A Promised Value Approach to Optimal Monetary Policy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Promised Value Approach to Optimal Monetary Policy Timothy S. Hills, Taisuke Nakata, and Takeki Sunakawa 2018-083 Please cite this paper as: Timothy S. Hills, Taisuke Nakata, and Takeki Sunakawa (2018). “A Promised Value Approach to Optimal Monetary Policy,” Finance and Economics Discussion Series 2018-083. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2018.083. 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.

A Promised Value Approach to Optimal Monetary Policy∗ Timothy Hills† Taisuke Nakata‡ Takeki Sunakawa§ New York University Federal Reserve Board Kobe University First Draft: April 2018 This Draft: August 2018 Abstract This paper characterizes optimal commitment policy in the New Keynesian model using a novel recursive formulation of the central bank’s infinite horizon optimization problem. In our recursive formulation motivated by Kydland and Prescott (1980), promised inflation and output gap—as opposed to lagged Lagrange multipliers—act as pseudo-state variables. Using three well known variants of the model—one featuring inflation bias, one featuring stabilization bias, and one featuring a lower bound constraint on nominal interest rates— we show that the proposed formulation sheds new light on the nature of the intertemporal trade-off facing the central bank. JEL: E32, E52, E61, E62, E63 Keywords: Commitment, Inflation Bias, Optimal Policy, Ramsey Plans, Stabilization Bias, Zero Lower Bound. ∗The views expressed in this paper, as well as all errors and omissions, should be regarded as those of the authors,andarenotnecessarilythoseoftheBoardofGovernorsoftheFederalReserveSystemortheFederal Reserve System. We thank Thomas Sargent for introducing us to the promised value approach to solving theRamseyproblemandYuichiroWakiandseminarparticipantsatTohokuUniversityforusefulcomments. Satoshi Hoshino and Donna Lormand provided excellent research and editorial assistance, respectively. †Stern School of Business, New York University; Email: hills.timoteo@gmail.com. ‡Board of Governors of the Federal Reserve System, Division of Research and Statistics, 20th Street and Constitution Avenue NW Washington, DC 20551; Email: taisuke.nakata@frb.gov. §Kobe University; Email: takeki.sunakawa@gmail.com. 1

1 Introduction Optimal commitment policy is a widely adopted approach among economists and policymakerstostudyingthequestionofhowtobestconductmonetarypolicy. Forexample, atthe Federal Reserve, the results of optimal commitment policy analysis from the FRB/US model have for some time been regularly presented to the Federal Open Market Committee to help inform its policy decisions (Brayton, Laubach, and Reifschneider (2014)). Most recently, in many advanced economies where the policy rate was constrained at the effective lower bound (ELB), the insights from the optimal commitment policy in a stylized New Keynesian model have played a key role in the inquiry on how long the policy rate should be kept at the ELB (Bullard (2013), Evans (2013), Kocherlakota (2011), Plosser (2013), Woodford (2012)). Accordingly, a deep understanding of optimal commitment policy is as relevant as ever. In this paper, we contribute to a better understanding of optimal commitment policy in the New Keynesian model—a workhorse model for analyzing monetary policy—by providing anovelmethodtocharacterizeit. Ourmethodusespromisedvaluesofinflationandoutputas pseudo-statevariablesinthespiritofKydlandandPrescott(1980)insteadoflaggedLagrange multipliers as in the standard method of Marcet and Marimon (2016). We describe our recursive approach—which we will refer to as the promised value approach—in three variants of the New Keynesian model that have been widely studied in the literature: the model with inflation bias, the model with stabilization bias, and the model with an ELB constraint. In each model, we define the infinite-horizon problem of the Ramsey planner, provide the recursive formulations of the Ramsey planner’s problem via the promised value approach, and describe the tradeoff facing the central bank in determining the optimal commitment policy. Theideaofusingpromisedvaluesaspseudo-statevariablestorecursifytheinfinite-horizon problem of the Ramsey planner was first suggested by Kydland and Prescott (1980) in the context of an optimal capital taxation problem. Later, Chang (1998) and Phelan and Stacchetti (2001) formally described, as an intermediate step toward characterizing sustainable policies, the recursive formulation of the Ramsey planner’s problem using promised marginal utility in models with money and with fiscal policy, respectively. However, because their focus was on characterizing sustainable policies, they did not solve for the Ramsey policy. To our knowledge, we are the first to formulate and solve the Ramsey policy using the promised value approach.1 Our aim is not to argue that readers should use the promised value approach instead of the Lagrange multiplier approach. Rather, our aim is to show that the promised value approach can be a useful analytical tool to supplement the analysis based on the standard Lagrange multiplier approach. Both approaches should be able to find the same allocation; we indeed find that both approaches reliably compute the optimal commitment policies in 1The only exception is a recent lecture note by Sargent and Stachurski (2018) which characterizes the Ramseypolicyinthelinear-quadraticversionofthemodelofCagan(1956)usingthepromisedvalueapproach. 2

the New Keynesian model. However, the Ramsey policies are often history-dependent in complex ways, and it is not always straightforward for researchers to understand the tradeofffacingthecentral bank. Accordingly, itisusefulforresearcherstohaveanalternativeway to analyze the Ramsey policy, as it may provide new insights on the optimal commitment policy. One difficulty associated with the promised value approach is that it requires researchers tocomputethesetoffeasiblepromisedvalues(seediscussioninMarcetandMarimon(2016)). We find that the extent to which this computation poses a challenge depends on the model. For the model with inflation bias and the model with stabilization bias, the promised rate of inflation is the only pseudo state variable, and we analytically show that the set of feasible promised inflation rates are identical to the set of feasible actual inflation rates—which is a primitive of the models—under nonrestrictive conditions. For the model with the ELB constraint, the set of feasible promised inflation-output pairs cannot be found analytically, and one needs a computationally nontrivial method described in Chang (1998) and Phelan and Stacchetti (2001), among others, to find the set. In our numerical example, we find that the set is large and does not represent a binding constraint for the control variables in the Bellman equation. Thus, if one wants to casually use the promised value approach, abstracting from the task of characterizing the set of feasible promises is unlikely to be harmful. InadditiontoChang(1998)andPhelanandStacchetti(2001)whorecursifiedtheRamsey planner’s problem using promised values, our paper is closely related to the large literature onoptimalpolicyinNewKeynesianmodels. Optimalcommitmentpoliciesinthemodelwith inflation bias and in the model with stabilization bias have been studied by many, including Clarida, Gali, and Gertler (1999), Gal´ı (2015), and Woodford (2003). Optimal commitment policy in the model with the ELB constraint has been studied by Eggertsson and Woodford (2003), Jung, Teranishi, and Watanabe (2005), Adam and Billi (2006), and Nakov (2008), among others. All of these papers—often implicitly but sometimes explicitly—rely on the method of Marcet and Marimon (2016) and use the lagged Lagrange multipliers as pseudostate variables to characterize optimal commitment policies; our contribution is to provide an alternative method to characterize them. The rest of the paper is organized as follows. Sections 2, 3, and 4 study the model with inflation bias, the model with stabilization bias, and the model with the ELB constraint, respectively. In each section, we first present the infinite-horizon problem of the Ramsey planner and describe how the infinite-horizon problem is made recursive under the promised value approach. We then discuss the dynamics of the Ramsey equilibrium, describe the key trade-off the central bank faces, and contrast the promised value approach with the standard Lagrange approach. Section 5 concludes. 3

2 Model with inflation bias Our first model is the one with inflation bias, which is a version of the standard New Keynesian model in which the inefficiency associated with monopolistic competition in the product market is not offset by a production subsidy. As the model is standard, we refer interested readers to Woodford (2003) and Gal´ı (2015) for more detailed descriptions. The economy starts at time one. The model is loglinearized around its deterministic steady state. Its private sector equilibrium conditions at time t are given by σy = σy +π −r +r∗, (1) t t+1 t+1 t π = κy +βπ , (2) t t t+1 where y , π , and r are the output gap, inflation, and the policy rate, respectively. σ, κ, β t t t are the inverse intertemporal elasticity of substitution, the slope of the Phillips curve, and the time discount rate, respectively. r∗ is the long-run natural rate of interest. Equations (1) and (2) are referred to as the Euler equation and the Phillips curve, respectively. In the model with inflation bias, we abstract from the ELB constraint on r . With this abstraction, t the Euler equation does not constrain the allocations the central bank can choose; it merely pins down the policy rate given the sequence of inflation and output. This abstraction is a common practice in the literature. We assume that y ∈ K and π ∈ K where K and K are closed intervals on the real t Y t Π Y Π line, R. For any variable x, let us denote {x }∞ by a bold font x. We say (y,π) (that is, t t=1 {y ,π }∞ ) is a competitive outcome if equation (2) is satisfied for all t ≥ 1, and use CE to t t t=1 denote the set of all competitive outcomes. The sequence of values, {V }∞ , associated with a competitive outcome, {y , π }∞ , is t t=1 t t t=1 given by ∞ (cid:88) V = βk−tu(y ,π ), t k k k=t where u(·,·), the payoff function, is given by 1 u(y,π) = − [π2+λ(y−y∗)2]. (3) 2 This quadratic payoff function can be derived as the second-order approximation to the household welfare.2 The presence of y∗ in this objective function captures the inefficiency associatedwithmonopolisticcompetitionintheproductmarket. TheproblemoftheRamsey planner is to choose a competitive outcome that maximizes the time-one value as follows: V = max V . (4) ram,1 1 (y,π)∈CE 2See, for example, Gal´ı (2015) for the derivation. 4

The Ramsey outcome is defined as the solution to this optimization problem and is denoted by {y , π }∞ . The value sequence associated with the Ramsey outcome is denoted ram,t ram,t t=1 by {V }∞ . ram,t t=1 2.1 Promised value approach Under the promised value approach, the infinite-horizon optimization problem of the Ramseyplannergivenbyequation(4)isdividedintotwosteps. Inthefirststep,thefollowing constrained infinite-horizon Ramsey problem is formulated: ∞ 1 (cid:88) (cid:104) (cid:105) w∗(η) = max − βt−1 π2+λ(y −y∗)2 , (y,π)∈Γ(η) 2 t t t=1 where Γ(η) is the set of competitive outcomes in which the initial inflation, π , is η. This 1 set is formally defined in Appendix A. In the second step, the Ramsey planner chooses the initial inflation promise, η, that maximizes w∗(η). That is, V = max w∗(η), (5) ram,1 η∈Ω where Ω is the set of time-one inflation rates consistent with the existence of a competitive outcome. This set is formally defined and computed analytically in Appendix A. By the standard dynamic programming argument, it can be shown that w∗(η) satisfies the following functional equation: w(η) = max u(y,π)+βw(η(cid:48)) (6) y∈K Y,π∈K Π,η(cid:48)∈Ω subject to π = η π = κy+βη(cid:48), where η is the promised rate of inflation for the current period from the previous period, and η(cid:48) is the promised rate of inflation for tomorrow. Conversely, if a bounded function, w : Ω → R, satisfies this functional equation, then w = w∗.3 Let {w (·), y (·), π (·), η(cid:48) (·)} be the value and policy functions associated with PV PV PV PV this Bellman equation. The Ramsey value sequence and the Ramsey outcome are obtained by iterating over these functions with the time-one inflation rate set to the argmax of w∗(η) in equation (5). 3The proof is closely related to the proof of the Bellman optimality principle. See Chang (1998). 5

2.2 Lagrange multiplier approach It is useful to contrast the recursive formulation of the promised value approach with that of the more standard Lagrange multiplier approach of Marcet and Marimon (2016). In the Lagrange multiplier approach, a saddle-point functional equation is used to recursify the infinite-horizon optimization problem of the Ramsey planner.4 In the model with inflation bias, it is given by W(φ) = min max f(y,π,φ,φ(cid:48))+βW(φ(cid:48)), (7) φ(cid:48) y∈K Y,π∈K Π where f(·), the modified payoff function, is given by f(y,π,φ,φ(cid:48)) = u(y,π)+φ(cid:48)(π−κy)−φπ. Let {y (·), π (·), φ(cid:48) (·)} be the policy functions associated with this saddle-point func- LM LM LM tional equation. One can find the Ramsey outcome by iterating over these policy functions with the initial Lagrange multiplier set to zero. 2.3 Analysis of optimal policy For both the Bellman equation of the promised value approach and the saddle-point functional equation of the Lagrange multiplier approach, the payoff function is quadratic and the constraints are linear. This linear-quadratic structure allows us to solve the model analytically for both approaches.5 However, to describe how the promised value approach works in a transparent way, we use a numerical example in the main text and relegate the closed-form solutions to Appendix D. The parameter values used in the numerical example are from Woodford (2003) and are shown in Table 1. Table 1: Parameters —Model with Inflation Bias— β y∗ λ κ 0.9925 0.01 0.003 0.024 Figure 1 shows the policy functions for the promised rate of inflation in the next period and the output gap in the current period as well as the value function associated with the 4Inmanypapers,authorscasuallyrefertothetheoryofMarcetandMarimon(2016)tojustifytherecursive characterizationoftheRamseypolicywithlaggedLagrangemultipliers,andthesaddle-pointfunctionequation is rarely explicitly formulated. For examples of papers explicitly formulating the saddle-point functional equation associated with the infinite-horizon optimization problem of the Ramsey planner in the context of sticky-pricemodels,seeKhan,King,andWolman(2003),AdamandBilli(2006),Svensson(2010),andNakata (2016). 5If the upper and lower bounds of the two closed intervals, K and K , are binding constraints, the Y Π problem is not linear-quadratic. We confirmed that they are not binding constraints in our model. 6

Bellman equation. Figure 1: Policy Functions from the Promised Value Approach —Model with Inflation Bias— '( ) y( ) w( ) 0.4 2 -0.18 0.2 1 -0.2 0 0 -0.22 -0.2 -1 -0.4 -2 -0.24 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 Note: η is the rate of inflation that was promised in the previous period and needs to be delivered in the currentperiod. η(cid:48) isthepromisedraetofinflationforthenextperiod. Theseratesareexpressedinannualized percent. w is the value associated with the Bellman equation (equation (6)). In the promised value approach, the initial inflation rate is given by the argmax of the value function associated with the Bellman equation—shown in the right panel of Figure 1. According to the panel, the initial inflation rate—indicated by the dashed vertical line—is slightly below 0.2 percent. Once the initial inflation rate is determined, the dynamics of the economy are sequentially pinned down by the policy functions linking the promised rate of inflation in the current period (η) to the promised rate of inflation in the next period (η(cid:48)) and output in the current period (y), shown in the left and middle panels, respectively. For example, the time-two inflation rate is determined by the policy function for the promised rate of inflation evaluated at the initial inflation rate and is shown by the pentagram in the left-panel. The time-one output is determined by the policy function for output evaluated at the initial rate of inflation and is shown by the pentagram in the middle-panel. The black dots in the policy functions trace the dynamics of the economy afterward. The implied dynamics of the economy are shown in Figure 2. The central bank has an incentive to generate a positive inflation rate at time one, which is associated with a level of output gap that is above zero but below y∗. Inflation and output converge eventually to zero, a well-known feature of the optimal commitment policy in this model.6 Tounderstandthetrade-offassociatedwiththeBellmanequation(equation(6)),weshow in Figure 3 the objective function to be maximized and its two subcomponents—today’s 6In Appendix G, we contrast the Ramsey equilibrium to the Markov perfect equilibrium and the valuemaximizing pair of inflation and output. 7

