Reputation and liquidity traps

Accepted Manuscript Reputation and Liquidity Traps

Taisuke Nakata

PII: DOI: Reference:

S1094-2025(17)30072-8 http://dx.doi.org/10.1016/j.red.2017.09.001 YREDY 839

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Review of Economic Dynamics

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Please cite this article in press as: Nakata, T. Reputation and Liquidity Traps. Review of Economic Dynamics (2017), http://dx.doi.org/10.1016/j.red.2017.09.001

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Reputation and Liquidity Traps∗ Taisuke Nakata† Federal Reserve Board First Draft: March 2013 This Draft: September 2017 Abstract Can the central bank credibly commit to keeping the nominal interest rate low for an extended period of time in the aftermath of a deep recession? By analyzing credible plans in a sticky-price economy with occasionally binding zero lower bound constraints, I find that the answer is yes if contractionary shocks hit the economy with sufficient frequency. In the best credible plan, if the central bank reneges on the promise of low policy rates, it will lose reputation and the private sector will not believe such promises in future recessions. Even with a very small probability of future contractionary shocks, the incentive to maintain reputation outweighs the short-run incentive to close consumption and inflation gaps and keeps the central bank on the originally announced path of low nominal interest rates. JEL: E32, E52, E61, E62, E63 ∗

I would like to thank the Editor, two anonymous referees, Klaus Adam, Roberto Chang, John Cochrane, Gauti Eggertsson, George Evans, Mark Gertler, Keiichiro Kobayashi, Takushi Kurozumi, Olivier Loisel, Nick Moschovakis, Toshihiko Mukoyama, Juan Pablo Nicolini, Georgio Primiceri, Sebastian Schmidt, Takeki Sunakawa, Pierre Yared, my colleagues at the Federal Reserve Board, and seminar participants at 16th Macro Conference at Keio University, 2014 Annual Meeting of Society for Economic Dynamics, 2015 European Economic Association Meeting, 2015 North American Winter Meetings of the Econometric Society, 2015 Spring Midwest Macro, 2015 World Congress, 9th Dynare Conference, 20th International Conference on Computing in Economics and Finance, Bank of Japan, Bank of England, European Central Bank, Federal Reserve Board of Governors, NYU Alumni Conference, Philadelphia Fed, Sargent Reading Group Alumni Conference, St. Louis Fed, and University of Tokyo for thoughtful comments and/or helpful discussions. Timothy Hills and Paul Yoo provided excellent research assistance. The views expressed in this paper, and all errors and omissions, should be regarded as those of the author, and are not necessarily those of the Federal Reserve Board of Governors or the Federal Reserve System. † Division of Research and Statistics, Board of Governors of the Federal Reserve System, 20th Street and Constitution Avenue N.W., Washington, D.C. 20551; Email: [email protected].

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Keywords: Commitment, Credible Policy, Forward Guidance, Liquidity Trap, Reputation, Sustainable Plan, Time Consistency, Trigger Strategy, Zero Lower Bound.

1

Introduction Statements about the period during which the short-term nominal interest rate

is expected to remain near zero were an important feature of recent monetary policy in the United States. The FOMC stated that a highly accommodative stance of monetary policy will remain appropriate for a considerable time after the economic recovery strengthens.1 With the current policy rate at its effective lower bound, the expected path of short-term rates is a prominent determinant of the long-term interest rates, which affects the decisions of households and businesses. Thus, the statement expressing the FOMC’s intention to keep the policy rate low for a considerable period likely did much to keep the long-term nominal rates low, thereby stimulating economic activities. Some policymakers and economists have debated whether these statements should be interpreted as a commitment to optimal time-inconsistent policy.2 In the New Keynesian model—a widely-used model of monetary policy at central banks—in response to a large contractionary shock, the central bank equipped with commitment technology promises to keep the nominal interest rate low even after the contractionary shock disappears. Such a promise reduces the long-term real interest rate and stimulates household spending.3 However, in the model, if the central bank were to re-optimize again after the shock disappears, it would renege on the promise and raise the rate to close consumption and inflation gaps. In other words, the policy of an extended period of low nominal interest rates is time-inconsistent. In reality, no central bank has an explicit commitment device to bind its future policy decisions. Thus, while the theory of optimal commitment policy can explain why the central bank should promise an extended period of low policy rates, it can neither explain why the central bank should fulfill such a promise, nor why the private sector should believe it. This paper provides a theory that explains why the central bank may want to fulfill the promise of keeping the nominal interest rate low even after the economic recovery strengthens. The theory is based on credible plans in a stochastic New Keynesian 1

See the FOMC statements from September 2012 to January 2014. For alternative perspectives on the degree of commitment implied by the FOMC’s statement, see Bullard (2013), Dudley (2013), and Woodford (2012). 3 See, for example, Eggertsson and Woodford (2003); Jung, Teranishi, and Watanabe (2005); Adam and Billi (2006); and Werning (2012). 2

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economy in which the nominal interest rate is subject to the zero lower bound (ZLB) constraint and contractionary shocks hit the economy occasionally. Credible plans can capture rich interactions between the government action and the private sector’s belief. I use this equilibrium concept to ask under what conditions, if any, the policy of keeping the nominal interest rate low even after the economic recovery strengthens is time-consistent.4 I find that the policy of keeping the nominal interest rate low for long can be made time-consistent by a particular trigger-type strategy—the revert-to-discretion plan—if the frequency of contractionary shocks is sufficiently high. In the revert-to-discretion plan, if the central bank reneges on its promise to keep the nominal interest rate low, it will lose reputation and the private sector will never believe such promises in the face of future contractionary shocks. If the private sector does not believe the promise of an extended period of low nominal interest rates, the contractionary shock will cause large declines in consumption and inflation. Large consumption collapse and deflation in the future liquidity traps reduce welfare even during normal times because the central bank cares about the discounted sum of future utility flows. Thus, the potential loss of reputation gives the central bank an incentive to fulfill the promise. When the frequency of shocks is sufficiently high, this incentive to maintain reputation outweighs the short-run incentive to raise the rate to close consumption and inflation gaps, keeping the central bank on the originally announced path of low nominal interest rates. In my baseline parameterization intended to capture the severity of the Great Recession, the threshold crisis frequency above which the Ramsey outcome can be made time-consistent by the revert-to-discretion plan is very small (0.04 percent). In the United States, two large shocks pushed the policy rate to the effective lower bound over the past 100 years since the creation of the Federal Reserve System. Thus, a na¨ıve estimate of the crisis frequency parameter would be 0.5 percent (=2/400) at a quarterly frequency. The threshold crisis frequency of 0.04 under the baseline parameterization is comfortably below this na¨ıve estimate. Even when I reduce the power of reputation by making the punishment duration finite, the threshold crisis frequency remains small: When the punishment duration is 10 years, the threshold crisis frequency is 0.3 percent. In section 5, I consider how the threshold crisis frequency depends on alternative parameter choices, the introduction of cost-push shocks, and the use of a fully nonlinear model. I find that, for a wide range of parameter specifications and 4

The equilibrium concept has been referred to with many different names, including sustainable plans, reputational equilibria, sequential equilibria, and subgame Markov-perfect equilibria.

