Fiscal Multipliers at the Zero Lower Bound: The Role of Policy Inertia

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Fiscal Multipliers at the Zero Lower Bound: The Role of Policy Inertia Timothy S. Hills and Taisuke Nakata 2014-107 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Fiscal Multipliers at the Zero Lower Bound: The Role of Policy Inertia∗ Timothy Hills† Taisuke Nakata‡ Federal Reserve Board Federal Reserve Board First Draft: April 2014 This Draft: November 2014 Abstract The presence of the lagged shadow policy rate in the interest rate feedback rule reduces the government spending multiplier nontrivially when the policy rate is constrained at the zero lower bound (ZLB). In the economy with policy inertia, increased inflation and output due to higher government spending during a recession speed up the return of the policy rate to the steady state after the recession ends. This in turn dampens the expansionary effects of the government spending during the recession via expectations. In our baseline calibration, the output multiplier at the ZLB is 2.5 when the weight on the lagged shadow rate is zero, and 1.1 when the weight is 0.9. JEL: E32, E52, E61, E62, E63 Keywords: Fiscal Policy, Government Spending Multipliers, Interest Rate Smoothing, Liquidity Trap, Zero Lower Bound. ∗WewouldliketothankMarkGertler,ChristopherGust,LenaKo¨rber,JesperL´ınde,MatthiasPaustian, Steven Pennings, John Roberts, Sebastian Schmidt, Michael Siemer, and seminar participants at theBoardofGovernorsoftheFederalReserveSystem,Bundesbank-KielJointWorkshoponFiscalPolicy andMacroeconomicPerformance,Fall2014MidwestMacroMeeting,andthe10thDynareConferencefor usefulsuggestionsandcomments. Theviewsexpressedinthispaper,andallerrorsandomissions,should be regarded as those solely of the authors, and are not necessarily those of the Federal Reserve Board of Governors or the Federal Reserve System. †Division of Research and Statistics, Federal Reserve Board, 20th Street and Constitution Avenue N.W. Washington, D.C. 20551; Email: tim.hills@frb.gov ‡Corresponding author; Division of Research and Statistics, Federal Reserve Board, 20th Street and Constitution Avenue N.W. Washington, D.C. 20551; Email: taisuke.nakata@frb.gov 1

1 Introduction Recent theoretical literature on the positive effects of fiscal policy has emphasized that the government spending multiplier on output can be quite large when the nominal interest rate is constrained at the zero lower bound (ZLB). For example, seminal papers by Christiano, Eichenbaum, and Rebelo (2011); Eggertsson (2011); and Woodford (2011) report a multiplier that is substantially greater than one in plausibly parameterized standard New Keynesian models. Asreviewedindetaillater,theliteraturehasmostlyexaminedthisissueusingamodel in which the policy rate is given by a truncated Taylor rule without policy inertia—that is, without an interest rate smoothing term. The goal of this paper is to examine the sensitivityofthegovernmentspendingmultiplierattheZLBtothedegreeofpolicyinertia in the interest rate feedback rule. Our main focus is a version of the inertial policy rule in which today’s shadow policy rate—a hypothetical policy rate that would prevail were it not for the ZLB constraint—depends on the lagged shadow policy rate. However, we will also examine an alternative version of the inertial Taylor rule in which the shadow policy rate today depends on the lagged actual policy rate. The main result of our paper is that the presence of the lagged shadow policy rate in the interest rate feedback rule reduces the government spending multiplier at the ZLB in a quantitatively important way. Under a baseline parameterization of the model intended to mimic the Great Depression, the output multiplier at the ZLB is 1.1 with an inertial parameter of 0.9 and 2.5 without any inertia. The reason for the smaller multiplier under the inertial Taylor rule is as follows. Independent of policy inertia, an increase in government spending during a recession increases inflation and output during the recession. In a model with policy inertia, this improved allocation during the recession speeds up the return of the policy rate to the steady state after the recession ends, which in turn dampens the expansionary effects of the government spending during the recession via expectations. The multiplier at the ZLB approaches the multiplier away from the ZLB from above as the inertia approaches unity, and it can go below one with a sufficiently high inertial parameter. However, for a plausible range of weights on the lagged policy rate, the ZLB multiplier remains above one. By contrast, when the nominal interest rate is away from the ZLB, policy inertia increases the fiscal multiplier. In the presence of policy inertia, the nominal interest rate is slow to respond to an exogenous change in government spending. Thus, the expansionary effects of government spending are larger. However, the effects of policy inertia are quantitatively much smaller away from the ZLB than at the ZLB, and the multiplier remains below one in the former case. The claim in the existing literature that the government spending multiplier is larger at the ZLB than away from it is thus robust to policy inertia. With the version of the inertial policy rule where the shadow rate today depends on thelaggedactual policyrate,thedegreeofinertiadoesnotaffectthegovernmentspending 2

multiplier at the ZLB at all in our baseline environment with a two-state crisis shock. In this version of the model, what matters for the policy path after the recession is the actual policy rate at the end of the recession. Since the actual policy rate at the end of the recession is zero regardless of the government spending shock, the policy path—and thusthepathsofinflationandoutput—aftertherecessionisunalteredbythegovernment spending shock during the recession. Accordingly, the inertial parameter does not affect the expansionary effects of the government spending shock during the recession in the economy with this version of the inertial policy rule. To emphasize the importance of the endogeneity of post-recession policy rates in generatingalowergovernmentspendingmultiplier, wealsoconsiderthemultipliersunder other history-dependent monetary policy specifications: a price-level targeting rule, the Reifschneider-Williams (2000) rule, and optimal commitment policy. We find that the government spending multipliers are smaller under these three specifications of monetary policy determination than under the baseline non-inertial Taylor rule. Under all of these policies, government spending shocks during a recession lead to tighter policy after the recession, implying lower multipliers. Thus, the mechanism by which the shadow-rate version of the inertial Taylor rule reduces the multiplier is relevant to understanding why the government spending multiplier is lower under other widely studied policy rules. Our focus on the inertial Taylor rule is motivated by two considerations. First, as reviewed in detail in Section 2, there is strong empirical evidence for the presence of the lagged policy rate in the nominal interest rate rule. When the policy rule is estimated in the context of structural models, the weight on the lagged policy rate is often estimated to be positive and large. Estimation of the reduced-form policy rule also lends support to the presence of policy inertia. Second, the presence of policy inertia is consistent with recent FOMC statements indicatingthat the Committee intendsto keep the federalfunds rate near zero for a considerable period of time after the economic recovery strengthens. Introducing the lagged policy rate to the policy rule is one way, albeit not the only one, to characterize this policy of an extended period of low nominal interest rates. Our results narrow the gap between theoretical multipliers in the New Keynesian model and recent empirical estimates of the multiplier at the ZLB. Using a dataset for the U.S. extending back to 1889, Ramey and Zubairy (2014) find no evidence that the government spending multipliers on output are larger when the policy rate is constrained at the ZLB than when it is not, and that the multiplier is less than one at the ZLB. Using a dataset for the U.K. during the 1930s, Crafts and Mills (2012) find the multiplier below unity when the interest rate is constrained at the ZLB. Our analyses show that policy inertia can bring the prediction of the New Keynesian model closer to these empirical estimates. Policy inertia reduces the multiplier by reducing the effects of government spending shocks on expected inflation; this reduced response of inflation expectations is also consistent with the empirical evidence from Dupor and Li (2013), who find no evidence that the American Recovery and Reinvestment Acts pushed up inflation expectations in the Survey of Professional Forecasters. 3

This paper is related to work on the government spending multiplier that emphasizes the role of the ZLB constraint. Earlier work by Christiano, Eichenbaum, and Rebelo (2011); Eggertsson (2011); and Woodford (2011) have emphasized that the multiplier can be substantially larger than one in plausibly parameterized New Keynesian models. Since then, many authors have examined the government spending multiplier at the ZLB in various settings. Examples include Albertini, Poirier, and Roulleau-Pasdeloup (2014); Braun, K¨orber, and Waki (2013); Braun and Waki (2010); Corsetti, Kuester, Meier, and Muller (2010); Carlstrom, Fuerst, and Paustian (2014); Denes, Eggertsson, and Gilbukh (2013); Bouakez, Guillard, and Roulleau-Pasdeloup (2014); Fern´andez- Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ram´ırez (2012); Mertens and Ravn (2014); and Roulleau-Pasdeloup (2014). All of these papers consider only the truncated Taylor rule without any inertia. While the majority of the literature focuses on the non-inertial policy rule, several authors have considered policy rules with an interest rate smoothing term. Examples include Aruoba, Cuba-Borda, and Schorfheide (2014) and Erceg and Lind´e (2014), who consider an inertial policy rule with the lagged actual interest rate, and Cogan, Cwik, Taylor, and Wieland (2010) and Drautzburg and Uhlig (2013), who briefly consider the model with inertial policy rules with the lagged shadow policy rate.1 Coenen, Erceg, Freedman, Furceri, Kumhof, Lalonde, Laxton, Lind, Mourougane, Muir, Mursula, de Resende, Roberts, Roeger, Snudden, Trabandt, and in’t Veld (2012) examine the effects of various fiscal shocks—including the government spending shock—in several structural models used at policy institutions. Some of the models considered in their paper have either lagged actual or shadow policy rates in the interest rate feedback rule. However, none of these papers have analyzed how the degree and type of policy inertia affect the government spending multiplier at the ZLB. OneexceptionisabriefsensitivityanalysisbyCarrilloandPoilly(2013)thatreportsa smaller government spending multiplier with a larger weight on the lagged shadow policy rate in a model with financial frictions. Our analyses not only show the generality of their result in a wide range of models and parameter configurations, but also clarify the mechanism by which the lagged shadow rate reduces the multiplier by showing that the type of policy inertia matters. Another exercise close to ours is a sensitivity analysis by Erceg and Lind´e (2014) that considers the government spending multiplier when the nominal interest rate is determined according to price-level targeting (PLT). They find that the government spending multiplier is smaller under PLT than under a truncated (non-inertial) Taylor rule. In our setup and theirs, the key feature of the policy rule that leads to a smaller fiscal multiplier is that the policy rate after the recession depends on the government spending shock during the recession. While the way the government spendingshockaltersthepost-recessionpolicyratesdiffersbetweenourmodelandtheirs, our in-depth analysis of the fiscal multiplier in the presence of policy inertia nevertheless 1They assume an exogenous ZLB duration for most of the analysis. 4

