⁄erentiated good. Each intermediate goods producer faces a demand function for its output good that varies inversely with its output price P (f); and directly with aggregate demand Y : t t (1+(cid:18)p) P t (f) (cid:0) (cid:18)p Y (f) = Y (B.3) t t P t (cid:20) (cid:21) 54

Each intermediate goods producer utilizes capital services K (f) and a labor index L (f) (det t (cid:133)ned below) to produce its respective output good. The form of the production function is Cobb- Douglas: Y (f) = K (f)(cid:11)L (f)1 (cid:11) (B.4) t t t (cid:0) Firms face perfectly competitive factor markets for hiring capital and the labor index. Thus, each (cid:133)rm chooses K (f) and L (f), taking as given both the rental price of capital R and the t t Kt aggregate wage index W (de(cid:133)ned below). Firms can costlessly adjust either factor of production. t Thus, the standard static (cid:133)rst-order conditions for cost minimization imply that all (cid:133)rms have identical marginal cost per unit of output. The prices of the intermediate goods are determined by Calvo-Yun style staggered nominal contracts. Ineachperiod,each(cid:133)rmf facesaconstantprobability,1 (cid:24) ,ofbeingabletoreoptimize p (cid:0) its price P (f). The probability that any (cid:133)rm receives a signal to reset its price is assumed to t be independent of the time that it last reset its price. If a (cid:133)rm is not allowed to optimize its price in a given period, we follow Christiano, Eichenbaum and Evans (2005) by assuming that it adjusts its price by a weighted combination of the lagged and steady state rate of in(cid:135)ation, i.e., P (f) = (cid:25) (cid:19)p (cid:25)1 (cid:19)pP (f) where 0 (cid:19) 1: A positive value of (cid:19) introduces structural inertia t t (cid:0) 1 (cid:0) t (cid:0) 1 (cid:20) p (cid:20) p into the in(cid:135)ation process. B.1.2. Households and Wage Setting We assume a continuum of monopolistically competitive households (indexed on the unit interval), each of which supplies a di⁄erentiated labor service to the production sector; that is, goodsproducing (cid:133)rms regard each household(cid:146)s labor services N (h), h [0;1], as an imperfect substitute t 2 for the labor services of other households. It is convenient to assume that a representative labor aggregator combines households(cid:146)labor hours in the same proportions as (cid:133)rms would choose. Thus, the aggregator(cid:146)s demand for each household(cid:146)s labor is equal to the sum of (cid:133)rms(cid:146)demands. The labor index L has the Dixit-Stiglitz form: t 1 1+(cid:18)w 1 L t = N t (h)1+(cid:18)w dh (B.5) (cid:20)Z0 (cid:21) where (cid:18) > 0. The aggregator minimizes the cost of producing a given amount of the aggregate w labor index, taking each household(cid:146)s wage rate W (h) as given, and then sells units of the labor t index to the production sector at their unit cost W : t 1 (cid:18)w 1 (cid:0) W t = W t (h)(cid:18)(cid:0) w dh (B.6) (cid:20)Z0 (cid:21) It is natural to interpret W as the aggregate wage index. The aggregator(cid:146)s demand for the labor t hours of household h (cid:150)or equivalently, the total demand for this household(cid:146)s labor by all goodsproducing (cid:133)rms (cid:150)is given by 1+(cid:18)w W t (h) (cid:0) (cid:18)w N (h) = L (B.7) t t W t (cid:20) (cid:21) 55

The utility functional of a typical member of household h is Et 1 (cid:12)j 1 C t+j (h) {C t+j 1 (cid:23) c (cid:23) t 1 (cid:0) (cid:27) (cid:31) 0 N t+j (h)1+(cid:31) (B.8) f1 (cid:27) (cid:0) (cid:0) (cid:0) g (cid:0) 1+(cid:31) g j=0 (cid:0) X where the discount factor (cid:12) satis(cid:133)es 0 < (cid:12) < 1: The period utility function depends on household h(cid:146)s current consumption C (h), as well as lagged aggregate per capita consumption to allow for the t possibility of external habit persistence (Smets and Wouters 2003). As in the simple model considered in the previous section, a positive taste shock (cid:23) raises the marginal utility of consumption t associated with any given consumption level. The period utility function also depends inversely on hours worked N (h): t Household h(cid:146)s budget constraint in period t states that its expenditure on goods and net purchases of (cid:133)nancial assets must equal its disposable income: 1 (I (h) I (h))2 t t 1 