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model variants, the threshold crisis frequency remains very small. The recent recession has shown that the ZLB can be a binding constraint in many advanced economies. Some argue that the ZLB is likely to bind more frequently in the future than in the past.5 Accordingly, developing effective strategies to mitigate the adverse consequences of the ZLB is an important task for macroeconomists. While many researchers have shown theoretically the value of optimal commitment policy in limiting the adverse consequences of the ZLB constraints, there are concerns about the effectiveness of adopting this policy in reality, partly based on the notion that it is time-inconsistent.6 The result of this paper—reputational force, combined with a very small crisis probability, can make the commitment policy time-consistent—can be interpreted as alleviating such concerns. This paper is related to the literature examining whether reputation can make the Ramsey equilibrium time-consistent in various macroeconomic models. In early contributions, Barro and Gordon (1983) and Rogoff (1987) asked whether reputation can overcome inflation bias in models with a short-run trade-off between inflation and output. Chari and Kehoe (1990), Phelan and Stacchetti (2001), and Stokey (1991) have studied the time-consistency of the Ramsey policy in models of fiscal policy, while Chang (1998), Ireland (1997), and Orlik and Presno (2013) have studied the time-consistency of the Friedman rule in monetary models. More recently, Kurozumi (2008), Loisel (2008) and Sunakawa (2015) studied the time-consistency of the Ramsey policy in New Keynesian models with cost-push shocks but without the ZLB constraint. The paper is closely related to other works examining how the government can improve allocations at the ZLB in the absence of commitment technology. Eggertsson (2006) and Bhattarai, Eggertsson, and Gafarov (2013) have showed that, if the government has access to nominal debt, it chooses to issue nominal debt during the period of contractionary shocks so as to give the future government an incentive to lower the nominal interest rate and create inflation, and that this goes a long way toward achieving the Ramsey allocations. Jeanne and Svensson (2007) demonstrated that, if the government has concerns about its balance sheet, it can attain the Ramsey allocations by managing the balance sheet so as to give the future government an incentive to depreciate its currency and thus create inflation. This paper contributes to this body of work by proposing a new mechanism by which the central bank can attain the Ramsey allocations without a commitment technology. The proposed mechanism 5

See, for example, Ball (2013). See Nakata (2015), a non-technical summary of this paper, for a list of speeches in which policymakers pointed out the time-inconsistency of the Ramsey policy as the reason why they did not adopt the Ramsey policy at the ZLB. 6

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in this paper is novel in that it does not involve any additional policy instruments. The paper is also related to Bodenstein, Hebden, and Nunes (2012) who study the consequence of imperfect credibility in the context of a New Keynesian economy with occasionally binding ZLB constraints. While both papers are motivated by the idea that credibility of the central bank may be a key factor in understanding the effectiveness of the forward guidance policy, our approaches and the questions we ask are different. Their analysis is positive. They model imperfect credibility in a specific way—randomizing the timing of central bank’s optimization in a way reminiscent of the Calvo-pricing model—and ask how imperfect credibility affects output and inflation at the ZLB. On the other hand, my analysis is normative. I ask why the central bank may want to fulfill the promise and under what conditions the Ramsey outcome can prevail. The rest of the paper is organized as follows. Section 2 describes the model, defines the competitive equilibria, the discretionary and the Ramsey outcomes, and discusses their key features. Section 3 defines a plan and credibility, and the revert-to-discretion plan that induces the Ramsey outcome. Section 4 presents the main results on the credibility of the Ramsey outcome, and section 5 explores the quantitative implications of the model. Section 6 discusses additional results and a final section concludes.

2

Model and competitive outcomes

The model is given by a standard New Keynesian economy. The environment features a representative household, monopolistic competition among a continuum of intermediate-goods producers, and sticky prices. The model abstracts from capital. Since this is a workhorse model, I will start with the well-known equilibrium conditions of the economy.7 Following the majority of the previous literature on the ZLB, I will conduct the analysis in a partially log-linearized version of the model, in which the equilibrium conditions are log-linearized except for the ZLB constraint on the nominal interest rate.8 The only exogenous variable of the model is st , interpreted as “the natural rate of interest.” I will also refer to st as the contractionary shock, the crisis shock, or the state. st takes two values, H and L. H will be set to the steady-state real interest rate, and L will be assigned to a negative value so that the nominal interest rate that would 7

See, for example, Woodford (2003). See, for example, Cochrane (2016); Eggertsson and Woodford (2003); Jung, Teranishi, and Watanabe (2005); and Werning (2012). 8

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keep inflation and consumption at the steady-state level is negative. When st = H, the economy is said to be in the high state or the normal state. When st = L, I will say that the economy is in the low state, or the economy is hit by the contractionary or crisis shock. A lower L will be interpreted as the shock being more severe. I will use st to denote a history of states up to period t (i.e. st := {sk }tk=1 ) and S to denote the set of values st can take, i.e., S := {L, H}. The natural rate of interest rate evolves according to a two-state Markov process. Transition probabilities are given by Prob(st+1 = L|st = H) = pH

(1)

Prob(st+1 = L|st = L) = pL

(2)

pH is the probability of moving to the low state next period when the economy is in the normal state today, and will be referred to as the frequency of the contractionary shocks. pL is the probability of staying in the low state when the economy is in the low state today, and will be referred to as the persistence of the contractionary shocks. One key exercise of this paper will be to examine the credibility of the Ramsey policy in various economies with different values of pH and pL . I refer to the state-contingent sequence of consumption, inflation, and the nominal interest rate, {ct (st ), πt (st ), rt (st )}∞ t=1 , as an outcome. Given a process of st , an outcome is said to be competitive if, for all t ≥ 1 and st ∈ St , (i) ct (st ) ∈ R, πt (st ) ∈ R, rt (st ) ∈ R≥0 where R denotes a set of real numbers and R≥0 denotes a set of nonnegative real number and (ii)   χc ct (st ) = χc Et ct+1 (st+1 ) − rt (st ) − Et πt+1 (st+1 ) + st πt (st ) = κct (st ) + βEt πt+1 (st+1 )

(3) (4)

where ct (st ) and πt (st ) are consumption and inflation expressed as the log deviation from the deterministic steady-state and rt (st ) is the nominal interest rate.9 χc is the inverse intertemporal elasticity of substitution of the representative household, and κ is the slope of the Phillips curve. The ZLB constraint on the nominal interest rate is imposed by the restriction that rt (st ) ∈ R≥0 . 9

In the model without investment and government spending, consumption equals output. It is common in the New Keynesian literature to replace consumption with output in the Euler equation. However, I will depart from the common practice in presenting the model. Formulating credible plans requires us to specify who chooses what and when, and it is more natural to think of the household as choosing consumption instead of output.

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The government’s objective function at period t is given by wt (s ) := Et t

∞ 

β j u(πt+j (st+j ), ct+j (st+j ))

(5)

j=0

where the utility flow at each period is given by the following function.  1 2 2 u(π, c) := − π + λc 2

(6)

This objective function can be obtained as the second-order approximation to the household’s welfare.10 For any outcome, there is an associated state-contingent sequence of values, {wt (st )}∞ t=1 , which will be referred to as the value sequence.

