Conservatism and Liquidity Traps

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Conservatism and Liquidity Traps Taisuke Nakata and Sebastian Schmidt 2014-105 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.

Conservatism and Liquidity Traps∗ Taisuke Nakata† Sebastian Schmidt‡ Federal Reserve Board European Central Bank First Draft: October 2014 This Draft: November 2014 Abstract Appointing Rogoff’s (1985) conservative central banker improves welfare if the economy is subject to large contractionary shocks and the policy rate occasionally falls to the zero lower bound (ZLB). In an economy with occasionally binding ZLB constraints, the anticipation of future ZLB episodes creates a trade-off between inflation and output stabilization. As a consequence, inflation systematically falls below target even when the policy rate is above zero. A conservative central banker mitigates this deflationary bias away from the ZLB, improving allocations both at and away from the ZLB through expectations. Keywords: Discretion, Inflation Conservatism, Inflation Targeting, Liquidity Traps, Zero Lower Bound JEL-Codes: E52, E61 ∗WewouldliketothankMatthiasBurgert,LenaKo¨rber,JohnRoberts,YuichiroWaki,andseminarparticipants at the European Central Bank for comments. We would also like to thank Timothy Hills for his excellent research assistance. Theviewsexpressedinthispaper,andallerrorsandomissions,shouldberegardedasthosesolelyofthe authors,andarenotnecessarilythoseoftheFederalReserveBoardofGovernors,theFederalReserveSystem,orthe European Central Bank. †Board of Governors of the Federal Reserve System, Division of Research and Statistics, 20th Street and Constitution Avenue N.W. Washington, D.C. 20551; Email: taisuke.nakata@frb.gov. ‡European Central Bank, Monetary Policy Research Division, 60640 Frankfurt, Germany; Email: sebastian.schmidt@ecb.int.

1 Introduction Over the past few decades, a growing number of central banks around the world have adopted inflation targeting as a policy framework. The performance of inflation targeting in practice has been widely considered a success.1 However, some economists and policymakers have voiced the need to re-examine central banks’ monetary policy frameworks in light of the liquidity trap conditions currently prevailing in many advanced economies.2 As shown in Eggertsson and Woodford (2003) among others, the zero lower bound (ZLB) on nominal interest rates severely limits the ability of inflation-targeting central banks to stabilize the economy absent an explicit commitment technology.3 Some argue that the ZLB is likely to bind more frequently and that liquidity trap episodes might hit the economy more severely in the future than they have in the past.4 Understanding the implications of the ZLB for the conduct of monetary policy is therefore of the utmost importance for economists and policymakers alike. In this paper, we contribute to this task by examining the desirability of Rogoff’s (1985) conservative central banker in a standard New Keynesian model in which large contractionary shocks occasionally push the policy rate to the ZLB. Rogoff (1985) showed that in a model where a lack of commitment leads to an inflation bias, society can be better off if the central bank is less concerned with output gap stability relative to inflation stability than is society. To focus on the role of the ZLB, we abstract from the original inflation bias by assuming that the steady-state distortions are eliminated by appropriate subsidies. Society’s welfare is then given by the negative of the weighted sum of inflation and output volatility. We analyze how the economy behaves under a discretionary central banker with an alternative weight on output volatility, and we compute the optimal weight that maximizes society’s welfare. We find that the appointment of a fully conservative central banker—that is, a banker who places zero weight on output stabilization—is optimal in our baseline model, which features only a demand shock. That is, society’s welfare is maximized when the central bank focuses exclusively on inflation stabilization. The mechanism behind our result is as follows. In the economy in which future shocks can push the policy rate to the lower bound, the anticipation of lower inflation and outputgivesforward-lookinghouseholdsandfirmsincentivestoreduceconsumptionandpriceseven when the policy rate is above the ZLB. The central bank cannot fully counteract these incentives. Whenthecentralbankisconcernedwithbothinflationandoutputstabilization, itfacesatrade-off between the two objectives, implying deflation and a positive output gap in those states where the ZLB is not binding. Following the terminology of Nakov (2008), we will refer to this deflation when the policy rate is above zero as deflationary bias. A central banker who puts comparatively more weight on inflation stabilization mitigates the deflationbiasawayfromtheZLBatthecostofapotentiallyhigheroutputgap. Viewedinisolation, 1See, for instance, Walsh (2009) and Svensson (2010), and the references therein. 2See, for example, Blanchard, Dell’ariccia, and Mauro (2010); Tabellini (2014); Williams (2014). 3See also Jung, Teranishi, and Watanabe (2005), Adam and Billi (2007), and Nakov (2008). 4See, for example, IMF (2014) and Chung, Laforte, Reifschneider, and Williams (2012). 2

thisiswelfare-reducingbecauseitshiftsinflationandoutputgaprealizationsawayfromthewelfareimplied target criteria. However, lower deflation and higher output gaps away from the ZLB also reduce expected real interest rates and increase the expected marginal costs at the ZLB, mitigating deflation and output declines there. This in turn allows the central bank to achieve zero inflation with a smaller positive output gap away from the ZLB, setting in motion a positive feedback loop. We prove analytically the optimality of placing zero weight on output stabilization for the baseline versionofourmodelinwhichthedemandshockfollowsatwo-stateMarkovprocess,andweconfirm this result numerically for a version of the model with a first order autoregressive shock process. Thedesirabilityofconservatismisrobusttointroducingcost-pushshocksintotheeconomy, but theoptimalweightonoutputstabilizationmaynolongerbezero. Inthemodelwithdemandshocks and cost-push shocks, the optimal weight would coincide with society’s weight in the absence of the ZLB. Accounting for the ZLB, the optimal weight lies between zero and society’s weight, as long as the cost-push shock is sufficiently small and the demand shock is the key driver of liquidity trap episodes. The greater the frequency of the ZLB episodes, the closer the optimal weight is to zero. This observation makes intuitive sense and is reminiscent of the finding in Coibion, Gorodnichenko, and Wieland (2012) that the effect of the ZLB on the optimal inflation target is larger when the ZLB constraint binds more frequently. Our result may initially strike some readers as counterintuitive. The desirability of assigning a higher weight on the inflation objective was shown originally in a framework in which the lack of an explicit commitment technology leads to inflation that is too high. The problem of the economy facing the ZLB constraint is the opposite—inflation that is too low—which may lead one to conjecture that the opposite prescription of assigning a lower weight on inflation would be desirable.5 Ouranalysesshowthatthisisnotthecase. Indescribingwhy,wetraceoutthebeneficial effects of stabilizing inflation expectations, which are central to understanding the desirability of the conservative central bank in the original model of Rogoff (1985) as well. A valuable byproduct of our analysis of conservatism is a closed-form characterization of the conditions that guarantee the existence of the standard Markov-Perfect equilibrium with occasionally binding ZLB constraints. Some researchers have reported difficulty in obtaining numerical convergence when solving the model with the ZLB and have suggested that equilibrium under some parameter configurations does not exist.6 Yet not much is known about the conditions for equilibrium existence. We prove that the equilibrium ceases to exist when the frequency and persistence of crisis shocks are sufficiently high, and we provide analytical expressions for the frequency and persistence thresholds at which this occurs. This result should be a useful reference for those who numerically solve the New Keynesian model with occasionally binding ZLB constraints. While our analysis focuses on the standard Markov-Perfect equilibrium that fluctuates around a positive nominal interest rate so that the ZLB constraint binds only occasionally, there exists a 5Tabellini (2014), for example, conjectures that a lower weight on the output stability objective is detrimental to stabilization policy in the model that incorporates the ZLB constraint. 6See,forexample,AdamandBilli(2007)andBilli(2013)forthenon-convergenceresultundertheMarkov-Perfect equilibrium and Richter and Throckmorton (2014) under the Taylor-rule equilibrium. 3

second Markov-Perfect equilibrium in which the nominal interest rate is at zero permanently. In the Appendix, we also provide analytical characterizations of the conditions for the existence of this deflationary Markov-Perfect equilibrium. Our paper is related to a set of papers that have examined various ways to improve allocations at the ZLB in time-consistent manners. Eggertsson (2006), Burgert and Schmidt (2014), and Bhattarai, Eggertsson, and Gafarov (2014) considered economies in which the government can choose the level of nominal debt and showed that an increase in government bonds during the liquidity trap improves allocations by creating incentives for future governments to inflate. In a model in which government spending is valued by the household, Nakata (2013) and Schmidt (2013) showed that a temporary increase in government spending can improve welfare whenever the policy rate is stuck at the ZLB.7 A key characteristic of these proposals is that they involve additional policy instruments and require perfect coordination of monetary and fiscal authorities. The approach studied in our paper only requires that the central bank is maximizing its assigned objective. A few recent papers examine other time-consistent ways to better stabilize inflation and output in the model with the ZLB constraint without relying on additional policy instruments. Nakata (2014) demonstrates that a reputational concern on the part of the central bank can make the promise of overshooting inflation and output time-consistent. Billi (2013) revisits the desirability of assigning a nominal-income stabilization objective to the central bank. In our ongoing work, we compare the relative benefits of various alternative objectives, including price-level stabilization, nominal-income stabilization, and interest-rate smoothing (Nakata and Schmidt, 2014). ThispaperisalsorelatedtoasetofpapersthatexaminethedesirabilityofRogoff’sconservative central banker in settings other than the original model with inflation bias. Clarida, Gali, and Gertler (1999) showed that the appointment of a conservative central banker is also desirable in a New Keynesian model, in which the presence of persistent cost-push shocks creates a stabilization bias in discretionary monetary policy—that is, an inferior short-run trade-off between inflation and output stabilization compared with the time-inconsistent Ramsey policy. Adam and Billi (2008), Adam and Billi (2014), and Niemann (2011) examined the benefit of conservatism in versions of New Keynesian models augmented with endogenous fiscal policy. However, all of these authors have abstracted from the ZLB constraint. Finally, ouranalysesoftheconditionsthatguaranteetheexistenceofstandardanddeflationary Markov-Perfect equilibria are related to the analyses by Tambakis (2014) and Armenter (2014). Tambakis (2014) characterizes the conditions that guarantee the existence of the standard Markov- Perfect equilibrium while assuming that the probability of the crisis shock does not depend on the stateoftheeconomy. WeextendhisresultsbyconsideringbothstandardanddeflationaryMarkov- Perfect equilibria and allowing for state dependence in the distribution of shocks. Armenter (2014) shows that the deflationary Markov-Perfect equilibrium exists in an economy with n-state Markov 7Schmidt (2014) examines what type of fiscal policymaker is best suited for dealing with liquidity traps in the absenceofpolicycommitment. Hefindsthatanactivistfiscalauthoritythatcareslessaboutgovernmentconsumption stability relative to output gap and inflation stability than society does is welfare-improving. 4