P t C t (h)+P t I t (h)+ 2 I P t I (cid:0) (h (cid:0) ) + t 1 (cid:0) P B B + (cid:24) B (h) B (h) (B.9) B;t G;t+1 G;t t;t+1 D;t+1 D;t (cid:0) (cid:0) s Z = (1 (cid:28) )W (h)N (h)+(1 (cid:28) )R K (h)+(cid:14)(cid:28) P K (h)+(cid:0) (h) T (h) N;t t t K K;t t K t t t t (cid:0) (cid:0) (cid:0) Thus, the household purchases the (cid:133)nal output good (at a price of P ); which it chooses either to t consume C (h) or invest I (h) in physical capital. The total cost of investment to each household t t h is assumed to depend on how rapidly the household changes its rate of investment (as well as on the purchase price). Our speci(cid:133)cation of investment adjustment costs as depending on the square of the change in the household(cid:146)s gross investment rate follows Christiano, Eichenbaum, and Evans (2005). Investment in physical capital augments the household(cid:146)s (end-of-period) capital stock K (h) according to a linear transition law of the form: t+1 K (h) = (1 (cid:14))K (h)+I (h) (B.10) t+1 t t (cid:0) Inadditiontoaccumulatingphysicalcapital,householdsmayaugmenttheir(cid:133)nancialassetsthrough increasing their government bond holdings (P B B ); and through the net acquisition B;t G;t+1 G;t (cid:0) of state-contingent bonds. We assume that agents can engage in frictionless trading of a complete set of contingent claims. The term (cid:24) B (h) B (h) represents net purchases of s t;t+1 D;t+1 (cid:0) D;t state-contingent domestic bonds, with (cid:24) denoting the state price, and B (h) the quantity t;t+R1 D;t+1 of such claims purchased at time t. Each member of household h earns after-tax labor income (1 (cid:28) )W (h)N (h), after-tax capital rental income of (1 (cid:28) )R K (h); and a depreciation N;t t t K K;t t (cid:0) (cid:0) allowance of (cid:14)(cid:28) P K (h). Each member also receives an aliquot share (cid:0) (h)of the pro(cid:133)ts of all K t t t (cid:133)rms, and pays a lump-sum tax of T (h) (this may be regarded as taxes net of any transfers). t In every period t, each member of household h maximizes the utility functional (B.8) with respect to its consumption, investment, (end-of-period) capital stock, bond holdings, and holdings 56

of contingent claims, subject to its labor demand function (B.7), budget constraint (B.9), and transition equation for capital (B.10). Households also set nominal wages in Calvo-style staggered contracts that are generally similar to the price contracts described above. Thus, the probability that a household receives a signal to reoptimize its wage contract in a given period is denoted by 1 (cid:24) . In addition, we specify a dynamic indexation scheme for the adjustment of the wages of w (cid:0) those households that do not get a signal to reoptimize, i.e., W (h) = !(cid:19)w (cid:25)1 (cid:19)wW (h); where t t 1 (cid:0) t 1 (cid:0) (cid:0) ! is gross nominal wage in(cid:135)ation in period t 1: Dynamic indexation of this form introduces t 1 (cid:0) (cid:0) some structural persistence into the wage-setting process. B.1.3. Fiscal and Monetary Policy and the Aggregate Resource Constraint Government purchases G are assumed to follow an exogenous AR(1) process with a persistence t coe¢ cient of 0:9. Government purchases have no e⁄ect on the marginal utility of private consumption, nor do they serve as an input into goods production. Government expenditures are (cid:133)nanced by a combination of labor, capital, and lump-sum taxes. The government does not need to balance its budget each period, and issues nominal debt to (cid:133)nance budget de(cid:133)cits according to: P B B = P G T (cid:28) W L (cid:28) (R (cid:14)P )K : (B.11) B;t G;t+1 G;t t t t N;t t t K K;t t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) In eq. (B.11), all quantity variables are aggregated across households, so that B is the aggregate G;t 1 stock of government bonds and K is the aggregate capital stock, and T = ( T (h)dh) aggregate t t 0 t lump-sum taxes. In our benchmark speci(cid:133)cation, the lump-sum and capital tax rate is held (cid:133)xed, R andlump-sumtaxesadjustendogenouslyaccordingtoataxratereactionfunctionthatallowstaxes to respond to debt (as in Section 2.5) subject to smoothing. In log-linearized form: (cid:28) (cid:28) = (’ )((cid:28) (cid:28) )+(1 ’ )’ ~b ~b ; (B.12) N;t N (cid:28) N;t 1 