sheds light on the mechanism behind their result. Several authors have recently emphasized that the government spending multiplier can be small or modest even at the ZLB. Kiley (2014) finds the government spending multiplier at the ZLB to be below unity in the sticky information model. Cochrane (2014) argues that the multiplier can be small in alternative nonrecursive equilibria, and Mertens and Ravn (2014) find the multiplier below unity when the ZLB is triggered by a belief shock. Braun, K¨orber, and Waki (2013) demonstrate that the multiplier is only modestly above one under a variety of plausible parameter configurations, and Albertini,Poirier,andRoulleau-Pasdeloup(2014)obtainsmallmultipliersinamodelwith productive government spending. Our paper finds that a simple and empirically plausible modification to the standard setup in this literature—introducing the lagged shadow rate intothetruncatedTaylorrule—goesalongwaytowardreducingthegovernmentspending multiplier at the ZLB. Therestofthepaperisorganizedasfollows. Section2reviewstheempiricalliterature on policy inertia. Section 3 describes the model and Section 4 defines the government spending multiplier. Sections 5 and 6 present the results. Section 7 discusses the sensitivity of the main results, and Section 8 concludes. 2 Empirical Evidence for Policy Inertia 2.1 Evidence from Structural Estimation Structural estimation of dynamic stochastic general equilibrium (DSGE) models provides substantial evidence toward an intrinsic adjustment of nominal interest rates in the central bank’s reaction function. The level of inertia suggested by these estimates is typically high. Posterior estimates of postwar, pre–Great Recession U.S. data by Christiano, Trabandt, and Walentin (2010) suggest a persistence parameter between 0.85 and 0.91; Del Negro, Schorfheide, Smets, and Wouters (2007) place estimates between 0.7 and 0.86; Justiniano, Primiceri, and Tambalotti (2010) suggest a value between 0.757 to 0.819; and Smets and Wouters (2007) estimate a range from 0.77 to 0.85. Smets and Wouters (2003) also show that for the Euro area, smoothing levels can be particularly high, with parameters ranging from 0.93 to 0.97, suggesting that there is partial adjustment of nominal interest rates in other advanced economies as well. Gust, L´opez-Salido, and Smith (2012) estimate a model with a truncated inertial Taylor rule using U.S. data that includes the periods in which the nominal interest rates were at the effective lower bound. They find that the level of smoothed nominal adjustments is still high, with an estimated inertial parameter between 0.78 and 0.92. 2.2 Evidence from Reduced-Form Estimation The reduced-form estimation of the interest rate feedback rule also provides evidence in favor of an interest rate smoothing component. Clarida, Gali, and Gertler (2000) show 5

that the weights of inertia have increased since the pre-Volcker period and that levels as high as 0.91 are plausible. Orphanides (2003) finds that Greenbook and Survey of Professional Forecasters (SPF) survey forecasts suggest inertial parameters in the range of 0.75 to 0.91. Clarida, Gali, and Gertler (1998) show that the weights on the lagged policyrateareintherangeof0.87to0.97fortheU.S.andotheradvancedeconomies,with the majority of estimates suggesting a high level of inertia above 0.9. These international estimates again suggest that intrinsic policy smoothing is not a phenomenon unique to the U.S. alone. However, the interpretation of the reduced-form estimates has been a subject of debate. Rudebusch (2006) has argued that policy rates are contingent on both data that is incoming and changes in the economic outlook. Thus, policy is defined as more of an extrinsic reaction rather than an intrinsic adjustment—that is, inertia is more likely a reflection of omitted variables in the Fed’s reaction function than an intrinsic decision made on the part of the policymakers. Coibion and Gorodnichenko (2012) have argued that a formal statistical test favors the interest rate smoothing hypothesis once one allows for higher-order smoothing, as opposed to the commonly used restriction of a first-order autoregressive process. Coibion and Gorodnichenko (2012) find that autoregressive parametersinthepolicyrateerrortermbecomeinsignificantorevennegative,thusproviding evidence against the argument that inertia is merely a mirage of persistent shocks. 2.3 Relation with Forward Guidance Policies Aninterestratefeedbackrulethatincludesthelaggedpolicyratehascertainelements in common with the recent forward guidance policy in the U.S. whereby the Federal Reserve has stated that the policy rate will be kept at zero for an extended period of time even after the economic recovery strengthens. If the central bank adjusts its policy rate based only on the current economic conditions, it would raise the policy rate as the economic recovery strengthens. On the other hand, if the policy rule features a large weight on the lagged policy rate, then the central bank would raise the policy rate slowly. Policy inertia is not the only way to model an extended period of low nominal interest rates. For example, larger response coefficients on the economic conditions are also consistentwithstayingattheZLBforalongperiod, astheywouldlowerthenominalinterest rate when inflation and output gaps are negative. Modelling the central bank as following optimal commitment policy is another way to make the ZLB bind for a prolonged period. 2.4 Lagged Shadow Versus Actual Policy Rates These empirical considerations point to the validity of introducing the lagged policy rate into the Taylor rule. At the ZLB, however, one must also consider which version of the inertial policy rule—the one with the lagged shadow rate or the one with the lagged actual rate—to introduce. As we will see shortly, it is easier to generate persistent ZLB episodes with the shadow 6

rate version than with the actual rate version. Thus, our conjecture is that the inertial Taylor rule with the lagged shadow rate is likely to fit the recent prolonged liquidity trap episode in the U.S. better. Also, some policymakers have characterized the Federal Reserve’s “low-for-long” policy partially as an attempt to compensate for the fact that the policy rate was constrained at the ZLB in the past, which is also consistent with the shadow rate version of the policy rule.2 However, making a convincing case for one version of the truncated inertial policy rule over the other requires a formal econometric analysis, a task we leave for future research. We simply note that our result—that these two versions imply substantially different fiscal multipliers—suggests the importance of such econometric analyses.3 3 Model 3.1 Model Description We use a standard New Keynesian model with Rotemberg price adjustments. The economy is formulated in discrete time with an infinite horizon, and it is populated with a representative household, a final good producer, a continuum of intermediate goods producers with measure one, and the government. 3.1.1 Household The representative household chooses its consumption level, amount of labor, and bond holdings so as to maximize the expected discounted sum of utility in future periods. As is common in the literature, the household enjoys consumption and dislikes labor. Assuming that period utility is separable, the household problem can be defined by (cid:88) ∞ (cid:34) (cid:89) t−1 (cid:35)(cid:34) C1−χc N1+χn (cid:35) max E βt−1 δ t − t , (1) 1 s {Ct,Nt,Bt}∞ t=1 t=1 s=0 1−χ c 1+χ n subject to the budget constraint P C +R−1B ≤ W N +B +P Φ +P T (2) t t t t t t t−1 t t t t or, equivalently, B B t t−1 C + ≤ w N + +Φ +T , (3) t t t t t R P P t t t 2See, for example, Evans (2012) and Bullard (2013). This feature is also consistent with other policy rules that have been shown to be effective in mitigating the adverse consequences of the ZLB constraint, such as price-level targeting and the Reifschneider-Williams (2000) rule, as well as with optimal commitment policy. 3Inongoingwork,weusevariousmeasuresoftheexpectedZLBdurationbasedonsurveyandfinancial data to distinguish which version of the inertial policy rules fits the data better. 7

where C is consumption, N is the labor supply, P is the price of the consumption good, t t t W (w ) is the nominal (real) wage, Φ is the profit share (dividends) of the household t t t from the intermediate goods producers, B is a one-period risk-free bond that pays one t unit of money at period t+1, T is lump-sum taxes, and R−1 is the price of the bond. t t The discount rate at time t is given by βδ , where δ is the discount factor shock t t altering the weight of future utility at time t+1 relative to the period utility at time t. Following Eggertsson (2011) and Woodford (2011) among many others, we assume that δ is governed by an exogenous state s , which follows a two-state Markov shock process. t t s takes two values, H and L, and the transition probability is given by t Prob(s = L|s = L) = 1 (4) t+1 t and Prob(s = H|s = H) = µ. (5) t+1 t Itcanbeseenthats = Lisanabsolvingstate. Thevalueofδ dependsontherealization t t of s as follows. t  δ = 1 if s = L ss t δ = (6) t δ if s = H H t An increase in δ increases the relative valuation of future utility flows, making the houset hold more willing to save for tomorrow and less willing to consume today. While we work with this two-state Markov structure in the baseline, we will consider a case in which this shock follows an AR(1) process. 3.1.2 Firms There is a final good producer and a continuum of intermediate goods producers indexed by i ∈ [0,1]. The final good producer purchases the intermediate goods Y at i,t the intermediate price P and aggregates them using CES technology to produce and i,t sell the final good Y to the household and government at price P . Its problem is then t t summarized as (cid:90) 1 max P Y − P Y di, (7) t t i,t i,t Yi,t,i∈[0,1] 0 subject to the CES production function (cid:34) (cid:35) θ (cid:90) 1 θ−1 θ−1 Y = Y θ di . (8) t i,t 0 Intermediate goods producers use labor to produce the imperfectly substitutable intermediate goods according to a linear production function (Y = N ) and then sell the i,t i,t product to the final good producer. Each firm maximizes its expected discounted sum of 8

future profits by setting the price of its own good.4 We further assume that any price changes are subject to quadratic adjustment costs: ∞ (cid:34)t−1 (cid:35) (cid:34) (cid:35) max}∞ E (cid:88) βt−1 (cid:89) δ λ P Y −W N −P ϕ(cid:104) P i,t −1 (cid:105)2 Y , (9) {Pi,t t=1 1 t=1 s=0 s t i,t i,t t i,t t 2 P i,t−1 t such that (cid:20) P (cid:21)−θ Y = i,t Y .5 (10) i,t t P t (cid:34) (cid:35) λ istheLagrangemultiplieronthehousehold’sbudgetconstraintattimetandβt−1 (cid:81)t−1 δ λ t s=0 s t isthemarginalvalueofanadditionalprofittothehousehold. Thepositivetimezeroprice is the same across firms (i.e., P = P > 0). i,0 0 3.1.3 Government We assume that the nominal interest rate is determined according to the following truncated inertial Taylor rule: R = max (cid:2) 1,R∗(cid:3) (11) t t R∗ = 1 (cid:32) R t ∗ −1 (cid:33)ρr (cid:32) Πφπ (cid:16) Y t (cid:17)φy (cid:33)1−ρr , (12) t β R t Y ss ss where Π ≡ Pt and R and Y are the steady-state levels of the nominal interest rate t Pt−1 ss ss and output. R∗ is the hypothetical policy rate that would prevail were it not for the ZLB t constraint at time t, and will be referred to as the shadow, notional, or recommended nominal interest rate. In Section 6, we will also consider three alternative assumptions about the determination of the nominal interest rate: (i) that it follows a version of the inertial Taylor rule in which today’s shadow rate depends on the lagged actual policy rate, (ii) that it uses price-level targeting, and (iii) that it is chosen optimally by the government with commitment. In the baseline exercise, we assume that the supply of the one-period risk-free bond is zero and that lump-sum taxes are available to finance the government spending. Thus, the government budget constraint is given by G = T . (13) t t 4Each period, as it is written below, is in nominal terms. However, we want each period’s profits in real terms so the profits in each period will be divided by that period’s price level P . t 5This expression is derived from the profit maximizing input demand schedule when solving for the final good producer’s problem above. Plugging this expression back into the CES production function (cid:34) (cid:35) 1 1−θ implies that the final good producer will set the price of the final good P = (cid:82)1P1−θdi . t 0 i,t 9