shocks if and only if the standard Markov-Perfect equilibrium exists, but he is silent about the conditions under which these equilibria exist. We focus on an economy with two-state Markov shocks and provide a complete characterization of the conditions for the existence of both types of Markov-Perfect equilibria.8 The rest of the paper is organized as follows. Section 2 describes the model and the government’s optimization problem, and defines the welfare measure. Section 3 presents the main results. Section 4 extends the analysis to a model with both demand and cost-push shocks and to a continuous-state model. The final section concludes. 2 The model This section presents the model, lays down the policy problem of the central bank and defines the equilibrium. 2.1 Private sector The private sector of the economy is given by the standard New Keynesian structure formulated in discrete time with infinite horizon as developed in detail in Woodford (2003) and Gali (2008). A continuum of identical, infinitely-living households consumes a basket of differentiated goods and supplies labor in a perfectly competitive labor market. The consumption goods are produced by firms using (industry-specific) labor. Firms maximize profits subject to staggered price-setting as in Calvo (1983). Following the majority of the literature on the ZLB, we put all model equations except for the ZLB constraint in semi-loglinear form. This allows us to derive closed-form results. The equilibrium conditions of the private sector are given by the following two equations: π = κy +βE π (1) t t t t+1 and y = E y −σ(i −E π −r∗)+d . (2) t t t+1 t t t+1 t where π is the inflation rate between period t − 1 and t, y denotes the output gap, i is the t t t level of the nominal interest rate between period t and t + 1, and d is an exogenous demand t shock capturing fluctuations in the natural real rate of interest, r := r∗ + 1d . Equation (1) is t σ t a standard New Keynesian Phillips curve and equation (2) is the consumption Euler equation. 8Several studies have characterized the conditions for the existence of Taylor-rule equilibria in models with the ZLB.Eggertsson(2011)andBraun,Ko¨rber,andWaki(2013)characterizetheconditionsguaranteeingtheexistence of a Taylor-rule equilibrium in a semi-loglinear model, assuming that the economy eventually reverts back to an absorbing state and the ZLB does not bind in the absorbing state. Christiano and Eichenbaum (2012) analyzed the existence and multiplicity of Taylor-rule equilibria in a fully nonlinear model, again assuming the eventual return to the steady-state where the ZLB does not bind. Mendes (2011) characterizes the conditions for the existence of the standard and deflationary Taylor-rule equilibria in a fully stochastic semi-loglinear New Keynesian economy, assuming that the process for the natural rate of interest has no persistence. In an early contribution, Benhabib, Schmitt-Grohe, and Uribe (2001) show the existence of two steady-states in a sticky-price economy that abstracts from fundamental shocks. 5

The parameters are defined as follows: β ∈ (0,1) denotes the representative household’s subjective discount factor, σ > 0 is the intertemporal elasticity of substitution in consumption, and r∗ = 1−1 β isthedeterministicsteadystateofthenaturalrealrate. κrepresentstheslopeoftheNewKeynesian Phillips curve and is related to the structural parameters of the economy as follows: κ = (1−α)(1−αβ) (cid:0) σ−1+η (cid:1) , (3) α(1+ηθ) where α ∈ (0,1) denotes the share of firms that cannot reoptimize their price in a given period, η > 0 is the inverse of the elasticity of labor supply, and θ > 1 denotes the price elasticity of demand for differentiated goods. We assume that the demand shock d follows a two-state Markov process, as in Eggertsson and t Woodford (2003) and others, which allows us to reveal the underlying mechanism in a simple and intuitive way. In particular, d takes the value of either d or d where we refer to d > −σr∗ as t H L H the high state and d < −σr∗ as the low state. The transition probabilities are given by L Prob(d = d |d = d ) = p (4) t+1 L t H H and Prob(d = d |d = d ) = p . (5) t+1 L t L L p is the probability of moving to the low state in the next period when the economy is in the H high state today and will be referred to as the frequency of the contractionary shocks. p is the L probability of staying in the low state when the economy is in the low state today and will be referred to as the persistence of the contractionary shocks. We will also refer to high and low states as non-crisis and crisis states, respectively. In Section 4, we extend the analysis to a continuous-state model in which the demand shock follows a stationary autoregressive process. 2.2 Society’s objective and the central bank’s problem We assume that society’s value, or welfare, at time t is given by the expected discounted sum of future utility flows, V = u(π ,y )+βE V , (6) t t t t t+1 wheresociety’scontemporaneousutilityfunction,u(·,·),isgivenbythestandardquadraticfunction of inflation and the output gap, u(π,y) = − 1 (cid:0) π2+λ¯y2(cid:1) . (7) 2 This objective function can be motivated by a second-order approximation to the household’s preferences. In such a case, λ¯ is a function of the structural parameters and is given by λ¯ = κ. θ The value for the central bank is given by 6

VCB = uCB(π ,y )+βE VCB. (8) t t t t t+1 where the central bank’s contemporaneous utility function, uCB(·,·), is given by uCB(π,y) = − 1 (cid:0) π2+λy2(cid:1) . (9) 2 Note that, while the central bank’s objective function resembles the private sector’s, the relative weight that it attaches to the stabilization of the output gap, λ ≥ 0, may differ from λ¯. We assume that the central bank does not have a commitment technology. Each period t, the central bank chooses the inflation rate, the output gap, and the nominal interest rate in order to maximize its objective function subject to the behavioral constraints of the private sector, with the policy functions at time t+1 taken as given. The problem of the central bank is thus given by VCB(d ) = max uCB(π ,y )+βE VCB(d ). (10) t t t t t t+1 t+1 πt,yt,it subject to the zero lower bound constraint, i ≥ 0, (11) t and the private-sector equilibrium conditions (1) and (2) described above. A Markov-Perfect equilibrium is defined as a set of time-invariant value and policy functions {VCB(·), y(·), π(·), i(·)} that solves the central bank’s problem above, together with society’s value function V(·), which is consistent with y(·) and π(·). As discussed in Armenter (2014) and Nakata (2014), there are two Markov-Perfect equilibria in this economy: One fluctuates around a positive nominal interest rate and zero inflation/output (the standard Markov-Perfect equilibrium), and the other fluctuates around a zero nominal interest rate and negative inflation/output (the deflationary Markov-Perfect equilibrium). While the deflationary Markov-Perfect equilibrium is interesting, we focus on the standard Markov-Perfect equilibrium in this paper. In most economies that have recently faced a liquidity trap, long-run inflation expectations have been well anchored to some positive rate and various survey data strongly suggests that private-sector agents expect the central bank to eventually raise the policy rate. Thus, the standard Markov-Perfect equilibrium seems to be more relevant on empirical grounds.9 The main exercise of the paper will be to examine the effects of λ on welfare. We quantify the welfare of an economy by the perpetual consumption transfer (as a share of its steady state) that would make a household in the economy indifferent to living in the economy without any fluctuations. This is given by W := (1−β) θ (cid:0) σ−1+η (cid:1) E[V]. (12) κ 9An exception is Japan where the policy rate has been at the ZLB for more than a decade. 7

where the mathematical expectation is taken with respect to the unconditional distribution of d . t 3 Results AfterprovidingconditionsfortheexistenceofthestandardMarkov-Perfectequilibrium,thissection shows how output and inflation in the two states depend on the central bank’s relative weight on output stabilization λ and shows that λ = 0 is optimal. The second part of this section provides a numerical illustration. ThestandardMarkov-Perfectequilibriumisgivenbyavector{y ,π ,i ,y ,π ,i }thatsolves H H H L L L the following system of linear equations— y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (13) H H H H L H H H L H H (cid:2) (cid:3) π = κy +β (1−p )π +p π , (14) H H H H H L 0 = λy +κπ , (15) H H y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (16) L L H L L L H L L L L (cid:2) (cid:3) π = κy +β (1−p )π +p π , (17) L L L H L L and i = 0, (18) L —and satisfies the following two inequality constraints: i > 0 (19) H and λy +κπ < 0. (20) L L For any variable x, x denotes the value of that variable in the k state where k ∈ {H,L}. The k first inequality constraint checks the nonnegativity of the nominal interest rate in the high state, and the second checks nonpositivity of the Lagrangean multiplier on the ZLB constraint in the low state. The model can be solved in closed form. We first prove key properties of the model and then move on to numerical analyses. 3.1 Analytical results Proposition 1: The standard Markov-Perfect equilibrium exists if and only if p ≤ p∗(Θ ) L L (−pL) 8

