Strategic deviations in optimal monetary policy

This paper investigates the circumstances under which a central bank is more or less likely to deviate from the optimal monetary policy rule. The research question is addressed in a simple New Keynesian dynamic stochastic general equilibrium (DSGE) model in which monetary policy deviations occur endogenously. The model solution suggests that higher future central bank credibility attenuates the current period policy trade-off between a stable inflation rate and a stable output gap. Together with the loss of credibility after a policy deviation, this provides the central bank with an incentive to implement past policy commitments. The result is valid even if the central bank may recover credibility with some probability after a policy deviation. My main finding is that the central bank is willing to implement past policy commitments if a sufficient fraction of agents is not aware of the exact end date of the policy commitment. The result challenges the time-inconsistency argument against monetary policy commitments and provides a potential explanation for the repeated implementation of monetary policy commitments in reality.


Introduction
Central banks have recently used more or less explicit policy commitments to manage public expectations. For example, the Swiss National Bank (SNB) promised to defend a EUR/CHF exchange rate floor with " utmost determination" (September 6, 2011). Somewhat less explicit, the Federal Reserve Bank (Fed) " anticipate[d] (...) exceptionally low levels of the federal funds rate for some time" (December 16, 2008). Similarly, the European Central Bank (ECB) " expecte [d] the key ECB interest rates to remain at present or lower levels for an extended period of time" (Juli 4, 2013).
An open question is, however, under which circumstances a central bank is more or less likely to deviate from the announced policy path. Moreover, it is unclear how future central bank credibility interacts with the incentives to implement past policy commitments. The two research questions are: First, how does future central bank credibility affect the optimal monetary policy rule? Second, under which circumstances is it optimal for the central bank to implement past policy commitments?
This paper connects to various strands of the literature, in particular the literature on rules versus discretion, the literature on limited commitment in monetary policy, and the literature on forward guidance. The basic conceptual framework dates back to the work of Kydland and Prescott (1977) and Barro and Gordon (1983a) on rules versus discretion. Their work studies the permanent temptation to deviate from a policy rule that prescribes a stateindependent, pre-announced inflation rate. Inflation surprises are beneficial because they reduce the natural unemployment rate towards the time-invariant efficient unemployment rate which is below the natural unemployment rate. Both Kydland and Prescott (1977) and Barro and Gordon (1983a) find that policy rules are, in general, not enforceable (i.e. time-inconsistent), unless a commitment technology is assumed. In an extension, Barro and Gordon (1983b) investigate enforceable policy rules when the central bank looses reputation from a policy deviation. They find that under such circumstances, policy rules may be enforceable if they are sufficiently close to the discretionary policy prescription.
The current debate on time-inconsistency of monetary policy rules is related to the central bank's optimal response to an exogenous inflation shock. Such a cost-push shock drives a temporary wedge between the natural output level and the efficient output level. In other words, the debate shifted away from the central bank's permanent temptation to overstimulate as in Kydland and Prescott (1977) and Barro and Gordon (1983a) towards the central bank's temporary temptation to deviate from the optimal monetary policy rule. More specifically, following Galí (2015), Woodford (2005), and Clarida et al. (1999), a central bank faces a policy trade-off in the presence of a cost-push shock: Either it stabilizes the inflation rate or it stabilizes the output gap. The optimal response to a cost-push shock is to smooth the response of the inflation rate and the output gap over time. Under full commitment, the central bank can deliver such an outcome, even though the optimal policy path may be timeinconsistent. 1 In contrast, under discretion a central bank lacks the credibility to effectively commit to a future policy path. Being constrained by that, the central bank must (sub-optimally) react more forcefully in the period when the cost-push shock hits.
Problematically, both full commitment and discretion are implausible on theoretical and empirical grounds. Concerning full commitment, it is unclear how a central bank can prevent itself from a favorable policy deviation once time passes. Monetary policy decisions are usually taken by a committee in which individual members serve for some years only. Consequently, later cohorts can overturn commitments of earlier cohorts. Discretion, on the other hand, has become a less appealing concept in light of recent monetary policy conduct ("forward guidance"): It seems implausible to assume that a central bank issues a statement regarding its future policy conduct without any intention to deliver. For example, Woodford (2012) argues that any form of forward guidance is in part interpreted as a policy commitment with some, but limited commitment.