Further on, we will consider cases in which a distortionary tax is available and the supply of the government bond is allowed to be nonzero. Government spending, G , is governed by the state variable s introduced earlier: t t  G if s = L ss t G = (14) t G if s = H. H t Our assumption of perfect correlation between the preference shock and government spending shock follows the existing literature. Later, we will consider a case in which the government spending shock follows an AR(1) process. We will use γ to denote the steady-state share of government spending to output (i.e., γ ≡ Gss). We will use g to denote the log deviation of G from G (i.e., g ≡ log (cid:0)GH (cid:1) ). Yss H ss Gss 3.1.4 Market Clearing Conditions The market clearing conditions for the final good, labor, and government bond are given by (cid:34) (cid:35)2 (cid:90) 1 ϕ P i,t Y = C + −1 Y di+G , (15) t t t t 2 P 0 i,t−1 (cid:90) 1 N = N di, (16) t i,t 0 and B = 0. (17) t 3.1.5 Nonlinear Equilibrium Conditions Given P and a two-state Markov shock process, s , governing the evolution of δ and 0 t t G ,anequilibriumisdefinedasallocations{C ,N ,N ,Y ,Y ,G }∞ ,prices{W ,P ,P }∞ , t t t i,t t i,t t t=1 t t i,t t=1 and a policy instrument {R }∞ , such that (i) given the determined prices and polit t=1 cies, allocations solve the household problem; (ii) P solves the problem of firm i; (iii) i,t P = P ∀i (cid:54)= j; (iv) R follows a specified rule; and (v) all markets clear. i,t j,t t Combiningalloftheresultsfrom(i)–(v),asymmetricequilibriumcanbecharacterized recursively by {C ,N ,Y ,w ,Π ,R }∞ satisfying the following equilibrium conditions: t t t t t t t=1 C−χc = βδ R E C−χcΠ−1 , (18) t t t t t+1 t+1 w = NχnCχc, (19) t t t (cid:34) (cid:35) Y Y t t+1 ϕ(Π −1)Π −(1−θ)−θw = βδ E ϕ(Π −1)Π , (20) Cχc t t t t t Cχc t+1 t+1 t t+1 ϕ Y = C + [Π −1]2Y +G , (21) t t t t t 2 Y = N , (22) t t 10

and equations (11) and (12). Equation (18) is the consumption Euler Equation, equation (19) is the intratemporal optimality condition of the household, equation (20) is the optimal condition of the intermediate good-producing firms (forward-looking Phillips Curve) relating today’s inflation to real marginal cost today and expected inflation tomorrow, equation (21) is the aggregate resource constraint capturing the resource cost of price adjustment, and equation (22) is the aggregate production function. 3.2 Semi-Loglinear Equilibrium Conditions Following the majority of the existing literature, we will mainly work with a semiloglinear version of the economy in which all the equilibrium conditions are log-linearized except for the ZLB constraint on the policy rate. An equilibrium in the semi-loglinear economy is characterized by {Cˆ ,Yˆ,Πˆ ,i ,i∗}∞ satisfying the following conditions: t t t t t t=1 Yˆ = E Yˆ −σ(i −E Πˆ +δˆ −r¯)+γ(Gˆ −E Gˆ ), (23) t t t+1 t t t+1 t t t t+1 Πˆ = κYˆ −κψσ−1γGˆ +βE Πˆ , (24) t t t t t+1 Yˆ = (1−γ)Cˆ +γGˆ , (25) t t t i = max (cid:2) 0,i∗(cid:3) , (26) t t (27) and i∗ = r¯+ρ (i∗ −r¯)+(1−ρ )(φ Πˆ +φ Yˆ), (28) t r t−1 r π t y t where σ ≡ 1−γ, κ ≡ (θ−1)(χn+σ−1) , ψ ≡ 1 , r¯≡ 1−β, and i ≡ Rˆ +r¯, χc ϕ χn+σ−1 t t  0 if s = L δˆ = t (29) t δˆ if s = H, H t and  0 if s = L Gˆ = t (30) t g ≡ log (cid:0)GH (cid:1) if s = H. Gss t A recursive competitive equilibrium of this semi-loglinear economy is given by a set of policy functions, {Yˆ(·,·), Cˆ(·,·), Πˆ(·,·), i(·,·), i∗(·,·)}, that satisfies the functional equations above. These policy functions are functions of the lagged shadow nominal interest rate, i∗ , and the state, s ∈ {H,L} . We use a time-iteration method from t−1 t Coleman(1991)tofindthemnumerically. Thedetailsofthesolutionmethodaredescribed in the Appendix. Some have argued that the fiscal multipliers computed in the semi-loglinear economy 11

maybeapoorapproximationtothoseinthefullynonlineareconomy. Accordingly,wewill examine the robustness of our results to solving the model fully nonlinearly in Section 7. 3.3 Parameterization As our baseline, we use parameter values consistent with Denes, Eggertsson, and Gilbukh (2013) to match U.S. data during the Great Depression. The values are listed in Table 1. There are a few parameters that are not in the Denes, Eggertsson, and Gilbukh (2013) model but are present in ours. For the steady-state ratio of government spending to output, we choose γ ≡ Gss = 0.2. For the coefficients on inflation and output in the Yss inertial Taylor rule, we choose (φ ,φ ) = (1.5,0.25). Our main exercise is to vary the π y weight on the lagged shadow interest rate. We consider 100 weights that are multiples of 0.01from0to0.99.6 Wewillexaminetherobustnessofourresulttoalternativeparameter values in Section 7. Table 1: Great Depression (Baseline) Parameterization β χ χ γ θ ϕ φ φ µ c n π y 0.997 0.9224 1.53 0.20 12.70 3444.8512 1.50 0.25 0.902 4 Definition of Government Spending Multipliers We define the government spending multiplier function, GM(i∗ ,g,δˆ ), as follows: −1 H Yˆ(i∗ ,H;g,δˆ )−Yˆ(i∗ ,H;0,δˆ ) GM(i∗ ,g,δˆ ) := −1 H −1 H , (31) −1 H γg where y(·,·;g,δˆ ) is a policy function for output indexed by (g,δˆ ). This function mea- H H surestheaverageincreaseinoutputinresponsetoanincreaseinthegovernmentspending of size g in the high (crisis) state when the lagged shadow rate is r¯and the state and the size of the discount factor shock is δˆ . H We call the government spending multiplier function evaluated at (i∗ ,g,δˆ ) = (r¯, −1 H 0.01, 0) the non-crisis multiplier. It measures the average effect of a one percent increase in government spending on output in the economy without any preference shocks. We will also refer to the non-crisis multiplier as the government spending multiplier away from the ZLB, or non-ZLB multiplier. We call the government spending multiplier function evaluated at (i∗ ,g,δˆ ) = (r¯, −1 H 0.01, δˆ ) the crisis multiplier when δˆ > 0. This measures the average effect of a one H H percent increase in government spending on output in the economy when the preference shock hits. We will be interested mostly in a case in which δˆ is sufficiently large that H i(i∗ ,H;0,δˆ ) = 0. Our definitions are consistent with those in the literature (e.g., −1 H 6We will not consider the case with ρ =1 because this case violates the Taylor principle. r 12

Eggertsson (2011) and Woodford (2011)). We will also refer to the crisis multiplier as the government spending multiplier at the ZLB or the ZLB multiplier.7 Our definition of the multiplier may appear unnecessarily confusing to some readers, but our definition is the same as that used in the papers on which our work builds— Eggertsson (2011), Woodford (2011), and Denes, Eggertsson, and Gilbukh (2013)—even though the multiplier is often not explicitly defined in these papers. The definition looks complicated because of the perfect correlation assumption on two shocks as well as our desire to be explicit about the dependence of the effects of government spending shocks on the initial condition and the magnitude of shocks.8 We have decided to be precise at the risk of initially appearing confusing to some readers. Our government spending multiplier measures the effects of the government spending shock on output at time one. In other words, our baseline multiplier is an impact multiplier. An alternative concept of the government spending multiplier that is also common in the literature is the present value multiplier, which aims to capture the average effects of government spending shocks on output over time. Our focus on the impact multiplier is motivated by our desire to remain simple and close to the work of the aforementioned papers we build upon. In particular, we find it useful to make the government spending multiplier in our model without policy inertia identical to the multiplier reported in Denes, Eggertsson, and Gilbukh (2013), as we use their parameter values. We do think that the present-value multiplier is also a useful object, and we examine how the policy inertia affects the present value multiplier in the Appendix. We find that the main results of the paper are robust to the alternative definition of the multiplier. The main exercise of this paper is to compare the fiscal multipliers in economies with various degrees of policy inertia. When comparing the effects of policy inertia on the fiscal multiplier, we adjust the size of the preference shock, δˆ , so that the initial declines H in output are the same across different values of the inertia parameter. That is, the government spending multiplier in the economy with ρ is computed as r Yˆ(i∗ ,H;g,h(ρ ))−Yˆ(i∗ ,H;0,h(ρ )) GM(i∗ ,g,h(ρ )) := −1 r −1 r , (32) −1 r γg whereforanyρ > 0,theadjustedshocksize,h(ρ ),iscomputedsothatYˆ(i∗ ,H;0,h(ρ )) r r −1 r in the economy with policy inertia is the same as Yˆ(i∗ ,H;0,δˆ ) in the economy with −1 H ρ = 0. We make this adjustment because we want to understand the effects of the fiscal r multiplier in a comparable situation. In the absence of this adjustment, the initial decline in output will be smaller in the economy with policy inertia than in the economy without it, and one would have difficulty understanding whether the difference in the fiscal 7Whentheweightonthenominalinterestrateissufficientlylarge,theZLBdoesnotnecessarilybind at time one in response to the preference shock. Thus, in this sense, using the terms the crisis multiplier and the ZLB multiplier interchangeably can be misleading. We will be clear whenever this happens to avoid any misunderstandings. 8Without the prefect correlation assumption, the multiplier is small even at the ZLB. See Woodford (2011). 13

multipliers across two economies is driven by the difference in the policy rule or by the differences in the severity of the recession across two economies. A similar adjustment is made in sensitivity analyses conducted by Braun, K¨orber, and Waki (2013) and Erceg and Lind´e (2014). 5 Results Figure 1 shows how the fiscal multiplier varies with the policy inertia parameter. The black and red lines are for the government spending multipliers when the nominal interest rate is constrained at the ZLB and when it is not, respectively. 5.1 Policy Inertia and Multipliers at the ZLB The black line shows two features of the fiscal multipliers at the ZLB with inertia. First, the fiscal multiplier is smaller in the economy with policy inertia than in the economy without it. While the multiplier without inertia is 2.5, the multiplier with an inertia of 0.9 is 1.1. As the inertia increases, the ZLB multiplier approaches the non-ZLB multiplier from above. The multiplier is substantially above those away from the ZLB and remains above one for a plausible range of the weights. A second finding is that the policy inertia affects the fiscal multiplier in a non-monotonic way. As the inertia increases from 0 to 0.03, the multiplier decreases monotonically. However, at an inertia level of 0.04, the multiplier jumps up; after the jump, the multiplier declines monotonically until it jumps up again. Figure 1: Policy Inertia and the Government Spending Multiplier 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF ZLB does ZLB binds not bind at t = 1 at t = 1 ZLBepisode NoZLBepisode 14