and p ≤ p∗ (Θ ), H H (−pH) where i) for any parameter x, Θ denotes the set of parameter values excluding x, and ii) the (−x) cutoff values p∗(Θ ) and p∗ (Θ ) are given in Appendix A. L (−pL) H (−pH) See Appendix A for the proof. The two conditions guarantee the nonpositivity of the Lagrange multiplier in the crisis state and the nonnegativity of the nominal interest rate in the non-crisis state, respectively. When the frequency of the contractionary shock, p , is high, the central H bank reduces the nominal interest rate aggressively to mitigate the deflation bias, which will be described shortly. Thus, for the policy rate to be positive in the high state, p must be sufficiently H low. With p > p∗(Θ ), inflation and output in the low state are positive when they satisfy L L (−pL) the consumption Euler equation and the Phillips curve. Though this is somewhat unintuitive, it makes sense. When the persistence of the crisis, p , is high, inflation and output in today’s low L state are largely dependent on households’ and firms’ expectations of inflation and output in the next period’s low state. Thus, positive inflation and output in the low state can be self-fulfilling. However, such positive inflation and output cannot be an equilibrium because the central bank would have incentives to raise the nominal interest rate from zero in the low state. This incentive manifests itself in the positive Lagrangean multiplier in the low state when inflation and output are positive.10 When the conditions for the existence of the equilibrium hold, the signs of the endogenous variables are unambiguously determined. Proposition2: Foranyλ ≥ 0,π ≤ 0,y > 0,i < r ,π < 0,andy < 0. Withλ = 0,π = 0. H H H H L L H See Appendix A for the proof. In the low state, the ZLB constraint becomes binding, and output and inflation are below target. In the high state, a positive probability of entering the low state in the next period reduces expected marginal costs of production and leads firms to lower prices in anticipation of future crises events. This raises the expected real rate faced by the representative household and gives it an incentive to postpone consumption. The central bank chooses to lower the nominal interest rate to mitigate these anticipation effects. In equilibrium, inflation and output in the high state are negative and positive, respectively, and the non-crisis policy rate is below the real interest rate. These analyticalresults are consistent with the numericalresults in the literature (see Nakov (2008), among others). In particular, negative inflation away from the ZLB has been referred to as deflationary bias. This proposition provides the first analytical underpinning for the deflation bias.11 10TheconditionsfortheexistenceoftheotherMarkov-Perfectequilibriumturnouttobeidenticaltothoseforthe existence of the standard Markov-Perfect equilibrium; see Appendix E. 11Note that the deflation bias vanishes when the high state is an absorbing state—i.e., p = 0. p = 0 is often H H assumedinliteratureontheZLBthatworkswithdiscrete-statemodels. See,forinstance,EggertssonandWoodford (2003) and Christiano, Eichenbaum, and Rebelo (2011). 9

We now establish several results on how the degree of conservatism affects endogenous variables in both states. Proposition 3: For any λ ≥ 0, ∂πH < 0, ∂πL < 0, and ∂yL < 0. For any λ ≥ 0, ∂yH < 0 if and (cid:16) ∂λ ∂λ (cid:17) ∂λ ∂λ only if βp −(1−β) 1−pL(1−βp +βp )−p < 0. H κσ L H L See Appendix A for the proof. ∂πH < 0 means that, as the central bank cares more about inflation, ∂λ inflation in the high state is higher (i.e., the deflation bias in the high state is smaller). Since a lower rate of deflation in the high state increases output and inflation in the low state via expectations, inflation and output in the low state both increase with the degree of conservatism (∂πL < 0 ∂λ and ∂yL < 0). The effect of conservatism on output in the high state is ambiguous. On the one ∂λ hand, a more conservative central bank is willing to tolerate a larger overshooting of output given the same inflation expectations. On the other hand, higher inflation in both states improves the trade-off between inflation and output stabilization implied by the Phillips curve, making it possible to reduce the overshooting of output in the non-crisis state. Proposition 3 demonstrates that (cid:16) (cid:17) theformereffectdominatesthelatterifandonlyifβp −(1−β) 1−pL(1−βp +βp )−p < 0. H κσ L H L Proposition 4: Suppose that p and p are sufficiently low so that p ≤ p∗(Θ ) and L H L L (−pL) p ≤ p∗ (Θ ) for all λ in [0,λ¯]. Then, welfare is maximized at λ = 0. H H (−pH) See Appendix A for the proof. As demonstrated in Proposition 3, deflation in the high state is smaller and inflation and output decline less in the low state with a smaller λ. These forces work (cid:16) (cid:17) to improve society’s welfare. If βp −(1−β) 1−pL(1−βp +βp )−p > 0, then output in H κσ L H L the high state becomes smaller with a smaller λ and the optimality of zero weight is obvious. If (cid:16) (cid:17) βp −(1−β) 1−pL(1−βp +βp )−p < 0, then a smaller λ increases the already positive H κσ L H L output gap and thus has ambiguous effects on welfare. Proposition 4 demonstrates that, even in this case, the beneficial effects of a smaller λ on π , π , and y dominate the adverse effect on H L L y .12 H 3.2 Numerical illustration We now illustrate the aforementioned properties of the model with specific parameter values. The structural parameters are calibrated using the parameter values from Eggertsson and Woodford (2003), as listed in Table 1. The frequency of the crisis shock is chosen so that the ZLB episode occursonceperfivedecades,onaverage. Thepersistenceof0.875meansthattheexpectedduration of the crisis is two years. The size of the shock is chosen so that the decline in output during the crisis is 10 percent. Figure 1 shows how the output gap, inflation, and the nominal interest rate in both states vary 12Whiletheoptimalλisalwayszero,thewelfaregainsfromconservatismdodependontheseparameters. Seethe analysis in Appendix B. 10

Table 1: Parameterization (Two-state shock model) Parameter Value Economic interpretation β 0.99 Subjective discount factor σ 0.5 Intertemporal elasticity of substitution in consumption η 0.47 Inverse labor supply elasticity θ 10 Price elasticity of demand α 0.8106 Share of firms per period keeping prices unchanged d 0 Demand shock in the high state H d -0.0113 Demand shock in the low state L p 0.005 Frequency of contractionary demand shock H p 0.875 Persistence of contractionary demand shock L Figure 1: Output gap, inflation, and nominal interest rate High State 0.36 0.35 0.34 0 1 2 3 4 −3 x 10 paG tuptuO )%( Low State −9.5 −10 −10.5 0 1 2 3 4 λ λ −3 x 10 0 −0.2 −0.4 0 1 2 3 4 −3 x 10 noitalfnI )% dezilaunnA( −5.5 −6 −6.5 0 1 2 3 4 λ λ −3 x 10 4 3.5 3 0 1 2 3 4 −3 x 10 etaR tseretnI lanimoN )% dezilaunnA( 1 0 −1 0 1 2 3 4 λ λ −3 x 10 Note: The figure displays how the output gap, the inflation rate, and the nominal interest rate in both states vary with λ. The dashed vertical lines indicate society’s weight, λ¯. 11

Figure 2: Welfare −0.5 −0.55 −0.6 −0.65 −0.7 0 1 2 3 4 λ −3 x 10 )%( erafleW Note: Thefiguredisplayshowwelfarevarieswithλinthetwo-stateshockmodel. Thedashedverticallinesindicate society’s weight, λ¯. with the weight on output stabilization, λ. The dashed vertical lines show society’s weight, λ¯. Consistent with Proposition 2, output and inflation in the high state are positive and negative, respectively, for any λ. The nominal interest rate is below the natural rate of interest, which is 4 percent. In the low state, output and inflation are negative, and the nominal interest rate is zero. Consistent with Proposition 3, as λ decreases (i.e., as the central bank becomes more conservative), the deflation bias in the high state is reduced. This comes at the cost of a higher positive output gap in the high state, but a smaller deflation bias in the high state mitigates the decline in inflation and output in the low state. The benefits of the smaller rate of deflation in the high state and larger output and inflation in the low state dominate the negative effect of a larger output gap in the high state. Accordingly, welfareincreaseswiththedegreeofconservatism,asshowninFigure2. ConsistentwithProposition 4, the optimal weight is zero. In this case, the welfare gain is about 0.05 percent of the efficient level of consumption. Given that the welfare costs of business cycle fluctuations tend to be very small in this class of representative agent models, this number is non-trivial. Further analysis in Appendix B shows that the size of the welfare gain increases with the frequency, persistence, and size of the crisis shock. 4 Extensions In this section, we show that the desirability of inflation conservatism is robust to two model extensions. In the first extension, we augment the baseline discrete-state model with an additional 12

shock that affects the supply side of the economy via the New Keynesian Phillips curve. In the second extension, we consider a continuous-state variant of the baseline model. 4.1 A model with demand and supply shocks Thus far, the analysis has focused on an economy in which demand shocks are the only source of uncertainty. We now extend the analysis to an economy that is subject to both demand and supply shocks. In this case, the New Keynesian Phillips curve becomes π = κy +βE π +u , (21) t t t t+1 t where u is a cost-push shock. We assume that the cost-push shock takes two values, u = c ≥ 0 t H and u = −c, with probability 0.5 regardless of the state today. L Figure 3: Optimal weight in the model with cost-push shocks −3 x 10 3 2 1 0 0 2 4 6 −4 x 10 λ lamitpO −3 x 10 3 2 1 0 0 0.005 0.01 c λ lamitpO p H −3 x 10 3 2 1 0 0 0.5 1 λ lamitpO −3 x 10 2 1.5 1 0.5 0 −0.015 −0.01 −0.005 p L λ lamitpO d L Note: Thefiguredisplayshowtheoptimalweightonoutputstabilizationλdependsonthesizeofthecost-pushshock, and the frequency, persistence and size of the demand shock. Solid vertical lines indicate the baseline calibration. Thetop-leftpanelofFigure3showshowtheoptimalweightvarieswiththesizeofthecost-push shock. While the optimal weight remains zero when the size of the cost-push shock is small, it is positive for a sufficiently large shock size and increases with the size of the shock. There are two reasons why the optimal weight increases with the size of the cost-push shock. First, the optimal weight is equal to the social weight in the model with cost-push shocks but without demand shocks or the ZLB (see, for example, Clarida, Gali, and Gertler (1999)), while it is zero in the model without cost-push shocks but with demand shocks and the ZLB. Thus, a 13