As a consequence, researchers have recently started to study limited commitment in optimal monetary policy, as for example Debortoli et al. (2014), Debortoli and Lakdawala (2014), and Schaumburg and Tambalotti (2007). In their models, the central bank is exogenously selected to deviate from past policy commitments with a time-invariant, state-independent probability. My work suggests that such a time-invariant limited commitment scenario is not fully compatible with strategic policy decisions: In fact, central bank credibility (which is defined as the probability with which the central bank commitment is expected to be implemented in the future) is either time-varying or equal to one of the two extreme cases (full credibility or zero credibility).
1 Barro and Gordon (1983a, 599-600) argue that it is "deceptive" to term a policy rule "time-inconsistent" when "policymakers [have] incentives to deviate from the rule when agents expect it to be followed." They claim that "the incentives to deviate from the rule are irrelevant, since commitments are assumed to be binding. Thus, the time-inconsistency of the optimal solution is (...) irrelevant when commitments are feasible." Somewhat less restrictive, Clarida et al. (1999) define time-consistency as the absence of "incentives to change its plans in an unexpected way." I will use the latter definition of time-consistency.
Some of the literature on forward guidance also considers with monetary policy commitments. Bodenstein et al. (2012), for example, define forward guidance as the explicit commitment to implement policy in accordance with the optimal monetary policy rule under time-invariant limited credibility. Similar to Debortoli et al. (2014), the timing of a policy deviation is, however, exogenous, provoking outcomes which are inconsistent with basic economic logic, namely policy deviations when the implementation of past policy commitments would have delivered a higher welfare. Such outcomes are ruled out in my model. Haberis et al. (2014) model forward guidance as an imperfectly credible interest rate peg. They assume that the central bank's credibility increases with a (time-varying) fixed cost associated to a policy deviation. My model is more transparent about the nature of this cost: A policy deviation is costly because it is associated to higher future macroeconomics volatility, arising from the assumption that the central bank looses all its credibility forever as soon as it first deviates from the announced policy path. Furthermore, in Haberis et al. (2014), the actual decision of whether or not to implement past policy commitments is (yet again) just a coin-toss. It is hence subject to the critique that this may force the central bank to deviate even though it would have preferred to deliver.
In sum, my paper investigates the effects of future central bank credibility on the optimal monetary policy rule, as well as the circumstances under which a central bank is more or less likely to deviate from the announced policy path. It applies a simple New Keynesian dynamic stochastic general equilibrium (DSGE) model in which the central bank decides strategically whether or not to honor past policy commitments. Endogenous policy deviation come with a permanent and complete loss of central bank credibility, i.e. higher macroeconomic volatility in the future.
My results show that higher future expected central bank credibility attenuates the current period policy trade-off between a stable inflation rate and a stable output gap. This provides the central bank with an incentive to implement past policy commitments. I find that the central bank is willing to implement past policy commitments if a sufficient fraction of agents is not aware of the exact end date of the policy commitment.
The remaining of this paper is organized as follows. Section 2 derives the solution to the optimal monetary policy problem under limited credibility. Furthermore, it introduces the notion of strategic policy decisions. Section 3 presents the results of the analysis. Section 4 concludes.

Model Environment And Optimal Monetary Policy
I analyze optimal monetary policy with strategic policy deviations and limited commitment in a simple New Keynesian dynamic stochastic general equilibrium (DSGE) model similar to Galí (2015).
The representative household maximizes its utility function (equation 1) which is additively separable in a consumption bundle (C t ) and leisure (1 − N t ). β ∈ (0, 1) is the discount factor of the household, σ ≥ 0 the constant relative risk aversion or, equivalently, the inverse intertemporal elasticity of substitution, and ϕ ≥ 0 the inverse of the Frisch labor supply elasticity, i.e. the inverse wage elasticity of hours worked (N t ). A taste for variety over intermediary consumption goods C it enters via the Dixit-Stiglitz (Dixit and Stiglitz (1977)) aggregator (equation 2) in which the elasticity of substitution is given by ∈ (1, ∞).
The household problem is subject to a sequence of dynamic budget constraints (equation 3) and a No-Ponzi condition. P t denotes the price level, Q t the period t price of a risk-free security B t+1 which pays one unit in period t + 1, W t the nominal wage, and D t the (aggregated) nominal firm profit.
There is a continuum of monopolistically competitive firms on the interval [0, 1], each producing a single differentiated intermediary output good Y it with a linear technology Y it = N it (with N it being the labor demand of firm i). As in Calvo (1983), in every period a firm decides on a new price P ti of its output good with probability 1 − θ. Y t+k|t is the output of a firm that last reset its price in period t, C t+k (Y t+k|t ) the associated nominal cost function, and Λ t,T ≡ β T −t U C,T U C,t the stochastic discount factor.