2.1

The discretionary outcome

At each time t ≥ 1, the discretionary government chooses today’s consumption, inflation, and nominal interest rate in order to maximize its objective function, taking as given the value function and policy functions for consumption, inflation, and the nominal interest rate in the next period.11 The Bellman equation of the government’s problem is given by wt (st ) =

max

{ct ∈R,πt ∈R,rt ∈R≥0 }

u(ct , πt ) + βEt wt+1 (st+1 )

(7)

where the optimization is subject to the equations characterizing the competitive equilibria (i.e., equations (3) and (4)). Let {wd (·), cd (·), πd (·), rd (·)} be the set of timeinvariant value function and policy functions for consumption, inflation, and the nominal interest rate that solves this problem in which the ZLB only binds in the low state.12 The discretionary outcome is defined as, and denoted by, the state-contingent sequence 10

See, for example, Woodford (2003) and Gal´ı (2015). Following the literature, I assume that the discretionary government acts as a planner and chooses the policy instrument and allocations without being explicit about the within-period timing assumption of the government and the private sector. It is important to note that, by assuming that the planner picks the allocation, I implicitly abstract from the task of specifying how the government behaves in an off-equilibrium path if the private sector ever deviates from the equilibrium behavior. See Bassetto (2005) and Atkeson, Chari, and Kehoe (2010) for careful discussions on the undesirability of being silent on the government’s behavior in off-equilibrium paths in the context of the Ramsey equilibrium, and for an alternative framework in which the government’s behavior is specified for all possible histories of actions by the private sector. 12 In online appendix D, I demonstrate the existence of a time-invariant solution to this discretionary government’s problem in which the ZLB binds in both states. See also Armenter (2016) and Nakata and Schmidt (2014) for extensive analyses of such deflationary Markov-perfect equilibrium. 11

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of consumption, inflation, and the nominal interest rate, {cd,t (st ), πd,t (st ), rd,t (st )}∞ t=1 t t t such that cd,t (s ) := cd (st ), πd,t (s ) := πd (st ), and rd,t (s ) := rd (st ) and the discretionary t value sequence is defined as, and denoted by, {wd,t (st )}∞ t=1 such that wd,t (s ) := wd (st ). Figure 1 shows the discretionary outcome and value sequence for a particular realization of s10 ∈ S10 in which s1 = L, and st = H for 2 ≤ t ≤ 10. It also plots the sequence of contemporaneous utility, {u(cd,t , πd,t )}∞ t=1 associated with the consumption and inflation sequence. Parameter values are listed in table 1.13 The magnitude of the shock, L, is chosen to generate the decline in output in the low state under the discretionary outcome -7 percent to capture the severity of the Great Recession.14 In each panel, solid black and dashed red lines are for the economy with pH = 0.02 and pH = 0. As described in online appendix D, first-order necessary conditions of the discretionary government’s problem is given by a system of linear equations, and thus the discretionary outcome and values can be computed by linear algebra. Table 1: Baseline Parameter Values Parameter β χc κ λ H L pH pL

Description Discount rate Inverse IES The slope of the Phillips curve the relative weight on output volatility The natural rate of interest in the high state The natural rate of interest in the low state Crisis frequency Crisis persistence

Parameter Value 1 1+0.0075 ≈ 0.9925 1 0.012 0.0012 1 β − 1 (=0.0075) −0.0107∗ [0, 0.02] 0.8

*The value of L is chosen so that consumption is minus 7 percent in the low state under the discretionary outcome.

In the model without commitment, as soon as the contractionary shock disappears, 13

β is chosen so that the nominal interest rate is 3 percent in the normal state when pH = 0. χc = 1 is standard in the literature (see, for example, Erceg and Lind´e (2014) and Fern´ andez-Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ram´ırez (2015)). κ = 0.012 is consistent with the slope of the Phillips curve obtained from a loglinearized New Keynesian model with Calvo pricing in which the probability of firms unable to optimize their prices each period is 81 percent (81 percent is the estimate from a version of the medium-scale DSGE model of Smets and Wouters (2007), as reported by Del Negro, Giannoni, and Schorfheide (2015)). See online appendix K for a complete mapping between the reduced-form parameters of the semi-loglinear model and the structural parameters of the Calvo model. When the objective function is seen as the second-order approximation to the household’s welfare, λ = κ/θ, where θ is the elasticity of substitution among intermediate goods. With θ = 10, κ = 0.012 implies λ = 0.0012. Finally, H is chosen to be consistent with the value of β. 14 Boneva, Braun, and Waki (2016) similarly choose the magnitude of the shock to generate a 7 percent decline in the output gap in the crisis state. They also introduce another shock, a productivity shock, to the model and aim for a one-percentage point decline in inflation in the crisis state.

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Figure 1: The discretionary outcome and value sequence rt: Nominal Interest Rate (Ann. %) 4 2

πt: Inflation

ct: Consumption 5

1

0

0

−5

−1

0 0

5 Time u(πt,ct): Utility flow

10

−10 0

0

0

−0.5

−10

−1

−20

0

5 Time wt: Value

10

−2 0

5 Time

10

(pH,pL) = (0.02,0.8) (pH,pL) = (0,0.8)

5 10 0 5 10 Time Time Note: Solid blue vertical lines show the period with contractionary shocks.

the government raises the nominal interest rate in order to stabilize consumption and inflation. In the model where the high state is an absorbing state (i.e., pH =0), the government raises the nominal interest rate to H, and consumption and inflation are fully stabilized at zero at time 2. Accordingly, the contemporaneous utility is zero as well. In the model with a positive pH , the household and firms will have an incentive to lower consumption and prices in the normal period, as they expect that consumption and inflation will decline in some states tomorrow. The government tries to prevent those declines by reducing the nominal interest rates from the deterministic steady-state level, and in equilibrium, consumption and inflation are respectively slightly positive and negative. As a result, the contemporaneous utility flows are slightly negative. One key feature of the model with recurring shocks is that the discretionary value remain negative even after the shock disappears, as captured in the dashed red line bottom-left panel. The discretionary value stays negative even during the normal times for two reasons. First, consumption and inflation are slightly positive and negative due to the anticipation effects described above, pushing down contemporaneous utility flow below zero in the high state. Second, and more quantitatively importantly, the possibility that consumption and inflation will decline in response to the future contractionary shock tomorrow lowers the discretionary value by reducing the continuation value of the government. This is in a sharp contrast to the economy in which 9

the contractionary shock never hits after the initial shock. In such a model, the discretionary value becomes zero after the shock disappears, as shown in the red line in the bottom-right panel. This feature of the economy with recurring shocks—that the discretionary values remain negative even after the shock disappears—will be important in understanding the reputational force present in the credible plans.15

2.2

The Ramsey outcome

The Ramsey planner chooses a state-contingent sequence of consumption, inflation, and the nominal interest rate in order to maximize the expected discounted sum of future utility flows at time one. For each s1 ∈ S, the Ramsey planner’s problem is given by w1 (s1 ) (8) max (c(s1 ),π(s1 ),r(s1 ))∈CE(s1 )

where CE(s1 ) denotes the set of competitive outcomes with the initial state s1 and a bold-font x(s1 ) denotes a state-contingent sequence with the initial state s1 for any variable x. The optimization is subject to the equations characterizing the competitive equilibria (i.e., equations (3) and (4)). The Ramsey outcome is defined as the statecontingent sequence of consumption, inflation, the nominal interest rate that solves the problem above.16 In other words, the Ramsey outcome is a competitive outcome with the highest time-one value. I will denote the Ramsey outcome by {cram,t (st ), πram,t (st ), rram,t (st )}∞ t=1 . The value sequence associated with the Ramsey outcome is denoted by {wram,t (st )}∞ t=1 , and will be referred to as the Ramsey value sequence. Solid black lines in figure 2 show the Ramsey outcome and value sequence in the economy with pH = 0.01 for a particular realization of s10 ∈ S10 , together with the sequence of contemporaneous utility, {u(cram,t (st ), πram,t (st ))}∞ t=1 , associated with the outcome sequence. The Ramsey problem is solved numerically by first recursifying the infinite-horizon problem into a saddle-point functional equation by Marcet and Marimon (2016) and then applying a time-iteration method described Coleman (1991) to the saddle-point functional equation.17 Details of the saddle-point functional equation and the solution method are described in online appendix E. Parameter values are 15

See Nakata and Schmidt (2014) for various analytical results on the discretionary outcome in this model. 16 An alternative way to define the Ramsey outcome would be to let the central bank optimize before they observe s1 , provided that the distribution of s1 is not degenerate. 17 To my knowledge, no analytical characterization of the Ramsey outcome is available in the literature for the stochastic New Keynesian model with the ZLB constraint. Werning (2012) provides some analytical results for the Ramsey outcome in a deterministic New Keynesian model with the ZLB constraint.