key determinant of the optimal weight in the model with both demand and cost-push shocks is the relative importance of these two shocks: the larger the size of the cost-push shock, the closer the optimal weight is to the social weight. Second, a lower weight on the output gap implies larger fluctuations in the nominal interest rate in states in which the demand shock is positive, d = d . t H If the size of the cost-push shock is sufficiently large, then for small weights on output stabilization λ the ZLB can bind in the high-demand shock state as well. In such a case, the central bank cannot achieve zero inflation in both of the high-demand shock states and reducing λ further will lead to a larger deflation bias. In the particular parameterization shown in the figure, the optimal weight increases with the size of the cost-push shock for the second reason. One consequence of this result is that the optimal weight decreases with the frequency, persistence, and severity of the demand shock. This can be seen in the top-right, bottom-left, and bottom-right panels of Figure 3, which show how the optimal weight varies with the frequency, persistence, and severity of the demand shock when the size of the cost-push shock is held at 0.02. 100 Theoptimalweightcoincideswiththesocialweightwhentheseparametersarezero, anditdeclines as these parameters, and thus the relative importance of the demand shock, increase. Interestingly, the effects of these parameters on the optimal weight are not monotonic. This is because reducing the weight on output stabilization can reduce welfare when doing so leads to excessive fluctuations in the nominal interest rate and causes the ZLB to bind in one of the high-demand shock states. For sufficiently small values of p , p , and |d |, the optimal weight can be so large that the ZLB H L L does not bind anywhere near the optimal weight. Marginal increases in these parameters mean more severe deflation bias and optimal weights are lower as a result. For intermediate values of p , p , and |d |, marginal increases in these parameters do not lead to a reduction in the optimal H L L weight because a reduction in the weight placed on output stabilization causes the ZLB to bind in one of the high-demand shock states. When p , p , and |d | are sufficiently large, the benefit of H L L reducing the deflation bias dominates the adverse consequences of hitting the ZLB in one of the high-demand shock states, and the optimal weight on output stabilization becomes zero. Appendix C extends the analysis to a model with persistent cost-push shocks. 4.2 A continuous-state model We next examine whether the results from the analysis of the baseline model with a two-state Markov process for the demand shock also hold true when the shock is allowed to assume a continuum of values. Specifically, we assume that d follows a stationary AR(1)-process t d = ρ d +(cid:15)d, (22) t d t−1 t where the parameter ρ represents the persistence of the shock and (cid:15)d is an i.i.d. N (cid:0) 0,σ2(cid:1) innovad t (cid:15) tion. We employ a projection method to approximate the policy functions numerically. The details of the solution algorithm are described in Appendix D. The parameters are calibrated using the values from Adam and Billi (2007), as shown in Table 2. 14

Table 2: Parameterization (Continuous-state model) Parameter Value Economic interpretation β 0.9913 Subjective discount factor σ 6.25 Intertemporal elasticity of substitution in consumption η 0.47 Inverse labor supply elasticity θ 7.66 Price elasticity of demand α 0.66 Share of firms per period keeping prices unchanged ρ 0.8 AR-coefficient demand shock d σ 1.524 Standard deviation demand shock innovation (in %) (cid:15) Figure 4 shows the approximated policy functions for two alternative central bank regimes. Underthefirstregime,thecentralbankfocusessolelyoninflationstabilization(i.e.,theconservative regime) as was shown to be optimal in the two-state model. Under the second regime, the central bank’s preferences are identical to those of society as a whole (i.e., the benchmark regime). Figure 4: Approximated policy functions (Continuous-state model) 2 0 −2 −4 −6 −8 −1 0 1 2 3 4 paG tuptuO )%( 0.5 0 −0.5 −1 −1.5 −2 −1 0 1 2 3 4 400×r t noitalfnI )% dezilaunnA ( 400×r t 4 3 2 1 0 −1 0 1 2 3 4 etaR tseretnI lanimoN )% dezilaunnA( 4 3 2 1 0 −1 0 1 2 3 4 400×r t etaR tseretnI laeR )% dezilaunnA( 400×r t Note: The figure displays the approximated policy functions for λ=0 (solid lines) and λ=λ¯ (dashed lines). When the economy is in a liquidity trap and the natural real rate is negative, inflation and the output gap in both the conservative regime and the benchmark regime are negative, but as in the two-state model, the decline in the two target variables is less severe if the central bank is headed by a conservative policymaker. The larger the adverse shock, the more pronounced is the difference in the equilibrium responses between the two regimes. Note, however, that in the continuous-state model,theZLBisbindingnotonlywhenthenaturalrealrateofinterestisnegativebutalsowhenit 15

Figure 5: Conditional expectations (Continuous-state model) ZLB not binding 0.08 0.05 0.02 0 1 2 3 4 −3 x 10 paG tuptuO )%( ZLB binding −0.5 −1.25 −2 0 1 2 3 4 λ λ −3 x 10 0 −0.03 −0.06 0 1 2 3 4 −3 x 10 noitalfnI )% dezilaunnA( −0.2 −0.4 −0.6 0 1 2 3 4 λ λ −3 x 10 3.7 3.625 3.55 0 1 2 3 4 −3 x 10 etaR tseretnI lanimoN )% dezilaunnA( 1 0 −1 0 1 2 3 4 λ λ −3 x 10 Note: Thefiguredisplayshowtheconditionalaveragesoftheoutputgap,theinflationrateandthenominalinterest rate vary with λ. The dashed-dotted line indicates λ¯. 16

is close to zero and positive.13 In these states, inflation remains below zero but the output gap can be either negative or positive. The central bank can offset the direct effect of the natural rate shock butrunsintotheZLBwhentryingtocounteractthecombinedimpactofthenaturalrateshockand the downward bias in agents’ expectations. Finally, away from the ZLB, the conservative central banker perfectly stabilizes inflation at zero. Since inflation expectations are negative in all states of the world, inflation stability requires a positive output gap. In contrast, under the benchmark regime the economy is plagued by deflationary bias (i.e., negative inflation rates). Figure 5 shows how the average inflation rate, output gap and nominal interest rate vary with the central banker’s weight on output gap stabilization in a state in which the ZLB is not binding (left column) and in which the ZLB is binding (right column). The dashed vertical lines represent the averages associated with λ = λ¯. The results are very similar to those in the two-state model; however, unlike in the numerical example for the two-state model, the average output gap away from the ZLB is reduced as λ decreases, reflecting a strong feedback mechanism between policy actions and stabilization outcomes away from the ZLB and stabilization outcomes at the ZLB. Figure 6: Welfare (Continuous-state model) −3 x 10 −1 −1.5 −2 −2.5 0 1 2 3 4 λ −3 x 10 )%( erafleW Note: The figure displays how welfare as defined in (12) varies with λ. The dashed-dotted line indicates λ¯. Finally,Figure6showshowwelfareasdefinedin(12)dependsonthecentralbanker’spreference parameterλ. Thebenchmarkregimewithλ = λ¯ isindicatedbythedashed-dottedline. Thewelfare resultsfromthebaselinemodelcontinuetohold. First,thepresenceoftheoccasionallybindingZLB makesitdesirableforsocietytoappointaconservativecentralbanker. Second, thebest-performing central banker puts zero weight on output gap stability. 13Inthetwo-statemodelthiscasewasruledoutbytheassumptionthatthenominalinterestrateisstrictlypositive when the economy is in the normal state. 17

5 Conclusion We have demonstrated, both analytically and numerically, that an economy that experiences occasional ZLB episodes can improve welfare by appointing a conservative central banker who is more concernedwithinflationstabilizationrelativetooutputstabilizationthansocietyis. Intheabsence of policy commitment, optimal monetary policy suffers from a deflationary bias. Inflation stays below target even when the policy rate is positive because households and firms anticipate that the ZLB can be binding in the future. Subdued inflation rates away from the ZLB in turn exacerbate the decline in output and inflation when the economy is in a liquidity trap. A conservative central banker counteracts this vicious cycle by mitigating the deflationary bias away from the ZLB, thereby improving stabilization outcomes at and away from the ZLB. As a byproduct of our analysis, we provide a closed-form characterization of the conditions that guarantee the existence of the standard as well as the deflationary Markov-Perfect equilibrium for the discrete-state version of our model. 18

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Appendix A Proofs In this section, we will provide details of the proofs in the main text. Since the proofs are algebraically intensive, we will have to omit some details in this section. A.1 Proof of Proposition 1 The standard Markov-Perfect equilibrium is given by a vector {y ,π ,i ,y ,π ,i } that solves H H H L L L the following system of linear equations— y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d (A.1) H H H H L H H H L H H (cid:2) (cid:3) π = κy +β (1−p )π +p π (A.2) H H H H H L 0 = λy +κπ (A.3) H H y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d (A.4) L L H L L L H L L L L (cid:2) (cid:3) π = κy +β (1−p )π +p π (A.5) L L L H L L and i = 0 (A.6) L —and satisfies the following two inequality constraints: i > 0 (A.7) H and φ < 0. (A.8) L φ denotes the Lagrangean multiplier on the ZLB constraint in the low state: L φ := λy +κπ . (A.9) L L L We first prove four preliminary propositions (Propositions 1.A–1.D), then use them to prove the main proposition (Proposition 1) on the necessary and sufficient conditions for the existence of the standard Markov Perfect equilibrium. 22

Let A(λ) := −βλp , (A.10) H B(λ) := κ2+λ(1−β(1−p )), (A.11) H (1−p ) L C := (1−βp +βp )−p , (A.12) L H L σκ (1−p ) L D := − (1−βp +βp )−(1−p ) = −1−C, (A.13) L H L σκ and E(λ) := A(λ)D−B(λ)C. (A.14) Assumption 1.A: E(λ) (cid:54)= 0. Throughout the proof, we will assume that Assumption 1.A holds. Proposition 1.A: There exists a vector {y ,π ,i ,y ,π ,i } that solves (A.1)–(A.6). H H H L L L Proof: Rearranging the system of equations (A.1)–(A.6) and eliminating y and y , we obtain two H L unknowns for π and π in two equations: H L (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) A(λ) B(λ) π 0 L = C D π r H L (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) π 1 D −B(λ) 0 L ⇒ = , (A.15) π A(λ)D−B(λ)C −C A(λ) r H L where r = r∗+ 1d . L σ L Thus, A(λ) π := r (A.16) H L E(λ) and −B(λ) π := r . (A.17) L L E(λ) From the Phillips curves in both states, we obtain βκp H y = r (A.18) H L E(λ) and (1−βp )κ2+(1−β)(1+βp −βp )λ L H L y = − r . (A.19) L L κE(λ) 23