Figure 2: Dynamics —Model with Inflation Bias— y w 0.2 0.8 -0.197 0.15 0.6 -0.198 0.1 0.4 -0.199 0.05 0.2 0 0 -0.2 1 10 20 1 10 20 1 10 20 time time time Note: The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. payoff, u(·,·), and the discounted continuation value, βw(·)—at t = 20 when the economy has essentially converged to its steady state of zero inflation so that η = 0. Note that two arguments for the payoff function, inflation and output, are functions of η and η(cid:48). Thus, the payoff function u(π,y) can be transformed to an indirect payoff function, u∗(η,η(cid:48)). Figure 3: Trade-off under the Promised Value Approach —Model with Inflation Bias— w( ') at =0 u*( ') at =0 w'( ') at =0 -0.15 0 -0.18 -0.2 -0.02 -0.2 -0.25 -0.04 -0.22 -0.3 -0.06 -0.24 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 ' ' ' Note: η is the rate of inflation that was promised in the previous period and needs to be delivered in the currentperiod. η(cid:48) isthepromisedrateofinflationforthenextperiod. Theseratesareexpressedinannualized percent. Thefactthatinflationiszeroatthesteadystateiscapturedbythefactthattheobjective 8

function evaluated at η = 0, shown by the left panel, is maximized at η(cid:48) = 0. The optimality of promising zero inflation in the next period when the promised inflation rate for the current period is zero reflects two competing forces. The first force is how the promised inflation rate affects today’s payoff. Given that the central bank needs to deliver zero inflation today, the lower the promised inflation rate is for next period, the higher the output today has to be in order to satisfy the Phillips curve.7 Because of the presence of y∗ in the payoff function, a higher output (a lower inflation) means a higher payoff as long as output is below y∗. Thus, the central bank has an incentive to promise some deflation next period, as captured by the middle panel of Figure 3 which shows that today’s utility is maximized at η(cid:48) < 0. The second force is how the promised inflation affects the discounted continuation value. As shown in the right panel of Figure 3, a higher promised inflation rate is associated with a higher continuation value up to a certain point, as a higher future inflation is associated with a higher future level of output that is closer to y∗. The optimality of promising a zero inflation rate reflects these two competing effects of adjusting the inflation promise on the today’s payoff and on the discounted continuation value. Figure 4: Policy Functions from the Lagrange-Multiplier Approach —Model with Inflation Bias— ( ) y( ) ( ) -1 -1 10 -3 -1 0.6 2 2 0.4 1 1 0.2 0 0 0 -0.2 -1 -1 -2 0 2 -2 0 2 -2 0 2 -3 -3 -3 10 10 10 -1 -1 -1 Note: φ is the lagged Lagrange multiplier, whereas φ is the Lagrange multplier in the current period. The −1 rate of inflation is expressed in annualized percent. The output gap is expressed in percent. We will close the section by examining the policy functions from the standard Lagrange multiplier approach. Figure 4 shows the policy functions for inflation, output, and the Lagrange multiplier associated with the saddle-point functional equation (7). Unlike in the promised value approach, these functions are functions of the lagged Lagrange multiplier, φ . Time-one allocations are given by the policy functions evaluated at the initial lagged −1 Lagrange multiplier of zero and are indicated by the pentagram. The black dots trace the 7To see this, set η=0 in the two constraints in the Bellman equation (6). 9

dynamics of inflation, output, and the Lagrange multiplier after the first period. According to the right panel, the Lagrange multiplier eventually converges to a positive value. As the Lagrange multiplier converges, inflation and output also converge to zero, as shown in the left and middle panels, respectively. The dynamics of inflation and output derived from the Lagrange multiplier approach are of course identical to those implied by the promised value approachshowninFigure2. InAppendixD,weprovideanalyticalprooffortheirequivalence. 3 Model with stabilization bias Our second model is the model with stabilization bias. The private sector equilibrium conditions in this model at time t are given by σy (st) = σE y (st+1)+E π (st+1)−r (st)+r∗, t t t+1 t t+1 t π (st) = κy (st)+βE π (st+1)+s . t t t t+1 t The key difference between this model and the model in the previous section is that in this model,thereisacost-pushshock,denotedbys ,thatadditivelyentersintothePhillipscurve. t The cost-push shock follows an N-state Markov process and its possible values are given by the set, S := {e ,e ,...,e }. The probability of moving from state i to state j is denoted by 1 2 N p(e |e ). st denotes the history of shocks up to time t. That is, st := {s }t . Because there j i h h=1 is uncertainty, the allocations are state-contingent and depend on st. As in the model with inflation bias and consistent with common practice in the literature on stabilization bias, we abstract from the ELB constraint on the policy rate, which in turn allows us to abstract from the Euler equation. We assume that y ∈ K and π ∈ K , where t Y t Π K and K are closed intervals on the real line, R. For any variable x, let us denote its Y Π state-contingent sequence {x (st)}∞ by x (bold font) and its state-contingent sequence with t t=1 the time-one state s = s by x(s). We say (y,π) is a competitive outcome if the Philips 1 curve is satisfied for all t ≥ 1. We use CE to denote the set of all competitive outcomes and use CE(s) to denote the set of competitive outcomes in which the initial state s is s. 1 The sequence of values {V (st)}∞ associated with a competitive outcome is given by t t=1 ∞ (cid:88) (cid:88) V (st) = βk−t µ(sk|st)u(y (sk),π (sk)), t k k k=t sk|st where µ(sk|st) is the conditional probability of observing sk after observing st. The payoff function, u(·,·), is given by 1 u(y,π) = − [π2+λy2]. (8) 2 The Ramsey problem is to choose the state-contingent sequences of inflation and output to 10

maximize the time-one value for each s ∈ S. That is, V (s) = max V (s). ram,1 1 (y(s),π(s))∈CE(s) The Ramsey outcome is defined as the state-contingent sequences of inflation and output that solve this optimization problem and is denoted by {y (st), π (st)}∞ . The value ram,t ram,t t=1 sequence associated with the Ramsey outcome is denoted by {V (st)}∞ . ram,t t=1 3.1 Promised value approach As in the model with inflation bias, the infinite-horizon optimization problem of the Ramseyplannerisdividedintotwostages. Inthefirststage,theconstrainedRamseyproblem is formulated as follows: ∞ w∗(η,s) = max − 1 (cid:88) βt−1 (cid:88) µ(st|s = s) (cid:2) π2+λy2(cid:3) , (y(s),π(s))∈Γ(η,s) 2 1 t t t=1 st|s1=s where Γ(η,s) is the set of competitive outcomes with the initial state s = s in which the 1 initial inflation is η. This set is formally defined in Appendix B. In the second stage, the Ramsey planner chooses the initial inflation to maximize w∗(η,s): V (s) = max w∗(η,s), (9) ram,1 η∈Ω(s) where Ω(s) is the set of time-one inflation rates consistent with the existence of a competitive outcome with the initial state s = s. This set is formally defined and computed analytically 1 in Appendix B. The Bellman equation associated with the first-stage constrained Ramsey problem is given by N (cid:88) w(η ,e ) = max u(y,π)+β p(e |e )w(η(cid:48),e ) (10) i i j i j j y∈K Y,π∈K Π,{η j (cid:48)∈Ωj}N j=1 j=1 subject to π = η i N (cid:88) π = κy+β p(e |e )η(cid:48) +e , j i j i j=1 where we are now explicit about the specifics of the shock (recall that S := {e ,e ,...,e }). 1 2 N Note that the control variables include η(cid:48) for each j ∈ {1,2,...,N}. j Let {w (·), y (·), π (·), {η(cid:48) (·)}N } be the value and policy function associated PV PV PV PV,j t=1 with the Bellman equation. Note that there are N promised inflation rates that have to be 11

chosen.8 The Ramsey value sequence and the Ramsey outcome can be obtained by iterating over these functions with the time-one inflation set to the argmax of w∗(η,s) in equation (9). 3.2 Lagrange multiplier approach Thesaddle-pointfunctionalequationassociatedwiththeRamseyplanner’sproblemabove is given by N (cid:88) W(φ,e ) = min max f(y,π,φ,φ(cid:48),e )+β p(e |e )W(cid:48)(φ(cid:48),e ) i i j i j φ(cid:48) y∈K Y,π∈K Π j=1 where f(·), the modified payoff function, is given by f(y,π,φ,φ(cid:48),e ) = u(y,π)+φ(cid:48)(π−κy−e )−φπ. i i Let {y (·), π (·), φ(cid:48) (·)} be the policy functions associated with this saddle-point func- LM LM LM tional equation. One can find the Ramsey outcome by iterating over these policy functions with the initial Lagrange multiplier set to zero. 3.3 Analysis of optimal policy Given the linear-quadratic structure of the model, the solutions to the Bellman equation from the promised value approach and the saddle-point functional equation can be obtained analytically.9 However, we will again use a numerical example to illustrate the mechanics of the promised value approach in a transparent way; the analytical results are provided in Appendix E. To make the exposition as transparent as possible, we will assume that (i) there are only two states (high and normal), (ii) the economy starts in the high state, (iii) the economy will move to the normal state with certainty in period 2, and (iv) the normal state is absorbing. In the remainder of this section, we will use the notation e and e , instead of h n e and e , to refer to the high and normal states, respectively. Parameter values are shown 1 2 in Table 2. The values for the parameter governing the private sector behavior are the same as in the previous section. Table 2: Parameters and Transition Probabilities —Model with Stabilization Bias— β λ κ e e p(e |e ) p(e |e ) p(e |e ) p(e |e ) h n h h n h h n n n 0.9925 0.003 0.024 0.001 0 0 1 0 1 8As a result, the larger the number of exogenous states is, the larger the number of policy functions to solve for is. However, because an increase in the number of exogenous states does not affect the state space, it does not necessarily lead to increased computational burden. 9As in the model with inflation bias, we confirm that the upper and lower bounds implied by the closed intervals on choice variables are not binding in equilibrium. 12

Figure 5: Policy Functions from the Promised Value Approach —Model with Stabilization Bias— '( ,e ) y( ,e ) w( ,e ) h h h 0 0 0 -0.01 -0.2 -1 -0.02 -0.4 -2 -0.03 -0.6 -3 -0.04 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 '( ,e ) y( ,e ) w( ,e ) n n 10 -3 n 0.2 2 5 0.1 1 0 0 0 -5 -0.1 -1 -0.2 -2 -10 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 Note: η is the rate of inflation that was promised in the previous period and needs to be delivered in the currentperiod. η(cid:48) isthepromisedrateofinflationforthenextperiod. Theseratesareexpressedinannualized percent. w is the value associated with the Bellman equation (equation (10)). Figure 5 shows the policy functions for the promised inflation rate in the next period and output in the current period as well as the value function associated with the Bellman equation of the promised value approach. The top and bottom panels are for the high state and the normal state, respectively. The initial inflation rate is given by the argmax of the value function from the high state, shown by the top-right panel of the figure. The initial inflation—indicated by the dashed vertical line—is about 0.25 percent. Once the time-one inflation rate is determined, the timetwo inflation rate (π ) and the time-one output (y ) are determined by the high-state policy 2 1 functions shown in the top-left and top-middle panels, respectively. Subsequent sequences of 13

inflation and output—shown by the black dots—are determined by the normal-state policy functions shown in the bottom panels. Figure 6 shows the implied dynamics of inflation, output, and the value. A well-known feature of the optimal commitment policy in the model with stabilization bias is that, in the initial period, the central bank promises to undershoot its inflation target once the shock disappears. Relative to the equilibrium under the Markov perfect policy—shown by the dashedlines—inwhichthecentralbankdoesnothaveacommitmenttechnology,suchpromise of undershooting improves the trade-off between inflation and output stabilization at t = 1 through expectations when the economy is buffeted by the cost-push shock, allowing the central bank to achieve a higher period-one value.10 The undershooting of inflation and output will fade gradually, and inflation and output will eventually converge to zero. Figure 6: Dynamics —Model with Stabilization Bias— y w 0.4 0.2 0.002 Ramsey 0.3 Markov perfect 0 0 0.2 -0.2 -0.002 0.1 -0.4 0 -0.6 -0.004 -0.1 -0.8 1 5 10 1 5 10 1 5 10 time time time Note: The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. To understand the trade-off the central bank faces in choosing to create deflation in the second period, we show in Figure 7 the objective function associated with the Bellman equation—shown in the left-panel—and its two subcomponents—shown in the middle and right panels—at time one when the cost-push shock is present. Consistent with Figures 5 and 6, the value of the objective function, w(·), is maximized at η(cid:48) < 0. To understand why some deflation is optimal, we need to examine how η(cid:48) affects today’s payoff as well as the discounted continuation value, shown in the middle and right panels of Figure 7, respectively. On the one hand, because the inflation rate in the current period has been chosen in the previous period, today’s payoff is maximized when today’s output is zero. Conditional on the initial promised inflation rate of η = η = 0.26/400 and the cost-push shock of 1 10Appendix H formulates the optimization of the discretionary central bank and solves for the Markov perfect policy. 14

Figure 7: Trade-off under the Promised Value Approach —Model with Stabilization Bias— w( ') at u( ') at w'( ') at = = = 1 1 10 -3 1 0 0 0 -0.01 -2 -0.02 -0.02 -4 -0.04 -0.03 -6 -0.06 -0.04 -8 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 ' ' ' Note: η is the rate of inflation that was promised in the previous period and needs to be delivered in the currentperiod. η(cid:48) isthepromisedrateofinflationforthenextperiod. Theseratesareexpressedinannualized percent. e = e = 0.1/100, the Phillips curve implies that zero output today is achieved only by h promising some deflation for the next period, as indicated by the dashed vertical line in the middle panel of Figure 7. On the other hand, the discounted continuation value is maximized if the central bank chooses to promise zero inflation for the next period (η(cid:48) = 0), as shown by the dashed vertical line in the right panel of Figure 7. The promised inflation rate of zero maximizes the discounted continuation value because it is associated with fully stabilized paths of inflation and output in the future and thus with the highest possible continuation value of zero. The optimal rate of promised inflation—indicated by the solid vertical line— balances these two forces. All told, the overall objective function—shown in the left-panel of Figure 7—is maximized at η(cid:48) < 0. Finally, Figure 8 shows the policy functions for inflation, output, and the Lagrange multiplier associated with the saddle-point functional equation (3.2) of the Lagrange multiplier approach. The top and bottom panels are for the high state and the normal state, respectively. The time-one inflation, output, and Lagrange multiplier are indicated by the pentagrams in the top panels, which are the high-state policy functions evaluated at the initial lagged Lagrange multiplier of zero. Thereafter, the dynamics of the economy are governed by the normal-state policy functions shown in the bottom panels—because the cost-push shock is assumed to disappear after the first period—and are traced by the black dots. In Appendix E, we analytically verify the equivalence of the dynamics of the economy obtained from the promised value and Lagrange multiplier approaches. 15

Figure 8: Policy Functions from the Lagrange-Multiplier Approach —Model with Stabilization Bias— ( ,e ) -1 h ( ,e ) y( ,e ) -1 h -1 h 10 -3 1 1 2 0.5 0 1 0 -1 0 -0.5 -2 -1 -2 0 2 -2 0 2 -2 0 2 -3 -3 -3 10 10 10 -1 -1 -1 ( ,e ) -1 n ( ,e ) y( ,e ) -1 n -1 n 10 -3 0.4 2 2 0.2 1 1 0 0 0 -0.2 -1 -1 -0.4 -2 -2 -2 0 2 -2 0 2 -2 0 2 -3 -3 -3 10 10 10 -1 -1 -1 Note: φ isthelaggedLagrangemultiplier,whereasφistheLagrangemultiplierinthecurrentperiod. The −1 rate of inflation is expressed in annualized percent. The output gap is expressed in percent. 4 Model with the ELB Our final model features the ELB constraint on nominal interest rates and a natural rate shock. The private sector equilibrium conditions at time t are given by σy (st) = σE y (st+1)+E π (st+1)−r (st)+r∗+s (11) t t t+1 t t+1 t t π (st) = κy (st)+βE π (st+1) (12) t t t t+1 r ≥ r (13) t ELB 16

where s is a natural rate shock following a N-state Markov process and r is the ELB t ELB constraint on the policy rate. Possible values of the demand shock are given by the set, S := {δ ,δ ,...,δ }. The probability of moving from state i to state j is denoted by p(δ |δ ). 1 2 n j i st denotes the history of shocks up to time t. That is, st := {s }t . Because there is h h=1 uncertainty, the allocations are state-contingent and depend on st. We assume that y ∈ K and π ∈ K where K and K are closed intervals on the real t Y t Π Y Π line, R. We also assume that r ∈ K := [r ,r ], where r is set to r to respect t R min max min ELB the ELB constraint (13). For any variable x, let us denote its state-contingent sequence {x (st)}∞ by a bold font x and its state-contingent sequence with the time-one state s = s t t=1 1 by x(s). We say (y,π,r) is a competitive outcome if equations (11) and (12) are satisfied for all t ≥ 1. We use CE to denote the set of all competitive outcomes and use CE(s) to denote the set of competitive outcomes in which the initial state s is s. 1 The value sequence, {V (st)}∞ associated with a competitive outcome is given by t t=1 ∞ (cid:88) (cid:88) V (st) = βk−t µ(sk|st)u(y (sk),π (sk)), t k k k=t sk|st where µ(sk|st) is the conditional probability of observing sk after observing st. The payoff function, u(·,·), is given by equation (8) from the previous section. The Ramsey planner’s problem is to choose the state-contingent sequences of inflation and output to maximize the time-one value for each s ∈ E. That is, V (s) = max V (s). (14) ram,1 1 (y(s),π(s),r(s))∈CE(s) TheRamseyoutcomeisdefinedbythesolutiontothisproblemandisdenotedby{y (st), ram,t π (st),r (st)}∞ . ThevaluesequenceassociatedwiththeRamseyoutcomeisdenoted ram,t ram,t t=1 by {V (st)}∞ . ram,t t=1 4.1 Promised value approach As in the previous two models, the infinite-horizon optimization problem of the Ramsey planner is divided into two stages. In the first stage, the constrained Ramsey problem is formulated as follows: ∞ w∗(η ,η ,s) = max − 1 (cid:88) βt−1 (cid:88) µ(st|s = s) (cid:2) π (st)2+λy (st)2(cid:3) , 1 2 1 t t (y(s),π(s),r(s))∈Γ(η1,η2,s) 2 t=1 st|s1=s where Γ(η ,η ,s) is the set of competitive outcomes with the initial state s = s in which 1 2 1 the initial output and inflation are η and η , respectively. This set is more formally defined 1 2 in Appendix C. In the second stage, the Ramsey planner chooses the initial inflation and 17