Smaller Multiplier To understand the first result, it is useful to examine how the government spending shock affects the economy at the ZLB with and without inertia. The left column of Figure 2 shows realizations of consumption, output, inflation, and the nominal interest rate in the economy without inertia when the preference shock hits the economy at time one and lasts for eight quarters. The right column of Figure 2 shows the same set of impulse response functions for the economy with ρ = 0.85.9 In each panel, the black line r is the case without an increase in government spending, and the red line is for the case witha25percentincreaseingovernmentspending.10 Output, consumption, andinflation decline substantially and the nominal interest rate is at the ZLB during the crisis period in economies both with and without policy inertia. Figure 2: IRFs at the ZLB—with and without policy inertia— 0 −10 −20 −30 0 10 20 30 40 )%( t ˆY ρr = 0 0 −10 w/oG −20 w/G −30 0 10 20 30 40 )%( t ˆY ρr = 0.85 0 −20 −40 0 10 20 30 40 )%( t ˆC 0 −20 −40 0 10 20 30 40 )%( t ˆC 0 −5 −10 0 10 20 30 40 )% .nna( t Πˆ 0 −5 −10 0 10 20 30 40 )% .nna( t Πˆ 1.5 1 0.5 0 0 10 20 30 40 )% .nna( t i 1.5 1 0.5 0 0 10 20 30 40 )% .nna( t i 0 −20 −40 0 10 20 30 40 )% .nna( ∗i t 0 −20 −40 0 10 20 30 40 )% .nna( ∗i t Without policy inertia, the economy reverts back to the steady state as soon as the shockdisappears, regardlessofwhethergovernmentspendingvaries. Intheeconomywith policy inertia, the nominal interest rate stays at the ZLB even after the shock disappears. This is because the shadow rate is negative just before the shock disappears. Eventually, the nominal interest rate lifts off and gradually returns to the steady-state level. The 9ThisvalueisclosetotheposteriormodeofthisparameterfromGust,L´opez-Salido,andSmith(2012) whoestimateasticky-pricemodelusingtheU.S.dataoverthesampleincludingtherecentZLBepisode. 10Readersshouldbearinmindthatwesetthesizeofthegovernmentspendingshocktobe25percent inthefiguresothattheeffectsarevisuallyclearandthatthegovernmentspendingmultiplieriscomputed based on a 1 percent increase in government spending as discussed in Section 4. 15

nominal interest rate path is thus lower relative to the path under the economy without inertia after the shock disappears. This extended period of low nominal interest rates causes the economy to overshoot in consumption, inflation, and output. This feature of theeconomyistrueregardlessofwhetherornotthegovernmentspendingshockispresent or not. In the economy without inertia, an exogenous increase in government spending during a recession raises the demand for the final good, which in turn increases output and inflation. WiththenominalinterestratestuckattheZLB,anincreaseininflationreduces the real interest rate, which in turn boosts consumption. With a positive consumption responsetothegovernmentspendingshock, themultiplieronoutputisaboveone. Notice that,inthiseconomywithoutinertia,thegovernmentspendingshockduringtherecession hasnoimplicationsfortheeconomyaftertherecession. Theeconomyisbackatthesteady state after the shock dissipates regardless of the government spending shock. In the economy with policy inertia, an exogenous increase in government spending during the crisis has effects on the economy both during and after the recession. An exogenous increase in the government spending during the recession increases consumption, inflation, and output during the recession, as it does in the economy without inertia. In the inertial economy, the higher inflation path during the recession implies a higher path of shadow policy rates relative to the path in the absence of the government spending shock. As a result, the nominal interest rate returns to the steady state more quickly in the presence of the government spending shock than in its absence, and the overshooting of inflation and output is slightly smaller after the recession with the fiscal policy than without. Lower inflation and a higher nominal interest rate after the recession reduces consumption during the crisis since the household is forward-looking and cares about the expected future real rate. Lower output after the recession reduces prices during the crisis since firms are forward-looking and care about the expected future marginal costs of production. Since consumption is ultimately determined by the undiscounted sum of future expected real interest rates, we can better understand the effect of government spending shocks on consumption, and thus output, by examining the effects of government spending shocks on the path of expected real interest rates. For that purpose, Figure 3 shows expected nominal interest rates, inflation, and real interest rates with and without government spending shocks. The left panels show the economy without policy inertia while the right panels depict the economy with policy inertia. By inspecting the differential effects of government spending on the path of expected nominal interest rates, inflation, and real interest rates, the differences in the government spending multipliers with and without policy inertia become clearer. AsshownintheleftpanelsofFigure3,andconsistentwiththepreviousdiscussion,the government spending shock does not affect the expected path of nominal interest rates in the economy without policy inertia. However, an increase in government spending increases inflation expectations at all forecast horizons because government spending is 16

Figure 3: Time-one Mean Forecasts at the ZLB—with and without policy inertia— 1 0.5 0 0 10 20 30 40 ]t i[1 E ρr =0 1 0.5 w/oG w/G 0 0 10 20 30 40 ]t i[1 E ρr =0.85 0 −5 −10 0 10 20 30 40 ]t Πˆ[ 1 E 0 −5 −10 0 10 20 30 40 ]t Πˆ[1 E 10 8 6 4 2 0 10 20 30 40 ] 1+t Πˆ-t i[ 1 E 10 8 6 4 2 0 10 20 30 40 ]1+t Πˆ-t i[1 E expected to remain elevated as long as the preference shock lasts, and an elevated level of government spending is associated with smaller deflation. Accordingly, the expected real interest rates are lower with government spending than without it at all forecast horizons. In the economy with policy inertia, an increase in government spending during the recession pushes up the path of shadow nominal interest rates and the nominal interest rate moves back to the steady-state level more quickly for any realizations of preference shocks. Thus, the expected nominal interest rate is higher with government spending shocks than without them at all forecast horizons, as shown in the top-right panel. The tighter monetary policy implies that the expansionary effects of the government spending shock on inflation are mitigated. As shown in the middle-right panel, the expected path of inflation is higher with government spending, but not by as much as in the economy without policy inertia. A higher path of expected nominal interest rates and subdued increases in the inflation expectations mean that the government spending shock reduces the expected real interest rate by less in the economy with policy inertia than in the economywithoutit. Accordingtothebottompanels,theeffectsofagovernmentspending shock on expected real interest rates, as captured by the grey area, is smaller in the inertial economy than in the non-inertial economy. As a result, consumption rises less in the inertial economy, leading to a smaller output multiplier. Non-Monotonicity Whilethemultiplierissmallerintheeconomywithpolicyinertiathanintheeconomy without it, the effects of policy inertia on the multiplier is not monotonic. In particular, there are several points in ρ = [0,1) where the multiplier increases in response to a r marginal increase in ρ . r 17

To understand where this non-monotonicity comes from, notice that the duration of the ZLB depends on the inertia parameter. In particular, an increase in the policy inertia can increase the duration of the ZLB episode given a certain path of the crisis shock and the expected duration of the ZLB that results. When it does, the increase occurs in a discrete way, since the duration of the ZLB is a discrete variable as shown in Figure 4. The fiscal multiplier depends on the duration of the ZLB episode, since the duration determines the periods during which the expansionary effects of the government spending are not offset by a corresponding increase in the nominal interest rate, as emphasized in the work of Eggertsson (2011), Woodford (2011), and Erceg and Lind´e (2014). The longer the ZLB duration, the larger the multiplier. The jumps in the fiscal multiplier in Figure 1 correspond to the jumps in the expected duration of the ZLB in Figure 4. Figure 4: Policy Inertia and Expected ZLB Duration 40 35 30 25 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρr noitaruD BLZ ZLB duration by shock duration Expected ZLB duration 5.2 Policy Inertia and Multipliers Away from the ZLB The effect of policy inertia on the government spending multiplier is qualitatively different when the nominal interest rate is away from the ZLB. The red line in Figure 5 shows that the government spending multiplier increases with the weight on the lagged policy rate if the nominal interest rate is away from the ZLB. The effect of inertia on the multiplier is much smaller when the nominal interest rate is away from the ZLB than when it is at the ZLB. Away from the ZLB, the multiplier increases by 0.1 (from 0.8 to 0.9) as the inertia increases from zero to 0.85. Intheeconomywithinertia,thenominalinterestrateadjustsslowlytothegovernment spending shock. Thus, output expands by more in the economy with inertia than in the economy without it. Later, as the policy rate gradually increases, output and inflation 18

decline. Overall,theeffectsofthegovernmentspendingshockontheexpectedrealinterest rateissmallerandconsumptiondecreasesbylessattimeoneintheinertialeconomythan in the non-inertial economy. Figure 5: IRFs away from the ZLB—with and without policy inertia— 2 1 0 0 10 20 30 40 )%( t ˆY ρr = 0 w/oG 2 w/G 1 0 0 10 20 30 40 )%( t ˆY ρr = 0.85 0 −2 −4 0 10 20 30 40 )%( t ˆC 0 −2 −4 0 10 20 30 40 )%( t ˆC 0 −0.2 −0.4 0 10 20 30 40 )% .nna( t Πˆ 0 −0.2 −0.4 0 10 20 30 40 )% .nna( t Πˆ 2.5 2 1.5 0 10 20 30 40 )% .nna( t i 2.5 2 1.5 0 10 20 30 40 )% .nna( t i 6 Additional results 6.1 Alternative Policy Rules An Alternative Version of the Inertial Taylor Rule Thusfar,wehavefocusedontheversionoftheinertialpolicyruleinwhichtheshadow policy rate today depends on the lagged shadow policy rate. An alternative formulation would be an inertial rule in which the shadow policy rate depends on the actual policy rate, as follows: R = max (cid:2) 1,R∗(cid:3) (33) t t (cid:32) (cid:33)ρr (cid:32) (cid:33)1−ρr R∗ = 1 R t−1 Πφπ (cid:16) Y t (cid:17)φy . (34) t β R t Y ss ss The top-left panel of Figure 6 shows how the government spending multiplier varies with the policy inertia parameter in the economy with this version of the inertial Taylor rule. The multiplier does not vary with the inertial parameter unless the inertia is very 19

closetoone. Thereasonforthisinsensitivityisthat,inthisspecificationofthepolicyrule, the policy rate after the recession depends on the actual policy rate in the last quarter of the recession, which is zero no matter how dramatically inflation and output decline during the recession. Thus, increased inflation and output during the recession due to the government spending shock do not have any influence on the path of the policy rate after the recession, unless the policy inertia is very close to zero. This can be seen in the first column of Figure 7, which shows the evolution of output, consumption, inflation, and the nominal interest rate with and without government spending shocks. The figure shows that the government spending shock does not alter at all the path of policy rates after the shock disappears. Accordingly, the government spending multiplier is unchanged from that in the economy without policy inertia. Figure 6: Alternative History-Dependent Policy Rules Alt. InertialTaylor Rule 2.5 2 1.5 1 0.5 0 0 0.5 1 ρ r reilpitluMlacsiF Price-LevelTargeting Rule 2.5 2 1.5 1 0.5 0 0 0.5 1 φ p reilpitluMlacsiF 2.5 2 1.5 1 0.5 0 0 0.5 1 α z reilpitluMlacsiF Reifschneider & Williams (2000)Rule ZLB episode No ZLB episode Price-Level Targeting A key mechanism by which the inertia in the policy rule reduces the government spending multiplier is that government spending today affects the evolution of the economy after the shock disappears. This mechanism is not a unique feature of the model with the inertial policy rule studied above; it is also present in the economies where the nominal interest rate is determined according to other history-dependent policy rules. To make this point, we examine the fiscal multipliers in the economy in which the nominal interest rate is determined according to price-level targeting. The nominal interest rate in the price-level targeting regime is determined by iPT = max (cid:2) 0,r¯+φ p +φ Πˆ +φ Yˆ(cid:3) , (35) t p t π t y t 20