Proposition 1.B: Suppose (A.1)–(A.6) are satisfied. Then φ < 0 if and only if E(λ) < 0. L Proof: Notice that (1−βp )κ2+(1−β)(1+βp −βp )λ −B(λ) L H L φ = −λ r +κ r L L L κE(λ) E(λ) (cid:20) (cid:21) = − λ(cid:2) (1−βp )κ2+(1−β)(1+βp −βp )λ (cid:3) +κB(λ) r L . (A.20) L H L κ E(λ) Notice also that r < 0, (1−βp )κ2 > 0, (1−β)(1+βp −βp )λ ≥ 0, and κB(λ) > 0. Thus, if L L H L φ < 0, then E(λ) < 0. Similarly, if E(λ) < 0, then φ < 0. L L Proposition 1.C: E(λ) < 0 if and only if p∗ < (Θ ). L −pL Proof: It is convenient to view E(·) as a function of p and p instead of λ for a moment. H L (cid:20) (cid:21) 1−p L E(p ,p ) = βλp −Γ (1−βp +βp )−p H L H L H L σκ (cid:20) (cid:21) 1 = βλp −Γ (1−βp +βp −p +βp2 −βp p )−p H σκ L H L L H L L (cid:20) (cid:21) 1 1 1 = −Γ βp2 +Γ (1+β+βp )+1 p +βλp −Γ (1+βp ) σκ L σκ H L H σκ H := q p2 +q p +q , (A.21) 2 L 1 L 0 where Γ := κ2+λ(1−β) and 1 q := βλp −Γ (1+βp ), (A.22) 0 H H σκ (cid:2) 1 (cid:3) q := Γ (1+β +βp )+1 > 0, (A.23) 1 H σκ and 1 q := −Γ β < 0. (A.24) 2 σκ This function, E(·,·), has the following properties. Property 1: E(p ,1) > 0 for any 0 ≤ p ≤ 1. H H 24

Proof: (cid:20) (cid:21) 1 1 1 E(p ,1) = −Γ β +Γ (1+β +βp )+1 +βλp −Γ (1+βp ) H H H H σκ σκ σκ = Γ+βλp > 0 (A.25) H Property 2: E(p ,p ) is maximized at p > 1 for any 0 ≤ p ≤ 1. H L L H Proof: ∂E(p ,p ) H L = 2q p∗ +q = 0 ∂p 2 L 1 L q ⇔ p∗ = − 1 L 2q 2 Γ (cid:2) 1 (1+β +βp )+1 (cid:3) = σκ H 2Γ 1 β σκ (cid:2) 1 (2β+(1−β)+βp )+1 (cid:3) = σκ H > 1. (A.26) 2 1 β σκ These two properties imply i) one root of E(·,p ) is below 1 and ii) E(·,p ) < 0 below this root. L L Let’s call this root p∗(Θ ). p∗(Θ ) is given by L −pL L −pL (cid:112) −q + q2−4q q p∗(Θ ) := 1 1 2 0 . (A.27) L −pL 2q 2 If E(λ) < 0, then p < p∗(Θ ). Similarly, if p < p∗(Θ ), then E(λ) < 0. This completes L L −pL L L −pL the proof of Proposition 1.C. Note that Proposition 1.C holds independently of whether the system of linear equations (A.1)–(A.6) is satisfied or not. Proposition 1.D: Suppose (A.1)–(A.6) are satisfied and E(λ) < 0. Then i > 0 if and H only if p < p∗ (Θ ). H H −pH Proof: First, notice that i is given by H 25

1(cid:2) (cid:3) (cid:2) (cid:3) i =r + −p y +p y + (1−p )π +p π H H H H H L H H H L σ 1 −(1−βp )κ−(1−β)(1+βp −βp )λ/κ−βκp L H L H =r + p r H H L σ E(λ) A(λ) −B(λ) +(1−p ) r +p r H L H L E(λ) E(λ) (cid:20) (cid:21) r βΓ r (1−βp )Γ =− L p2 − L L +κ2+λ p +r . (A.28) E(λ)σκ H E(λ) σκ H H Since E(λ) < 0, i > 0 requires H (cid:18) (cid:19) βΓ (1−βp )Γ r p2 +r L +κ2+λ p −r E(λ) > 0 L σκ H L σκ H H (cid:20) (cid:18) (cid:19) (cid:21) βΓ (1−βp )Γ 1−p ⇔ r p2 + r L +κ2+λ −r βλ+r Γβ L p L σκ H L σκ H H σκ H (cid:18) (cid:19) 1−p L +r Γ (1−βp )−p > 0. (A.29) H L L σκ Dividing both sides by Γ and by −r , we obtain L − β p2 − (cid:34) (1−βp L )+(1−p L )βr r H L + κ2+(1−βr r H L )λ (cid:35) p σκ H σκ Γ H (cid:20) (cid:21) 1−p r L H − (1−βp )−p > 0. L L σκ r L (A.30) Let P(p ) := φ p2 +φ p +φ , (A.31) H 2 H 1 H 0 where (cid:20) (cid:21) 1−p r L H φ := − (1−βp )−p (A.32) 0 L L σκ r L (1−βp )+(1−p )βrH κ2+(1−βrH)λ φ := − L L rL − rL (A.33) 1 σκ Γ and β φ := − < 0. (A.34) 2 σκ 26

Property 1: φ > 0 0 Proof: Notice that i = r > 0 when p = 0. Since E(λ) < 0, the sign of i is the same as the H H H H sign of φ p2 +φ p +φ . Thus, φ > 0. This completes the proof of Property 1. 2 H 1 H 0 0 φ > 0 and φ < 0 imply that one root of (A.31) is non-negative and i > 0 if and only if p is 0 2 H H below this non-negative root, given by (cid:112) −φ − φ2−4φ φ p∗ (Θ ) := 1 1 0 2 . (A.35) H −pH 2φ 2 This completes the proof of Proposition 1.D. With these four preliminary propositions (1.A–1.D), we are ready to prove our Proposition 1. Proposition 1: There exists a vector {y ,π ,i ,y ,π ,i } that solves the system of H H H L L L linear equations (A.1)–(A.6) and satisfies φ < 0 and i > 0 if and only if p < p∗(Θ ) L H L L −pL and p < p∗ (Θ ). H H −pH Proof of “if” part: Suppose that p < p∗(Θ ) and p < p∗ (Θ ). According to Proposition L L −pL H H −pH 1.Athereexistsavector{y ,π ,i ,y ,π ,i }thatsolves(A.1)–(A.6). AccordingtoPropositions H H H L L L 1.B and 1.C, E(λ) < 0 and φ < 0. According to Proposition 1.D and the fact that E(λ) < 0, L i > 0. This completes the proof of the “if” part. H Proof of “only if” part: Suppose that φ < 0 and i > 0. According to Proposition 1.A there L H exists a vector {y ,π ,i ,y ,π ,i } that solves (A.1)–(A.6). According to Propositions 1.B and H H H L L L 1.C, E(λ) < 0 and p < p∗(Θ ). According to Proposition 1.D and the fact that E(λ) < 0, L L −pL p < p∗ (Θ ). This completes the proof of the “only if” part. H H −pH A.2 Proof of Proposition 2 Proposition 2 characterizes the sign of inflation and output in both states. Using i) the restriction onE(λ)(i.e. E(λ) < 0),ii)r < 0,andiii)inequalitiesonA(λ),B(λ),C,andD givenbyequations L (A.10)–(A.13), it is straightforward to check that: 27

A(λ) π = r ≤ 0, (A.36) H L E(λ) −B(λ) π = r < 0, (A.37) L L E(λ) βκp H y = r > 0, (A.38) H L E(λ) and (1−βp )κ2+(1−β)(1+βp −βp )λ L H L y = − r < 0. (A.39) L L κE(λ) Inflation and output are negative in the low state. Inflation in the high state is negative, which is what we call deflation bias. Positive output in the high state is consistent with negative inflation in the high state and the optimality condition of the central bank (i.e., equation (15)). A.3 Proof of Proposition 3 Proposition 3 characterizes how λ affects inflation and output in both states. ∂π A(cid:48)(λ)E(λ)−A(λ)E(cid:48)(λ) H := r ∂λ E(λ)2 L A(λ)B(cid:48)(λ)−A(cid:48)(λ)B(λ) = Cr E(λ)2 L −βp λ(1−β +βp )+βp (cid:0) κ2+(1−β+βp )λ (cid:1) H H H H = Cr E(λ)2 L βp κ2 H = Cr < 0, (A.40) E(λ)2 L where A(cid:48)(λ) and B(cid:48)(λ) denote the partial derivatives of A(·) and B(·) with respect to λ. ∂π −B(cid:48)(λ)E(λ)+B(λ)E(cid:48)(λ) L := r ∂λ E(λ)2 L A(cid:48)(λ)B(λ)−A(λ)B(cid:48)(λ) = Dr E(λ)2 L βp κ2 H = − Dr < 0 (A.41) E(λ)2 L 28

∂y −βκp E(cid:48)(λ) H H := r ∂λ E(λ)2 L βκp (A(cid:48)(λ)D−B(cid:48)(λ)C) H = − r E(λ)2 L βκp H = − [βp −(1−β)C]r E(λ)2 H L (cid:20) (cid:18) (cid:19)(cid:21) βκp 1−p H L = − βp −(1−β) (1−βp +βp )−p r (A.42) E(λ)2 H κσ L H L L ∂y (1−β)(1−βp +βp )E(λ)− (cid:0) (1−βp )κ2+(1−β)(1−βp +βp )λ (cid:1) E(cid:48)(λ) L L H L L H := − r ∂λ κE(λ)2 L (cid:20) (1−β)(1−βp +βp )(A(λ)D−B(λ)C) L H = − κE(λ)2 (cid:0) (1−βp )κ2+(1−β)(1−βp +βp )λ (cid:1) (A(cid:48)(λ)D−B(cid:48)(λ)C)(cid:21) L L H − r κE(λ)2 L βκp H = [(1−β)C +(1−βp )]r < 0 (A.43) E(λ)2 L L A.4 Proof of Proposition 4 Proposition 4 states that welfare is maximized at λ = 0. Society’s unconditional expected value is given by (cid:20) (cid:21) 1 1−p p L H EV(λ) = u(π ,y )+ u(π ,y ) . (A.44) H H L L 1−β 1−p +p 1−p +p L H L H To show that E[V(λ)] is maximized at λ = 0, we show that ∂EV(λ) < 0 (A.45) ∂λ for all λ ≥ 0. The derivative of the unconditional expected value is given by (cid:20) (cid:21) ∂EV 1 1−p ∂u(π ,y ) p ∂u(π ,y ) L H H H L L = + . (A.46) ∂λ 1−β 1−p +p ∂λ 1−p +p ∂λ L H L H The partial derivatives of society’s utility are given by 29