output promises that maximize w∗(η ,η ,s): 1 2 wr(s) = max w∗(η ,η ,s), (15) 1 2 (η1,η2)∈Ω(s) where Ω(s) is the set of time-one inflation rates consistent with the existence of a competitive outcome with the initial state s = s. This set is formally defined and computed numerically 1 in Appendix C. The Bellman equation associated with the first-stage constrained Ramsey problem is given by n (cid:88) w(η ,η ,δ ) = max u(y,π)+β p(δ |δ )w(η(cid:48) ,η(cid:48) ,δ ) (16) 1,i 2,i i j i 1,j 2,j j y∈K Y,π∈K Π,r∈K R,{(η 1 (cid:48) ,j ,η 2 (cid:48) ,j )∈Ωj}N j=1 j=1 subject to y = η 1,i π = η 2,i N (cid:88) σy = p(δ |δ )[ση(cid:48) +η(cid:48) ]−r+δ j i 1,j 2,j i j=1 N (cid:88) π = κy+β p(δ |δ )η(cid:48) , j i 2,j j=1 wherewearenowexplicitaboutthespecificsoftheshock(recallthatS := {δ ,δ ,...,δ }). Let 1 2 n {w (·), y (·), π (·), r (·), {η(cid:48) (·), η(cid:48) (·)}N } be the value and policy functions PV PV PV PV 1,j,PV 2,j,PV j=1 associated with this Bellman equation. Note that there are N promises for both inflation and output that have to be chosen. The Ramsey value sequence and the Ramsey outcome can be obtained by iterating over the policy functions for inflation, output, and the policy rate found in the first step with time-one output and inflation set to the argmax of w∗(η ,η ,s) 1 2 in equation (15). 4.2 Lagrange multiplier approach The saddle-point functional equation associated with the infinite-horizon optimization problem of the Ramsey planner above (equation (14)) is given by N (cid:88) W(φ ,φ ,δ ) = min max f(y,π,r,φ ,φ(cid:48),φ ,φ(cid:48),δ )+β p(δ |δ )W(φ(cid:48),φ(cid:48),δ ), 1 2 i 1 1 2 2 i j i 1 2 j φ(cid:48) 1 ,φ(cid:48) 2 y∈K Y,π∈K Π,r∈K R j=1 (17) 18

where f(·), the modified payoff function, is given by f(y,π,r,φ ,φ(cid:48),φ ,φ(cid:48),δ ) 1 1 2 2 i 1 =u(y,π)+φ(cid:48)(r+σy+δ )− (σy+π)+φ(cid:48)(π−κy)−φ π. 1 i β 2 2 Let {y (·), π (·), r (·), φ(cid:48) (·), φ(cid:48) (·)} be the policy functions associated with this LM LM LM 1,LM 2,LM saddle-point functional equation. As in the previous two models, one can find the Ramsey valueandoutcomebyiteratingoverthesepolicyfunctionswiththeinitialLagrangemultiplier set to zero. 4.3 Analysis of optimal policy Unlike the first two models, the model with the ELB constraint cannot be solved analyticallyundereitherapproach. Thus, wesolvethemodelnumerically. Thesolutionmethodsare standard and their details are described in Appendix F. As in the model with stabilization bias, we will simplify the shock structure in order to describe the mechanics of the promised value approach in a transparent way. In particular, we assume that (i) there are only two states (crisis and normal), (ii) the economy starts in the crisis state, (iii) the economy will move to the normal state with certainty in the second period, and (iv) the normal state is absorbing. In the remainder of the section, we will use the notation δ and δ , instead of δ c n 1 and δ , to refer to the crisis and normal states, respectively. Parameter values are shown in 2 Table 3. The values for the parameters governing the private sector behavior are the same as those in the previous sections. Table 3: Parameters and Transition Probabilities —Model with ELB— β λ κ r δ δ p(δ |δ ) p(δ |δ ) p(δ |δ ) p(δ |δ ) ELB c n c c n c c n n n 0.9925 0.003 0.024 0 0.001 0 0 1 0 1 Figure 9 shows the policy functions for the promised inflation and output in the next period as well as the value function associated with the Bellman equation from the promised value approach, while Figure 10 shows the dynamics of inflation, output, the policy rate, and the value implied by these functions. The pair of the initial inflation rate and output is given by the argmax of the crisis-state valuefunction—showninthetop-rightpanelofFigure9—andisindicatedbythesolidvertical line. Theinitialinflationrateandoutputareaboutminus0.01percentandminus0.6percent, respectively. Once the initial inflation rate and output are determined, the dynamics of the economy are governed by the normal-state policy functions linking the promised inflation rate and output today to the promised inflation rate and output next period, shown by the 19

Figure 9: Policy Functions from the Promised Value Approach —Model with ELB— ' ( , , ) ' ( , , ) w( , , ) 1 1 2 c 2 1 2 c 1 2 c 0.1 2 0 0 1 -0.01 -0.1 -0.2 0 -0.02 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.4 -0.1 -0.4 -0.1 -0.4 -0.8 -0.8 -0.8 2 1 2 1 2 1 ' ( , , ) ' ( , , ) w( , , ) 1 1 2 n 2 1 2 n 1 2 n 10-3 0.1 0.5 0 0 -2 0 -0.1 -4 -0.2 -0.5 -6 0.05 0.05 0.05 0.8 0.8 0.8 0 0.4 0 0.4 0 0.4 -0.05 0 -0.05 0 -0.05 0 2 1 2 1 2 1 Note: η andη aretheoutputgapandtherateofinflation,respectively,thatwerepromisedintheprevious 1 2 periodandneedtobedeliveredinthecurrentperiod. η(cid:48) andη(cid:48) arethepromisedoutputgapandthepromised 1 2 rateofinflation,respectively,forthenextperiod. wisthevalueassociatedwiththeBellmanequation(equation (16)). The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. bottom panels. The dots in the policy functions trace the dynamics of the economy. The economy’s dynamics are shown in Figure 10. The key feature of the optimal commitment policy in the model with the ELB constraint is that in the initial period, the central bank promises to overshooting inflation and output once the crisis shock disappears in the second period—a feature well-known in the literature (Eggertsson and Woodford (2003); Jung, Teranishi, and Watanabe (2005); Adam and Billi (2006)). The overshooting commitment mitigates the declines in inflation and output at the ELB via expectations. After the second period, inflation and output gradually approach to their steady state values of zero. Note that, under the optimal discretionary policy— shown by the dashed lines—there is no overshooting in the aftermath of the crisis shock, and the declines in inflation and output are larger during the crisis than under the optimal commitment policy.11 11WeformulatetheoptimizationproblemofthediscretionarycentralbankandsolvefortheMarkovperfect 20

Figure 10: Dynamics —Model with ELB— y 0.05 1 0 0 -0.05 -1 Ramsey -0.1 Markov perfect -0.15 -2 2 4 6 8 10 2 4 6 8 10 time time r w 4 0.001 3 0 2 -0.001 1 -0.002 0 -0.003 2 4 6 8 10 2 4 6 8 10 time time Note: The rate of inflation and the policy rate are expressed in annualized percent. The output gap is expressed in percent. Figure 11 shows the trade-off associated with the Bellman equation, given by equation 16, in the first period when the economy is in the crisis state today but is expected to return to the normal state in the next period. Given the initial inflation rate and output the central bank has to deliver today (η and η ), the Phillips curve pins down the promised inflation 1,c 2,c in the next period. Thus, the only control variable available for the central bank to adjust is the promised output for the next period, η(cid:48). The left panel shows how the overall objective 1 function varies with the promised output, whereas the middle and right panels show how the two subcomponents of the overall objective function, today’s payoff and the discounted continuation value, vary with the promised output. Asshowninthemiddlepanel,today’spayoffisconstant,asitdependsonlyonthecurrentperiod inflation rate and output that were promised in the previous period and thus does not depend on the promised output for the next period. Thus, what maximizes the discounted continuationvalue—shownintherightpanel—alsomaximizestheoverallobjectivefunction— equilibrium in Appendix I. 21

Figure 11: Trade-off under the Promised Value Approach (Time-One Value) —Model with ELB— w( ' ) at u( ' ) at w'( ' ) at 1 1 1 ( , , )=( , , ) ( , , )=( , , ) ( , , )=( , , ) 1 2 1,1 2,1 c 1 2 1,1 2,1 c 1 2 1,1 2,1 c 10-3 10-3 0 1 0 0.5 -1 -1 0 -2 -2 -0.5 -3 -1 -3 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 ' ' ' 1 1 1 Note: η(cid:48) and η(cid:48) are the promised output gap and the promised rate of inflation for the next period. Grey 1 2 shades indicate the range of η(cid:48) consistent with negative nominal interest rates. 1 shown in the left panel. As indicated by the solid vertical line in that panel, the discounted continuation value is maximized at around η(cid:48) = 0.2, meaning that it is optimal to promise 1 an output overshoot. Theoptimalityofapositivetime-twooutputforthediscountedcontinuationvaluereflects the following two competing forces. On the one hand, because the time-two inflation is given—it is implied by the time-one inflation and output, as discussed above—promising the time-two inflation rate of zero maximizes the time-two payoff, as shown in the middle panel of Figure 12. On the other hand, because the time-two inflation rate is positive, a promise of zero time-two output means that the time-three inflation has to be positive and even slightly higher than the time-two inflation because of the time-two Phillips curve constraint.12 By promisingahigheroutputfortimetwo, thecentralbankensuresthatthetime-threeinflation is closer to zero, which is desirable because it is associated with a higher value, as shown by the right panel of Figure 12. The optimality of promising a positive time-two output is the outcome of the intertemporal trade-off between these two forces. While η(cid:48) = 0.2 maximizes the time-one value, this is not the level of output the central 1 bank ends up promising for t = 2 because of the ELB constraint on the policy rate. The ELB constraint on the policy rate puts a lower bound on the promised output the central bank can choose due to the Euler equation; given today’s output and the rate of inflation in the next period, the policy rate needs to be sufficiently low in order to support a low level of output in the next period. In Figure 12, any promised output below the dashed vertical line is associated with a negative policy rate in the current period. The maximum is attained 12With y =0, the time-two Phillips curve implies that π =π /β. 2 3 2 22

Figure 12: Trade-off under the Promised Value Approach (Time-Two Value) —Model with ELB— w'( ' ) at u'( ' ) at w''( ' ) at 1 1 1 ( , )=( , ) ( , )=( , ) ( , )=( , ) 1 2 1,1 2,1 1 2 1,1 2,1 1 2 1,1 2,1 10-3 10-3 10-4 0 0 0 -0.5 -2 -1 -1 -4 -2 -1.5 -6 -3 -2 -8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 ' ' ' 1 1 1 Note: η(cid:48) and η(cid:48) are the promised output gap and the promised rate of inflation, respectively, for the next 1 2 period. Grey shades indicate the range of η(cid:48) consistent with negative nominal interest rates. 1 when the promised output is at its lower bound and the policy rate is zero. Figure 13: Policy Functions from the Lagrange-Multiplier Approach —Model with ELB— ( , ) y( , ) ( , ) ( , ) 1,-1 2,-1 1,-1 2,-1 1 1,-1 2,-1 2 1,-1 2,-1 0.05 -0.4 0 10 0 -0.6 -2 0 -0.05 -0.8 -4 -10 -2 -2 -2 -2 -1 5 -1 5 -1 5 -1 5 0 0 0 0 0 -5 0 -5 0 -5 0 -5 ( , ) y( , ) ( , ) ( , ) 1,-1 2,-1 1,-1 2,-1 1 1,-1 2,-1 2 1,-1 2,-1 0.1 1 1 20 0 0 0 0 -0.1 -1 -1 -20 -2 -2 -2 -2 -1 5 -1 5 -1 5 -1 5 0 0 0 0 0 -5 0 -5 0 -5 0 -5 Note: φ and φ are the lagged Lagrange multipliers, whereas φ and φ are the Lagrange multipliers 1,−1 2,−1 1 2 in the current period. The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. Turning to the Lagrange multiplier approach, we show in Figure 13 the policy functions and value function associated with the saddle-point functional equation (17). The time-one 23

inflation, output, and Lagrange multipliers are given by the crisis-state policy functions— shown in the top panels—evaluated at (φ ,φ ) = (0,0) and are indicated by the penta- 1,−1 2,−1 grams. Thereafter, the dynamics of the economy are determined by the normal-state policy functions shown in the bottom panels and are traced by the black dots. The dynamics of inflation and output derived from the Lagrange multiplier approach are identical to those implied by the promised value approach, up to the accuracy of the numerical methods used. In Appendix F, we contrast the dynamics obtained from the promised value and Lagrange multiplierapproachesandshowthatthedifferencesareofamagnitudeinlinewiththenumerical errors associated with the global solution methods. 5 Conclusion In this paper, we characterized optimal commitment policies in three well-known versions of the New Keynesian model using a novel recursive approach—which we called the promised value approach—inspired by Kydland and Prescott (1980). Under the promised value approach, promised inflation and output act as pseudo state variables, as opposed to thelaggedLagrangemultipliersunderthestandardapproachofMarcetandMarimon(2016). The Bellman equation from the promised value approach sheds new light on the trade-off facing the central bank and provides fresh perspectives on optimal commitment policies. The promised value approach can serve as a useful analytical tool for those economists interested in analyzing optimal monetary policy. References Adam, K., and R. Billi (2006): “Optimal Monetary Policy Under Commitment with a Zero Bound on Nominal Interest Rates,” Journal of Money, Credit, and Banking, 38(7), 1877–1905. Brayton, F., T. Laubach, and D. Reifschneider (2014): “Optimal-Control Monetary Policy in the FRB/US Model,” FEDS Notes 2014-11-21, Board of Governors of the Federal Reserve System (U.S.). Bullard, J. (2013): “Monetary Policy in a Low Policy Rate Environment,” OMFIF Golden Series Lecture, London, United Kingdom. Cagan, P. (1956): “The Monetary Dynamics of Hyperinflation,” Studies in the Quantity Theory of Money, pp. 25–117. Chang, R. (1998): “Credible Monetary Policies in an Infinite Horizon Model: Recursive Approaches,” Journal of Economic Theory, 81, 431–461. 24

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Technical Appendix for Online Publication This technical appendix is organized as follows: • Sections A, B, and C describe the technical details of the promised value approach for the three models considered in the paper. • Sections D and E present analytical results for the model with inflation bias and the model with stabilization bias, respectively. • Section F describes the global solution methods used to solve the model with the effective lower bound (ELB) constraint and presents the accuracy of the solution. • Section G presents a few additional results for the model with inflation bias. • Sections H and I characterize Markov perfect equilibria in models with stabilization bias and with the ELB constraint on nominal interest rates, respectively. A Details of the promised value approach for the model with inflation bias Notations closely follow Chang (1998). For any variable, x , let x ≡ {x }∞ . (y,π) is t t t=1 said to be a competitive outcome if, for all t ≥ 1, y ∈ K , π ∈ K , and t Y t Π π = κy +βπ . t t t+1 Let CE denote the set of all competitive outcomes. That is, CE ≡ {(y,π)|(y,π)is a competitive outcome}. The sequence of values, {V }∞ , associated with a competitive outcome, {y , π }∞ , is given t t=1 t t t=1 by ∞ (cid:88) V = βk−tu(y ,π ) t k k k=t where u(·,·), the payoff function, is given by 1 u(y,π) = − [π2+λ(y−y∗)2] 2 The Ramsey problem is to choose a competitive outcome that maximizes the time-one value: V = max V . (18) ram,1 1 (y,π)∈CE A.1 Recursive formulation A recursive treatment of the Ramsey problem entails the use of promised inflation made in period t given by η = π . Hence, η is period t+1’s inflation rate that is promised t+1 t+1 t+1 by the equilibrium in period t. 27