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given in table 1. The figure shows that the Ramsey planner keeps the nominal interest rate at zero even after the contractionary shock disappears. An extended period of low nominal interest rates, together with consumption boom and above-trend inflation at time 2, mitigates the declines in consumption and inflation during the period of the contractionary shock. Figure 2: The Ramsey outcome and value sequence

rt: Nominal Interest Rate (Ann. %)

πt: Inflation

ct: Consumption

4 2

1

0.1

0

0

−1 0 0

−0.1

−2 5 Time u(πt,ct): Utility flow

10

0

5 Time wt: Value

0 0 −0.02

0

5 Time

10

The Ramsey outcome and value sequence The outcome the Ramsey planner would choose in the hypothetical reoptimization at t=2

−0.5

−0.04 0

10

−1 5 10 0 5 10 Time Time Note: Solid blue vertical lines show the period with contractionary shocks.

Since the contemporaneous utility flow is maximized when consumption and inflation are stabilized at zero, the consumption boom and above-trend inflation are undesirable ex post. Thus, if the Ramsey planner was hypothetically given an opportunity to re-optimize again after the shock disappears, the planner would choose to stabilize consumption and inflation. This is captured in the dashed red lines which show the sequence of consumption, inflation and the nominal interest rate the Ramsey planner would choose in the hypothetical reoptmization at time 2. The planner would renege on the promise of the low nominal interest rate and raise the rate in order to stabilize consumption and inflation. This discrepancy between the pre-announced policy path (solid black lines) and the policy path the government would like to choose in the future (dashed red lines) captures the time-inconsistency of the Ramsey policy. In the rest of the paper, we will study credible plans, which allow the private sector’s belief to shift if the government reneges on the promise it has made in the past. By 11

allowing the private sector’s belief to depend on the history of policy actions, credible plans can give the government an incentive to fulfill the promise of the low nominal interest rate, and can make the Ramsey policy time-consistent.

3

Definition of a plan and credibility This section defines a plan, credibility, and the revert-to-discretion plan. The defi-

nitions closely follow Chang (1998).

3.1

Plan

A government strategy, denoted by σg := {σg,t }∞ t=1 , is a sequence of functions that maps a history of the nominal interest rates up to the previous period and a history of states up to today into today’s nominal interest rate. Formally, σg,t is given by t 18 Given a particular σg,1 : S → R≥0 and σg,t : Rt−1 ≥0 × S → R≥0 for all t ≥ 2. realization of {st }∞ t=1 , a sequence of nominal interest rates will be determined recursively by r1 = σg,1 (s1 ) and rt = σg,t (rt−1 , st ) for all t > 1 and for all st ∈ St . A government strategy is said to induce a sequence of the nominal interest rates. A private-sector strategy, denoted by σp := {σp,t }∞ t=1 , is a sequence of functions mapping a history of nominal interest rates up to today and a history of states up to today into today’s consumption and inflation. Formally, σp,t is given by σp,t : Rt × St → (R, R) for all t. Given a government and private-sector strategy, a sequence of consumption and inflation will be determined recursively by (ct , πt ) = σp,t (rt , st ) for all t ≥ 1 and for all st ∈ St . A private sector strategy, together with a government strategy, is said to induce a sequence of consumption and inflation.19 A plan is defined as a pair of government and private sector strategies, (σg ,σp ). Notice that a plan induces an outcome—a state-contingent sequence of consumption, inflation, and the nominal interest rate. As discussed earlier, there is a value sequence {wt (st )}∞ t=1 , associated with any outcome.

3.2

Credibility

Let us use CEtR (s) to denote a set of state-contingent sequences of the nominal interest rate consistent with the existence of a competitive equilibrium when 18

The first period is a special case, as there is no previous policy action. Note that, while the nominal interest rate today depends on the history of nominal interest rates up to the previous period, consumption and inflation today depend on the history of nominal interest rates up to today. The implicit within-period-timing protocol behind this setup is that the government moves before the private sector does. 19

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st = s. Formally, for each s ∈ S, CEtR (s) := {rt (s) ∈ R∞ | ∃ (ct (s), πt (s)) s.t. (ct (s), πt (s), rt (s)) ∈ CEt (s)}. σg is said to be admissible if, after any history of policy actions, rt−1 , and any history of states, st , rt (s) induced by the continuation of σg belongs to CEtR (st ). A plan, (σg , σp ), is credible if (i) σg is admissible, (ii) after any history of policy actions, rt , and any history of states, st , the continuation of σp and σg induce a (ct (st ), πt (st ), rt (st )) ∈ CEt (st ), and (iii) after any history rt−1 and st , rt (st ) induced by σg maximizes the government’s objective over CEtR (st ) given σp . In plain languages, a plan is said to be credible if neither the private sector nor the government has incentive to deviate from the strategies associated with it. An outcome is said to be credible if there is a credible plan that induces it. When a certain plan A is credible and the plan A induces a certain outcome α, we say that the outcome α can be made credible, or time-consistent, by the plan A.

3.3

The revert-to-discretion plan

I now define a key object of this paper, the revert-to-discretion plan, and discuss the condition under which this plan is credible. The revert-to-discretion plan, (σgrtd , σprtd ), consists of (i) the following government rtd rtd t−1 t = rram,1 (s1 ) for any s1 ∈ S, σg,t (r , s ) = rram,t (st ) if rj = rram,j (sj ) strategy: σg,1 rtd t−1 t (r , s ) = rd,t (st ) otherwise, and (ii) the following privatefor all j ≤ t − 1, and σg,t rtd t t (r , s ) = (cram,t (st ), πram,t (st )) if rj = rram,j (sj ) for all j ≤ t, sector strategy: σp,t rtd t t (r , s ) = (cbr (st , rt ), πbr (st , rt )) otherwise,20 where σp,t cbr (st , rt ) = Et cd,t+1 (st+1 ) −

  1  rt − Et πd,t+1 (st+1 ) − st χc

πbr (st , rt ) = κcbr (st , rt ) + βEt πd,t+1 (st+1 )

(9) (10)

The government strategy instructs the government to choose the nominal interest rate consistent with the Ramsey outcome, but chooses the interest rate consistent with the discretionary outcome if it has deviated from the Ramsey outcome at some point in the past. The private sector strategy instructs the household and firms to choose consumption and inflation consistent with the Ramsey outcome as long as the government has never deviated from the Ramsey outcome. If the government has ever deviated from the nominal interest rate consistent with the Ramsey outcome, the private sector strategy instructs the household and firms to choose consumption and inflation today based 20

Subscript br stands for best r esponse.