(cid:20) (cid:21) ∂u(π ,y ) ∂ 1 H H := − (λ¯y (λ)2+π (λ)2) H H ∂λ ∂λ 2 (cid:18) βκp (cid:18) βκp E(cid:48)(λ) (cid:19) A(λ) A(cid:48)(λ)E(λ)−A(λ)E(cid:48)(λ) (cid:19) = − λ¯ H r − H r + r r E(λ) L E(λ)2 L E(λ) L E(λ)2 L λ¯β2κ2p2 E(cid:48)(λ)+λβ2p2 E(λ)−λ2β2p2 E(cid:48)(λ) = − H H H r2 E(λ)3 L β2κ2p2 = H (cid:0) λ¯E(cid:48)(λ)+λC (cid:1) r2 E(λ)3 L β2κ2p2 = H (cid:2) λ¯(βp −(1−β)C)+λC (cid:3) r2. (A.47) E(λ)3 H L Note that we have already shown that the sign of the first term in square brackets, βp −(1−β)C, H determines the sign of ∂yH. If βp −(1−β)C > 0, then ∂yH > 0 and ∂u(πH,yH) > 0. If instead ∂λ H ∂λ ∂λ βp −(1−β)C < 0, then the sign of ∂u(πH,yH) is ambiguous. H ∂λ (cid:20) (cid:21) ∂u(π ,y ) ∂ 1 L L := − (λ¯y (λ)2+π (λ)2) L L ∂λ ∂λ 2 (cid:18) = βp H λ¯(cid:2) (1−βp )κ2+(1−β)(1+βp −βp )λ (cid:3)(cid:2) (1−β)C +(1−βp ) (cid:3) E(λ)3 L H L L (cid:19) +κ2(cid:2) κ2+λ(1−β(1−p )) (cid:3) (1+C) r2 H L := βp H (cid:0) λ¯Φ (λ)+Φ (λ) (cid:1) r2 < 0, (A.48) E(λ)3 1 2 L where Φ (λ) := Φ +Φ λ, (A.49) 1 1,1 1,2 Φ (λ) := Φ +Φ λ, (A.50) 2 2,1 2,2 and Φ := (1−βp )κ2(cid:2) (1−β)C +(1−βp ) (cid:3) > 0, (A.51) 1,1 L L (cid:2) (cid:3) Φ := (1−β)C +(1−βp ) (1−β)(1+βp −βp ) > 0, (A.52) 1,2 L H L Φ := κ4(1+C) > 0, (A.53) 2,1 and Φ := κ2(1−β(1−p ))(1+C) > 0. (A.54) 2,2 H Hence, 30

∂EV = (1−β)−1r L 2 (cid:18) (1−p ) β2κ2p2 H (cid:2) λ¯(βp −(1−β)C)+λC (cid:3) +p βp H (cid:0) λ¯Φ (λ)+Φ (λ) (cid:1) (cid:19) ∂λ 1−p +p L E(λ)3 H H E(λ)3 1 2 L H (1−β)−1βp2 r2 = H L (cid:0) βκ2(1−p ) (cid:2) λ¯(βp −(1−β)C)+λC (cid:3) + (cid:0) λ¯Φ (λ)+Φ (λ) (cid:1)(cid:1) .(A.55) (1−p +p )E(λ)3 L H 1 2 L H Let Ω(λ) := βκ2(1−p )(βp −(1−β)C)λ¯+βκ2(1−p )Cλ+λ¯(Φ +Φ λ)+Φ +Φ λ. (A.56) L H L 1,1 1,2 2,1 2,2 If Ω(λ) > 0 for all λ ≥ 0, then ∂EV(λ) < 0 for all λ ≥ 0. Notice that Ω(cid:48)(λ) is positive since the ∂λ coefficients on λ are all positive. Thus, we only need to show Ω(0) > 0 to show that Ω(λ) > 0 for all λ ≥ 0. Ω(0) = βκ2(1−p ) (cid:2) βp −(1−β)C (cid:3) λ¯+λ¯Φ +Φ L H 1,1 2,1 = (cid:2) βκ2(1−p ) (cid:2) βp −(1−β)C (cid:3) +Φ (cid:3) λ¯+Φ L H 1,1 2,1 = (cid:2) β2κ2(1−p )p +(1−β)2κ2C +(1−βp )2κ2(cid:3) λ¯+κ4(1+C) > 0, (A.57) L H L given that C > 0 for the equilibrium to exist and λ¯ > 0. This completes the proof. B Welfare gains of conservatism In the baseline calibration of the two-state shock model, the welfare gain of appointing a fully conservative central banker is about 0.05 percent of the efficient level of consumption. This number clearly depends on the frequency, persistence, and severity of the ZLB episodes. Figure 7 shows this dependency. The top-left panel shows that the welfare gain of conservatism increases with the frequency of the shock, and it reaches about 0.3 percent when the frequency is 1.5 percent. According to the top-right panel, the welfare gain increases sharply with persistence, exceeding 2 percent at p = 0.9. Finally, the welfare gain increases with the severity of the shock (the absolute L value of d ), as shown in the bottom-left panel. L C The model with persistent cost-push shocks In the main text, we analyzed how the introduction of i.i.d. cost-push shocks affects the optimal weight placed on the output gap stability term. In this section, we relax the i.i.d. assumption to consider persistent cost-push shocks. In particular, we consider cases in which the probability of staying at the high (or low) cost-push state tomorrow when today’s cost-push state is high (or low) is either 0.6 or 0.8, as opposed to 0.5 in the baseline i.i.d. case. In the top-left panel of Figure 8, the blue and red dashed lines show how the optimal weight on 31

Figure 7: Welfare gains of conservatism (Two-state shock model) 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 )%( niag erafleW p p H L 4 Baseline 3 2 1 0 0.75 0.8 0.85 0.9 )%( niag erafleW d L 0.3 0.2 0.1 0 −0.0175 −0.0142 −0.0108 −0.0075 )%( niag erafleW Note: The figure displays how the welfare gains of conservatism (λ = 0) relative to the baseline regime (λ = λ¯) depend on the frequency, persistence and size of the demand shock. the output gap volatility term varies with the size of the cost-push shocks in the economies with persistence of 0.6 and 0.8, respectively. The black line is for the optimal weight in the baseline economy with non-persistent cost-push shocks. The panel shows that the optimal weight is lower when cost-push shocks are more persistent. The reason is as follows. In the model without the ZLB, the optimal weight is the same as the true weight if the shock is not persistent. When costpush shocks are persistent, the optimal weight is smaller than the true weight because inflation is expected to be non-zero in the future and a smaller weight on the output gap stabilization term reduces the deviation of expected inflation from zero. As such, the more persistent the cost-push shocks are, the smaller the optimal weight is. As we saw in the main text, the introduction of the ZLB makes it desirable to place a lower weight on the output gap stabilization term. The fact that the optimal weight is lower when cost-push shocks are more persistent in the model with the ZLB is inherited from the same feature in the model without the ZLB. The top-right and bottom two panels shows how the optimal weight varies with the frequency, persistence, and size of the crisis shock. Consistent with the baseline model with non-persistent cost-push shocks, the optimal weight declines as the frequency, persistence, and severity of the crisis increases with the same non-monotonicity discussed in the main text. Consistent with our discussion in the previous paragraph, the optimal weight is generally lower when cost-push shocks are more persistent. However, due to the non-monotonicity, there are regions of parameter values under which the optimal weight is larger with more persistent cost-push shocks. For example, the 32

Figure 8: Optimal weight in the model with persistent cost-push shocks −3 x 10 4 3 2 1 0 0 2 4 6 −4 x 10 λ lamitpO −3 x 10 2.5 2 1.5 1 0.5 0 0 0.005 0.01 0.015 c p H λ lamitpO −3 x 10 2.5 2 1.5 1 0.5 0 0 0.5 1 λ lamitpO −3 x 10 2 1.5 1 0.5 0 −0.025 −0.015 −0.005 p L λ lamitpO Baseline Moderate persistence High persistence d L Note: Thefiguredisplayshowtheoptimalweightonoutputstabilizationλdependsonthesizeofthecost-pushshock, and the frequency, persistence and size of the demand shock. Under the baseline case, the probability of staying at the high (low) cost-push state tomorrow when today’s cost-push state is high (low) is 0.5. Under the “moderate” and “high” persistence cases, that probability is 0.6 and 0.8., respectively. 33