Figure 7: IRFs at the ZLB with Three Alternative Policy Rules 10 0 −10 −20 −30 0 20 40 )%( ˆY t PLT (φ = 1.0) p 10 0 −10 −20 −30 0 20 40 )%( ˆC t 2 0 −2 −4 −6 0 20 40 )% .nna( Πˆ t 1 0.5 0 0 20 40 )% .nna( i t 0 −10 −20 −30 0 20 40 )%( ˆY t AIT (ρ = 0.85) r 0 −10 −20 −30 0 20 40 )%( ˆC t 0 −5 −10 0 20 40 )% .nna( Πˆ t 1 0.5 0 0 20 40 )% .nna( i t 10 0 −10 −20 −30 0 20 40 )%( ˆY t RW (2000) (α =1) z w/oG w/G 0 −20 −40 0 20 40 )%( ˆC t 5 0 −5 0 20 40 )% .nna( Πˆ t 1 0.5 0 0 20 40 )% .nna( i t wherep isthelog-deviationofthepricelevelP fromatargetlevelP∗ > 0. Theevolution t t of p is given by t p = p +Πˆ . (36) t t−1 t The top-right panel of Figure 6 shows the government spending multipliers from the economy with price-level targeting. According to the black line, the government spending multiplier declines with the weight on the price-level stabilization term in the policy rule. Forasufficientlylargeweightonthepricelevel,theZLBmultiplierdeclinesbelowone. As showninthemiddlepanelsofFigure7, undertheprice-leveltargetingregime, anincrease in government spending during the recession speeds up the return of the policy rate to the steady state and reduces the overshooting of output, consumption, and inflation after the shock disappears. These developments after the recession mitigate the expansionary effects of the government spending shocks during the recession via expectations. Thus, price-level targeting lowers the government spending multiplier in the same manner as our baseline inertial policy rule. Reifschneider-Williams (2000) Rule Anotherwell-knownhistory-dependentpolicyruleistheReifschneider-Williams(2000) rule. The rule is defined by the following set of equations: iRW = max(0,iTR−α Z ), (37) t t z t 21

where iTR = r¯+φ Πˆ +φ Yˆ (38) t π t y t is the unconstrained Taylor rule and Z = Z +(iRW −iTR) (39) t t−1 t−1 t−1 is the sum of all past deviations of the realized nominal interest rate rule from the rate suggestedbytheunconstrainedTaylorrule. Settingα = 0yieldsthebaseline. Thelonger z and/or the more severe the contractionary shock is, the larger the cumulative deviation is; as a result, the policy rate is kept at the ZLB for a longer period. As demonstrated in the bottom-left panel of Figure 6, the government spending multiplier is lower under the Reifschneider-Williams (2000) rule than under the non-inertial truncated Taylor rule. With a sufficiently large weight on the cumulative deviation variable, the multiplier is about 0.5. As in the inertial rule with lagged shadow rates and the price-level targeting rule, the endogeneity of post-recession policy rates to the government spending shock during a recession is responsible for the lower multiplier. With the Reifschneider-Williams (2000) rule, positive government spending shocks during a recession increase inflation, output, and shadow policy rates. Thus, the cumulative deviation oftheactualpolicyratesfromtheshadowpolicyrateissmallerwithgovernmentspending shocks than without them. In the impulse response function shown in the right column of Figure 7, the policy path is tighter, with government spending partially offsetting the expansionary effects of government spending shocks during the recession.11 Optimal Commitment Policy Investigationsofthethreealternativepolicyrulesabovemakeitclearthatthekeyfactor lowering the government spending multiplier is the endogeneity of the post-recession evolution of policy rates on the economic performance during the recession. Such endogeneity is a hallmark of optimal commitment policy, under which the policy rate is kept at the ZLB even after the contractionary shock disappears and the ZLB duration is endogenous to the outcomes of the economy during the recession. In this sense, our baseline inertial Taylor rule and the price-level targeting rule are similar to optimal commitment policy. When we compute the government spending multiplier under optimal commitment policy, we find that the multiplier is 0.85 at the ZLB. Thus, one way to interpret our main result—that the government spending multiplier declines with policy inertia—is that the closer the policy rule is to optimal commitment policy, the lower the multiplier is. This observation is reminiscent of the result in Nakata (2013) and Schmidt (2013) that the optimal increase in government spending at the ZLB is smaller in the model with commitment than in the model without it. 11SeealsoBundick(2014)whoanalyzesvariousmultipliersattheZLBundertheReifschneider-Willaims (2000) rule. 22

6.2 Is the Government Spending Still Self-Financing? Thus far, this paper has abstracted from debt dynamics. Denes, Eggertsson, and Gilbukh (2013) and Erceg and Lind´e (2014) have argued that an increase in government expenditure can be self-financing at the ZLB, as an expansion in output increases tax revenue. We have shown that the multiplier can be substantially smaller in the economy with policy inertia. Thus, a natural question is whether or not the government spending increase is still self-financing. To answer this question, we follow Erceg and Lind´e (2014) by augmenting our model with a simple rule for debt dynamics and analyzing the evolution of debt in that setup. The new government budget constraint is given by γGˆ +bG = βbG+γ(χ +1)Yˆ +γχ Cˆ +τ , (40) t t−1 t n t c t t where bG = Bt and τ = Tt are government debt and lump-sum taxes, respectively, as t PtYss t Yss sharesofnominal(real)trendinoutput; theyareexpressedaspercentagepointdeviations from their steady-state values, which are 0. Following Erceg and Lind´e (2014), we let τ = φ bG be the reaction function defining lump-sum tax adjustments each period, and t b t−1 we set the tax rule parameter φ = 0.01. As is shown above, we have implicitly fixed a b labor income tax, τw, such that government spending is solely financed by this labor tax in the steady state (γY = τw Yχn+1Cχc ⇒ τw = γθ ). ss 1−τw ss ss θ−1 Figure 8 shows how differently the debt-to-GDP ratio evolves in the economies with and without policy inertia. Under this parameterization of the fiscal policy rule, the government spending is indeed self-financing in the absence of inertia. As seen in the left panel of Figure 8, the debt ratio declines on average in response to a small government stimulus due to the higher government spending multiplier that results in an economy constrained by the ZLB. This is the same result that Erceg and Lind´e (2014) and Denes, Eggertsson, and Gilbukh (2013) reach in their experiments. However, as inertia in monetary policy increases—resulting in a decline in the government spending multiplier—the reduction in the debt share becomes small, and after a certain point, as seen in the right panel of Figure 8, the increase in the government spending increases debt. Because a higher degree of policy inertia implies a smaller fiscal multiplier, this diminishes the effect of the increased financing derived from the distortionary labor tax, which is due to an increaseinoutput. Asaresult, thissourceoffinancingisunabletofullyoffsettheupward pressure of government spending on debt. Thus we see that the financing of government spending in this environment depends significantly on the degree of policy inertia. 23

Figure 8: Evolution of the Debt-to-GDP ratio 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 Gb t ρr=0 1.8 1.6 1.4 1.2 1 w/oGatZLB w/GatZLB w/GawayZLB 0.8 0.6 0.4 0.2 0 0 10 20 30 40 Gb t ρr=0.85 7 Sensitivity Analyses 7.1 Shock Size In the model without inertia, the size of the preference shock, and thus the severity of the recession, does not affect the fiscal multiplier. However, in the economy with inertia, the size of the shock affects the fiscal multiplier to some degree. According to the top-left panel of Figure 9, the fiscal multiplier is lower when the recession is less severe. Thefiscalmultiplierislowerintheless-severerecessionbecausethedurationforwhich the policy rate is zero after the shock disappears is shorter for any given realization of the preference shock. With a very severe recession, the actual policy rate is kept at zero for a long period. Since an increase in the shadow policy rate due to a government spending shock at the end of the recession gradually diminishes after the recession, the difference between shadow rates with and without government spending shocks becomes small by the time the shadow rate is positive. Thus, the differences in the actual policy rate path is smaller when the recession is severe and the ZLB is expected to bind for a long time. On the other hand, when the recession is not severe, the lift-off occurs soon after the shock disappears. Under this circumstance, the increase in the shadow rate at the end of the recession is still large, leading to a larger increase in the actual policy rate. A larger increase in the actual policy rate is associated with a higher path of real interest rates, making the fiscal multiplier lower when the recession is less severe. 7.2 Output Response Coefficient The top-right panel of Figure 9 studies the effect of the output gap coefficient on the governmentspendingmultiplier. Accordingtothefigure,thecoefficientontheoutputgap 24

Figure 9: Sensitivity Analyses Shock size (calib. δ ⇒ -∆y) H 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF Output response coefficient (φ ) y 2.5 2 ZLBEp. -30% ∆y ZLBEp. -20% ∆y 1.5 ZLBEp. -10% ∆y No ZLBEp. 1 0.5 0 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF ZLBEp. φ y = 0.0 ZLBEp. φ y = 0.25 No ZLBEp. φ y = 0.0 No ZLBEp. φ y = 0.25 Initial condition (i∗ ) 0 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF Great Recession vs. Great Depression 2.5 2 ZLBEp.,i∗ = r¯ 0 ZLBEp.,i∗ = 0 0 1.5 No ZLBEp.,i∗ = r¯ 0 1 0.5 0 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF ZLBEp.,GD ZLBEp.,GR No ZLBEp.,GR No ZLBEp.,GD in the policy rule does not affect the fiscal multiplier when the policy inertia parameter is zero. However, the fiscal multiplier is generally lower when the coefficient on the output gap is larger. Thecoefficientontheoutputgapdoesnotaffectthemultiplierintheeconomywithout policy inertia because it does not affect the path of actual policy rates; regardless of the coefficient, the policy rate goes back to its steady-state value once the shock disappears. In the economy with policy inertia, the output gap coefficient affects the path of actual policy rates by affecting the path of shadow rates. With a positive output gap coefficient, an increased output during the recession also contributes to the increase in the shadow rates, speeding up the return of the policy rates to the steady-state level. With a tighter policy path, the effects of the government spending shock are more muted when there is a larger response coefficient on the output gap. 7.3 Initial Condition In our baseline exercise, we compute the government spending multiplier under the scenario in which the initial lagged nominal rate is at the steady-state level, r¯. In this environment, for a sufficiently large weight on the lagged policy rate, the ZLB does not 25

bind at time one. An alternative worth entertaining when computing the crisis multiplier is to assume that the initial lagged policy rate is zero so that the nominal interest rate is zero at time one for any degree of policy inertia. We find that the initial condition does not materially alter the government spending multiplier, as shown in the bottomleft panel of Figure 9. This is because the shadow rates decline substantially during the recession and the initial difference of r¯ is negligible relative to the effect of government spending shocks on the path of shadow rates. 7.4 The Great Recession Calibration The size of the fiscal multiplier is of course sensitive to the choice of structural parameters. For example, the multiplier is 1.2 in the Great Recession calibration of Denes, Eggertsson, and Gilbukh (2013) shown in Table 2, compared to 2.5 in the Great Depression calibration without policy inertia. However, the result that the policy inertia reduces the government spending multiplier at the ZLB is robust, as seen in the bottom-right panel of Figure 9. Table 2: Great Recession (Alternative) Parameterization β χ χ γ θ ϕ φ φ µ c n π y 0.9970 0.9760 1.69 0.20 13.23 4748.7605 1.50 0.25 0.857 7.5 Distortionary Taxation Our baseline model assumes that an increase in government spending is financed by a corresponding increase in the lump-sum tax. In this section, we assess the robustness of our result to the economy in which government spending is financed by a labor income tax with no debt. As shown in the top-left panel of Figure 10, when the inertia is small, replacing the lump-sum tax with the labor income tax reduces the government spending multiplier. The reason is as follows. Since the government spending shock increases output by more than one for one, the labor income tax rate must decline in order to balance the budget. As pointed out by Eggertsson (2011), among others, a decline in the labor income tax reduces output when the nominal interest rate is zero, as it reduces the marginal cost for firms, exacerbates deflation, and raises the real interest rate. As a result, the government spending multiplier on output is lower in the economy with a labor income tax than in the economy with lump-sum taxes. However, in the presence of policy inertia, the government spending shock does not increase output as much and the decline in the labor income tax rate required to balance the budget is smaller. As a result, the wedge between the multipliers in the economies with lump-sum taxation and labor income taxation is smaller, and the fiscal multiplier of the economy with a labor income tax exceeds that 26