13

on the belief that the government in the future will choose the nominal interest rate consistent with the discretionary outcome. By construction, the revert-to-discretion plan induces the Ramsey outcome, and the implied value sequence is identical to the Ramsey value sequence. It is relatively straightforward to show that the revert-to-discretion plan is credible if and only if wram,t (st ) ≥ wd,t (st ) for all t ≥ 1 and all st ∈ St . See online appendix B for proof. The condition that wram,t (st ) ≥ wd,t (st ) for all t ≥ 1 and all st ∈ St makes sure that the government does not have an incentive to deviate from the instruction given by the government strategy after any history rt−1 and st in which the Ramsey policy has been followed. Notice that, after the history in which the Ramsey policy has been followed, wram,t is the value of following the instruction given by the revert-to-discretion plan and wd,t is the best value the government can attain if the government deviates from the instruction. Since wram,t and wd,t are the sum of today’s utility flows (u(cram,t , πram,t ), u(cd,t , πd,t )) and the discounted continuation values (βEt wram,t+1 , βEt wd,t+1 ), the restriction wram,t ≥ wd,t can be written as u(cram,t , πram,t ) + βEt wram,t+1 ≥ u(cd,t , πd,t ) + βEt wd,t+1     ⇔ βEt wram,t+1 − wd,t+1 ≥ u(cd,t , πd,t ) − u(cram,t , πram,t ) .

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The left-hand and right-hand sides of this last inequality constraint respectively capture the loss in the continuation value and the gain in today’s utility flow if the government deviates from the Ramsey policy.21 Thus, the aforementioned condition for credibility can be restated as “If the loss in the continuation value caused by the deviation from the Ramsey prescription is larger than the gain in today’s utility flow, the revert-todiscretion plan is credible.”

4

Results

In this section, I solve the discretionary and Ramsey value sequences for various combinations of parameter values and characterize the circumstances under which wram,t (st ) ≥ wd,t (st ) for all t ≥ 1 and all st ∈ St , and thus the revert-to-discretion 21

Note that what I call the gain in today’s utility flow can be negative in some states (for example, in crisis states, or in normal states after the effects of the crisis shock on the Ramsey outcome have dissipated), potentially making my terminology misleading. However, according to my numerical simulations, in the aftermath of the crisis featuring the overshooting of consumption and inflation, u(cd,t , πd,t ) − u(cram,t , πram,t ) is positive and captures the short-run temptation of the government to renege on the promise.

14

plan is credible. In particular, I analyze the set of (pH , pL ) under which the revert-todiscretion plan is credible, given the baseline values for other parameters of the model (i.e. L, β, χc , κ, λ) as listed in table 1. In online appendix J and K, I examine how this set varies when other parameters take alternative values. For each set of parameter values considered, I simulate the model until I observe one million episodes of contractionary shocks, and decide that “wram,t (st ) ≥ wd,t (st ) for all t ≥ 1 and all st ∈ St ” if the simulated Ramsey values are always above the simulated discretionary values. Figure 3: Credibility of the revert-to-discretion plan 1 0.9

The revert−to−discretion plan is not credible The revert−to−discretion plan is credible The discretionary outcome does not exist

0.8

100pH (shock frequency)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

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60 100pL (shock persistence)

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Figure 3 shows whether the revert-to-discretion plan is credible or not for the set of (pH , pL ) ∈ PH × PL where PH is 101 equally spaced grid points between [0, 0.01] and PL is 51 equally spaced grid points between [0, 1]. Blank areas indicate combinations of (pH , pL ) for which the revert-to-discretion plan is credible. Blue dots indicate the combinations of (pH , pL ) for which the revert-to-discretion plan is not credible. Black dots indicate the combinations of (pH , pL ) for which the revert-to-discretion plan is not defined because the discretionary outcome does not exist.22 Result 1 (Frequency): For any given pL , there exists p∗H such that the revert-to22

Online appendix D explains in detail why the solution does not exist for certain combinations of (pH , pL ).

15

discretion plan is credible if p∗H ≤ pH ≤ pmax and not credible if pH ≤ p∗H where H is the maximum value of the crisis frequency consistent with the existence of the pmax H discretionary outcome. In other words, for any given pL ∈ PL , the revert-to-discretion plan is credible if and only if the contractionary shock hits the economy sufficiently frequently. To gain insights on this result, figure 4 compares particular realizations of the discretionary and Ramsey outcomes/value sequences for two economies—one with frequent shocks (i.e., a small pH ) and the other with infrequent shock (i.e., a large pH ). In this figure, st = L for 1 ≤ t ≤ 4 and st = H for 5 ≤ t ≤ 15. The left column shows the realization of the Ramsey and discretionary outcomes/value sequences in the model with infrequent shocks, while the right column shows the realization in the model with frequent shocks. Figure 4: The discretionary and Ramsey outcomes/value sequences: Frequent vs. infrequent shocks

2 0 5

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Solid black line: The Ramsey outcome and value sequence Dashed red line: The discretionary outcome and value sequence

16

Top three rows show that the discretionary and Ramsey outcomes are very similar across two models with infrequent and frequent shocks. However, according to the bottom row, the discretionary and Ramsey value sequences behave differently when the shock frequencies are different. In particular, in the model with frequent shocks, the discretionary value stays below the Ramsey value at time 5—right after the crisis shock disappears—and remains so afterward. In contrast, in the model with infrequent shocks, the discretionary value exceeds the Ramsey value at time 5. Thus, the revertto-discretion plan is not credible when the contractionary shock occurs infrequently. To understand why the discretionary value stays below the Ramsey value in the model with frequent shocks, it is useful to examine how the loss in the continuation value and the gain in today’s utility flow at time 5 vary with the frequency of the shock.23 Since the frequency of the shock does not substantially affects the discretionary and Ramsey outcomes at time 5, the gain in today’s utility flow of deviating from the Ramsey policy are essentially unaltered by the shock frequency, as seen in the constant red line. However, the frequency of the shock does alter the loss in the continuation value associated with the deviation from the Ramsey prescription. In particular, the loss in the continuation value increases with frequency shocks. This result arises because the contractionary shock leads to a larger decline in consumption and inflation in the discretionary outcome than in the Ramsey outcome in the face of contractionary shock. Thus, a higher probability of contractionary shocks reduce the expected discounted sum of future utility flows associated with the discretionary outcome by more than that associated with the Ramsey outcome, making the loss in the continuation value an increasing function of the frequency. All told, for sufficiently frequent shocks (i.e., sufficiently large pH ), the losses in the continuation value becomes larger than the short-run temptation, making the revert-to-discretion plan credible.24 Result 2 (Persistence): For a sufficiently high pH , the revert-to-discretion plan is credible for any pL below pmax L —the maximum value of the crisis persistence consistent with the existence of the discretionary outcome. For a sufficiently small pH , there and is exists p∗L such that the revert-to-discretion plan is credible if p∗L ≤ pL ≤ pmax L not credible if 0 ≤ pL ≤ p∗L . For an intermediate range of pH , there exists p∗L such that and is not credible if the revert-to-discretion plan is credible if 0 < p∗L ≤ pL ≤ pmax L 23

Note that the short-run incentive to deviate is maximal right after the crisis shock disappears and declines gradually afterward, as the magnitude of the overshooting of consumption and inflation are the largest right after the crisis shock disappears. 24 See online appendix I for more detailed analyses on how the crisis frequency affects the short-run gain and the long-run loss from deviating from the Ramsey promise after the crisis shock disappears.