N (cid:28) b G;t G (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) where ~b BG;t: In the working paper version of the paper (cid:150)Erceg and Linde (2010) - we G;t (cid:17) 4PtY assumed lump-sum taxes were used to stabilize debt, but as Section 2.5 highlighted that the di⁄erence between lump-sum and distortionary tax (cid:133)nancing can potentially be important in long-lived liquidity traps, we choose to work with distortionary tax (cid:133)nancing which we think is more (cid:0) empirically plausible in this version of the paper.B.1 (cid:0) B.1Intheworkingpaperversion(ErcegandLinde,2010),wecomparethee⁄ectswithalternative(cid:133)nancingschedules (lump-sumanddistortionarylabor-incometaxes),and(cid:133)ndthatthee⁄ectsaremodestforan8quarterliquiditytrap: In this version of the paper, it turns out that the (cid:133)nancing assumption does not matter much even in 10-11 quarter liquidity trap, as we consider a tax rule that is somewhat less responsive to debt (following the empirical evidence in Traum and Yang, 2011, and our own simple regression of quarterly federal income tax rate on debt over the 1960-2011period). Itisworthpointingoutthattheformofthetaxratereactionfunctionforlump-sumtaxeshasno e⁄ectonequilibriumallocationswhenthatallagentsareRicardian,butwhenashareofhouseholdsare(cid:147)Keynesian(cid:148) and consume directly their current after-tax disposable income, even the coe¢ cients in a lump-sum rule matter for equilibrium allocations. 57

Monetary policy is assumed to be given by a Taylor-style interest rate reaction function similar to equation (3) except allowing for a smoothing coe¢ cient (cid:13) : i i = max( i;(1 (cid:13) )((cid:13) (cid:25) +(cid:13) x )+(cid:13) i ) (B.13) t i (cid:25) t x t i t 1 f (cid:0) (cid:0) (cid:0) g Finally, total output of the service sector is subject to the resource constraint: Y = C +I +G + (B.14) t t t t I;t where is the adjustment cost on investment aggregated across all households (from eq. B.9, I;t I;t (cid:17) 2 1 I (It(h I ) t (cid:0) I 1 t ((cid:0)h 1 ) (h))2 ). (cid:0) B.1.4. Keynesian Households In the full with non-Ricardian households, we assume that a fraction s of the population consists kh of (cid:147)Keynesian(cid:148)households whose members consume their current after-tax income each period, and set their wage equal to the average wage of the optimizing households. Because all households face the same labor demand schedule, each Keynesian household works the same number of hours as the average optimizing household. Thus, the consumption of Keynesian households CK(h) is t simply determined as P CK(h) = (1 (cid:28) )W (h)N (h) T ; t t Nt t t t (cid:0) (cid:0) where T denotes (net) lump-sum taxes. Consumption of the non-Keynesian households is given t the consumption Euler equation derived by maximizing (B.8) subject to (B.9): B.1.5. Production of capital services We build on the model described above by incorporating a (cid:133)nancial accelerator mechanism following the basic approach of Bernanke, Gertler and Gilchrist (1999). Thus, the intermediate goods producers rent capital services from entrepeneurs (at the price R ) rather than directly from Kt households. Entrepeneurs purchase capital from competitive capital goods producers, with the latter employing the same technology to transform investment goods into (cid:133)nished capital goods as described by equations B.10) and B.9). To (cid:133)nance the acquisition of physical capital, each entrepreneur combines his net worth with a loan from a bank, for which the entrepreneur must pay an external (cid:133)nance premium (over the risk-free interest rate set by the central bank) due to an agency problem. We follow Christiano, Motto and Rostagno (2008) by assuming that the debt contract between entrepreneurs and banks is written in nominal terms (rather than real terms as in Bernanke, Gertler and Gilchrist, 1999). Banks obtain funds to lend to the entrepreneurs by issuing deposits to households at the interest rate set by the central bank. By assuming perfect competition and free entry among banks and that all bank portfolios are well diversi(cid:133)ed (i.e., that each bank lends out to a continuum of entrepreneurs, whose default risk is independently distributed), 58