of the economy with only lump-sum taxation when there is a sufficiently high degree of policy inertia. Figure 10: Sensitivity Analyses cont... 2.5 2 1.5 1 0.5 0 0 0.5 1 ρr reilpitluM lacsiF Distortionary tax (τˆt w) 1.1 1 0.9 ZLB Ep., no τˆt w ZLB Ep., τˆt w 0.8 No ZLB Ep., no τˆt w No ZLB Ep., τˆt w 0.7 0.6 0.5 0 0.5 1 ρr reilpitluM lacsiF Nonlinear model (NL) ZLB Ep., L ZLB Ep., NL No ZLB Ep., NL No ZLB Ep., L 2.5 2 1.5 1 0.5 0 0 0.5 1 ρr reilpitluM lacsiF Autoregressivemodel (AR(1)) ZLB Ep., AR(1) No ZLB Ep., AR(1) 7.6 A Fully Nonlinear Model Some authors have shown that the commonly used semi-loglinear approximation can bepoorwhenthenominalinterestrateisattheZLB,astheeconomytendstobefaraway from the steady state. In particular, Braun and Waki (2010) and Braun, K¨orber, and Waki(2013)haveshownthatthefiscalmultipliercomputedintheapproximatedeconomy often overstates the multiplier in the underlying fully nonlinear economy. Accordingly, we analyze how policy inertia affects the multiplier in a fully nonlinear economy. The government spending multiplier in the fully nonlinear economy is defined similarly to those in the semi-loglinear economy. In this exercise, we mainly follow Braun, K¨orber, and Waki (2013) and calibrate the parameters to attain a 7 percent decline in output and 2.5 percent deflation in the low state.12 Thetop-right panelofFigure10 showshow the government spending multipliervaries 12TheequilibriumdoesnotexistunderthebaselineGreatDepressioncalibrationinthefullynonlinear economy. 27

Table 3: Great Recession (Nonlinear) Parameterization β χ χ γ θ ϕ φ φ µ c n π y 0.9970 1.00 0.2790 0.20 7.67 458.40 1.50 0.25 0.75 withpolicyinertiaunderthiscalibration. Solidanddashedblacklinesarerespectivelythe semi-loglinear and fully nonlinear models. As with the government spending multiplier in the semi-loglinear economy, the multiplier in the nonlinear economy declines with the policyinertiaparameter. ConsistentwithBraun, K¨orber, andWaki(2013), themultiplier is modestly smaller in the nonlinear economy than in the semi-loglinear economy when in the absence of policy inertia. However, the nonlinear multiplier declines by less as the policy inertia increases, and the multiplier in the nonlinear economy becomes larger than that in the semi-loglinear economy with a sufficiently high ρ . r 7.7 AR(1) Shock Throughout the paper, we followed the majority of the literature and assumed that boththegovernmentspendingandpreferenceshocksfollowedtwo-stateMarkovprocesses. We now modify the semi-loglinear model so that both shocks follow AR(1) processes, as follows: δ −1 = ρ (δ −1) (41) t δ t−1 and G −G = ρ (G −G ), (42) t ss g t−1 ss where we have assumed perfect foresight, following Erceg and Lind´e (2014). In this exercise, we set ρ = 0.9 and ρ = 0.9. We parameterize the reduced-form parameters δ g of our AR(1) shock model based on the values used in Erceg and Lind´e (2014).13 δ is 1 initializedsoastogenerateaninitial30percentdeclineinoutputintheabsenceofafiscal stimulus (i.e. G = G ) and we then consider the effects of a one percent increase in 1 ss the initial government spending (i.e., G = 1.01G ). The bottom-left panel of Figure 10 1 ss demonstrates that the main result ofthe paper—that the government spending multiplier issmallerintheeconomywithpolicyinertia—stillholdsunderthisalternativeassumption on the shock process. 13In particular, the discount factor is set to 0.995, the slope of the Phillips curve is set to 0.0611, the coefficientongovernmentspendinginthePhillipscurveissetto-0.0205,andtheinterestrateelasticityof theoutputgapissetto0.792. Thesearetheimpliedparametervalueswhentheirsticky-priceparameter ξ issetto0.9. Thesteady-stateshareofgovernmentspendingandallparametersinthemonetarypolicy P reaction function are set to our baseline calibration. 28

7.8 Extended Models Throughout the paper, we focus on a stylized New Keynesian model without any additional features. In the Appendix, we demonstrate that the main result of the paper— that the presence of the lagged shadow rate reduces the government spending multiplier at the ZLB—is robust to the introducion of features such as consumption habits, price indexation, sticky wages, capital, and hand-to-mouth households into the model. 7.9 Sensitivity Analyses for the Policy Rule with the Lagged Actual Policy Rate Throughout this section, we have demonstrated the robustness of the result that the presence of the lagged shadow policy rate reduces the spending multiplier at the ZLB to various model specifications. Another key result of the paper is that the presence of the lagged actual policy rate has little or no effect on the multiplier at the ZLB. For the sake of brevity, the robustness of this second claim to various model specifications is examined in the Appendix. This claim is indeed robust to various specifications considered in this section. 8 Conclusion This paper has studied how the presence of the lagged nominal interest rate in the policyruleaffectsthefiscalmultiplierattheZLB.Wehaveshownthatthefiscalmultiplier is nontrivially smaller in the presence of the lagged shadow rate than in the absence thereof. For the Great Depression calibration of Denes, Eggertsson, and Gilbukh (2013), the ZLB multiplier is 1.1 with an inertia parameter of 0.9, as opposed to 2.5 with an inertia parameter of 0. However, the ZLB multiplier remains above one for a plausible range of weights on the lagged policy rate. The claim that the ZLB multiplier is larger than the non-ZLB multiplier is robust. We have also shown that the presence of the lagged actual policy rate has little or no effect on the multiplier. Our result shows the importance of understanding the conduct of monetary policy in understanding the effects of fiscal policy. Different rules for the nominal interest rate affect the economy differently even at the ZLB because they influence future expectations dissimilarly. While we focused on fiscal multipliers, we believe that this message is more general. For example, the specification of the nominal interest rate policy is likely to matter when one tries to understand the effects of unconventional monetary policies.14 Thus, coming up with a good characterization of monetary policy in the recent ZLB episode is a high priority for future research. 14Nakata (2013) shows that the effect of uncertainty is smaller with a larger weight on the lagged shadow rate. Bundick (2014) shows that the effects of productivity and mark-up shocks can be even qualitatively differentattheZLBifthepolicyrateischosenaccordingtoaruleproposedbyReifschneider andWilliams(2000),whichisakintothebaselinetruncatedinertialTaylorruleconsideredinthispaper. 29

References Albertini, J., A. Poirier, and J. Roulleau-Pasdeloup (2014): “The Composition of Government Spending and the Multiplier at the Zero Lower Bound,” Economics Letters, 122(1), 31–35. Aruoba, B. S., P. Cuba-Borda, and F. Schorfheide (2014): “Macroeconomic Dynamics Near the ZLB: a Tale of Two Countries,” Working Paper. Bouakez, H., M. Guillard, and J. Roulleau-Pasdeloup(2014): “PublicInvestment,Time to Build, and the Zero Lower Bound,” Mimeo. Braun, A. R., L. M. Ko¨rber, and Y. Waki (2013): “Small and Orthodox Fiscal Multipliers at the Zero Lower Bound,” Working Paper 2013-13, Federal Reserve Bank of Atlanta. Braun, A. R., and Y. Waki (2010): “On the Size of the Fiscal Multiplier When the Nominal Interest Rate is Zero,” Working Paper. Bullard, J. (2013): “Perspectives on the Current Stance of Monetary Policy,” Speech delivered at NYU Stern Center for Global Economy and Business. Bundick, B. (2014): “Real Fluctuations at the Zero Lower Bound,” Mimeo. Carlstrom, C. T., T. S. Fuerst, and M. Paustian (2014): “Fiscal Multipliers Under an Interest Rate Peg of Deterministic vs. Stochastic Duration,” Journal of Money, Credit and Banking, Forthcoming. Carrillo, J. A., and C. Poilly (2013): “How Do Financial Frictions Affect the Spending Multiplier During a Liquidity Trap?,” Review of Economic Dynamics, 16(2), 296–311. Christiano, L., M. Eichenbaum, and S. Rebelo(2011): “WhenistheGovernmentSpending Multiplier Large?,” Journal of Political Economy, 119(1), 78–121. Christiano, L., M. Trabandt, and K. Walentin (2010): “Involuntary Unemployment in a Business Cycle Model,” NBER Working Paper. Chugh, S. K. (2006): “Optimal Fiscal and Monetary Policy with Sticky Wages and Sticky Prices,” Review of Economic Dynamics, 9, 683–714. Clarida, R., J. Gali, and M. Gertler (1998): “Monetary Policy Rules in Practice: Some International Evidence,” European Economic Review, 42(6), 1033–1067. Clarida, R. H., J. Gali, and M. Gertler (2000): “Monetary Policy Rules and MacroeconomicStability: EvidenceandSomeTheory,”QuarterlyJournalofEconomics,115(1),147–180. Cochrane, J. H. (2014): “The New-Keynesian Liquidity Trap,” Mimeo. Coenen, G., C. J. Erceg, C. Freedman, D. Furceri, M. Kumhof, R. Lalonde, D. Laxton, J. Lind, A. Mourougane, D. Muir, S. Mursula, C. de Resende, J. Roberts, W. Roeger, S. Snudden, M. Trabandt, and J. in’t Veld (2012): “Effects of Fiscal Stimulus in Structural Models,” American Economic Journal: Macroeconomics, 4(1), 22–68. 30

Cogan, J. F., T. Cwik, J. B. Taylor, and V. Wieland (2010): “New Keynesian versus Old Keynesian Government Spending Multipliers,” Journal of Economic Dynamics and Control, 34(3), 281–295. Coibion, O., and Y. Gorodnichenko (2012): “Why Are Target Interest Rate Changes So Persistent?,” American Economic Journal: Macroeconomics, 4(4), 126–62. Coleman, W. J. (1991): “Equilibrium in a Production Economy with an Income Tax,” Econometrica, 59(4), 1091–1104. Corsetti, G., K. Kuester, A. Meier, and G. J. Muller (2010): “Debt Consolidation and Fiscal Stabilization of Deep Recessions,” American Economic Review: Papers & Proceedings, 100(2), 41–45. Crafts, N., and T. C. Mills (2012): “Fiscal Policy in a Depressed Economy: Was There a Free Lunch in 1930s’ Britain?,” Mimeo. Del Negro, M., F. Schorfheide, F. Smets, and R. Wouters (2007): “On the Fit of New Keynesian Models,” Journal of Business & Economic Statistics, 25(2), 123–143. Denes, M., G. Eggertsson, and S. Gilbukh (2013): “Deficits, Public Debt Dynamics and Tax and Spending Multipliers,” Economic Journal, 123(566), 133–163. Drautzburg,T.,andH.Uhlig(2013): “FiscalStimulusandDistortionaryTaxation,”Working Paper. Dupor, B., and R. Li (2013): “The 2009 Recovery Act and the Expected Inflation Channel of Government Spending,” mimeo. Eggertsson, G. (2011): “What Fiscal Policy is Effective at Zero Interest Rates?,” NBER Macroeconomic Annual 2010, 25, 59–112. Erceg, C.J., andJ.Linde´(2014): “IsThereaFiscalFreeLunchinaLiquidityTrap?,”Journal of the European Economic Association, 12(1), 73–107. Evans, C.(2012): “MonetaryPolicyinChallengingTimes,”RemarksattheC.D.HoweInstitute. Ferna´ndez-Villaverde, J., G. Gordon, P. A. Guerro´n-Quintana, and J. Rubio- Ram´ırez (2012): “Nonlinear Adventures at the Zero Lower Bound,” NBER Working Paper Series 18058, National Bureau of Economic Research. Gal´ı, J., J. D. Lope´z-Salido, and J. Valle´s (2007): “Understanding the Effects of Government Spending on Consumption,” Journal of the European Economic Association, 5, 227–270. Gust, C., D. Lo´pez-Salido, and M. Smith (2012): “The Empirical Implications of the Interest-Rate Lower Bound,” Working Paper. Ireland, P. N. (2007): “Changes in the Federal Reserve’s Inflation Target: Causes and Consequences,” Journal of Money, Credit, and Banking, 39(8), 1851–1882. Justiniano, A., G. Primiceri, and A. Tambalotti(2010): “InvestmentShocksandBusiness Cycles,” Journal of Monetary Economics, 57(2), 132–145. 31