Let Ω denote the set of all possible initial inflation promises that are consistent with a competitive outcome. That is, Ω ≡ {η ∈ R|η = π for some (y,π) ∈ CE}. 1 Let Γ(η) be the set of all possible competitive outcomes whose initial promise is given by η. That is, Γ(η) ≡ {(y,π) ∈ CE|π = η}. 1 Under the promised value approach, the problem of the Ramsey planner is divided into two steps. In the first step, the following constrained problem is formulated for all η ∈ Ω. ∞ 1 (cid:88) (cid:104) (cid:105) w∗(η) = max − βt−1 π2+λ(y −y∗)2 (y,π)∈Γ(η) 2 t t t=1 Inthesecondstep,theRamseyplannerchoosestheinitialinflationpromise,η,thatmaximize w∗(η). V = max w∗(η). ram,1 η∈Ω By the standard dynamic programing argument, it can be shown that w∗(η) satisfies the following functional equation: w(η) = max − 1 [π2+λ(y−y∗)2]+βw (cid:0) η(cid:48)(cid:1) y,π,η(cid:48) 2 such that (y,π,η(cid:48)) ∈ K ×K ×Ω, y π π = η, and π = κy+βη(cid:48). Conversely, it can be also shown that, if a bounded function, w : Ω → R, satisfies the above functional equation, then w = w∗. Since K and K are primitives of the model and are known, the object of interest Π Y becomes Ω. To find Ω, we define an operator, B, as follows. For Q ∈ R, let (cid:110) (cid:111) B(Q) = η ∈ R|∃ (cid:0) y,π,η(cid:48)(cid:1) ∈ K ×K ×Q, where π = η and π = κy+βη(cid:48) .13 Y Π It can be shown that (i) Q ⊆ B(Q) ⇒ B(Q) ⊆ Ω (a.k.a self generation), (ii) Ω = B(Ω) (a.k.a. factorization), and (iii) letting Q = (cid:2) η,η (cid:3) and Q = B(Q ), Q ⊇ Q and 0 n n−1 n n+1 Ω = ∩∞ Q . As a result, in order to find Ω, one can start from a closed interval, apply the n=0 n operator until it converges. The converged set is Ω. See Chang (1998) for the proof. A.2 Computing Ω with the B operator We can show that, if K is sufficiently large, Ω = K . Y Π Proof: Let Q := K . From the properties of B-operator described above, we can prove 0 Π Ω = K by showing Q ⊆ B(Q ). Let K := [π ,π ]. Take K := [y ,y ] with Π 0 0 Π min max Y min max 13See Appendix A.2 28

y ≤ (1 − β)π /κ and y ≥ (1 − β)π /κ. Take η ∈ Q . We want to show that min min max max 0 η ∈ B(Q ). Consider y = (1−β)η/κ, π = η, and η(cid:48) = η. By construction, y ≤ y ≤ y , 0 min max π ∈ Q := K , and η(cid:48) ∈ Q . That is, (y,π,η(cid:48)) ∈ K ×K ×Q Thus, η ∈ B(Q ). 0 Π 0 Y Π 0 0 B Details of the promised value approach for the model with stabilization bias Let s denote the exogenous shock of the model at time t and let st denote the history of t shocks up to time t. For any variable, x, with range X, let us denote its corresponding state-contingent sequence by x ≡ (cid:8) x (cid:0) δt(cid:1)(cid:9)∞ . That is, x is a sequence of functions mapping a history of states t t=1 into X: x : S → X 1 and x : St → X. t Foranyvariable, x, withrangeX, x(s)representsastate-contingentsequencewiths = s 1 defined by a sequence of functions mapping a history of states with s = s into X: 1 x : s → X 1 and x : St → X. t A state-contingent sequence of inflation and output, (y,π), is said to be a competitive outcome if, ∀t ≥ 1 and ∀st ∈ St, y (st) ∈ K , π (st) ∈ K , and t Y t Π (cid:88) π (st) = κy (st)+β µ(st+1|st)π (st+1)+s . t t t+1 t st+1|st For each s ∈ S, let CE denote the set of all competitive outcomes. That is, CE ≡ {(y,π) |(y,π)is a competitive outcome}. For each s ∈ S, let CE(s) denote the set of all competitive outcomes with s = s. That is, 1 CE(s) ≡ {(y(s),π(s)) |(y(s),π(s))is a competitive outcomewiths = s}. 1 The Ramsey planner’s problem is to choose the competitive outcome that maximizes the time-one value: V (s) = max V (s ) ram,1 1 1 (y(s),π(s))∈CE(s) B.1 Recursive formulation A recursive treatment of the Ramsey problem entails the use of state-contingent promised value(s) made in period t given by η (u |u ) = π (s |s ) given s ∈ S and s ∈ S. t+1 t+1 t t+1 t+1 t t t+1 Hence, η (s |s ) is period t+1’s inflation rate, for some state s ∈ S, that is promised t+1 t+1 t t+1 by the equilibrium in period t for a given s ∈ S. From now on, we will denote these promised t variables by ηs(cid:48) , s(cid:48) ∈ S, s = s(cid:48). t+1 t+1 29

For any s ∈ S, denote the set of all possible initial inflation promises consistent with a competitive outcome by Ω(s). That is, Ω(s) ≡ {η ∈ R|η = π for some(y(s),π(s)) ∈ CE(s)}. 1 For any s ∈ S, denote the set of all possible competitive outcomes whose initial promise is η by Γ(η,s). That is, Γ(η,s) ≡ {(y(s),π(s)) ∈ CE(s)|π = η}. 1 For a given s ∈ S, the recursive formulation takes two steps. In the first step, the constrained Ramsey problem is formulated. ∞ w∗(η,s) = max − 1 (cid:88) βt−1 (cid:88) µ(st|s = s) (cid:2) π (st)2+λy (st)2(cid:3) 1 t t (y(s),π(s))∈Γ(η,s) 2 t=1 st|s1=s In the second step, the Ramsey planner chooses the initial promise to maximize w∗(η,s). V (s) = max w∗(η,s). ram,1 η∈Ω(s) Bythestandarddynamicprogrammingargument, itcanbeshownthat, foragivens ∈ S, w∗(η,s) satisfies the functional equation: w(η,s) = max − 1 [π2+λy2]+β (cid:88) p(s(cid:48)|s)w( (cid:110) ηs(cid:48) (cid:111) ,s(cid:48) ∈ S) y,π,{ηs(cid:48)} s(cid:48)∈S 2 s(cid:48)∈S (cid:16) (cid:110) (cid:111) (cid:17) such that y,π, ηs(cid:48) ∈ K ×K ×{Ω(s(cid:48))} , s(cid:48)∈S Y Π s(cid:48)∈S π = η, and π = κy+β (cid:88) p(s(cid:48)|s)ηs(cid:48) +s. s(cid:48)∈S Conversely, it can be also shown that, if a bounded function, w : Ω(u)×U → R, satisfies the above functional equation, then w = w∗. Since K and K are already defined, the objects of interest become Ω(s), s ∈ S. To find π y Ω(s) for a given s ∈ S, we define an operator, B, as follows. For Q(s) ∈ R and s ∈ S, let (cid:110) (cid:16) (cid:110) (cid:111) (cid:17) B(Q)(s) = η |∃ y,π, ηs(cid:48) ∈ K ×K × (cid:8) Q(s(cid:48)) (cid:9) , s(cid:48)∈S π y s(cid:48)∈S where π = η and π = κy+β (cid:88) p(s(cid:48)|s)ηs(cid:48) +s (cid:111) . s(cid:48)∈S It can be shown that, for a given s ∈ S, (i) Q(s) ⊆ B(Q)(s) ⇒ B(Q)(s) ⊆ Ω(s) (a.k.a self generation), (ii) Ω(s) = B(Ω)(s) (a.k.a. factorization), and (iii) letting Q (s) = (cid:2) η,η (cid:3) and 0 Q (s) = B(Q )(s), Q (s) ⊇ Q (s) and Ω(s) = ∩∞ Q (s). n n−1 n n+1 n=0 n B.2 Computing Ω(s) with the B operator We can show that, if K is sufficiently large, Ω(e ) = Ω(e ) = K . Y h n Π 30

Proof: Let Q (e ) := K and Q (e ) := K . From the properties of B-operator described 0 h Π 0 n Π above, we can prove Ω(e ) = Ω(e ) = K by showing Q (e ) ⊆ B(Q )(e ) and Q (e ) ⊆ h n Π 0 h 0 h 0 n B(Q )(e ). LetK := [π ,π ]. TakeK := [y ,y ]withy ≤ [(1−β)π −e ]/κ 0 n Π min max Y min max min min h and y ≥ [(1−β)π −e ]/κ. max max n Take η ∈ Q (e ). We want to show that η ∈ B(Q )(e ). Consider y = [(1−β)π−e ]/κ, 0 h 0 h h π = η, η(cid:48) = η, and η(cid:48) = η. Note that the lowest possible y (when η = π ) is given by h n min y = [(1−β)π −e ]/κ ≥ y and the highest possible y (when η = π ) is given by min h min max y = [(1−β)π −e ]/κ ≤ [(1−β)π −e ]/κ = y . Thus, y ∈ K . π ∈ Q := K , max h max n max Y 0 Π η(cid:48) ∈ Q (e ), η(cid:48) ∈ Q (e ) := Q (e ). That is, (y,π,η(cid:48),η(cid:48)) ∈ K ×K ×Q (e )×Q (e ) h 0 h n 0 h 0 n h n Y Π 0 h 0 n Thus, η ∈ B(Q )(e ). 0 h Take η ∈ Q (e ). We want to show that η ∈ B(Q )(e ). Consider y = [(1−β)π−e ]/κ, 0 n 0 n n π = η, η(cid:48) = η, and η(cid:48) = η. Note that the lowest possible y (when η = π ) is given by h n min y = [(1−β)π −e ]/κ ≥ [(1−β)π −e ]/κ ≥ y and the highest possible y (when min n min h min η = π ) is given by y = [(1−β)π −e ]/κ ≤ y . Thus, y ∈ K . π ∈ Q := K , max max n max Y 0 Π η(cid:48) ∈ Q (e ), η(cid:48) ∈ Q (e ) := Q (e ). That is, (y,π,η(cid:48),η(cid:48)) ∈ K ×K ×Q (e )×Q (e ) h 0 h n 0 h 0 n h n Y Π 0 h 0 n Thus, η ∈ B(Q )(e ). 0 n C Details of the promised value approach for the model with ELB Notations, s , st, x, x(s), are the same as in the model with stabilization bias. t A state-contingent sequence of inflation, output, and the policy rate, (y,π,r), is said to be a competitive outcome if, ∀t ≥ 1 and ∀st ∈ St, y (st) ∈ K , π (st) ∈ K , r (st) ∈ K , and t Y t Π t R σy (st) = σE y (st+1)+E π (st+1)−r (st)+r∗+s t t t+1 t t+1 t t π (st) = κy (st)+βE π (st+1) t t t t+1 r ≥ r t ELB Let CE denote the set of all competitive outcomes. That is, CE ≡ {(y,π,r) |(y(s),π(s),r(s))is a competitive outcome}. For each s ∈ S, let CE(s) denote the set of all competitive outcomes with s = s. That is, 1 CE(s) ≡ {(y(s),π(s),r(s)) |(y(s),π(s),r(s))is a competitive outcomewiths = s}. 1 The Ramsey planner’s problem is to chooses the competitive outcome that maximizes the time-one value: V (s) = max V (s) ram,1 1 (y(s),π(s),r(s))∈CE(s) C.1 Recursive formulation A recursive treatment of the Ramsey problem entails the use of state-contingent promised value(s) made in period t given by η (s |s ) = π (s |s ) and η (s |s ) = 1,t+1 t+1 t t+1 t+1 t 2,t+1 t+1 t y (s |s ) given s ∈ S and s ∈ S. Hence, η (s |s ) and η (s |s ) are pet+1 t+1 t t t+1 1,t+1 t+1 t 2,t+1 t+1 t riod t+1’s inflation rate and consumption levels, respectively, for some state s ∈ S, that t+1 31

is promised by the equilibrium in period t for a given s ∈ S. From now on, we will denote t these promised variables by ηs(cid:48) and ηs(cid:48) , s(cid:48) ∈ S, s = s(cid:48). 1,t+1 2,t+1 t+1 For any s ∈ S, let Ω(s) denote the set of all possible pairs of initial inflation and output promises consistent with the existence of a competitive outcome. That is, Ω(s) ≡ {(η ,η ) ∈ R2 |η = π andη = y for some(y(s),π(s),r(s)) ∈ CE(s)}. 1 2 1 1 2 1 For any s ∈ S, let Γ(η ,η ,s) denote the set of all possible competitive outcomes whose 1 2 initial promise pair is given by (η ,η ). That is, 1 2 Γ(η ,η ,s) ≡ {(y(s),π(s),r(s)) ∈ CE(s)|y = η , andπ = η }. 1 2 1 1 1 2 For a given s ∈ S, the recursive formulation takes two steps. In the first step, the constrained Ramsey problem is formulated: ∞ w∗(η ,η ,s) = max − 1 (cid:88) βt−1 (cid:88) µ(st|s = s) (cid:2) π (st)2+λy (st)2(cid:3) 1 2 1 t t (y(s),π(s),r(s))∈Γ(η1,η2,s) 2 t=1 st|s1=s In the second step, the Ramsey planner chooses the initial inflation and output promises that maximize w∗(η ,η ,s). 1 2 V (s) = max w∗(η ,η ,s). ram,1 1 2 (η1,η2)∈Ω(s) By the standard dynamic programming argument, it can be shown that, for any s ∈ S, w∗(η ,η ,s) satisfies the functional equation: 1 2 w(η ,η ,s) = max − 1 [π2+λy2]+β (cid:88) p(s(cid:48)|s)w( (cid:110) ηs(cid:48) ,ηs(cid:48) (cid:111) ,s(cid:48) ∈ S) 1 2 y,π,r,{(η 1 s(cid:48),η 2 s(cid:48))} s(cid:48)∈S 2 s(cid:48)∈S 1 2 (cid:16) (cid:110) (cid:111) (cid:17) such that y,π,r, (ηs(cid:48),ηs(cid:48)) ∈ K ×K ×K ×{Ω(s(cid:48))} , 1 2 s(cid:48)∈S Y Π R s(cid:48)∈S y = η , 1 π = η , 2 r = 1 (cid:88) p(s(cid:48)|s)ηs(cid:48) + (cid:88) p(s(cid:48)|s)ηs(cid:48) − 1 y+s, σ 1 2 σ s(cid:48)∈S s(cid:48)∈S and π = κy+β (cid:88) p(s(cid:48)|s)ηs(cid:48) . 2 s(cid:48)∈S Conversely, if a bounded function, w : Ω(s)×S → R, satisfies the above functional equation, then w = w∗. Since {0,R+}, K and K are already defined, the objects of interest become Ω(s), s ∈ S. π y To find Ω(s) for a given s ∈ S, we define an operator, B, as follows. For Q(s) ∈ R2 and s ∈ S, 32

let (cid:110) (cid:16) (cid:110) (cid:111) (cid:17) B(Q(s)) = (η ,η )|∃ y,π,r, η ,ηs(cid:48) ∈ K ×K ×K × (cid:8) Q(s(cid:48)) (cid:9) , 1 2 1s(cid:48) 2 s(cid:48)∈S Y Π R s(cid:48)∈S where y = η ,π = η ,r = 1 (cid:88) p(s(cid:48)|s)ηs(cid:48) + (cid:88) p(s(cid:48)|s)ηs(cid:48) − 1 y+s, 1 2 σ 2 1 σ s(cid:48)∈S s(cid:48)∈S and π = κy+β (cid:88) p(s(cid:48)|s)ηs(cid:48) (cid:111) . 1 s(cid:48)∈S It can be shown that, for any s ∈ S, (i) Q(s) ⊆ B(Q)(s) ⇒ B(Q)(s) ⊆ Ω(s) (a.k.a self (cid:104) (cid:105) generation), (ii) Ω(s) = B(Ω)(s) (a.k.a. factorization), and (iii) letting Q (s) = η ,η × 0 1 1 (cid:104) (cid:105) η ,η and Q (s) = B(Q )(s), Q (s) ⊇ Q (s) and Ω(s) = ∩∞ Q (s). 2 2 n n−1 n n+1 n=0 n C.2 Computing Ω(s) with the operator B The obvious guess for Ω(δ ) and Ω(δ ) is K ×K . However, we can show that, with c n Y Π Q (δ ) := K ×K and Q (δ ) := K ×K , B(Q )(δ ) ⊂ Q (δ ) and B(Q )(δ ) ⊂ Q (δ ). 0 c Y Π 0 n Y Π 0 c 0 c 0 n 0 n Proof: Let Q (δ ) := K ×K and Q (δ ) := K ×K . Take (η ,η ) = (y ,π ) ∈ 0 c Y Π 0 n Y Π 1 2 max min Q (δ ). We want to show that (η ,η ) ∈/ B(Q )(δ ). Let y = η and π = η . Note that, in 0 n 1 2 0 n 1 2 order to satisfy the Phillips curve, η (δ ) = (η −κη )/β. Note that η (δ ) = (η −κη )/β = 2 n 2 1 2 n 2 1 (π − κy )/β < π − κy /β ≤ π . The second-to-last inequality follows from min max min max min the fact that π < 0. The last inequality follows from the fact that y > 0.14 Thus, min max (η ,η ) ∈/ B(Q )(δ ). Similarly, we can prove that (η ,η ) ∈/ B(Q )(δ ). 1 2 0 n 1 2 0 c Since B(Q )(δ ) ⊂ Q (δ ) := K ×K and B(Q )(δ ) ⊂ Q (δ ) := K ×K , Ω(δ ) (cid:54)= 0 c 0 c Y Π 0 n 0 n Y Π c K ×K and Ω(δ ) (cid:54)= K ×K . Thus, we have to apply the operator B repeatedly until it Y Π n Y Π converges to find Ω(δ ) and Ω(δ ). c n Analytically characterizing the sequence of {Q (δ ),Q (δ )}∞ seems daunting, if not j c j n j=0 infeasible. Thus, we will use a numerical method similar to Feng, Miao, Peralta-Alva, and Santos (2014) in order to numerically compute {Q (δ ),Q (δ )}∞ and their convergent sets, j c j n j=0 Ω(δ ) and Ω(δ ). c n C.2.1 Setup Let A = K ×K ×K be known as the action space. Due to the following for each Y Π R s ∈ S. (i) Make an initial guess for Ω(δ), i.e. Qˆ0(δ) = K ×K . Y Π (ii) Create an object, Qˆ (δ), by discretizing each dimension, η, of Qˆ(δ) into N equigrid η distant points. This results in (N −1) × (N −1) rectangles each denoted by Ξ —i = η1 η2 i 1,...,(N −1)×(N −1)—which in turn yields a position (j ,j ), j = 1,...,N −1 and η1 η2 η1 η2 η1 η1 j = 1,...,N −1, in the discretized state space, Qˆ (δ). η2 (cid:16) η2 (cid:17) (cid:110) (cid:16) grid (cid:17)(cid:111) (iii) Let G0 Qˆ (δ) = I (Ξ ),...,I Ξ be a vector of indicator δ grid 0,1 1 0,1 (Nη1 −1)×(Nη2 −1) functions indicating the inclusion of each rectangle, Ξ , where a value of 1 indicates inclusion i 14Sincewewanttoallow(y,π)totakethevalueof(0,0),bothK andK havetocover0. Inotherwords, Y Π π and y has to be strictly negative and π and y has to be strictly positive. min min max max 33