it follows that banks make zero pro(cid:133)ts in each state of the economy and that there is no credit risk to households associated with bank deposits.B.2 B.1.6. Solution and Calibration To analyze the behavior of the model, we log-linearize the model(cid:146)s equations around the nonstochastic steady state. Nominal variables, such as the contract price and wage, are rendered stationary by suitable transformations. To solve the unconstrained version of the model, we compute the reduced-form solution of the model for a given set of parameters using the numerical algorithmofAndersonandMoore(1985),whichprovidesane¢ cientimplementationofthesolution method proposed by Blanchard and Kahn (1980). When we solve the model subject to the non-linear monetary policy rule (B.13), we use the techniques described in Hebden, LindØ and Svensson (2009). An important feature of the Hebden, LindØ and Svensson algorithm is that the duration of the liquidity trap is endogenous, and is a⁄ected by the size of the (cid:133)scal impetus. Their algorithm consists of adding a sequence of current and future innvoations to the linear component of the policy rule to guarantee that the zero bound constraint is satis(cid:133)ed given the economy(cid:146)s state vector. The sequence of innovations is assumed to be correctly anticipated by private agents at each date. This solution method is easy to use, and well-suited to examine the implications of the zero bound constraint in models with large dimensional state spaces; moreover, it yields identical results to the method of Jung, Terinishi, and Watanabe (2005). As in Section 2, we set the discount factor (cid:12) = 0:995; and steady state (net) in(cid:135)ation (cid:25) = :005; implyingasteadystatenominalinterestrateof i = :01ataquarterlyrate. Thesubutilityfunction over consumption is logarithmic, so that (cid:27) = 1; and the parameter determining the degree of habit persistence in consumption { is set at 0:6 (similar to the empirical estimate of Smets and Wouters 2003). The Frisch elasticity of labor supply 1 of 0:4 is well within the range of most estimates from (cid:31) the empirical labor supply literature (see e.g. Domeij and FlodØn, 2006). The capital share parameter (cid:11) is set to 0:35: The quarterly depreciation rate of the capital stock (cid:14) = 0:025, implying an annual depreciation rate of 10 percent. We set the cost of adjusting investment parameter = 3, which is somewhat smaller than the value estimated by Christiano, I Eichenbaum, and Evans (2005) using a limited information approach; however, the analysis of Erceg, Guerrieri, and Gust (2006) suggests that a lower value may be better able to capture the unconditional volatility of investment. We maintain the assumption of a relatively (cid:135)at Phillips curve by setting the price contract duration parameter (cid:24) = 0:9. As in Christiano, Eichenbaum and Evans (2005), we also allow for p a fair amount of intrinsic persistence by setting the price indexation parameter (cid:19) = 0:9. It bears p emphasizing that our choice of (cid:24) does not necessarily imply an average price contract duration p B.2WerefertoBernanke,GertlerandGilchrist(1999)andChristiano,MottoandRostagno(2008)forfurtherdetails. An excellent exposition is also provided in Christiano, Trabandt and Walentin (2007). 59

of 10 quarters. Altig et al. (2011) show in a model very similar to ours that a low slope of the Phillips curve can be consistent with frequent price reoptimization if capital is (cid:133)rm-speci(cid:133)c, at least provided that the steady-state markup is not too high, and it is costly to vary capital utilization; both of these conditions are satis(cid:133)ed in our model, as the steady state markup is 10 percent ((cid:18) = :10), and capital utilization is (cid:133)xed. Speci(cid:133)cally, our choice of (cid:24) implies a Phillips p p curve slope of about 0:007. Given strategic complementarities in wage-setting across households, the wage markup in(cid:135)uences the slope of the wage Phillips curve. Our choices of a wage markup of (cid:18) = 1=3 and a wage contract duration parameter of (cid:24) = 0:85 along with a wage indexation W w (cid:0) parameter of (cid:19) = 0:9 - imply that wage in(cid:135)ation is about as responsive to the wage markup as w price in(cid:135)ation is to the price markup. The parameters