Kiley, M. T. (2014): “Policy Paradoxes in the New Keynesian Model,” Mimeo. Kim, J., and F. J. Ruge-Murcia (2011): “Monetary Policy When Wages are Downwardly Rigid: Friedman Meets Tobin,” Journal of Economic Dynamics and Control, 35, 2064–2077. Leeper, E. M., N. Traum, and T. B. Walker (2011): “Clearing Up the Fiscal Multiplier Morass,” NBER Working Paper Series 17444, National Bureau of Economic Research. Mertens, K., and M. Ravn (2014): “Fiscal Policy in an Expectations Driven Liquidity Trap,” Mimeo. Nakata, T.(2013): “UncertaintyattheZeroLowerBound,” FinanceandEconomicsDiscussion Series 2013-09, Board of Governors of the Federal Reserve System (U.S.). Orphanides, A. (2003): “Historical Monetary Policy Analysis and the Taylor Rule,” Journal of Monetary Economics, 50(5), 983–1022. Ramey, V. A., and S. Zubairy (2014): “GovernmentSpendingMultipliersinGoodTimesand in Bad: Evidence from U.S. Historical Data,” Mimeo. Reifschneider, D., and J. C. Williams (2000): “Three Lessons for Monetary Policy in a Low-Inflation Era,” Journal of Money, Credit and Banking, 32(4), 936–966. Roulleau-Pasdeloup, J. (2014): “The Government Spending Multiplier in a Recession with a Binding Zero Lower Bound,” Mimeo. Rudebusch, G. D. (2006): “Monetary Policy Inertia: Fact or Fiction?,” International Journal of Central Banking, 2(4), 85–135. Schmidt,S.(2013): “OptimalMonetaryandFiscalPolicywithaZeroBoundonNominalInterest Rates,” Journal of Money, Credit and Banking, 45(7), 1335–1350. Smets, F., and R. Wouters (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1(5), 1123–1175. (2007): “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach,” American Economic Review, 97(3), 586–606. Uhlig, H. (2011): “Some Fiscal Calculus,” American Economic Review, 100(2), 30–34. Woodford,M.(2011): “SimpleAnalyticsoftheGovernmentExpenditureMultiplier,”American Economic Journal: Macroeconomics, 3(1), 1–35. 32

Appendix A Solution Method The problem is to find a set of policy functions, {Yˆ(·,·), Cˆ(·,·), Πˆ(·,·), i(·,·), i∗(·,·)}, that solves the following system of functional equations. Yˆ(i∗ ,s ) = E Yˆ(i∗,s )−σ(i −E Πˆ(i∗,s )+δˆ −r¯)+γ(Gˆ −E Gˆ ) (A.1) t−1 t t t t+1 t t t t+1 t t t t+1 Πˆ(i∗ ,s ) = κYˆ(i∗ ,s )−κψσ−1γGˆ +βE Πˆ(i∗,s )) (A.2) t−1 t t−1 t t t t t+1 Yˆ(i∗ ,s ) = (1−γ)Cˆ(i∗ ,s )+γGˆ (A.3) t−1 t t−1 t t i(i∗ ,s ) = max (cid:2) 0,i∗(i∗ ,s ) (cid:3) (A.4) t−1 t t−1 t i∗(i∗ ,s ) = r¯+ρ (i∗ −r¯)+(1−ρ )(φ Πˆ(i∗ ,s )+φ Yˆ(i∗ ,s )) (A.5) t−1 t r t−1 r π t−1 t y t−1 t We use the standard time-iteration method to solve for the set of policy functions. Thetime-iterationmethodstartsbyspecifyingaguessforpolicyfunctions. LetX(·,·) be a vector of policy functions that solves the functional equations above and let X(0) be the initial guess of such policy functions.15 At the s-th iteration and at each point of the state space, we solve the system of nonlinear equations given by equations (A.1)– (A.5) to find today’s consumption, output, inflation, and the actual and shadow policy rates, given that X(s−1)(·,·) is in place for the next period. In solving the system of nonlinear equations, the expectation terms in the consumption Euler equation and the Phillips curve are evaluated based on the probability distribution of tomorrow’s state, which is conditional on the current state of the economy; the value of future variables not on the grid points are evaluated with linear interpolation.16 The system is solved numerically by using a nonlinear equation solver, dneqnf, provided by the IMSL Fortran Numerical Library. If the updated policy functions are sufficiently close to the guessed policy functions, then the algorithm ends. Otherwise, using the updated policy functions justobtainedastheguessforthenextperiod’spolicyfunctions, weiterateonthisprocess until the difference between the guessed and updated policy functions is sufficiently small ( (cid:13) (cid:13)vec(Xs(δ)−Xs−1(δ)) (cid:13) (cid:13) < 1e-10 is used as the convergence criteria). The lower and ∞ upper bounds on the endogenous state, the lagged shadow policy rate, are chosen so that, when we simulate the model for a long sample, the simulated path of the shadow rate stays within the bounds. We used 101 equally spaced grid points on the chosen interval of the lagged shadow rate.17 15For all models and all variables, we use flat functions at the deterministic steady-state values as the initial guess. 16The expectations of the model that assumes that both shocks follow an AR(1) shock process are solved using perfect foresight. 17We have also experimented with larger numbers of grid points, and the results are unchanged. 33

B Results from Extended Models In the main text, we focused on a stylized model without any endogenous variables. In this section, we consider the robustness of our main result to models with various additional features. The additional features we consider are consumption habits, price indexation, sticky wages, capital, and hand-to-mouth households. Since our solution method is computationally intensive, we will examine a series of models with each additional feature one at a time, as opposed to one model with all additional features. Due to space considerations, the description of the model and the results will be brief and casual. Further details are available upon request. B.1 Consumption Habits Following Smets and Wouters (2007), we introduce consumption habits by altering the utility function of the household in the following way. U (cid:0) C ,Ca ,N (cid:1) = (cid:0) C t −ζC t a −1 (cid:1)1−χc − N t 1+χn (B.1) t t t−1 t 1−χ 1+χ c n Households take as given last period’s aggregate consumption level, Ca . The habit t−1 parameter, ζ, is set to 0.7. The top-left panel of Figure B.1 shows how the multiplier varies with the degree of policy inertia. Solid lines are for the baseline model under the Great Recession calibration and the dashed lines are for the model with consumption habits. The introduction of consumption habits decreases the ZLB multiplier when there is no policy inertia. The multiplier is 1.1 with consumption habits compared to 1.2 in the baseline. This is because the change in the expected path of real rates induced by the government spending shock has fewer effects on the household consumption decision, as the household’s consumption decision is not fully forward-looking in the model with habits. This also leads the fiscal multiplier away from the ZLB to be higher in the model with consumption habits since the crowding out effect derived from increased government demand is mitigated. The multiplier is 0.8 with consumption habits compared to 0.4 in the baseline. Policy inertia reduces the government spending multiplier at the ZLB, but the change is less pronounced in the model with consumption habits; the multiplier is essentially unchanged until the inertia parameter increases to 0.8. From there, the multiplier noticeably decreases and converges to the non-ZLB multiplier as the inertia approaches one. B.2 Price Indexation Following Ireland (2007), we introduce price indexation to the model by modifying the adjustment cost specification facing the intermediate good producers, as follows. 34

Figure B.1: Extended Models 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Consumption Habits 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF PriceIndexation 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Sticky Wage 3 2.5 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Capital 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Hand-to-Mouth Consumer Baseline, ZLB ep. Extended, ZLB ep. Baseline, No ZLB ep. Extended, No ZLB ep. ϕ(cid:104) P (cid:105)2 i,t P −1 Y (B.2) t 2 Πα P t t−1 i,t−1 The indexation parameter, α, is set to 0.5. The top-right panel of Figure B.1 shows how the multiplier varies with the degree of policy inertia. Solid lines are for the baseline model under the Great Recession calibration, and the dashed lines are the model with price indexation. The introduction of price indexation increases the ZLB multiplier when there is no policy inertia. The multiplier is 1.6 with price indexation compared to 1.2 in the baseline. Since firms are now backward-looking, this implies that previous expected paths of real marginal costs figure into their current pricing decision by a multiple of α. An increase in demand for the final good will filter into subsequent pricing decisions in the future, leading to an increase in the expected path of inflation that is marginally higher than the baseline. Consequently, the decrease in the expected path of real interest rates is more dramatic, resulting in a higher fiscal multiplier in the model with price indexation. Policy inertia reduces the government spending multiplier. This reduction is more dramatic in the model with price indexation, since the multiplier still converges to the non-ZLB 35

multiplier as the inertia approaches one. B.3 Sticky Wages FollowingChugh(2006)andKimandRuge-Murcia(2011),stickywagesareintroduced through quadratic adjustment costs. The household chooses consumption, wage, and a one-period risk-free bond to maximize the expected discounted future utility flows, ∞ (cid:34)t−1 (cid:35)(cid:34) C1−χc N1+χn (cid:35) max E (cid:88) βt−1 (cid:89) δ h,t − h,t , (B.3) 1 s C h,t ,w h,t ,B h,t t=1 s=0 1−χ c 1+χ n subject to the following labor demand schedule and budget constraint: (cid:20) w (cid:21)−θw h,t N = N (B.4) h,t t w t and (cid:34) (cid:35)2 B ϕ w B h,t w h,t h,t−1 C + ≤ w N −w Π −1 N + +Φ −T , (B.5) h,t h,t h,t t t t t t R P 2 w P t t h,t−1 t where w = W h,t is the real wage set by the household and w = Wt is the aggregate h,t Pt t Pt real wage. A labor packer buys labor N from households at their monopolistic wage and sells h,t the packaged labor N to intermediate goods producers at W . The packer’s problem is t t given by (cid:90) 1 maxW N − W N dh, (B.6) t t h,t h,t N h,t 0 subject to the following CES technology: (cid:34) (cid:35) θw (cid:90) 1 θw−1 θw−1 N = N θw dh . (B.7) t h,t 0 Thefirst-orderconditionyieldsthelabordemandschedulethathouseholdsaresubjectto. This adjustment cost setup is less common than the Calvo-type setup of the nominal wage rigidity, but the resulting semi-loglinear dynamics are identical across these two alternative modelling approaches. The linearized model implies a composite parameter, κ = θw−1 (the slope of the w ϕw New Keynesian Wage Phillips Curve), which we set to 0.02. The middle-left panel of Figure B.1 shows how the multiplier varies with the degree of policy inertia. Solid lines are for the baseline under the Great Recession calibration and dashed lines are for the model with sticky wages. 36