17

pL ≤ p∗L or pL = 0.25 This result says that, even when the frequency of shock is small, the revert-to-discretion plan is credible if the contractionary shock is sufficiently persistent. For example, when pH = 0.001, the revert-to-discretion plan is credible if pL > 0.72, but is not credible otherwise. Another way of phrasing this result is that the threshold value of pH above which the revert-to-discretion plan is credible is decreasing in pL . To understand the mechanism behind this result, figure 5 compares particular realizations of the discretionary and Ramsey outcomes/value sequences for two economies— one with transient shocks (i.e., a small pL ) and the other with persistent shock (i.e., a large pL ). In this figure, st = L for 1 ≤ t ≤ 4 and st = H for 5 ≤ t ≤ 15. The left column shows the Ramsey and discretionary outcomes/value sequences in the model with transient shocks, while the right column shows those in the model with persistent shocks. When the persistence is high, the household and firms expect to stay in the low state for long. Since marginal costs and inflation are low in the low state, such expectation implies lower expected marginal costs and higher expected real interest rates. Accordingly, the household and firms in the low state choose lower consumption and inflation. However, the Ramsey planner can mitigate this effect by promising a higher inflation, a larger consumption boom, and a longer period of zero nominal interest rates after the shock disappears. Thus, the declines in consumption and inflation from marginal increases in persistence is larger in the discretionary outcome than in the Ramsey outcome, as captured in the second and third rows in figure 5. As a result, the continuation value of reverting back to the discretionary plan declines more rapidly with pL than that of staying with the Ramsey outcome, implying that the long-run loss of reverting back to a discretionary plan is higher with more persistent shocks. In the meantime, the promise of higher inflation and consumption increases in the economy with persistent shocks means that the short-run temptation to deviate from the promise is larger. Quantitatively, the long-run loss increases more rapidly than the short-run temptation for large values of pL , the former exceeds the latter for sufficiently large values of pL . 25

The statements in Result 2 are more complicated than those in Result 1. The reason is that the threshold crisis frequency is discontinuous at pL = 0. In particular, the threshold crisis frequency drops suddenly as pL approaches zero from above, which can be seen in figure 3. Online appendix M explains why this discontinuity arises. Note that, while the threshold crisis frequency is discontinuous in pL , the credibility of the revert-to-discretion plan is continuous in pH ; as stated in Result 1, holding all other parameters constant—including pL —there is a threshold crisis probability above which the revert-to-discretion plan is credible and below which it is not credible.

18

persistent shocks (100 pL=50)

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Figure 5: The discretionary and Ramsey outcomes/value sequences: Transient vs. persistent shocks

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Solid black line: The Ramsey outcome and value sequence Dashed red line: The discretionary outcome and value sequence

5

Quantitative exploration

In my baseline parameterization designed to capture the severity of the Great Recession, the threshold crisis frequency above which the revert-to-discretion plan is credible is very small, provided that the crisis persistence is sufficiently high. For example, when the crisis persistence is 0.8—which is near the lower end of the values used in the literature—the threshold crisis frequency is only about 0.04 percent.26 The threshold crisis frequency of 0.04 percent means that the crisis shock only needs to hit the economy once in 600 years on average to make the revert-to-discretion plan credible. 26

In the literature, the crisis persistence parameter is typically set to a very high number in the range of 0.8 to 0.9 to capture the fact that the most recent lower-bound episode in the United States lasted for an extended period.

19

In the United States, two large shocks have pushed the policy rate to the effective lower bound over the past 100 years since the creation of the Federal Reserve System. Thus, a na¨ıve estimate of the frequency parameter would be 0.5 percent (= 2/400) at quarterly frequency. The threshold crisis frequency of 0.04 percent computed under the baseline parameterization is comfortably below this na¨ıve estimate. Taken at face value, this result means that the revert-to-discretion plan can make the Ramsey policy of keeping the policy rate low for long credible under a realistic crisis frequency. In this section, I examine the sensitivity of the threshold crisis frequency to various alterations of the baseline model. I first analyze how the threshold crisis frequency varies when I relax the assumption that the economy falls into the discretionary path forever after the government’s deviation, allowing the punishment duration to be finite (section 5.1). I then compute the threshold crisis frequency—and how it depends on the punishment duration—in various models with alternative parameter values or alternative features (section 5.2 to 5.5).

5.1

The revert-to-discretion(N) plan

One may feel that the private sector’s punishment strategy of reverting to the discretionary outcome forever after the government’s deviation may be too harsh and unrealistic. In reality, households and firms do not live forever. The head of the central bank also changes with some frequency. Even if the same central banker is in charge for an extended period, central bank doctrines can change over the course of the tenure.27 Based on these considerations, I define and analyze the revert-to-discretion(N) plan in which the punishment regime lasts for a finite period of time (N) and the economy reverts back to the Ramsey outcome afterward. As a formal definition of this plan is involved, I relegate it to online appendix G for the sake of brevity.28 Here, I report the main results from the analysis. The solid line in figure 6 shows how the threshold crisis frequency varies with the number of punishment periods (in quarters), and the dashed line shows the threshold crisis frequency under the baseline case in which the punishment lasts forever. The range of the punishment duration shown is from 10 years (40 quarters) to 50 years (200 quarters). The shortest duration of 10 years is close to the average tenure of the head of the central bank in the United States, Canada, and the United Kingdom in the post 27

For example, consider the gradual move toward transparency during the tenure of Alan Greenspan at the Federal Reserve. 28 Loisel (2008) considers a similar plan with finite-period punishment in a sticky-price model without the ZLB constraint. Note that there is the threat of a permanent punishment if the government deviates from the temporary punishment path.

20

Figure 6: Credibility of the revert-to-discretion(N) plans (i.e., plans with finite-period punishment) 0.3

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N (Punishment Length) World War II era.29 Not surprisingly, the threshold crisis frequency increases as the punishment duration becomes shorter. A shorter punishment duration is associated with a larger value for the government in the case of defection and therefore a smaller long-run loss from reneging on the Ramsey promise. With a less severe punishment, the contractionary shock needs to be more frequent in order to make the revert-todiscretion plan credible. Although allowing for a finite-period punishment limits the power of reputation, the threshold crisis frequency remains quantitatively small. Even when the punishment lasts for only 10 years, the threshold crisis frequency is 0.26 percent, comfortably below the na¨ıve estimate of the crisis frequency in the United States. A threshold frequency of 0.26 percent means that the crisis shock needs to hit the economy once in about 100 years on average to make the revert-to-discretion plan credible.

5.2

Sensitivity analysis

In this subsection, I examine the extent to which the deviation from the baseline parameterization increases the threshold crisis frequency. Online appendices J and K analyze how each parameter of the model affects the threshold crisis frequency in detail in the reduced-form parameter and structural parameter space, respectively. Here, we 29

The average tenures are 8.6 years, 9.7 years, and 8.5 years in the United States, Canada, and the United Kingdom, respectively.