of the monetary policy rule are set as (cid:13) = 0:7, (cid:13) = 3 and (cid:13) = 0:25: i (cid:25) x These parameter choices are supported by simple regression analysis using instrumental variables over the 1993:Q1-2008:Q4 period. This analysis suggests that the response of the policy rate to in(cid:135)ationandtheoutputgaphasincreasedinrecentyears, whichhelpsaccountforsomewhathigher response coe¢ cients than typically estimated when using sample periods which include the 1970s and 1980s. Overall, as noted in the main text, our calibration of the monetary policy rule and the Phillips Curve slope parameters tilts in the direction of reducing the sensitivity of in(cid:135)ation to macroeconomic shocks. WesetthepopulationshareoftheKeynesianhouseholdstooptimizinghouseholds, s , to0:47, kh which implies that the Keynesian households(cid:146)share of total consumption is about 1=3. This calibration perhaps overstates the role of non-Ricardian households in a⁄ecting consumption behavior, but seems useful to help put plausible bounds on how the multiplier may vary with the degree of non-Ricardian behavior in consumption (recognizing that the CEE/SW workhorse model is a specialcaseinwhichs =0andthereareno(cid:133)nancialfrictions). Ourcalibrationoftheparameters kh a⁄ecting the (cid:133)nancial accelerator follow BGG (1999). Thus, the monitoring cost, (cid:22), expressed as a proportion of entrepreneurs(cid:146)total gross revenue, is 0:12. The default rate of entrepeneurs is 3 percent per year, and the variance of the idiosyncratic productivity to entrepreneurs is 0:28: The share of government spending of total expenditure is set equal to 20 percent. The government debt to GDP ratio is 0:5, close to the total estimated U.S. federal government debt to output ratio at end-2009. The steady state capital income tax rate, (cid:28) , is set to 0:2; while the lump-sum K tax revenue to GDP ratio is set to 0:02. For simplicity, we set the depreciation allowance (cid:14)(cid:28) = 0. K Given these choices, the government(cid:146)s intertemporal budget constraint implies that labor income tax rate (cid:28) equals 0:27 in steady state. The parameters in the (cid:133)scal policy rule in equation (B.12) N are set to ’ = 0:92; ’ = 0:1 following the evidence in Traum and Yang (2011), implying that the (cid:28) b tax rule is not very aggressive. Importantly, given the low share of government revenue accounted for by lump-sum taxes, most of the variation in the government budget de(cid:133)cit re(cid:135)ects (cid:135)uctuations inrevenuefromthecapitalandlaborincometax(duetovariationsinthetaxbase), andtheservice cost of debt. 60

B.2. Additional Results in the Large-Scale Model Here we discuss some additional results referred to in Section 3 of the main text. B.2.1. Initial Economic Conditions (Section 3.0) The e⁄ects of the government spending increases considered in Sections 3.1 and 3.2 clearly depend on initial conditions which determine the depth and duration of the underlying liquidity trap. Under our baseline setting for these initial conditions, we assume that taste shocks (phased in over three quarters) generate a sharp fall in output and in(cid:135)ation as shown by the solid lines in Figure B.1.a, and cause the policy rate to decline to its lower bound of zero for eight quarters.B.3 The taste shocks are scaled to induce a maximum output contraction of 8 percent relative to baseline, which is similar to the fall in U.S. GDP relative to trend that occurred during the Great Recession Theimplicationthattheliquiditytraplastseightquartersseemsreasonablyconsistentwithmarket perceptions of the likely duration of the liquidity trap in early 2009 when large-scale (cid:133)scal stimulus was proposed. For example, the (cid:147)projected(cid:148)path for the federal funds rate implied by overnight indexedswapratesinearly2009(cid:150)showninthelowerleftpanellabelledB.1.b(cid:150)wasbelow1percent for a horizon extending eight quarters. One feature of the baseline that merits additional discussion is the behavior of in(cid:135)ation. Under our benchmark calibration, Figure B.1.a shows that baseline in(cid:135)ation path falls more than 2 percentage points below its steady state level of 2 percent for over a year. The magnitude of the in(cid:135)ation response is