and 0 does not. (cid:16) (cid:17) (iv) Set G0 Qˆ (δ) = 1 (a vector of ones). δ grid (cid:110) (cid:111) (v) Let Qˆ0 (δ) = Ξ ∈ Qˆ |G0(Ξ ) = 1 . grid i grid δ i C.2.2 Brute-force search algorithm At each iteration, n ≥ 0, and for each s ∈ S, do the following: (cid:16) (cid:17) (cid:16) (cid:17) (i) Given Qˆn (δ) and Gn Qˆ (δ) , we want to update Qˆn+1(δ) and Gn+1 Qˆ (δ) . grid δ grid grid δ grid (ii) For each Ξ ∈ Qˆn (δ), make a judicious selection of points to test.15 If for at least i grid (cid:16) (cid:110) (cid:111) (cid:17) (cid:110) (cid:111) one point, (η ,η ) ∈ Ξ , ∃ y,π,r, ηδ(cid:48),ηδ(cid:48) ∈ A× Qˆn (δ(cid:48)) such that y = η , 1 2 i 1 2 δ(cid:48)∈D grid δ(cid:48)∈D 1 π = η , r = 1 (cid:80) p(δ(cid:48)|δ)ηδ(cid:48) + (cid:80) p(δ(cid:48)|δ)ηδ(cid:48) − 1y+δ, and π = κy+β (cid:80) p(δ(cid:48)|δ)ηδ(cid:48), 2 σ δ(cid:48)∈D 1 δ(cid:48)∈D 2 σ δ(cid:48)∈D 2 set Gn+1(Ξ ) = 1. Otherwise, set Gn+1(Ξ ) = 0. δ i δ i (cid:110) (cid:111) (iii) Update Qˆn+1(δ) = Ξ ∈ Qˆ |Gn+1(Ξ ) = 1 . grid i grid δ i (iv) If Qˆn (δ) = Qˆn+1(δ) ∀δ ∈ D, stop the algorithm and set Ωˆ(δ) = Qˆn+1(δ), s ∈ D. grid grid grid Otherwise, repeat the algorithm. C.2.3 Results Figure 14 shows the set of feasible pairs of initial inflation and output promises, Ω(δ). Figure 14: The set of feasible pairs of initial (y,π)-promises According to the figure, for both high (crisis) and low (normal) states, (i) combinations of a very high output and a very low output—northwest corner—and (ii) combinations of a very low output and very high inflation—southeast corner—are not feasible. This makes sense because Phillips curve constraint requires that, all else equal, inflation today has to be lower when output is lower. The set of feasible pairs of initial inflation and output promises is large. In particular, for both states, a wide range of areas around the steady state of (π = 0,y = 0) is feasible. Thus, the boundary of this set does not pose any binding constraints on the optimization problem associated with the Bellman equation. 15We use a total of nine points: the vertices, the midpoints between the vertices, and the point in the middle of the rectangle. 34

D Analytical results for the model with inflation bias Inthissection,wefirstprovidetheanalyticalsolutionstothesaddle-pointfunctionalequation associated with the Lagrange multiplier approach and the Bellman equation associated withthepromisedvalueapproachforthemodelwithinflationbias(sectionD.1andD.2). We then prove that the allocations obtained from the Lagrange multiplier approach are identical to those obtained from the promised value approach (section D.3). D.1 Lagrange multiplier approach Guess that the solution to the saddle-point functional equation takes the following form: π = α +α φ , (19) 0,π 1,π −1 φ = α +α φ , (20) 0,φ 1,φ −1 y = α +α φ . (21) 0,y 1,y −1 We would like to find (α ,α ,α ,α ) such that the following FONCs associated with 0,π 1,π 0,φ 1,φ the saddle-point functional equation are satisfied: φ = π+φ , (22) −1 0 = −λ(y−y∗)−κφ, (23) π = κy+βπ(cid:48). (24) Substituting (19) and (20) into (22), we obtain α +α φ = α +α φ +φ . 0,φ 1,φ −1 0,π 1,π −1 −1 For this equation to hold for any φ , the following two equations must hold: −1 α = α , (25) 0,φ 0,π α = 1+α . (26) 1,φ 1,π Substituting (22), (19), (20), and (21) into (23), we obtain 0 = −λ(α +α φ −y∗)−κ(π+φ ), 0,y 1,y −1 −1 = −λ(α +α φ −y∗)−κ(α +α φ +φ ), 0,y 1,y −1 0,π 1,π −1 −1 = −λα +λy∗−κα −λα φ −κ(1+α )φ . 0,y 0,π 1,y −1 1,π −1 For this equation to hold for any φ , the following two equations must hold: −1 α = −λ−1κ(1+α ), (27) 1,y 1,π α = y∗−κλ−1α . (28) 0,y 0,π Substituting (19)-(21) into (24), we obtain α +α φ = κ(α +α φ )+β(α +α φ), 0,π 1,π −1 0,y 1,y −1 0,π 1,π = κ(α +α φ )+βα +βα (α +α φ ), 0,y 1,y −1 0,π 1,π 0,φ 1,φ −1 = κα +κα φ +βα +βα α +βα α φ . 0,y 1,y −1 0,π 1,π 0,φ 1,π 1,φ −1 35

For this equation to hold for any φ , the following two equations must hold: −1 α = κα +β(α +α α ), (29) 0,π 0,y 0,π 1,π 0,φ α = κα +βα α . (30) 1,π 1,y 1,π 1,φ Substituting (25) and (28) into (29), α = κy∗−κ2λ−1α +βα (1+α ), 0,π 0,π 0,π 1,π κy∗ =⇒ α = . 0,π 1+κ2λ−1−β(1+α ) 1,π Substituting (27) into (30), α = −λ−1κ2(1+α )+βα (1+α ), 1,π 1,π 1,π 1,π =⇒ βα2 − (cid:0) 1−β+λ−1κ2(cid:1) α −λ−1κ2 = 0. (31) 1,π 1,π Substituting (26) into (31) and arranging it, βα2 − (cid:0) 1+β+λ−1κ2(cid:1) α +1 = 0. 1,φ 1,φ The solution to this quadratic function is (cid:113) (cid:0) 1+β +λ−1κ2(cid:1) ± (1+β+λ−1κ2)2−4β . 2β Since 1+β +λ−1κ2 > 0, (cid:113) (cid:0) 1+β+λ−1κ2(cid:1) − (1+β+λ−1κ2)2−4β α = . 1,φ 2β Summary of coefficients (cid:113) (cid:0) 1+β +λ−1κ2(cid:1) − (1+β+λ−1κ2)2−4β α = 1,φ 2β κy∗ α = 0,π 1+κ2λ−1−βα 1,φ α = α 0,φ 0,π α = y∗−κλ−1α 0,y 0,π α = α −1 1,π 1,φ α = −λ−1κ(1+α ) 1,y 1,π D.2 Promised value approach Guess that the solution to the Bellman equation takes the following form: η(cid:48) = a η, (32) η y = a η. (33) y 36

We would like to find (a ,α ) such that the following FONCs associated with the Bellman η y equation are satisfied: −λ(y−y∗)−κω = 0, ∂W(η(cid:48)) β −ωβ = 0, ∂η(cid:48) η = κy+βη(cid:48). where ω is the Lagrange multiplier on the Phillips curve. The envelope condition associated with the Bellman equation is ∂W(η) = −η+ω. ∂η From above four equations, we have λ λ 0 = y−η(cid:48)− y(cid:48), (34) κ κ η = κy+βη(cid:48). (35) Substituting (32) and (33) into (35), we obtain 1−βa η a = . (36) y κ Substituting (32) and (33) into (34), we obtain λ λ 0 = a η−a η− a η(cid:48), y η y κ κ λ λ = a η−a η− a a η, y η y η κ κ λ λ = a −a − a a . (37) y η y η κ κ Finally, substituting (36) into (37), we obtain (cid:18) (cid:19) (cid:18) (cid:19) λ 1−βa λ 1−βa η η 0 = −a − a , η η κ κ κ κ = λ(1−βa )−κ2a −a λ(1−βa ), η η η η = λ−λβa −κ2a −a λ+a2λβ, η η η η = βλa2 − (cid:2) (1+β)λ+κ2(cid:3) a +λ, η η = βa2 − (cid:0) 1+β +λ−1κ2(cid:1) a +1. η η The solution to this quadratic function is (cid:113) (cid:0) 1+β+λ−1κ2(cid:1) ± (1+β +λ−1κ2)2−4β a = . η 2β 37

Since 1+β +λ−1κ2 > 0, (cid:113) (cid:0) 1+β+λ−1κ2(cid:1) − (1+β +λ−1κ2)2−4β a = . η 2β Summary of coefficients 1−βa η a = y κ (cid:113) (cid:0) 1+β +λ−1κ2(cid:1) − (1+β+λ−1κ2)2−4β a = η 2β In the promised value approach, one needs to know the value function associated with the Bellman equation to find the initial inflation. Guess that the value function takes the following form: 1 W (η) = µ +µ η+ µ η2. (38) PV PV,0 PV,1 PV,2 2 The value function satisfies W (η) = − 1 (cid:2) η2+λ(y−y∗)2(cid:3) +βW (η(cid:48)). (39) PV PV 2 Substituting (32), (33), and (38) into (39), we obtain (cid:20) (cid:21) W (η) = − 1 (cid:2) η2+λ(y−y∗)2(cid:3) +β µ +µ η(cid:48)+ 1 µ η(cid:48)2 , PV PV,0 PV,1 PV,2 2 2 = − 1 η2− 1 λ (cid:2) a2η2−2a ηy∗+[y∗]2(cid:3) +βµ +βµ a η+ 1 βµ a2η2, 2 2 y y PV,0 PV,1 η 2 PV,2 η 1 1 1 1 = − η2− λa2η2+λa y∗η− λ[y∗]2+βµ +βµ a η+ βµ a2η2, 2 2 y y 2 PV,0 PV,1 η 2 PV,2 η = βµ − 1 λ[y∗]2+(λa y∗+βµ a )η+ 1 (cid:2) βµ a2 −(1+λa2) (cid:3) η2. PV,0 2 y PV,1 η 2 PV,2 η y Comparing the constant terms and coefficients on η and η2, we obtain λ[y∗]2 µ = − , PV,0 2(1−β) λa y∗ y µ = , PV,1 1−βa η 1+λa2 y µ = − . PV,2 1−βa2 η The initial inflation is given by π = argmax W 1  PV 38

∂W PV =⇒ = 0 ∂η =⇒ π = η 1 µ PV,1 = − µ PV,2 λa y∗ 1−βa2 y η = . 1−βa 1+λa2 η y D.3 Equivalence We now prove that the allocations under the Lagrange multiplier and the promised value approaches are identical. We do so in two steps. First, we show that the initial inflation and output implied by the two approaches are identical. We then show that if the initial output is the same, the allocations from t = 2 on are identical. D.3.1 Equivalence of time-one allocations We will first show π = π and then y = y . lm,1 pv,1 lm,1 pv,1 In the promised value approach, the time-1 inflation is given by λα y∗ 1−βα2 y η π = , pv,1 1−βα 1+λα2 η y λy∗1−βα2 η = , κ 1+λα2 y with 1−βα η α = . y κ In Lagrange multiplier approach, the time-1 inflation is given by κy∗ π = . lm,1 1+κ2λ−1−βα η 39

π = π pv,1 lm,1 λ ⇐⇒ (1−βα2)(1+κ2λ−1−βα ) = κ(1+λα2), κ η η y (cid:32) (cid:33) λ (cid:20) 1−βα (cid:21)2 ⇐⇒ (1−βα2)(1+κ2λ−1−βα ) = κ 1+λ η , κ η η κ (cid:16) (cid:17) ⇐⇒ κλ(1−βα2)(1+κ2λ−1−βα ) = κ κ2+λ[1−βα ]2 , η η η ⇐⇒ λ(1−βα2)(1+κ2λ−1−βα ) = κ2+λ[1−βα ]2, η η η ⇐⇒ (1−βα2)(λ+κ2−βλα ) = κ2+λ[1−βα ]2, η η η ⇐⇒ λ+κ2−βλα −βλα2 −βκ2α2 +β2λα3 = κ2+λ−2λβα +λβ2α2, η η η η η η ⇐⇒ −βλα −βλα2 −βκ2α2 +β2λα3 = −2λβα +λβ2α2, η η η η η η 1 ⇐⇒ −1−α −κ2 α +βα2 = −2+βα , η λ η η η (cid:18) κ2(cid:19) ⇐⇒ βα2 − 1+β+ α +1 = 0. η λ η Note that α was constructed so that the last equality holds. Thus, the last equality holds η and π = π . lm,1 pv,1 Now, we will show y = y . In the promised value approach, the time-1 output is lm,1 pv,1 given by 1−βα η y = π pv,1 pv,1 κ 1−βα κy∗ η = κ 1+λ−1κ2−βα eta 1−βα = η y∗ 1+λ−1κ2−βα eta λ−1κ2 = [1− ]y∗ 1+λ−1κ2−βα eta In the Lagrange multiplier approach, the time-1 output is given by κ2 κy∗ y = y∗− lm,1 λ 1+κ2λ−1−βα η λ−1κ2 = [1− ]y∗ 1+λ−1κ2−βα η Thus, y = y . lm,1 pv,1 D.3.2 Equivalence of allocations from t = 2 on To show the equivalence of allocations from t = 2 on, we first express the allocation at t as a function of output at t−1 under the two approaches. We then show that the function is identical under the two approaches. Since we have already shown that time-one output is the same across the two approaches, it follows that the allocations from t = 2 on are the 40

same across them. Let the mapping from output in the previous period to today’s allocations under the Lagrange multiplier approach be given by: π = γ y , lm,t π lm,t−1 y = γ y , lm,t y lm,t−1 for t = 2,3,.... Using 0 = −λ(y −y∗)−κφ , we have lm,t lm,t π = α +α φ lm,t 0,π 1,π lm,t λ λ = α + α y∗− α y 0,π 1,π 1,π lm,t−1 κ κ λ = − α y , 1,π lm,t−1 κ and y = α +α φ lm,t 0,π 1,π lm,t λ λ = α + α y∗− α y 0,y 1,y 1,y lm,t−1 κ κ λ = − α y , 1,y lm,t−1 κ for t = 1,2,.... Note that α = −λα y∗ and α = −λα y∗ are implied by 0,π κ 1,π 0,y κ 1,y βα2 − (cid:0) 1+β+λ−1κ2(cid:1) α +1 = 0. 1,φ 1,φ Thus, λ γ = − α π 1,π κ λ γ = − α . y 1,y κ Now, let the mapping from output in the previous period to today’s allocations under the promised value approach be given by: π = c y , pv,t π pv,t−1 y = c y , pv,t y pv,t−1 for t = 2,3,.... Using the solution from the promised value approach, π = a π pv,t η pv,t−1 = a a−1y , η y pv,t−1 y = a π pv,t y pv,t = a a a−1y y η y pv,t−1 = a y . η pv,t−1 41