21

focus on the following three results; the threshold crisis frequency is higher with (i) a larger χc , (ii) lower κ, and (iii) higher λ in the reduced-form parameter space. The mechanism behind these findings is discussed in online appendix J. The focus in this subsection is to examine quantitatively how high the threshold crisis frequency can be under alternative values for these three key parameters. In the top-left panel, the inverse intertemporal elasticity of substitution (χc ) is 2—the value used in Christiano, Eichenbaum, and Rebelo (2011)—as opposed to the baseline value of 1. When the punishment duration is infinite, the threshold crisis frequency is about 0.09 percent with χc = 2, whereas it is 0.04 percent in the baseline. When the punishment duration is 40 quarters, the threshold crisis frequency is 0.4 percent, whereas it is 0.26 percent in the baseline. Figure 7: Sensitivity Analysis χc = 2

κ = 0.003 1

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The top-right panel shows the threshold crisis probabilities when the slope of the Phillips curve (κ) is 0.003, much flatter than in the baseline case of κ = 0.012. This value of κ is consistent with the Calvo probability of 90 percent, the value used in Erceg and Lind´e (2014). According to the panel, when the punishment duration is infinite, the threshold crisis frequency is 0.15 percent under this alternative choice of κ. Although the threshold crisis frequency exceeds the na¨ıve estimate of 0.5 percent 22

when N = 40, it remains below 0.5 percent as long as the punishment duration is longer than 15 years. Finally, the bottom-left panel shows the threshold crisis probabilities when λ = 1/16, which implies equal weights on the output gap volatility term and the (annualized) inflation volatility term in the government’s objective function. This weight is substantially larger than the baseline weight of λ = 0.0012 that is consistent with the baseline value of κ when the objective function is derived as the second-order approximation of the household welfare, but is sometimes used among practitioners for policy analyses (Ajello, Laubach, L´opez-Salido, and Nakata (2016)). Under this alternative value of λ, the threshold crisis frequency is 0.11 percent when the punishment duration is infinity. With N=40 quarters, the threshold frequency is 0.88 percent, exceeding the na¨ıve estimate of 0.5 percent. However, the threshold crisis frequency remains below the na¨ıve estimate as long as the punishment duration is larger than 15 years.

5.3

Results from a model with cost-push shocks

The baseline model has only one shock. There is a long tradition in the sticky-price literature documenting the value of commitment in the model with a cost-push shock. In this subsection, I examine how the introduction of a cost-push shock to the baseline model affects the threshold crisis frequency. The cost-push shock, et , shows up in the Phillips curve as follows, πt = κct + βEt πt+1 + et .

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I assume that et follows a three-state Markov process. For simplicity, the Markov process is chosen so that the distribution is independent across time and symmetric. In particular, et takes the value of −c, 0, and c, where c is a positive constant, and the frequency of being in any one of the three states tomorrow is 1/3 regardless of the state today. The value of c is chosen so that, in the normal state, when et = c (et = −c), the policy rate is 2 percentage points above (below) the policy rate when et = 0. The dashed horizontal black line in the bottom-right panel of figure 7 shows the threshold frequency in the model with both crisis and cost-push shocks when the punishment period is infinite, while the solid black line shows the threshold frequency for various finite punishment periods in the same model. In the panel, red lines are for the baseline model with only a crisis shock. According to the panel, the introduction of the cost-push shock raises the threshold crisis frequency when the punishment duration is sufficiently long, but lowers it otherwise. 23

There are two opposing forces behind this result. First, the introduction of the cost-push shock reduces the discretionary value in the normal state: Even if the crisis frequency is zero, the discretionary value is strictly negative because the cost-push shock generates fluctuations in consumption and inflation. All else being equal, the lower discretionary value means that the long-run loss from deviating from the Ramsey path is larger and that the incentive to maintain reputation is stronger. Second, the introduction of the cost-push shock can make the crisis more severe if a negative costpush shock hits the economy when the crisis shock is present. In the aftermath of a recession featuring both the crisis shock and a negative cost-push shock, the Ramsey policy involves larger overshootings of consumption and inflation, which means the short-run temptation to renege on the Ramsey promise is stronger. Because the first force works to lower the threshold crisis frequency while the second works to increase it, whether the introduction of cost-push shocks lowers and increases the threshold crisis frequency depends on parameter configurations.

5.4

Two alternative calibrations

In this subsection, I consider the credibility of the revert-to-discretion plan under two parameter configurations considered in Denes, Eggertsson, and Gilbukh (2013); one intended to capture the severity of the Great Recession and the other the severity of the Great Depression. In Denes, Eggertsson, and Gilbukh (2013), the Great Recession (GR) parameterization is chosen so that output and inflation decline by 10 percent and 2 percentage points, respectively, in the crisis state under the discretionary outcome with pH = 0.30 The Great Depression (GD) parameterization is chosen so that output and inflation decline by 30 percent and 10 percentage points, respectively, in the crisis state under the discretionary outcome with pH = 0.31 These values are listed in table 2. The left and right panels of figure 8 show the threshold crisis frequency under the GR and GD parameterizations, respectively. In each panel, the solid black line shows 30

While the Federal Reserve employed various unconventional monetary policies during and in the aftermath of the Great Recession, the discretionary outcome abstracts from these policies. Accordingly, it may be more sensible to choose the size of the shock so that the declines in inflation and output under the discretionary outcome are larger than 10 percent and 2 percentage points to capture the severity of the underlying shock that caused the Great Recession. In an unreported exercise, I find that increasing the size of the shock for the Great Recession scenario has little effect on the threshold crisis frequency, consistent with the result of the sensitivity analysis in online appendix J that the size of the shock has little effect on the credible region. 31 Denes, Eggertsson, and Gilbukh (2013), along with many other works using two-state Markov processes for the crisis shock, assume pH = 0.

24

Table 2: Parameter Values from Denes, Eggertsson, and Gilbukh (2013) Parameter β χc κ λ H L pL

Great Recession (GR) 0.997 1.220 0.0075 0.00057 0.003 -0.0129 0.857

Great Depression (GD) 0.997 1.153 0.0091 0.00072 0.003 -0.0107 0.902

Figure 8: GR and GD Calibrations of Denes, Eggertsson, and Gilbukh (2013) GR

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how the threshold frequency above which the revert-to-discretion plan is credible varies with the punishment duration, whereas the dashed horizontal line indicates the threshold frequency when the punishment duration is infinite. Under the GR parameterization, the threshold frequency is 0.015 percent. Under the GD parameterization, the threshold frequency is even lower, 0.003 percent. Even when the punishment lasts for 10 years, the threshold crisis frequencies remain small, 0.45 percent in the GR parameterization and 0.31 percent in the GD parameterization, respectively. Again, these numbers are comfortably below 0.5 percent, the na¨ıve estimate of the crisis frequency discussed earlier.32

32

It is interesting to note that, even though the crisis frequency is lower under the GD scenario than under the GR scenario when N is high, it is higher under the GR scenario than under the GD scenario when N is low. How quickly the threshold crisis frequency increases as N decreases depends on parameters of the model, with the two most important parameters being the crisis persistence and the crisis severity. See online appendix L for detailed analyses on this issue.