The introduction of sticky wage decreases the ZLB multiplier when there is no policy inertia. Thisisbecausestickywagesaddrigiditytotheexpectedpathofrealwages,which in turn reduces the responsiveness of expected inflation and thus the responsiveness of real interest rates to the government spending shock. The multiplier is 1.1 with sticky wages compared to 1.2 in the baseline. Policy inertia reduces the government spending multiplier,butthechangeislesspronouncedinthemodelwithstickywages;themultiplier is essentially unchanged until the inertia parameter increases to 0.7. From there, the multiplier noticeably decreases and converges to the non-ZLB multiplier as the inertia approaches one. B.4 Capital The household’s problem in the model with capital is given by (cid:88) ∞ (cid:34) (cid:89) t−1 (cid:35)(cid:34) C1−χc N1+χn (cid:35) max E βt−1 δ t − t , (B.8) 1 s Ct,Nt,Bt,Kt,It 1−χ c 1+χ n t=1 s=0 subject to the following budget constraint and capital accumulation equation: B B C +I + t ≤ w N +RkK + t−1 +Φ −T (B.9) t t R P t t t t−1 P t t t t t and (cid:18) (cid:18) (cid:19) (cid:19) K t K = 1−AC −δ K +I . (B.10) t k t−1 t K t−1 (cid:16) (cid:17) (cid:104) (cid:105)2 The adjustment cost function is given by AC Kt = ϕ k Kt −1 . Kt−1 2 Kt−1 The production technology of the intermediate goods producers is given by Y = (Ks )αN1−α, (B.11) f,t f,t f,t where Ks = K is the level of effective capital used by each firm. Their profit f,t f,t−1 maximization problem becomes ∞ (cid:34)t−1 (cid:35) (cid:34) (cid:35) maxE (cid:88) βt−1 (cid:89) δ λ P Y −W N −P RkKs −P ϕ p (cid:104) P f,t −1 (cid:105)2 Y , (B.12) P f,t 1 t=1 s=0 s t f,t f,t t f,t t t f,t t 2 P f,t−1 t subject to the usual demand schedule imposed by the final good producer. We set α = 0.2, ϕ = 17, and δ = 0.025. The middle-right panel of Figure B.1 shows k k how the multiplier varies with the degree of policy inertia. Solid lines are for the baseline under the Great Depression calibration and dashed lines are for the model with capital. TheintroductionofcapitalincreasestheZLBmultiplierwhenthereisnopolicyinertia. The multiplier is 3.0 with capital compared to 2.5 in the baseline. The larger multiplier can be explained by the fact that there are now two private channels that contribute to 37

demand for the final good: consumption and investment. While consumer demand still increases due to the lower expected path of real interest rates, this lower path also implies lower costs for making investments. The increase in demand for the final good means that expected revenue from investments increases. Lower costs and higher revenue signal the households to increase their investment demand. The resultant higher consumer and investment demand implies a multiplier greater than the baseline. Just as in the baseline, policy inertia reduces the multiplier in the model with capital. B.5 Hand-to-Mouth Households In the model with hand-to-mouth households, we assume that a fraction of the households do not have access to financial markets. Their optimization problem is given by C1−χc N1+χn k,t k,t max − , (B.13) C k,t ,N k,t 1−χ c 1+χ n subject to the following budget constraint: C = w N −T . (B.14) k,t t k,t k,t FollowingGal´ı,Lop´ez-Salido,andVall´es(2007),weassumethatthesteady-stateshare of the lump-sum tax is chosen so that the steady-state consumption level is equal across the two types of households. For the sake of simplicity, we also assume that an increase in lump-sum taxes are shared equally across households. That is, Tˆ = Tˆ = Tˆ in the r,t k,t t linearized version of the model. We set ω, the share of hand-to-mouth households, to 0.2. The bottom-left panel of Figure B.1 shows how the multiplier varies with the degree of policy inertia. Solid lines are for the baseline under the Great Recession calibration and the dashed lines are for the model with hand-to-mouth consumers. Not surprisingly, the introduction of the hand-to-mouth consumers increases the fiscal multipliers both at and away from the ZLB. The increase in the real wage, induced by the increase in the government spending, gives Keynesian households more incentive to work and spend. Policy inertia reduces the government spending multiplier at the ZLB as in the baseline model without hand-to-mouth consumers, but the effects of policy inertia are more pronounced in the model with hand-to-mouth consumers. The ZLB multiplier is 2.0 without policy inertia compared to 1.1 with policy inertia of 0.9. C Sensitivity Analyses for the Model with the Lagged Actual Policy Rate In Section 7 and Appendix B, we have considered the robustness of the result with respect to the lagged shadow rate specification of the inertial Taylor rule. Here, we demonstrate the robustness of the result that the presence of the lagged actual policy 38

rate have little or no effect on the fiscal multiplier to various specifications considered in Section 7 and Appendix B. Figure C.1: Sensitivity Analyses, Lagged Actual Policy Rate Shock size (calib. δ ⇒ -∆y) H 2.5 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF Output response coefficient (φ ) y 2.5 2 -30% ∆y -20% ∆y -10% ∆y 1.5 1 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF φ y = 0.0 φ y = 0.25 Initial condition (i∗ ) 0 2.5 2.4 2.3 2.2 2.1 2 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF Great Recession vs. Great Depression 2.5 2 i∗ = r¯ 0 i∗ = 0 0 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 ρ r reilpitluM lacsiF GD GR Figures C.1 and C.2 show how the coefficient on the lagged actual policy rate affects thegovernmentspendingmultiplierundervariousspecificationsofthestylizedmodelconsidered in Section 7. Consistent with what was observed under the baseline specification (as discussed in Section 6), the result that the presence of the lagged actual policy has no effect on the multiplier holds under alternative specifications. Figure C.3 shows how the coefficientonthelaggedactualpolicyrateaffectsthemultiplierunderaseriesofextended models considered in Appendix B. In all of the extended models, the lagged actual policy rate has little or no effect on the multiplier. D Present Value Multipliers Our baseline concept of the multiplier measures the effect of government spending shocks on output upon impact. We have used this concept as our baseline in order to stay close to the work of Denes, Eggertsson, and Gilbukh (2013) and also to be simple and transparent. An alternative concept also common in the literature is the present value 39

Figure C.2: Sensitivity Analyses, Lagged Actual Policy Rate cont... 2.5 2 1.5 1 0 0.5 1 ρr reilpitluM lacsiF Distortionary tax (τˆt w) 1.1 1 no τˆt w 0.9 τˆt w 0.8 0.7 0 0.5 1 ρr reilpitluM lacsiF Nonlinear model (NL) L NL 2.5 2 1.5 1 0.5 0 0.5 1 ρr reilpitluM lacsiF Autoregressivemodel (AR(1)) AR(1) multiplier, which aims to measure the average effects of government spending shocks on output over time. To define the present value multiplier in the context of our model with two-state Markov shocks and an endogenous state variable, let us first define the present value multiplier function as follows. GM (i∗ ,H;g,δˆ ) = 1 (cid:80)k j=0 R s − s jE 0 (cid:2) Yˆ(i∗ j−1 ,s j ;g,δˆ H )−Yˆ(i∗ j−1 ,s j ;0,,δˆ H )|i∗ −1 ,s 0 = H (cid:3) k −1 H γ (cid:80)k R−jE (cid:2) Gˆ |s = H (cid:3) j=0 ss 0 H 0 (D.1) The numerator contains the expected discounted sum of changes in output due to government spending shocks up to time k, and the denominator contains the expected discounted sum of government spending shocks up to time k. This is the same definition as thatinUhlig(2011).18 Noticethatwhenk iszero, thisdefinitionbecomesidenticaltoour baseline government spending multiplier function in the main text. Consistent with the definition of our baseline impact multiplier, this function evaluated at i∗ = r¯, g = 0.01, −1 18Some authors adjust the discount factor term R−j to reflect time variation in the household preferss ence. See, for example, Leeper, Traum, and Walker (2011) and Drautzburg and Uhlig (2013). 40

Figure C.3: Extended Models, Lagged Actual Policy Rate 1.2 1 0.8 0.6 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Consumption Habits 1.6 1.4 1.2 1 0.8 0.6 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF PriceIndexation 1.2 1 0.8 0.6 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Sticky Wage 3 2.8 2.6 2.4 2.2 2 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Capital 2 1.5 1 0 0.2 0.4 0.6 0.8 1 ρr reilpitluM lacsiF Hand-to-Mouth Consumer Baseline, ZLB ep. Extended, ZLB ep. and δˆ—which is chosen to generate a 30 percent decline in output at time one—will be referred to as the present value multiplier at the ZLB. This function evaluated at i∗ = r¯, −1 g = 0.01, and δˆ= 0 will be referred to as the present value multiplier away from the ZLB. Figure D.1 shows how the present value multipliers with three different cutoff dates (k = 8,20,40) vary with the degree of policy inertia of the lagged shadow rate type. According to the figure, the present value multiplier behaves similarly to our baseline government spending multiplier. E Average and Marginal Multipliers ThemultipliersweemphasizearewhatErcegandLind´e(2014)callaveragemultipliers; they measure the average increase in output in response to a given increase in government spending. Another interesting object is the marginal multiplier that measures the marginal increase in output to a further (cid:15)-increase in government spending when government spending of size g is already in place. Formally, the marginal government spending multiplier function, GM (i∗ ,g,δˆ ), is defined as follows: (cid:15) −1 H 41

Figure D.1: Present Value Multipliers 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 ρr )k MG( reilpitluM tnemnrevoG eulaV tneserP GM GM0 GM8 GM20 40 Yˆ(i∗ ,H;g+(cid:15),δˆ )−Yˆ(i∗ ,H;g,δˆ ) GM (i∗ ,g,δˆ ) := −1 H −1 H . (E.1) (cid:15) −1 H γ(cid:15) TheleftpanelinFigureE.1comparestheaverageandmarginalmultipliersattheZLB in the model without policy inertia, while the right panel compares those in the model with policy inertia. The horizontal axis contains g, the size of the government spending increase already in place when we increase government spending further by (cid:15) percent of G . In this figure, (cid:15) is set to 0.01. In our framework with two-state Markov shocks, the ss 100 average and marginal multipliers are identical at the ZLB in the absence of policy inertia, unless the government spending shock is extremely large so that the ZLB does not bind at all with the government spending shock. However, if there is inertia in the policy rule, the average and marginal multipliers differ even in this two-state Markov shock model. The reason why the marginal multiplier is lower than the average multiplier in the economy with policy inertia is as follows. The effects of the government spending shock dependimportantlyonhowlongthepolicyrateiskeptatzeroaftertheshockdisappears. The longer the policy rate is expected to remain zero, the larger the government spending multiplier is. The expected ZLB duration is shorter when there is already some govern- 42

Figure E.1: Average versus Marginal Multipliers 2.5 2 1.5 1 0.5 0 −100 −50 0 50 100 reilpitluMlacsiF ZLB Episode: ρr = 0 1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 MarginalFM 1.1 AverageFM NZLB1st Period 1.05 −100 −50 0 50 100 %Change in Gov’tSpending reilpitluMlacsiF ZLB Episode: ρr = 0.85 %Change in Gov’tSpending ment spending in place than when there is not. Thus, a marginal increase in government spending is smaller when there is already some government spending in place than otherwise. This logic is closely related to why the multiplier is smaller when the shock size is smaller, which was discussed in detail in Section 7. 43