Thus, c = a a−1 π η y c = a . y η We want to show γ = c and γ = c , which imply y = y and π = π for y y π π lm,t pv,t lm,t pv,t t = 2,3,... given that y = y . Let us first show lm,1 pv,1 γ = c . y y λ γ = − α , y 1,y κ λ (cid:16) κ (cid:17) = − − α , 1,φ κ λ = α , 1,φ (cid:112) 1+β +λ−1κ2− (1+β +λ−1κ2)2−4β = , 2β (cid:112) (1+β)λ+κ2− [(1+β)λ+κ2]2−4βλ2 = , 2βλ = a = c . η y Next, let us show γ = a a−1. π η y We have λ γ = − (a −1), π η κ and κa a a−1 = η . η y 1−βa η κa λ η = − (a −1) η 1−βa κ η ⇐⇒ κ2a = −λ(a −1)(1−βa ) η η η ⇐⇒ κ2a = −λ(a −1−βa2 +βa ) η η η η ⇐⇒ κ2a = −λa +λ+βλa2 −βλa η η η η ⇐⇒ βλa2 −(λ+βλ+κ2)α +λ = 0 η η κ2 ⇐⇒ βa2 −(1+β+ )a +1 = 0 η λ η Note that a was constructed so that the last equality holds. Thus, the last equality holds η by construction. 42

E Analytical results for the model with stabilization bias In this section, we first provide the analytical solutions to the saddle-point functional equation associated with the Lagrange multiplier approach and the Bellman equation associated with the promised value approach for the model with stabilization bias (section E.1 andE.2). WethenprovethattheallocationsobtainedfromtheLagrangemultiplierapproach are identical to those obtained from the promised value approach (section E.3). E.1 Lagrange multiplier approach Given our assumption that the cost-push shock disappears after t = 1, the solution of the saddle-point functional equation in this model is identical to that in the model with inflation bias from t = 2 on, and is given by: α (2) = α (2)−1 1,π 1,φ α (2) = −κλ−1(1+α (2)) 1,y 1,π (cid:112) 1+β+κ2λ−1− (1+β +κ2λ−1)2−4β α (2) = 1,φ 2β Turning our attention to t = 1 when the cost-push shock is present, the FONCs are given by φ(s ) = π(s )+φ , (40) 1 1 −1 0 = −λy(s )−κφ(s ), (41) 1 1 π(s ) = κy(s )+βπ(φ(s ),e )+e . (42) 1 1 1 2 1 Guess that the solution takes the following form: π(s ) = α (1)+α (1)φ , (43) 1 0,π 1,π −1 φ(s ) = α (1)+α (1)φ , (44) 1 0,φ 1,φ −1 y(s ) = α (1)+α (1)φ , (45) 1 0,y 1,y −1 Note that π(φ(s ),e ) = α (2)φ(s ) 1 2 1,π 1 = α (2)(α (1)+α (1)φ ) 1,π 0,φ 1,φ −1 = α (2)α (1)+α (2)α (1)φ . (46) 1,π 0,φ 1,π 1,φ −1 Substituting (43) and (44) into (40), α (1)+α (1)φ = α (1)+(1+α (1))φ . 0,φ 1,φ −1 0,π 1,π −1 Therefore, we have α (1) = α (1), (47) 0,φ 0,π α (1) = 1+α (1). (48) 1,φ 1,π 43

Substituting (40), (43), and (45) into (41), 0 = −λ(α (1)+α (1)φ )−κ(α (1)+α (1)φ +φ ). 0,y 1,y −1 0,π 1,π −1 −1 Therefore, we have α (1) = −κλ−1α (1), (49) 0,y 0,π α (1) = −λ−1κ(1+α (1)). (50) 1,y 1,π Substituting (43), (45), and (46) into (42), α (1)+α (1)φ = κ(α (1)+α (1)φ )+β(α (2)α (1)+α (2)α (1)φ )+e . 0,π 1,π −1 0,y 1,y −1 1,π 0,φ 1,π 1,φ −1 1 Therefore, we have α (1) = κα (1)+β(α (2)α (1))+e , (51) 0,π 0,y 1,π 0,φ 1 α (1) = κα (1)+βα (2)α (1). (52) 1,π 1,y 1,π 1,φ Substituting (47) and (49) into (51), we have α (1) = −κ2λ−1α (1)+βα (2)α (1)+e , 0,π 0,π 1,π 0,π 1 e 1 =⇒ α (1) = . (53) 0,π 1−βα (2)+κ2λ−1 1,π Substituting (48) and (50) into (52), we have α (1) = −κ2λ−1(1+α (1))+βα (2)(1+α (1)), 1,π 1,π 1,π 1,π βα (2)−κ2λ−1 1,π =⇒ α (1) = . (54) 1,π 1+κ2λ−1−βα (2) 1,π Summary of coefficients for t = 1 α (1) = α (1) 0,φ 0,π α (1) = 1+α (1) 1,φ 1,π α (1) = −κλ−1α (1) 0,y 0,π α (1) = −λ−1κ(1+α (1)) 1,y 1,π e 1 α (1) = 0,π 1−βα (2)+κ2λ−1 1,π βα (2)−κ2λ−1 1,π α (1) = 1,π 1+κ2λ−1−βα (2) 1,π E.2 Promised value approach AswiththeLagrangemultiplierapproach,givenourassumptionthattheshockdisappears after t = 1, the solution to the relevant Bellman equation in this model is the same as that 44

in the model with inflation bias, and is given by: 1−βa (2) 1,η a (2) = 1,y κ (cid:112) 1+β+λ−1κ2− [(1+β)+λ−1κ2]2−4β a (2) = 1,η 2β µ (2) = 0 pv,0 µ (2) = 0 pv,1 1+λa (2)2 1,y µ (2) = − pv,2 1−βa (2)2 1,η Turning our attention to t = 1 when the cost-push shock is present, the FONCs are given by λy(s )−λy(η(cid:48)(s ),e )−κη(cid:48)(s ) = 0, (55) 1 1 2 1 η = κy(s )+βη(cid:48)(s )+e . (56) 1 1 1 Guess that the solution takes the following form: η(cid:48)(s ) = a (1)+a (1)η, (57) 1 0,η 1,η y(s ) = a (1)+a (1)η, (58) 1 0,y 1,y y(η(cid:48)(s ),e ) = a (2)(a (1)+a (1)η). (59) 1 2 1,y 0,η 1,η Substituting (57)-(59) into (55), λ(a (1)+a (1)η)−λa (2)(a +a (1)η)−κ(a (1)+a (1)η) = 0. 0,y 1,y 1,y 0,η 1,η 0,η 1,η Therefore, we have λa (1)−λa (2)a (1)−κa (1) = 0, (60) 1,y 1,y 1,η 1,η λa (1)−λa (2)a (1)−κa (1) = 0. (61) 0,y 1,y 0,η 0,η Substituting (57) and (58) into (56), η = κ(a (1)+a (1)η)+β(a (1)+a (1)η)+e . 0,y 1,y 0,η 1,η 1 Therefore, we have 1 = κa (1)+βa (1), (62) 1,y 1,η 0 = κa (1)+βa (1)+e . (63) 0,y 0,η 1 From (62), 1−βa (1) 1,η a (1) = . (64) 1,y κ Substituting (64) into (60), (cid:18) (cid:19) 1−βa (1) 1,η λ −λa (2)a (2)−κa (1) = 0, 1,y 1,η 1,η κ 45

λ−λβa (1)−λκa (2)a (1)−κ2a (1) = 0, 1,η 1,y 1,η 1,η λ = (λβ +λκa (2)+κ2)a (1), 1,y 1,η λ =⇒ a (1) = . 1,η λβ +κ2+λκa (2) 1,y Furthermore, λ a (1) = 1,η κ2+λβ +λκa (2) 1,y λ = κ2+λβ +λκ 1−βa1,η(2) κ λ = κ2+λβ +λ−λβa (2) 1,η λ = √ 1+β+λ−1κ2− [(1+β)+λ−1κ2]2−4β λ(1+β +λ−1κ2)−λβ 2β 1 = √ 1+β+λ−1κ2− [(1+β)+λ−1κ2]2−4β 1+β +λ−1κ2− 2 1 = √ 1+β+λ−1κ2+ [(1+β)+λ−1κ2]2−4β 2 2 = (cid:112) 1+β +λ−1κ2+ [(1+β)+λ−1κ2]2−4β (cid:112) 2 1+β+λ−1κ2− [(1+β)+λ−1κ2]2−4β = (cid:112) (cid:112) 1+β +λ−1κ2+ [(1+β)+λ−1κ2]2−4β1+β+λ−1κ2− [(1+β)+λ−1κ2]2−4β (cid:112) 1+β +λ−1κ2− [(1+β)+λ−1κ2]2 = 2β = a (2) 1,η Note that a (1) = a (2) implies a (1) = a (2) 1,η 1,η 1,y 1,y From (61), κ a (1) = a (1)+a (2)a (1). (65) 0,y 0,η 1,y 0,η λ Substituting (65) into (63), κ2 0 = a (1)+κa (2)a (1)+βa (1)+e , 0,η 1,y 0,η 0,η 1 λ 0 = κ2a (1)+λκa (2)a (1)+βλa (1)+e , 0,η 1,y 0,η 0,η 1 (κ2+λκa (2)+βλ)a (1) = −λe , 1,y 0,η 1 λe 1 =⇒ a (1) = − . 0,η κ2+λβ +λκa (2) 1,y 46

Summary of coefficients κ a (1) = a (1)+a (2)a (1) 0,y 0,η 1,y 0,η λ a (1) = a (2) 1,y 1,y λe 1 a (1) = − 0,η κ2+λβ +λκa (2) 1,y a (1) = a (2) 1,η 1,η Now, we solve for the value function at t = 1. 1 W (s ) = − [η2+λy(s )2]+βW (η(cid:48)(s ),e ). (66) pv 1 1 pv 1 2 2 Guess that the solution takes the following form: 1 W (s ) = µ (1)+µ (1)η+ µ (1)η2. pv 1 pv,0 pv,1 pv,2 2 Then, 1 W (η(cid:48)(s ),e ) = µ (2)+µ (2)(a (1)+a (1)η)+ µ (2)(a (1)+a (1)η)2. (67) pv 1 2 pv,0 pv,1 0,η 1,η pv,2 0,η 1,η 2 Since µ (2) = µ (2) = 0, (67) can be rewritten as follows: pv,0 pv,1 1 W (η(cid:48)(s ,e )) = µ (2)(a (1)+a (1)η)2. (68) pv 1 2 pv,2 0,η 1,η 2 Substituting (58) and (68) into (66), we obtain 1 1 W (s ) = − [η2+λ(a (1)+a (1)η)2]+ βµ (2)(a (1)+a (1)η)2, pv 1 0,y 1,y pv,,2 0,η 1,η 2 2 1 = − (η2+λa (1)2+2λa (1)a (1)η+λa (1)2η2) 0,y 0,y 1,y 1,y 2 1 + βµ (2)(a (1)2+2a (1)a (1)η+a (1)2η2), pv,2 0,η 0,η 1,η 1,η 2 λ 1 = − a (1)2+ βµ (2)a (1)2 0,y pv,2 0,η 2 2 −λa (1)a (1)η+βµ (2)a (1)a (1)η 0,y 1,y pv,2 0,η 1,η 1 λ 1 − η2− a (1)2η2+ βµ (2)a (1)2η2. 1,y pv,2 1,η 2 2 2 Comparing the constant terms and coefficients on η and η2, we obtain λ 1 µ (1) = − a (1)2+ βµ (2)a (1)2, pv,0 0,y pv,2 0,η 2 2 µ (1) = βµ (2)a (1)a (1)−λa (1)a (1), pv,1 pv,2 0,η 1,η 0,y 1,y µ (1) = βµ (2)a (2)2−1−λa (1)2. pv,2 pv,2 1,η 1,y 47

E.3 Equivalence We now prove that the allocations under the Lagrange multiplier and the promised value approaches are identical in the model with stabilization bias. Since the model’s solution is the same across two approaches from t = 2 on if time-one output is the same, it is sufficient toshowthattheinitialinflationandoutputimpliedbythetwoapproachesareidentical(that is, π = π and y = y ). lm,1 pv,1 lm,1 pv,1 We will first show π = π and then y = y . lm,1 pv,1 lm,1 pv,1 In the promised value approach, time-one inflation is given by µ (1) pv,1 π = − pv,1 µ (1) pv,2 βµ (2)a (1)a (1)−λa (1)a (1) pv,2 0,η 1,η 0,y 1,y = − βµ (2)a (1)2−1−λa (1)2 pv,2 1,η 1,y βµ (2)(−a (1)e )a (1)−λ[−(κ +a (2))a e ]a (1) = − pv,2 1,η 1 1,η λ 1,y 1,η 1 1,y βµ (2)a (1)2−1−λa (1)2 pv,2 1,η 1,y −βµ (2)a (1)2+λ(κ +a (2))a a (1) = −e pv,2 1,η λ 1,y 1,η 1,y 1 βµ (2)a (1)2−1−λa (1)2 pv,2 1,η 1,y In the Lagrange multiplier approach, time-one inflation is given by π = α (1) lm,1 0,π e 1 = 1−βa (2)+λ−1κ2 1,π e 1 = 1−β(a (2)−1)+λ−1κ2 1,φ e 1 = 1+β+λ−1κ2−βa (2) 1,φ e 1 = 1+β+λ−1κ2−βa (2) 1,η a (2)e 1,η 1 = [1+β+λ−1κ2−βa (2)]a (2) 1,η 1,η = a (2)e 1,η 1 where the last inequality follows from the definition of a (2). Putting things together, 1,η showing π = π amounts to showing lm,1 pv,1 −βµ (2)a (1)2+λ(κ +a (2))a a (1) a (2) = − pv,2 1,η λ 1,y 1,η 1,y 1,η βµ (2)a (1)2−1−λa (1)2 pv,2 1,η 1,y =⇒ κ βµ (2)a (1)2a (2)−a (2)−λa (1)2a (2) = βµ (2)a (1)2−λ( +a (2))a a (1) pv,2 1,η 1,η 1,η 1,y 1,η pv,2 1,η 1,y 1,η 1,y λ =⇒ κ −a (2)−λa (1)2a (2) = βµ (2)a (1)2(1−a (2))−λ( +a (2))a a (1) 1,η 1,y 1,η pv,2 1,η 1,η 1,y 1,η 1,y λ 48

=⇒ −a (2) = βµ (2)a (1)2(1−a (2))−κa a (1) 1,η pv,2 1,η 1,η 1,η 1,y =⇒ 1+λa (2)2 −a (2) = −β 1,y a (1)2(1−a (2))−κa a (1) 1,η 1−βa (2)2 1,η 1,η 1,η 1,y 1,η Multiplying both sides by 1−βa (2)2, we obtain 1,η −a (2)(1−βa (2)2) = −β(1+λa (2)2)a (1)2(1−a (2))−κa a (1)(1−βa (2)2) 1,η 1,η 1,y 1,η 1,η 1,η 1,y 1,η Dividing both sides by a (2), we obtain 1,η −(1−βa (2)2) = −β(1+λa (2)2)a (1)(1−a (2))−κa (1)(1−βa (2)2) 1,η 1,y 1,η 1,η 1,y 1,η =⇒ 1−βa (1) −(1−βa (2)2) = −β(1+λa (2)2)a (1)(1−a (2))−κ 1,η (1−βa (2)2) 1,η 1,y 1,η 1,η 1,η κ =⇒ −(1−βa (2)2) = −β(1+λa (2)2)a (1)(1−a (2))−(1−βa (1))(1−βa (2)2) 1,η 1,y 1,η 1,η 1,η 1,η =⇒ −βa (1)(1−βa (2)2) = −β(1+λa (2)2)a (1)(1−a (2)) 1,η 1,η 1,y 1,η 1,η Dividing both sides by −βa (1), we obtain 1,η 1−βa (2)2 = (1+λa (2)2)(1−a (2)) 1,η 1,y 1,η 1−βa (2) = (1+λ[ 1,η ]2)(1−a (2)) 1,η κ λ = (1+ [1−2βa (2)+β2a (2)2])(1−a (2)) κ2 1,η 1,η 1,η λ = (1+ [1−2βa (2)+β(1+β +λ−1κ2)a (2)−β])(1−a (2)) κ2 1,η 1,η 1,η λ(1−β) λβ = (1+ + (−1+β+λ−1κ2)a (2))(1−a (2)) κ2 κ2 1,η 1,η λ(1−β) λβ = 1+ + (−1+β +λ−1κ2)a (2) κ2 κ2 1,η λ(1−β) λβ −(1+ )a (2)− (−1+β+λ−1κ2)a (2)2 κ2 1,η κ2 1,η Multiplying both sides by κ2, we obtain κ2−βκ2a (2)2 = κ2+λ(1−β)+λβ(−1+β+λ−1κ2)a (2) 1,η 1,η −(κ2+λ(1−β))a (2)−λβ(−1+β +λ−1κ2)a (2)2 1,η 1,η = κ2+λ(1−β)+(λβ2+κ2β −κ2−λ)a (2) 1,η +(λβ −λβ2−βκ2)a (2)2 1,η 49