optimal weight is larger with persistence of 0.8 than with persistence of 0.6 for the values of p H between 0.01 and 0.013 and for the values of d between -0.021 and -0.018. L D Computational algorithm for the continuous-state model Let Z = [π,y](cid:48) and Z(cid:101) = [Z(cid:48),i](cid:48). We approximate Z by a linear combination of n basis functions ψ i , i = 1,...,n. In matrix notation, Z(d) ≈ CΨ(d), (D.1) where   ψ (d) (cid:32) (cid:33) 1 C = cπ 1 ··· cπ n , Ψ(d) =   . . .  . cy ··· cy   1 n ψ (d) n The coefficients cj, i = 1,2,...,n; j ∈ {π,y}, are set such that (D.1) holds exactly at n selected i collocation nodes collected in vector d (cid:0) (cid:1) (cid:0) (cid:1) Z d = CΨ d , (k) (k) for k = 1,...,n, where d refers to the kth element of d. We use linear splines as basis functions (k) and choose the breakpoints such that they coincide with the collocation nodes. Theiterativesolutionalgorithmtoobtainthepolicyfunctionapproximationsthenworksasfollows. We start with an initial guess on the coefficient matrix C(0). For fixed C(s) in iteration s, we first update the expectations functions, m Eπ(s)(cid:0) d (cid:1) = (cid:88) (cid:36) C (s) Ψ (cid:0) ρ d +(cid:15) (cid:1) (k) l (1,:) d (k) (l) l=1 and m Ey(s)(cid:0) d (cid:1) = (cid:88) (cid:36) C (s) Ψ (cid:0) ρ d +(cid:15) (cid:1) , (k) l (2,:) d (k) (l) l=1 for k = 1,...,n. We use a Gaussian quadrature scheme to discretize the normally distributed random variable, where (cid:15) is a m×1 matrix of quadrature nodes and (cid:36) is a vector of length m containing the weights. Assuming first that the zero bound is not binding at any collocation node, the optimality conditions for the discretionary policy regime imply Z˜(s)(cid:0) d (cid:1) = A−1·B+A−1·F ·EZ(s)(cid:0) d (cid:1) +A−1·D·d , (k) (k) (k) 34

for k = 1,...,n, where         1 −κ 0 0 β 0 0 A =  0 1 σ , B =  σr∗ , F =  σ 1 , D =  1 .         κ λ 0 0 0 0 0 For those k for which the zero lower bound is violated—i.e., i(s)(cid:0) d (cid:1) < 0—matrix A in the update (k) is replaced by   1 −κ 0 Aˆ=  0 1 σ .   0 0 1 WethenupdateC(s+1) = (cid:2) Z(s)(cid:0) d (cid:1) ··· Z(s)(cid:0) d (cid:1)(cid:3) andcontinuetheiterationprocedureuntil (1) (n) (cid:13) (cid:13)vec (cid:0) C(s+1)−C(s)(cid:1)(cid:13) (cid:13) < δ. ∞ The collocation nodes are equally distributed with a support covering ± 4 unconditional standard deviations of the exogenous state variable. We use MATLAB routines from the CompEcon toolbox of Miranda and Fackler(2002) to obtainthe Gaussian quadrature approximation ofthe innovations to the demand shock, and to evaluate the spline functions. E Existence of other Markov-Perfect Equilibria While we focus on the standard Markov-Perfect equilibrium in which the ZLB constraint binds in the low state but not in the high state, there are potentially three other types of Markov-Perfect equilibria: i)oneinwhichtheZLBconstraintbindsinbothstates(the deflationary Markov-Perfect equilibrium),ii)oneinwhichtheZLBconstraintdoesnotbindinbothstates(the ZLB-free Markov- Perfect equilibrium), and iii) one in which the ZLB binds in the high state but not in the low state (the topsy-turvy Markov-Perfect equilibrium). In this section, we examine whether and under what conditions any of these other types of Markov-Perfert equilibria exist. Our main results are that i) the conditions for the existence of the deflationary Markov-Perfect equilibrium are the same as those for the existence of the standard Markov-Perfect equilibrium and ii) the other two types do not exist under any parameter configurations.14 14There is a continuum of sunspot equilibria which may randomly move between the standard and deflationary Markov-Perfect equilibria. Characterizing the conditions for the existence of such sunspot equilibria is outside the scope of the paper. 35

E.1 Existence of the deflationary Markov-Perfect equilibrium ThedeflationaryMarkov-Perfectequilibriumisgivenbyavector{y ,π ,i ,y ,π ,i }thatsolves H H H L L L the following system of linear equations— y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.1) H H H H L H H H L H H (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.2) H H H H H L i = 0, (E.3) H y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.4) L L H L L L H L L L L (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.5) L L L H L L and i = 0, (E.6) L —and satisfies the following two inequality constraints: φ < 0 (E.7) H and φ < 0. (E.8) L φ and φ denote the Lagrangean multipliers on the ZLB constraint in the high state and in the H L low state: φ := λy +κπ (E.9) H H H and φ := λy +κπ . (E.10) L L L ThefollowingpropositionstatesthattheconditionsfortheexistenceofthedeflationaryMarkov- Perfect equilibrium are identical to the conditions for the existence of the standard Markov-Perfect equilibrium. Proposition 5: The deflationary Markov-Perfect equilibrium exists if and only if p ≤ p∗(Θ ) L L (−pL) and p ≤ p∗ (Θ ), H H (−pH) where the cutoff values p∗(Θ ) and p∗ (Θ ) are defined by (A.27) and (A.35) in Appendix L (−pL) H (−pH) A. We first prove six preliminary propositions, then use them to prove Proposition 5. 36

Let (cid:16)p (cid:17) A˜:= − H (1−βp +βp )+p , (E.11) L H H σκ B˜ := −A˜−1, (E.12) and E˜ := A˜D−B˜C = −A˜+C, (E.13) where C and D < 0 are defined in (A.12) and (A.13). Assumption 5.A: E˜ (cid:54)= 0. Throughout the proof, we will assume that Assumption 5.A holds. Proposition 5.A: There exists a vector {y ,π ,i ,y ,π ,i } that solves (E.1)–(E.6). H H H L L L Proof: Rearranging the system of equations (E.1)–(E.6) and eliminating y and y , we obtain two H L unknowns for π and π in two equations: H L (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) A˜ B˜ π r L H = C D π r H L (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) π 1 D −B˜ r L H ⇒ = , (E.14) π E˜ −C A˜ r H L where r = r∗+ 1d and r = r∗+ 1d . L σ L H σ H Thus, A˜ C π := r − r (E.15) H E˜ L E˜ H and −B˜ D π := r + r . (E.16) L E˜ L E˜ H From the Phillips curves in both states, we obtain (1−β)A˜−βp (1−β)C −βp H H y = r − r (E.17) H κE˜ L κE˜ H 37

and (1−β)A˜+(1−βp ) (1−β)C +(1−βp ) L L y = r − r . (E.18) L κE˜ L κE˜ H Proposition 5.B: Suppose (E.1)–(E.6) are satisfied. Then φ < 0 only if E˜ > 0. L Proof by contradiction: First, notice that (cid:20) (cid:21) 1 λ λ λ λ φ = −(1+C)κr −(1−β)C r −(1−βp ) r +(1+A˜)κr + (1−β)A˜r + (1−βp )r . L E˜ H κ H L κ H L κ L κ L L (E.19) Suppose that E˜ < 0. From the equation above we know that, given E˜ < 0, φ < 0 if and only L if λ λ λ λ −(1+C)κr −(1−β)C r −(1−βp ) r +(1+A˜)κr + (1−β)A˜r + (1−βp )r > 0. (E.20) H H L H L L L L κ κ κ κ Collecting terms, this condition can be simplified to (cid:18) (cid:19) λ κ+ (1−βp ) [(1+A)r −(1+C)r ] > 0. (E.21) L L H κ From (E.13), we know that E˜ < 0 if and only if C < A˜, where A˜< 0. Furthermore, from (A.12) we know that C > −1. Suppose C → −; then A > −1, which proves that (E.21) cannot hold. Proposition 5.C: Suppose (E.1)-(E.6) are satisfied and E˜ > 0. Then φ < 0 if φ < 0. L H Proof: This follows directly from noticing that κ2+λ φ = φ + (r −r ). (E.22) L H κE˜ L H Proposition 5.D: Suppose (E.1)–(E.6) are satisfied and E˜ > 0. Then φ < 0 if and only H if p < p∗ (Θ ). H H −pH Proof: First, notice that 38

(cid:18) (cid:19) 1 λ λ λ λ φ = [−Cκ−(1−β) C +β p ]r +[κA˜+(1−β) A˜−β p ]r . (E.23) H E˜ κ κ H H κ κ H L Since E˜ > 0, φ < 0 requires H λ λ λ λ −Cκ−(1−β) C +β p ]r +[κA˜+(1−β) A˜−β p ]r < 0. (E.24) H H H L κ κ κ κ Multiplying both sides by κ 1 and collecting terms, we get ΓrL − β p2 − (cid:32) (1−βp L )+(1−p L )βr r H L + κ2+(1−βr r H L )λ (cid:33) p σκ H σκ Γ H (cid:18) (cid:19) 1−p r L H − (1−βp )−p > 0. (E.25) L L σκ r L Let P(p ) := φ p2 +φ p +φ : (E.26) H 2 H 1 H 0 where (cid:20) (cid:21) 1−p r L H φ := − (1−βp )−p 0 L L σκ r L (1−βp )+(1−p )βrH κ2+(1−βrH)λ φ := − L L rL − rL 1 σκ Γ β φ := − < 0, (E.27) 2 σκ which is similar to the definition in Appendix A. φ > 0 and φ < 0 imply that one root of (E.26) 0 2 is non-negative and φ < 0 if and only if p is below this non-negative root, given by H H (cid:112) −φ − φ2−4φ φ p∗ (Θ ) := 1 1 0 2 . (E.28) H −pH 2φ 2 This completes the proof of Proposition 5.D. Proposition 5.E: E˜ > 0 and p < p∗ (Θ ) only if E(λ) < 0. H H −pH Proof: Suppose that E˜ > 0 and p < p∗ (Θ ). Then E˜ +P(p ) > 0. H H −pH H 39