25

5.5

Results from nonlinear models

In this subsection, I compute the threshold crisis frequency in a fully nonlinear version of the baseline semi-loglinear New Keynesian model considered thus far. The nonlinear model features a representative household, a final good producer, and a continuum of intermediate goods producers with unit measure subject to quadratic price-adjustment costs, as in Rotemberg (1982). The exogenous shock is given by the time-varying discount factor of the representative household. As this model is commonly used in the literature,33 the details of the model as well as the formal definitions of the discretionary outcome, the Ramsey outcome, and other relevant concepts in the nonlinear setup are described in online appendix H. Table 3: Parameter Values for Three Nonlinear Models Parameter β χc χn θ ϕ δcrisis pC

Baseline 0.993 1 1 10 325 1.016 0.8

BBW(2016) 0.997 1 0.28 7.67 458 1.012 0.8

CE(2012) 0.993 1 1 3 100 1.011 0.8

Note: BBW(2016) and CE(2012) refer to Boneva, Braun, and Waki (2016) and Christiano and Eichenbaum (2012), respectively. χn is the inverse Frisch elasticity, θ is the substitutability of intermediate goods in production, ϕ is the price adjustment cost parameter, and pC is the persistence of the crisis shock. δcrisis is the size of the crisis shock and is chosen to generate a 7 percent decline in consumption in the crisis state under the discretionary outcome for each parameterization. See online appendix H for details of the fully nonlinear model.

I consider three alternative parameter configurations. In the first, I consider a set of parameter values such that the implied reduced-form parameters of the semi-loglinear model are consistent with the baseline parameter values shown in table 1. In the second and third configurations, I consider the parameter values considered in Boneva, Braun, and Waki (2016) and Christiano and Eichenbaum (2012), respectively. In all cases, I choose the size of the discount factor shock so that consumption declines by 7 percent in the crisis state under the discretionary outcome.34 According to the left panel of figure 9, the threshold crisis frequency is smaller in the nonlinear setup than 33

See, for example, Nakata (2017) and Gavin, Keen, Richter, and Throckmorton (2015) In the model of Boneva, Braun, and Waki (2016), there is a productivity shock, in addition to the discount factor shock. They choose the sizes of the productivity shock and the discount factor shock to achieve the declines in output and inflation of 7 percent and 1 percent, respectively. While I abstract from the productivity shock in the paper, in an unreported exercise, I also computed the threshold crisis frequency in the nonlinear model with both productivity and discount factor shocks 34

26

Figure 9: Nonlinear Models Baseline

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in the semi-loglinear setup for any given punishment duration. When the punishment duration is infinite, the threshold frequency is 0.02 percent in the nonlinear model versus 0.04 percent in the semi-loglinear model. When the punishment duration is 10 years, the threshold frequency is 0.15 percent in the nonlinear model versus 0.26 percent in the semi-loglinear model. According to the middle and right panels, the threshold crisis frequencies are similarly very small under the parameterizations of Boneva, Braun, and Waki (2016) and Christiano and Eichenbaum (2012). For the parameterization of Boneva, Braun, and Waki (2016), the threshold crisis frequencies are 0.01 percent, 0.03 percent, and 0.20 percent when the punishment duration is infinite, 50 years, and 10 years, respectively. For the parameterization of Christiano and Eichenbaum (2012), the threshold crisis frequencies are 0.003 percent, 0.006 percent, and 0.08 percent when the punishment duration is infinite, 50 years, and 10 years, respectively. It would be interesting to investigate the source of the quantitative discrepancy between the nonlinear model and the semi-loglinear model in this paper. Several authors have carefully examined the quantitative and qualitative differences in the dynamics of the economy at the ZLB between fully nonlinear models and their semi-loglinear versions. While some have found that the quantitative properties of the nonlinear Rotemberg model and the semi-loglinear model can be nontrivially different at the ZLB, recent work by Eggertsson and Singh (2016) shows that the semi-loglinear model approximates the nonlinear Calvo model quite well even at the ZLB. Thus, one useful and found that the threshold crisis frequency is very similar to those reported in the middle panel of figure 9.

27

avenue for future research is to extend the analysis of this section to the nonlinear Calvo model.35

5.6

Discussion

The key takeaway from this section is that, with a very small crisis frequency, the Ramsey policy can be made time-consistent by the revert-to-discretion plan under a wide range of parameterizations and model variants. However, it is useful to highlight that, while I consider various deviations from the baseline parameterization and model, my analyses are all conducted in a stylized model with stylized shocks. As a result, the quantitative results of the paper need to be interpreted with caution. It would be useful to extend the quantitative analysis of the paper further to investigate the credibility of the revert-to-discretion plan in medium-sized DSGE models, such as the ones considered in Christiano, Eichenbaum, and Rebelo (2011) and Del Negro, Giannoni, and Schorfheide (2015). However, solving optimal policy in these models with the ZLB constraint while properly taking into account uncertainty poses computational challenges, be it discretionary or Ramsey policy. Thus, I leave such investigation to future research.

6

The revert-to-deflation plan

Throughout the paper, I focus on the question of whether the Ramsey outcome can be made credible by the revert-to-discretion plan. However, there is an alternative plan that induces the Ramsey outcome and that is credible under a different set of conditions than those for the revert-to-discretion plan. In online appendix F, I formally construct a plan called the revert-to-deflation plan in which the government’s deviation from the Ramsey prescription is punished by reverting to the Markov-perfect equilibrium, where the ZLB binds in both states, and show that it induces the Ramsey outcome and is credible regardless of the parameter values. Because an outcome is defined to be credible if there is a credible plan that induces it, the result that the revert-to-deflation plan is credible regardless of the parameter values means that the Ramsey outcome is credible regardless of the parameter values. 35

Characterizing the discretionary outcome and the Ramsey outcome in a fully nonlinear Calvo model poses a computational challenge: For the discretionary outcome, there is one endogenous state variable (price dispersion). For the Ramsey outcome, there are two additional Lagrange multipliers (in addition to the two Lagrange multipliers associated with the consumption Euler equation and the Phillips curve) and one endogenous state variable (price dispersion).

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I focus on the revert-to-discretion plan, instead of the revert-to-deflation plan, for the following reason. In the revert-to-deflation plan, there is no short-run temptation to renege on the promise after the contractionary shock disappears. As the plan instructs the private sector to expect deflationary outcomes to persist in the future after the government’s deviation, forward-looking households and firms would respond to the government’s deviation by lowering consumption and inflation immediately in the period of deviation. When one thinks about the time-consistency of the Ramsey promise in this model, the premise is that the government can stabilize consumption and inflation in the period of defection by reneging on the Ramsey promise. If this short-run temptation does not exist in the specified plan, then such a plan is not economically interesting.

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Conclusion

Why should the central bank fulfill the promise of keeping the nominal interest rate low even after the economic recovery strengthens? What force will prevent the future central bank from reneging on this promise? To shed light on these questions, this paper has analyzed credible plans in a stochastic New Keynesian economy in which the nominal interest rate is subject to the ZLB constraint and contractionary shocks occasionally hit the economy. I have demonstrated that the policy of keeping the nominal interest rate low for long can be made credible by the revert-to-discretion plan if the contractionary shocks hit the economy sufficiently often. In the revert-to-discretion plan, if the central bank reneges on its promise to keep the nominal interest rate low, it will lose reputation and the private sector will not believe such promises in the face of future contractionary shocks. If the private sector does not believe the promise of an extended period of low nominal interest rates, the contractionary shock will cause large declines in consumption and inflation, which will reduce welfare even during normal times since the agents care about the discounted sum of future utility flows. Thus, the potential loss of reputation gives the central bank an incentive to fulfill the promise. When the frequency or severity of shocks is sufficiently large, this incentive to maintain reputation outweighs the short-run incentive to raise the rate to close consumption and inflation gaps, and keeps the central bank on the originally announced path of low nominal interest rates.

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