Subtracting κ2−βκ2a (2)2 from both sides, we obtain 1,η 0 = λ(1−β)+[λ(β −1)(β +1)+κ2(1−β)]a (2)+λβ(1−β)a (2)2 1,η 1,η Dividing both sides by λ(1−β), we obtain κ2 0 = −(1+β + )a (2)+βa (2)2 1,η 1,η λ This quadratic equation holds by the definition of a (2). Thus, π = π . 1,η lm,1 pv,1 Now, we will show y = y . In the promised value approach, the time-1 output is lm,1 pv,1 given by y = a +a π pv,1 0,y 1,y pv,1 κ = −[ +a (2)]a (1)e +a (1)π 1,y 1,η 1 1,y pv,1 λ κ = −[ +a (2)]π +a (2)π 1,y pv,1 1,y pv,1 λ κ = − π pv,1 λ In the Lagrange multiplier approach, the time-1 output is given by y = α (1) lm,1 0,y κ = − α 0,π λ κ = − π lm,1 λ Thus, y = y . lm,1 pv,1 F Global solution methods and their accuracy for the model with ELB F.1 Lagrange multiplier approach F.1.1 Marcet and Marimon’s recursive formulation MarcetandMarimon(2016)recursifytheRamseyproblemusingtheLagrangemultipliers as pseudo state variables: (cid:104) λ 1 W(φ ,φ ,δ ) = min max − y2− π2 1 2 i φ(cid:48),φ(cid:48) y,π,r≥0 2 2 1 2 +φ(cid:48) (r+σy+δ )−β−1φ (σy+π) 1 i 1  N (cid:88) +φ(cid:48) 2 (π−κy)−φ 2 π+β p(δ j |δ i )W(φ(cid:48) 1 ,φ(cid:48) 2 ,δ j ). j=1 50

The first-order necessary conditions (FONCs) are given by ∂y :−λy+σφ(cid:48) −σβ−1φ −κφ(cid:48) = 0 1 1 2 ∂π :−π−β−1φ +φ(cid:48) −φ = 0 1 2 2 N (cid:88) ∂φ(cid:48) :r+σy−δ − p(δ |δ )W (φ(cid:48),φ(cid:48),δ ) = 0 1 i j i 1 1 2 j j=1 N (cid:88) ∂φ(cid:48) :π−κy−β p(δ |δ )W (φ(cid:48),φ(cid:48),δ ) = 0 2 j i 2 1 2 j j=1 where W (φ(cid:48),φ(cid:48),δ ) = ∂W(φ(cid:48),φ(cid:48),δ )/∂φ(cid:48) and W (φ(cid:48),φ(cid:48),δ ) = ∂W(φ(cid:48),φ(cid:48),δ )/∂φ(cid:48). The 1 1 2 j 1 2 j 1 2 1 2 j 1 2 j 2 following Karush-Kuhn-Tucker conditions (KKTCs) must be satisfied as well φ(cid:48)r = 0, φ(cid:48) ≤ 0, and r ≥ 0. 1 1 The intitial conditions are such that φ = 0 and φ = 0. 1 2 F.1.2 Time iteration method Weexplicitlyconsideravectorofpolicyfunctionsς(ξ ) = [y(ξ ),π(ξ ),r(ξ ),φ(cid:48)(ξ ),φ(cid:48)(ξ )](cid:48) i i i i 1 i 2 i asfunctionsofthestatevariablesξ = (φ ,φ ,δ )fori = 1,...,N. Usingtheenvelopetheorem i 1 2 i (i.e., W (φ ,φ ,δ ) = σy(ξ )+π(ξ )andW (φ ,φ ,δ ) = π(ξ )−κy(ξ )), wehavethefollowing 1 1 2 i i i 2 1 2 i i i a system of functional equations σ e (ξ ) ≡−λy(ξ )+σφ(cid:48)(ξ )− φ −κφ(cid:48)(ξ ) = 0, LM,1 i i 1 i β 1 2 i e (ξ ) ≡−π(ξ )−β−1φ +φ(cid:48)(ξ )−φ = 0, LM,2 i i 1 2 i 2 N e (ξ ) ≡r(ξ )+σy(ξ )−δ − (cid:88) p(δ |δ ) (cid:2) σy(φ(cid:48)(ξ ),φ(cid:48)(ξ ),δ )+π(φ(cid:48)(ξ ),φ(cid:48)(ξ ),δ ) (cid:3) = 0, LM,3 i i i j j i 1 i 2 i j 1 i 2 i j j=1 N e (ξ ) ≡π−κy−β (cid:88) p(δ |δ ) (cid:2) κy(φ(cid:48)(ξ ),φ(cid:48)(ξ ),δ )+π(φ(cid:48)(ξ ),φ(cid:48)(ξ ),δ ) (cid:3) = 0. LM,4 i j i 1 i 2 i j 1 i 2 i j j=1 Algorithm The time iteration method takes the following steps: 1. Make an initial guess for the policy function ς(0)(ξ ) for i = 1,...,N. i 2. For k = 1,2,... (k is an index for the number of iteration), given the policy function 51

previously obtained ς(k−1)(ξ ) for each i, solve i σ −λy+σφ(cid:48) − φ −κφ(cid:48) = 0 1 β 1 2 −π−β−1φ +φ(cid:48) −φ = 0 1 2 2 r+σy−δ i N (cid:88) (cid:104) (cid:105) − p(δ |δ ) σy(k−1)(φ(cid:48),φ(cid:48),δ )+π(k−1)(φ(cid:48),φ(cid:48),δ ) = 0 j i 1 2 j 1 2 j j=1 N (cid:88) (cid:104) (cid:105) π−κy−β p(δ |δ ) κy(k−1)(φ(cid:48),φ(cid:48),δ )+π(k−1)(φ(cid:48),φ(cid:48),δ ) = 0 j i 1 2 j 1 2 j j=1 for (y,π,r,φ(cid:48),φ(cid:48)). 1 2 3. Update the policy function by setting y = y(k)(ξ ), π = π(k)(ξ ), r = r(k)(ξ ), φ(cid:48) = i i i 1 φ (cid:48)(k) (ξ ), φ(cid:48) = φ (cid:48)(k) (ξ ) for i = 1,...,N. 1 i 2 2 i (cid:13) (cid:13) 4. Repeat 2-3 until (cid:13)ς(k)(ξ i )−ς(k−1)(ξ i )(cid:13) is small enough. WeusethefollowingindicatorfunctionapproachasinGust,Herbst,L´opez-Salido,andSmith (2017), Nakata (2017), and Hirose and Sunakawa (2017). That is, for ς ∈ {y,π,φ(cid:48),φ(cid:48)}, 1 2 ς(ξ ) = I ς (ξ )+ (cid:0) 1−I (cid:1) ς (ξ ) i rNZLB(ξi)≥0 NZLB i rNZLB(ξi)≥0 ZLB i where ς (ξ ) is the policy function assuming that ZLB always does not bind and ς (ξ ) NZLB i ZLB i is the policy function assuming that ZLB always binds. I is the indicator function rNZLB(ξi)≥0 that takes the value of one when r (ξ ) ≥ 0, otherwise takes the value of zero. Then, in NZLB i Steps 2 and 3, the problem becomes finding a pair of policy functions, (ς (ξ ),ς (ξ )), NZLB i ZLB i asfollows(wedenoteatupleofvariablestobesolved(y ,π ,φ(cid:48) ,φ(cid:48) ,r ) NZLB NZLB 1,NZLB 2,NZLB NZLB in the non-ZLB regime and (y ,π ,φ(cid:48) ,φ(cid:48) ,r ) in the ZLB regime): (i) When ZLB ZLB 1,ZLB 2,ZLB ZLB we assume that ZLB does not bind, given the values of φ , φ and φ(cid:48) = 0, solve 1 2 1,NZLB φ(cid:48) = κ−1(cid:0) −λy −σβ−1φ (cid:1) 2,NZLB NZLB 1 π = φ(cid:48) −φ −β−1φ NZLB 2,NZLB 2 1 N (cid:88) (cid:104) (cid:105) π −κy −β p(δ |δ ) κy(k−1)(0,φ(cid:48) ,δ )+π(k−1)(0,φ(cid:48) ,δ ) = 0 NZLB NZLB j i 2,NZLB j 2,NZLB j j=1 for (y ,π ,φ(cid:48) ). Then we have NZLB NZLB 2,NZLB N (cid:88) (cid:104) (cid:105) r = −σy +δ + p(δ |δ ) σy(k−1)(0,φ(cid:48) ,δ )+π(k−1)(0,φ(cid:48) ,δ ) NZLB NZLB i j i 2,NZLB j 2,NZLB j j=1 52

(k) (k) (k) and set y = y (ξ ), π = π (ξ ), and r = r (ξ ). (ii) When we NZLB NZLB i NZLB NZLB i NZLB NZLB i assume that ZLB binds, given the values of φ , φ and r = 0, solve 1 2 ZLB φ(cid:48) = π+φ +β−1φ 2,ZLB 2 1 φ(cid:48) = σ−1κφ(cid:48) +σ−1λy +β−1φ 1,ZLB 2,ZLB ZLB 1 N (cid:88) (cid:104) (cid:105) π −κy − p(δ |δ ) κy(k−1)(φ(cid:48) ,φ(cid:48) ,δ )+π(k−1)(φ(cid:48) ,φ(cid:48) ,δ ) = 0 ZLB ZLB j i 1,ZLB 2,ZLB j 1,ZLB 2,ZLB j j=1 N (cid:88) (cid:104) (cid:105) −σy +δ +β p(δ |δ ) σy(k−1)(φ(cid:48) ,φ(cid:48) ,δ )+π(k−1)(φ(cid:48) ,φ(cid:48) ,δ ) = 0 ZLB i j i 1,ZLB 2,ZLB j 1,ZLB 2,ZLB j j=1 for (y ,π ,φ(cid:48) ,φ(cid:48) ) and set y = y (k) (ξ ) and π = π (k) (ξ ). ZLB ZLB 1,ZLB 2,ZLB ZLB ZLB i ZLB ZLB i When we solve the problem on a computer, we discretize a rectangle of the state space of (φ ,φ ). We use 21 points for each state variable. We set φ ∈ [−0.002,0] and φ ∈ 1 2 1 2 [−0.005,0.009], and divide the state space by evenly spaced grid points. We use piecewiselinear functions to approximate the policy functions off the grid points. Figure15showstheimpulseresponseoftheresidualfunctions(e ,e ,e ,e ). LM,1 LM,2 LM,3 LM,4 Note that (e ,e ) (the FONCs) hold with equality (up to the machine precision), as LM,1 LM,2 we use these equations to substitute variables other than the ones we solve for with the other equations. Figure 15: Euler errors: LM approach. e e 10-12 LM,1 10-12 LM,2 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 2 4 6 8 10 2 4 6 8 10 time time e e 10-9 LM,3 10-8 LM,4 1.5 0 1 -0.5 0.5 0 -1 2 4 6 8 10 2 4 6 8 10 time time 53

F.2 Promised-value approach F.2.1 Recursive formulation We substitute out η = π, η(cid:48) = π(cid:48), η = y, and η(cid:48) = y(cid:48). For each δ ∈ D, the problem 1 1,j j 2 2,j j i for the optimal commitment policy planner can be written as N w(y ,π ,δ ) = max − λ y2− 1 π2+β (cid:88) p(δ |δ )w (cid:0) y(cid:48),π(cid:48),δ (cid:1) i i i y,π,r,{π(cid:48),y(cid:48)} 2 2 j i j j j j j j=1 subject to y = y i π = π i N σy = (cid:88) p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48)(cid:3) −r+δ j i j j i j=1 N (cid:88) π = κy+β p(δ |δ )π(cid:48) j i j j=1 Let the Lagrange multipliers on the constraints ω and ω . The FONCs are given as follows: 1 2 ∂y(cid:48) :−σω +βw (cid:0) y(cid:48),π(cid:48),δ (cid:1) = 0 j 1 1 j j j ∂π(cid:48) :−ω −βω +βw (cid:0) y(cid:48),π(cid:48),δ (cid:1) = 0 j 1 2 2 j j j N ∂ω :σy+r−δ − (cid:88) p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48)(cid:3) = 0 1 i j i j j j=1 N (cid:88) ∂ω :π−κy−β p(δ |δ )π(cid:48) = 0 2 j i j j=1 forj = 1,...,N,wherew (y(cid:48),π(cid:48),δ ) = ∂w(y(cid:48),π(cid:48),δ )/∂y(cid:48) andw (y(cid:48),π(cid:48),δ ) = ∂w(y(cid:48),π(cid:48),δ )/∂π(cid:48). 1 j j j j j j j 2 j j j j j j j (cid:110) (cid:111)N Notethattheseequationsholdateachstatej duetothestate-contingentpromises y(cid:48),π(cid:48) . j j j=1 By using the envelope theorem and noting y = y and π = π , i i w (y ,π ,δ ) = −λy +σω −κω 1 i i i i 1 2 w (y ,π ,δ ) = −π +ω 2 i i i i 2 for i = 1,...,N. Then we have −λy(cid:48) −β−1σω +σω(cid:48) −κω(cid:48) = 0 j 1 1 2 −π(cid:48) −β−1ω +ω(cid:48) −ω = 0 j 1 2 2 for j = 1,...,N. As an important reminder, these equations yield a total of 2N FONCs since there are N states. The following KKTCs must be satisfied as well ω r = 0, ω ≤ 0, and r ≥ 0. 1 1 54

F.2.2 Time iteration method with simulated grid Weexplicitlyconsideravectorofpolicyfunctionsς˜(ξ˜) = [{y(cid:48)(ξ˜),π(cid:48)(ξ˜)}N ,r(ξ˜),ω (ξ˜), i j i j i j=1 i 1 i ω (ξ˜)](cid:48) as functions of the state variables ξ˜ = (y,π,δ ) for i = 1,...,N. Then we have the 2 i i i following a system of functional equations e (ξ˜) ≡−λy(cid:48)(ξ˜)+σω (y(cid:48)(ξ˜),π(cid:48)(ξ˜),δ )−σβ−1ω (ξ˜)−κω (y(cid:48)(ξ˜),π(cid:48)(ξ˜),δ ) = 0, PV,1,j i j i 1 j i j i j 1 i 2 j i j i j for j = 1,...,N, e (ξ˜) ≡−π(cid:48)(ξ˜)−β−1ω (ξ˜)+ω (y(cid:48)(ξ˜),π(cid:48)(ξ˜),δ )−ω (ξ˜) = 0, for j = 1,...,N, PV,2,j i j i 1 i 2 j i j i j 2 i N (cid:88) (cid:104) (cid:105) e (ξ˜) ≡r(ξ˜)+σy−δ − p(δ |δ ) σy(cid:48)(ξ˜)+π(cid:48)(ξ˜) = 0, PV,3 i i j j i j i j i j=1 N (cid:88) e (ξ˜) ≡π−κy−β p(δ |δ )π(cid:48)(ξ˜) = 0. PV,4 i j i j i j=1 We solve for the Ramsey equilibrium in a way that captures the full range of plausible values for y and π. Given the set of parameter values and the shock process we assume, some pairs (y,π) in a rectangle state space (as in Section F.1) may not be plausible in the Ramsey equilibrium. This makes solving for the policy and value functions with the rectangle state space impossible. In order to circumvent this problem, we adapt the approach of Maliar and Maliar (2015) (hereafter MM). That is, we solve for the policy functions on simulated grid points based on ergodic distribution of {y ,π ,δ }, which are presumably included in the t t t distribution of plausible promised pairs. EDS algorithm As in MM, we merge the simulation-based sparse grid and the time iteration method by the following steps: 1. Initialization: (a) Choose initial values ξ˜ = (y ,π ,δ ) and simulation length, T. 0 0 0 0 (b) Draw a sequence of {δ }T where δ ∈ D and fix the sequence throughout the t t=1 t iterations. (c) Choose approximating policy functions ς˜(ξ˜;θ) and make an initial guess of θ, i where θ is a vector of coefficients on a polynomial. 2. Construction of an EDS grid (a) Given {δ }T , use ς˜(ξ˜;θ) to simulate {y ,π }T . t t=1 i t t t=1 (b) Construct an EDS grid Γ(δ ) ≡ {y ,π ;δ }Mi for each i = 1,...,N. i m m i m=1 3. Computation of a solution on EDS grid, ς˜(ξ˜;θ), using the time iteration method i (a) Make an initial guess for the policy function ς˜(0). (b) Fork = 1,2,...(k isanindexforthenumberofiteration),giventhepolicyfunction 55