(cid:18) (cid:19) (cid:18) (cid:19) β (1−βp )+(1−p )β 1−p E˜ +P(p ) = p2 + 1+ L L p + L (1−βp )−p H σκ H σκ H σκ L L − β p2 − (cid:32) (1−βp L )+(1−p L )βr r H L + κ2+(1−βr r H L )λ (cid:33) p σκ H σκ Γ H (cid:18) (cid:19) 1−p r L H − (1−βp )−p L L σκ r L (cid:20) (cid:18) (cid:19)(cid:21)(cid:18) (cid:19) λ 1−p 1−p r L L H = β p −β p − (1−βp )−p −1 . (E.29) H H L L Γ σκ σκ r L (cid:16) (cid:17) Since rH −1 < 0, the following condition has to hold: rL (cid:18) (cid:19) λ 1−p 1−p L L β p −β p − (1−βp )−p < 0. (E.30) H H L L Γ σκ σκ Collecting terms, we get (cid:20) (cid:21) 1 1 1 −Γ βp2 +Γ (1+β+βp )+1 p +βλp −Γ (1+βp ) = E(λ) < 0. (E.31) σκ L σκ H L H σκ H This completes the proof of Proposition 5.E. Note that Proposition 5.E holds independently of whether the system of linear equations (E.1)–(E.6) is satisfied or not. Proposition 5.F: E(λ) < 0 only if E˜ > 0. Proof: This follows directly from noticing that E(λ) βλ 1 E˜ = − + p + (βp +1+β(1−p ))p . (E.32) H H L H Γ Γ σκ Note that Proposition 5.F holds independently of whether the system of linear equations (E.1)– (E.6) is satisfied or not. With these six preliminary propositions (5.A–5.F), we are ready to prove Proposition 5. Proposition 5: There exists a vector {y ,π ,i ,y ,π ,i } that solves the system of H H H L L L linear equations (E.1)–(E.6) and satisfies φ < 0 and φ < 0 if and only if p < p∗(Θ ) L H L L −pL and p < p∗ (Θ ). H H −pH Proof of “if” part: According to Proposition 5.A, there exists a vector {y ,π ,i ,y ,π ,i } that H H H L L L solves (E.1)–(E.6). Suppose that p < p∗(Θ ) and p < p∗ (Θ ). According to Proposition L L −pL H H −pH 1.C (which does not rely on the system of linear equations), E(λ) < 0. According to Proposition 40

5.F, then E˜ > 0. According to Proposition 5.D, this implies φ < 0. Finally, according to Propo- H sition 5.C, this implies φ < 0. This completes the proof of “if” part. L Proof of “only if” part: According to Proposition 5.A, there exists a vector {y ,π ,i ,y ,π ,i } H H H L L L that solves (E.1)–(E.6). Suppose that φ < 0 and φ < 0. According to Proposition 5.B, E˜ > 0. L H According to Proposition 5.D, then p < p∗ (Θ ). According to Proposition 5.E, this implies H H −pH E(λ) < 0. According to Proposition 1.C (which does not rely on the system of linear equations), p < p∗(Θ ). This completes the proof of the “only if” part. L L −pL E.2 Nonexistence of the topsy-turvy Markov-Perfect equilibrium Thetopsy-turvyMarkov-Perfectequilibriumisgivenbyavector{y ,π ,i ,y ,π ,i }thatsolves H H H L L L the following system of linear equations— y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.33) H H H H L H H H L H H (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.34) H H H H H L i = 0, (E.35) H y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.36) L L H L L L H L L L L (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.37) L L L H L L 0 = λy +κπ , (E.38) L L —and satisfies the following two inequality constraints: φ < 0 (E.39) H and i > 0 (E.40) L φ denotes the Lagrangean multiplier on the ZLB constraint in the high state: H φ := λy +κπ . (E.41) H H H Proposition 6: The topsy-turvy Markov-Perfect equilibrium does not exist. We first prove three preliminary propositions, then use them to prove Proposition 6. 41

Let Cˆ(λ) := κ2+λ(1−βp ), (E.42) L Dˆ(λ) := −βλ(1−p ), (E.43) L and Eˆ(λ) := A˜Dˆ(λ)−B˜Cˆ(λ), (E.44) where A˜ and B˜ are defined in (E.11) and (E.12). Assumption 6.A: Eˆ(λ) (cid:54)= 0. Throughout the proof, we will assume that Assumption 6.A holds. Proposition 6.A: There exists a vector {y ,π ,i ,y ,π ,i } that solves (E.33)–(E.38). H H H L L L Proof: Rearranging the system of equations (E.33)–(E.38) and eliminating y and y , we obtain two H L unknowns for π and π in two equations: H L (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) A˜ B˜ π r L H = Cˆ(λ) Dˆ(λ) π 0 H (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) π 1 Dˆ(λ) −B˜ r L H ⇒ = , (E.45) π Eˆ(λ) −Cˆ(λ) A˜ 0 H where r = r∗+ 1d . H σ H Thus, Cˆ(λ) π := − r (E.46) H Eˆ(λ) H and Dˆ(λ) π := r . (E.47) L Eˆ(λ) H From the Phillips Curves in both states, we obtain (1−β)Cˆ(λ)+βp Γ H y = − r (E.48) H κEˆ(λ) H and (1−βp )Dˆ(λ)+(1−p )βCˆ(λ) L L y = − r . (E.49) L κEˆ(λ) H 42

Proposition 6.B: Suppose (E.33)–(E.38) are satisfied. Then φ < 0 if and only if H Eˆ(λ) > 0. Proof: Notice that 1 (cid:16) (cid:17) φ = − (κ2+(1−β))Cˆ(λ)+βp Γ r , (E.50) H κEˆ(λ) H H where (κ2+(1−β))Cˆ(λ)+βp Γ > 0 and r > 0. Hence, φ < 0 if and only if Eˆ(λ) > 0. H H H Proposition 6.C: Suppose (E.33)–(E.38) are satisfied. Then i > 0 only if Eˆ(λ) < 0. L Proof: Notice that (cid:20) 1 1 i = r − −κp Dˆ(λ)+κ(1−p )Cˆ(λ)+ (1−p ) L L κEˆ(λ) L L σ L (cid:21) (cid:16) (cid:17) (1−β)Cˆ(λ)+βp Γ+(1−βp )Dˆ(λ)+(1−p )βCˆ(λ) r , (E.51) H L L H (cid:20) 1 1 = r − −κp Dˆ(λ)+κ(1−p )Cˆ(λ)+ (1−p ) L κEˆ(λ) L L σ L (cid:21) (cid:16) (cid:17) (1−βp )Cˆ(λ)+βp Γ+(1−βp )Dˆ(λ) , (E.52) L H L (cid:20) (cid:21) 1 1 = r − −κp Dˆ(λ)+κ(1−p )Cˆ(λ)+ (1−p )(βp +(1−βp ))Γ r , (E.53) L κEˆ(λ) L L σ L H L H where−κp Dˆ(λ)+κ(1−p )Cˆ(λ)+1(1−p )(βp +(1−βp ))Γ > 0, r > 0, andr < 0. Hence, L L σ L H L H L i > 0 only if Eˆ(λ) < 0. L With these three preliminary propositions (6.A-6.C), we are ready to prove Proposition 6. Proposition 6: There exists no vector {y ,π ,i ,y ,π ,i } that solves the system of H H H L L L linear equations (E.33)–(E.38) and satisfies i > 0, φ < 0. L H Proof by contradiction: According to Proposition 6.A, there exists a vector {y ,π ,i ,y ,π ,i } H H H L L L that solves (E.33)–(E.38). Suppose that φ < 0 and i > 0. According to Proposition 6.B, φ < 0 H L H implies Eˆ(λ) > 0. According to Proposition 6.C, i > 0 implies Eˆ(λ) < 0, which contradicts L (i > 0, φ < 0). L H 43

E.3 Nonexistence of the ZLB-free Markov-Perfect equilibrium The ZLB-free Markov-Perfect equilibrium is given by a vector {y ,π ,i ,y ,π ,i } that solves H H H L L L the following system of linear equations— y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.54) H H H H L H H H L H H (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.55) H H H H H L 0 = λy +κπ , (E.56) H H y = (cid:2) (1−p )y +p y (cid:3) +σ (cid:2) (1−p )π +p π −i +r∗(cid:3) +d , (E.57) L L H L L L H L L L L (cid:2) (cid:3) π = κy +β (1−p )π +p π , (E.58) L L L H L L (E.59) and 0 = λy +κπ , (E.60) L L —and satisfies the following two inequality constraints: i > 0 (E.61) H and i > 0. (E.62) L Proposition 7: The ZLB-free Markov-Perfect equilibrium does not exist. Proof: Let Eˆ = (cid:2) 1−β(1−p )+ κ2 (cid:3) (1−βp + κ2 )−β2p (1−p ). (E.63) H L H L λ λ Assumption 7.A: Eˆ (cid:54)= 0. Throughout the proof, we will assume that Assumption 7.A holds. Notice that i and i only appear in the consumption Euler equations. Thus, we can first find H L a vector of {y ,π ,y ,π } that satisfies the Phillips curves and the government’s optimality con- H H L L ditionin bothstates, thenuse the two consumptionEuler equationsto findi andi . Rearranging H L the system of equations (E.55), (E.56), (E.58), and (E.60) and eliminating y and y , we obtain H L two unknowns for π and π in two equations: H L κ2 (cid:2) (cid:3) π = − π +β (1−p )π +p π (E.64) H H H H H L λ 44

and κ2 (cid:2) (cid:3) π = − π +β (1−p )π +p π (E.65) L L L H L L λ (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) 1−β(1−p )+ κ2 −βp π 0 ⇒ H λ H H = (E.66) −β(1−p ) 1−βp + κ2 π 0 L L λ L (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) π 1 1−βp + κ2 β(1−p ) 0 0 ⇒ H = L λ L = . (E.67) π Eˆ βp 1−β(1−p )+ κ2 0 0 L H H λ From the Phillips curves in both states, we obtain y = 0 (E.68) H and y = 0. (E.69) L From the consumption Euler equations in both states, we obtain d i = r∗+ H > 0 (E.70) H σ and d i = r∗+ L < 0. (E.71) L σ These two inequalities hold because we assume that d > −σr∗ and d < −σr∗ (see Section 2 H L in the main text). Thus, the inequality condition for the policy rate in the low state is violated. Accordingly, there is no vector that solves (E.54)–(E.60) and satsifies both i > 0 and i > 0. H L 45