somewhat greater than what occurred during the (cid:133)nancial crisis. During that episode, even short-run in(cid:135)ation expectations (cid:150)for instance, as proxied by the path of expected in(cid:135)ation over the next six quarters from the Survey of Professional Forecasters (shown in the lower right panel) (cid:150)remained remarkably stable. Moreover, in(cid:135)ation expectations have continued to remain very stable since that time. Although such evidence raises the possibility that our benchmark may overstate the responsiveness of in(cid:135)ation to large and highly persistent shocks, and thus perhaps exaggerate the (cid:133)scal multiplier, it bears emphasizing that our benchmark calibration implies much less movement in in(cid:135)ation than other commonly-adopted calibrations. To highlight this, Figure B.1.a also reports results for two alternative calibrations. In the case labelled (cid:147)more (cid:135)exible p and w,(cid:148)the mean duration of price and wage contracts is reduced to four and (cid:133)ve quarters, respectively (i.e., (cid:24) = :75 P and (cid:24) = :80); while another alternative labelled (cid:147)looser rule(cid:148)adopts the standard Taylor rule W coe¢ cients in the monetary policy rule (i.e., (cid:13) = 1.5 and (cid:13) = 0:125). In(cid:135)ation declines by (cid:25) x considerably more under either of these alternative calibrations. B.3The scenario is generated by a sequence of three unanticipated negative taste shocks (cid:23) that cause the policy t rate to fall to zero after three quarters. 61

B.2.2. Sources of Fiscal Free Lunch (Section 3.2) ComplementingthediscussioninSection3.2, FigureB.2providesadditionaldetailusefulforunderstanding the channels through which higher government spending can generate a (cid:133)scal free lunch in a long-lived 10 quarter liquidity trap. In particular, the (cid:133)gure shows how a 1 percent of baseline GDPhikeinspendinga⁄ectsthegovernment/debtGDPratioathorizonsof1-5years,andprovides a decomposition of the e⁄ects on government debt. The improvement in the overall debt/GDP ratio of a bit more than 1.5 percent after 12 quarters (cid:150)the black dash-dotted line in the (cid:133)gure (cid:150) is consistent with the average response of the debt/GDP ratio shown in the upper right panel in Figure 6. Turning to the sources of improvement, the contribution of government spending (cid:150)the red solid bars (cid:150)to the debt/GDP ratio is positive as expected, and continues to rise through time given that the spending increase in persistent (recalling that positive bars in the (cid:133)gure imply a source of upward pressure on debt). Even so, higher labor income tax revenue alone (the light blue bars, holding the labor tax rate at steady state) is nearly su¢ cient to keep debt from expanding at a horizon of three years (the same horizon as shown in Figure 5); with higher capital income tax revenue and lower debt-servicing costs, government debt falls about 1.5 percentage points below its initial level after 12 quarters as noted previously. The debt/GDP ratio eventually moves back toward its initial level as the spending contribution grows a bit more, and because the labor tax rate actually falls (accounting for the positive contribution in the (cid:133)gure). 62

2 0 −2 −4 −6 −8 −10 −12 0 1 2 3 4 Quarter tnecreP Fig. B.1.a: Simulated and Actual Paths for Key Macroeconomic Variables in Full Model Real Output Inflation (YoY) 4 2 0 −2 −4 −6 0 1 2 3 4 ZLB − Benchmark ZLB − More flex p and w ZLB − Looser rule tnecreP Quarter Nominal Interest Rate (APR) 4 3 2 1 0 −1 0 1 2 3 4 tnecreP Quarter 4 2 0 −2 −4 −6 08Q3 09Q3 10Q3 11Q3 12Q3 13Q3 Quarter tnecreP Actual and Expected Inflation (YoY) 3 2.5 2 1.5 1 0.5 US economy 08Q3−11Q4 Expectations at time t 0 08Q3 09Q3 10Q3 11Q3 12Q3 13Q3 14Q3 Quarter tnecreP Figure B.1.b: Actual and Expected FFR and Core Inflation Rates Actual and Expected Fed Funds Rates US economy 08Q3−11Q4 Expectations Jan−02−09 Expectations Dec−15−09 Expectations Dec−15−10

3 2 1 0 −1 −2 −3 −4 −5 4 8 12 16 20 Quarters stnioP egatnecreP Figure B.2: Contribution to Debt Dynamics in Full Model in a Deep (10 quarter) Liquidity Trap Labor Taxes Spending Capital Income Labor Income Interest Rate Debt as share of trend GDP (sum of all bars) Debt as share of actual GDP