previously obtained ς˜(k−1), solve −λy(cid:48) +σω (k−1) (y(cid:48),π(cid:48),δ ;θ)−σβ−1ω −κω (k−1) (y(cid:48),π(cid:48),δ ;θ) = 0, for j = 1,...,N, j 1 j j j 1 2 j j j −π(cid:48) −β−1ω +ω (k−1) (y(cid:48),π(cid:48),δ ;θ)−ω = 0, for j = 1,...,N, j 1 2 j j j 2 N r+σy−δ − (cid:88) p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48)(cid:3) = 0, i j i j j j=1 N (cid:88) π−κy−β p(δ |δ )π(cid:48) = 0, j i j j=1 for ({y(cid:48),π(cid:48)}N ,r,ω ,ω ). j j j=1 1 2 (c) Update the policy function by setting y(cid:48) = y (cid:48)(k) (ξ˜) for j = 1,...,N, π(cid:48) = π (cid:48)(k) (ξ˜) j j i j j i for j = 1,...,N, r = r(k)(ξ˜), ω = ω (k) (ξ˜), ω = ω (k) (ξ˜). i 1 1 i 2 2 i (cid:13) (cid:13) (d) Repeat 2-3 until (cid:13)ς˜(k)−ς˜(k−1) (cid:13) is small enough. 4. Repeat 2-3 until convergence of the EDS grid. In Step 2, we construct an EDS grid Γ(δ ) indexed by δ ∈ {δ ,δ } (we assume N = 2 and i i n c δ ∈ {δ ,δ }hereafter)fromanessentiallyergodicset{y ,π ,δ }T . Giventhepolicyfunction i n c t t t t=1 ς˜(s ;θ) and the sequence of {δ }T , we first simulate the economy to obtain an essentially i t t=1 ergodic set. As we assume the normal state is absorbing, i.e., p(δ |δ ) = p(δ |δ ) = 0, in c c c n order to obtain samples in the crisis state, {δ }T is not necessarily consistent with the t t=1 true stochastic process. In other words, the ergodic set we obtain here is quasi-ergodic (see Figure 18). In constructing an EDS grid from the ergodic set, we do the following two step procedure (see MM for more details): 1. Selecting points within an essentially ergodic set (called Algorithm Aη in MM) 2. Constructing a uniformly spaced set of points that covers the essentially ergodic set (called Algorithm P(cid:15) in MM) There are two important parameters in this two step procedure: The interval of sampling, ι, and the threshold of density, (cid:15). We set these parameters depending on the exogenous state variable δ . We set (ι ,ι ) = (5,1) and ((cid:15) ,(cid:15) ) = (0.001,0.000001), considering the ergodic i n c n c set has fewer number of samples in the crisis state. The number of grid points is set to M = M = 40. n c In Step 3, as in the LM approach, we use the following indicator function approach. That is, for ς˜∈ {y(cid:48),π(cid:48),y(cid:48),π(cid:48),ω ,ω }, n n c c 1 2 (cid:16) (cid:17) ς˜(ξ˜) = I ς˜ (ξ˜)+ 1−I ς˜ (ξ˜). i rNZLB(ξ˜ i)≥0 NZLB i rNZLB(ξ˜ i)≥0 ZLB i Then, in Steps 3(b) and 3(c), the problem becomes finding a pair of policy functions, (cid:16) (cid:17) ς˜ (ξ˜),ς˜ (ξ˜) , as follows (we denote a tuple of variables (y(cid:48) , π(cid:48) , y(cid:48) , NZLB i ZLB i n,NZLB n,NZLB c,NZLB π(cid:48) , ω , ω , r ) in the non-ZLB regime and (y(cid:48) , π(cid:48) , y(cid:48) , π(cid:48) , c,NZLB 1,NZLB 2,NZLB NZLB n,ZLB n,ZLB c,ZLB c,ZLB ω , ω , r ) in the ZLB regime): (i) When we assume that ZLB does not bind, 1,ZLB 2,ZLB ZLB 56

given the values of y, π and ω = 0, we solve 1,NZLB −λy(cid:48) +σω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−σβ−1ω −κω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ) = 0, n,NZLB 1 n,NZLB n,NZLB n 1 2 n,NZLB n,NZLB n −λy(cid:48) +σω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−σβ−1ω −κω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ) = 0, c,NZLB 1 c,NZLB c,NZLB c 1 2 c,NZLB c,NZLB c −π(cid:48) +ω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−ω = 0, n,NZLB 2 n,NZLB n,NZLB n 2,NZLB −π(cid:48) +ω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−ω = 0, c,NZLB 2 c,NZLB c,NZLB c 2,NZLB π−κy−β (cid:2) p(δ |δ )π(cid:48) +p(δ |δ )π(cid:48) (cid:3) = 0, n i n,NZLB c i c,NZLB for (y(cid:48) ,π(cid:48) ,y(cid:48) ,π(cid:48) ,ω ) and n,NZLB n,NZLB c,NZLB c,NZLB 2,NZLB r = −σy+δ +p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48) (cid:3) +p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48) (cid:3) NZLB i n i n,NZLB n,NZLB c i c,NZLB c,NZLB andset0 = ω (k) (ξ˜),ω = ω (k) (ξ˜),andr = r (k) (ξ˜). (ii)Whenweassume 1,NZLB i 2,NZLB 2,NZLB i NZLB NZLB i that ZLB binds, given the values of y, π and r = 0, we solve ZLB −λy(cid:48) +σω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−σβ−1ω −κω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ) = 0, n,ZLB 1 n,ZLB n,ZLB n 1,ZLB 2 n,ZLB n,ZLB n −λy(cid:48) +σω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−σβ−1ω −κω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ) = 0, c,ZLB 1 c,ZLB c,ZLB c 1,ZLB 2 c,ZLB c,ZLB c −π(cid:48) +ω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−ω = 0, n,ZLB 2 n,ZLB n,ZLB n 2,ZLB −π(cid:48) +ω (k−1) (y(cid:48) ,π(cid:48) ,δ ;θ)−ω = 0, c,ZLB 2 c,ZLB c,ZLB c 2,ZLB −σy+δ +p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48) (cid:3) +p(δ |δ ) (cid:2) σy(cid:48) +π(cid:48) (cid:3) = 0, i n i c,ZLB n,ZLB c i c,ZLB c,ZLB π−κy−β (cid:2) p(δ |δ )π(cid:48) +p(δ |δ )π(cid:48) (cid:3) = 0, n i n,ZLB c i c,ZLB for (y(cid:48) ,π(cid:48) ,y(cid:48) ,π(cid:48) ,ω ,ω ) and set ω = ω (k) (ξ˜) and ω = ω (k) (ξ˜). n,ZLB n,ZLB c,ZLB c,ZLB 1 2 1 1,ZLB i 2 2,ZLB i We use second-order polynomials to approximate the policy functions off the grid points. That is, we fit a second-order polynomial, ς˜(ξ˜;θ) = θ +θ y +θ π +θ y2 +θ y π +θ π2 i i,(0,0) i,(1,0) m i,(0,1) m i,(2,0) m i,(1,1) m m i,(0,2) m (cid:16) (cid:17) foreachvariable y(cid:48) ,π(cid:48) ,y(cid:48) ,π(cid:48) ,ω ,ω ,r onthegridpointsΓ(δ ) ={y ,π ;δ }Mi . n,l n,l c,l c,l 1,l 2,l l i m m i m=1 l∈{NZLB,ZLB} Whenwefitpolynomials,weusetheLS-SVDalgorithmbyfollowingJudd,Maliar,andMaliar (2011) to avoid potential multicollinearlity problems. Figure 16 shows the impulse response of the residual functions (e , e , e , PV,1,n PV,1,c PV,2,n e , e , e ). Note that (e ,e ) (the consumption Euler equation and NKPC) PV,2,c PV,3 PV,4 PV,3 PV,4 holdwithequality(uptomachineprecision),asweusetheseequationstosubstitutevariables other than the ones we solve for with the other equations. 57

Figure 16: Euler errors: PV approach. e e e PV,1,N PV,2,N PV,3 -5 -3 -12 10 10 10 20 1 1 15 0 0.5 10 -1 0 5 -2 -0.5 0 -3 -1 5 10 5 10 5 10 time time time e e e PV,1,C PV,2,C PV,4 -4 -12 10 10 15 0.02 1 10 0.5 0 5 0 -0.02 0 -0.5 -5 -0.04 -1 5 10 5 10 5 10 time time time Figure 17 shows the difference of the impulse response of inflation, output and the policy rate under LM approach and PV approach. Figure 17: Difference in dynamics between LM and PV approaches. y -y - r -r LM PV LM PV LM PV 10 -3 0.01 2 0.1 1 0 0.05 0 -0.01 0 -1 -0.02 -2 -0.05 5 10 5 10 5 10 time time time Note: The rate of inflation and the policy rate are expressed in annualized percent. The output gap is expressed in percent. 58

Figure 18: Quasi-ergodic distribution and EDS grid of (y,π) G Additional results for the model with inflation bias An interesting feature of the Ramsey outcome in the model with inflation bias is that inflation and the output gap eventually converge to zero. To better appreciate this feature, it is useful to contrast this convergent point with two time-invariant pairs of inflation and the output gap. The first pair is the one that prevails in the Markov perfect equilibrium. The second pair is the one that maximizes the time-one value. The analysis of Markov perfect equilibriuminthemodelwithinflationbiashasbeenstudiedbymany. Thevalue-maximizing pairofconstantinflationandoutputisstudiedinWolman(2001)inasticky-pricemodelwith inflation bias.16 G.1 Markov perfect policy in the model with inflation bias The problem of the discretionary central bank is to choose {y ,π }, taking as given the t t future value, V , and inflation, π : t+1 t+1 1 V = max − [π2+λ(y −y∗)2]+βV t yt,πt 2 t t t+1 subject to the Phillips curve constraint. The Markov perfect equilibrium in the model with inflation bias is given by a set of time-invariant value and policy functions that solve this 16Wolman(2001)analyzesafullynonlinearmodelanddidnotprovideanalyticalresults,whereaswestudy a semi-loglinear model that permits analytical results. 59

Bellman equation and is denoted by {V , y , π , r }. They are given by MP MP MP MP (1−β)λ2 y = y∗ MP κ2λ+(1−β)λ2 λκ π = y∗ MP κ2+(1−β)λ 1 V = u(y ,π ) MP MP MP 1−β Proof: Let φ be the Lagrange multiplier on the Phillips curve constraint. Then, ∂V : 0 = −λ(y−y∗)−κφ ∂y ∂V : 0 = −π+φ. ∂y These equations imply κ y = y∗− π λ Putting this into the Phillips curve, κ π = κ(y∗− π)+βπ(cid:48). λ In equilibrium, π = π(cid:48), which we can call π . Thus, MP κ π = κ(y∗− π )+βπ MP MP MP λ κ2 ⇐⇒ (1+ −β)π = κy∗ MP λ κ2+(1−β)λ ⇐⇒ π = κy∗ MP λ λκ ⇐⇒ π = y∗. MP κ2+(1−β)λ With this π , MP κ y = y∗− π MP MP λ κ λκ = y∗− y∗ λκ2+(1−β)λ λκ2 = y∗− y∗ κ2λ+(1−β)λ2 λκ2 = (1− )y∗ κ2λ+(1−β)λ2 (1−β)λ2 = y∗ κ2λ+(1−β)λ2 60

Figure 19: Dynamics —Model with Inflation Bias— y w 0.5 0.8 0 0.4 0.6 -0.5 0.3 0.4 0.2 -1 Ramsey 0.2 0.1 Markov perfect Value-maximizing 0 0 -1.5 1 10 20 1 10 20 1 10 20 time time time Note: The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. The pair of output and inflation consistent with the Markov perfect equilibrium is shown by the dashed lines in Figure 19. As is well known in the literature, the discretionary central bank that takes the expected inflation as given will try to increase output by raising inflation today. A higher inflation today in turn worsens the inflation-output trade-off for the central bank in the previous period. In equilibrium, the economy ends up with positive inflation and output that is positive, but below y∗. G.2 A value-maximizing pair of constant inflation and output gap The value-maximizing pair of constant inflation and output gap, denoted by (π ,y ), VM VM is the pair of constant inflation and output gap that maximize the time-one value. That is, (π ,y ) := argmax V VM VM y,π 1 where the optimization is subject to π = κy+βπ. It is straightforward to show that κλ(1−β) π = y∗, VM κ+λ(1−β)2 π y = y∗−κ VM , VM λ(1−β) V = u(π ,y ) VM VM VM 61

Proof: Let φ be the Phillips curve constraint. Then, ∂V : 0 = −λ(y−y∗)−κφ ∂y ∂V : 0 = −π+φ(1−β) ∂y The second equation means π φ = 1−β 0 = −λ(y−y∗)−κφ π ⇐⇒ 0 = −λ(y−y∗)−κ 1−β π ⇐⇒ λ(y−y∗) = −κ 1−β π ⇐⇒ y = y∗−κ λ(1−β) (1−β)π = κy π ⇐⇒ (1−β)π = κ = κy∗−κ2 λ(1−β) κ ⇐⇒ [(1−β)+ ]π = κy∗ λ(1−β) κ+λ(1−β)2 ⇐⇒ π = κy∗ λ(1−β) κλ(1−β) ⇐⇒ π = y∗ κ+λ(1−β)2 The value-maximizing pair of output and inflation is shown by the dash-dotted lines in Figure 19. Because of the presence of y∗ > 0, the value maximizing pair features positive inflation and a positive output gap, as in the Markov perfect equilibrium. However, the magnitudes are much smaller than under the Markov perfect equilibrium. As shown in Figure 20, the time-one value associated with the value-maximizing pair is by construction lower than the time-one value under the Ramsey equilibrium. However, once inflation and output converge under the Ramsey equilibrium, the Ramsey value is lower than the value associated with the value-maximizing pair. H Markov perfect policy in the model with stabilization bias In the discussion of the optimal commitment policy for the model with stabilization bias, we contrasted the allocations under the optimal commitment policy to those under the Markov perfect policy to describe the benefit of commitment. In this section, we formulate the problem of the discretionary central bank and solve for the Markov perfect equilibrium. For each s ∈ S, the problem of the discretionary central bank is to choose {y ,π }, taking t t t as given the future value function, V , and the future policy function for inflation, π . t+1 t+1 62

Figure 20: Dynamics —Model with Inflation Bias— w -0.197 Ramsey Value-maximizing -0.198 -0.199 -0.2 1 10 20 time Note: The rate of inflation is expressed in annualized percent. The output gap is expressed in percent. That is, 1 V (s ) = max − [π2+λy2]+βE V (s ) (69) t t yt,πt 2 t t t t+1 t+1 subject to the Phillips curve constraint. The Markov perfect equilibrium in the model with stabilization bias is given by a set of time-invariant value and policy functions that solves this Bellman equation and is denoted by {V (·), y (·), π (·), r (·)}. For the simple MP MP MP MP shockcaseconsideredinthemaintext, thesolutioncanbefoundanalytically. Forthenormal state, we have y (e ) = 0 MP n π (e ) = 0 MP n V (e ) = 0 MP n because the normal state is an absorbing state. For the high state when the cost-push shock hits the economy, we have κe h y (e ) = − MP h λ+κ2 e h π (e ) = MP h 1+ κ2 λ V (e ) = u(y (e ),π (e )) MP h MP h MP h I Markov perfect policy in the model with the ELB In the discussion of the optimal commitment policy for the model with the ELB, we contrasted the allocations under the optimal commitment policy to those under the Markov perfect policy to describe the benefit of commitment. In this section, we formulate the 63

problem of the discretionary central bank and solve for the Markov perfect equilibrium. The problem of the discretionary central bank is to choose {y ,π }, taking as given the t t future value (V ) and inflation π : t+1 t+1 1 V (s ) = max − [π2+λy2]+βE V (s ) t t yt,πt 2 t t t t+1 t+1 subjecttothe Eulerequationand Phillips curve constraints. TheMarkov perfectequilibrium in the model with the ELB is given by a set of time-invariant value and policy functions that solves this Bellman equation and is denoted by {V (·), y (·), π (·), r (·)}. MP MP MP MP For the two-state shock case considered in the main text, the Markov Perfect equilibrium can be characterized analytically. For the normal state, we have y (δ ) = 0 MP n π (δ ) = 0 MP n r (δ ) = r∗ MP n V (δ ) = 0 MP n because of the absorbing state assumption. For the crisis state, we have r∗+s t y (δ ) = MP c σ r∗+s t π (δ ) = κ MP c σ r (δ ) = 0 MP c V (δ ) = u(y (δ ),π (δ )). MP c MP c MP c 64