If selected to change its price, firm i maximizes the period t market value of its current and discounted future expected real profits over P * t (expression 4), subject to a sequence of demand constraints (equation 5). The log-linearized non-policy equilibrium of the model is given by the dynamic IS equation (6) and the New Keynesian Philipps curve (7) where x t is the efficient output gap, π t the inflation rate, u t the cost-push shock, i t the nominal interest rate, ρ the steady state real interest rate, and The central bank is a benevolent planer who aims at maximizing the welfare of the representative household. Borrowing from Galí (2015) and Woodford (2005), the welfare loss function is approximated by if the central bank operates under full commitment, i.e. if commitments are honored with probability 1. The weight of the output gap in the welfare loss function is given by ϑ ≡ ξ (σ + ϕ). 3 Naturally, the central bank can only control the household's expectations in as far as the household anticipates the central bank to honor its commitments. This is important because the allocations off the path on which commitments are honored are exogenous to the central bank problem.

Optimal Monetary Policy under Limited Commitment
Building on the work of Debortoli and Lakdawala (2014) and Debortoli et al. (2014), who derive the welfare loss function under limited commitment, I additionally introduce time-variation in central bank credibility. 4 where γ t denotes the central bank's credibility in period t and −1 i=0 γ i = 1. Note that γ 0 (rather than γ 1 ) is associated to (x 1 , π 1 ) because the probability with which the household expects the period 0 commitment to be implemented in period 1 is governed by the central bank's credibility in period 0. The policy problem is subject to the New Keynesian Phillips curve where E t π t+1 is the inflation rate that is expected to prevail if commitments are honored in period t + 1 and E t π d t+1 the inflation rate that is expected to prevail if the central bank deviates from the announced policy path in period t + 1. Assume that the inflation rate which is expected to prevail if the central bank reneges on past policy commitment in t+1 is an arbitrary (linear) function of the state variable(s) in t + 1. Formally, assume E t π d t+1 = E t f t+1 (u t+1 ) with the (time-varying) functional form of f t+1 unknown. Expressed as a Lagrangian, the central bank problem is with λ t being the Lagrange multiplier associated to the New Keynesian Phillips curve in period t. 5 Combining and iterating on the first order conditions with respect to π t and x t yields if γ i > 0 ∀i ∈ {0, . . . , t − 1}. 6 By construction, deviations from the announced policy path in period t imply λ t−1 = 0, as in Debortoli et al. (2014). 7 Consequently, with t = 0 being the initial period of the policy plan wherep t ≡ π t +p t−1 andp −1 = 0. 8 For t > 0, the optimal output gap depends not only on the current inflation rate but also on lagged inflation rates. That is, there is a history dependence in the optimal output gap. This finding previews the result that under (limited) credibility it is both possible and optimal to commit to future policy responses when facing a current period cost-push shock. The reason is that such a commitment affects the household's expectations which in turn affect current period variables (in particular, the inflation rate). Consequently, less of a current period variability in the output gap is necessary to achieve the optimal inflation rate. This is beneficial because the welfare loss function is strictly convex in the inflation rate and the output gap. To solve the model, re-express the New Keynesian Phillips curve in terms ofp t .p with µ t ≡ ϑ ϑ(1+βγt)+κ 2 . Suppose u t ∼ AR(1) with E(ε u t ) = 0 and V (ε u t ) = σ 2 ε u t and guess the time-varying solution forp t to be a linear function ofp t−1 and u t .
6 With γ 0 = 0, the central bank's first order condition is x t = − κ ϑ π t . 7 Suppose λ −1 = 0. Then, from equation 146 and 148 it follows that x 0 = − κ ϑ π 0 which is the the optimality condition for the period in which the policy plan is first implemented (cf. Galí (2015, 130, 135)). In other words, setting the non-physical λ −1 = 0 is akin to a deviation from the announced policy path in period 0.
8 In the period of the policy implementation (t = 0), the ratio between the inflation rate and the output gap is independent of central bank credibilities. However, the level of the inflation rate and the output gap changes with {γ t } T t=0 (cf. equation 20).
Further guess the time-varying (linear) solution π d t =ĥ t u t withĥ t unknown. Finally, assume that the central bank implements the discretionary solution with certainty in T + i ∀i ≥ 1. Analyzing a finite period model (T < ∞) is useful because it allows to solve the model by backward iteration without any meaningful loss of economic substance. In particular, as of period T , the only optimal monetary policy rule consistent with rational expectations and perfect information is the discretionary optimal monetary policy rule. This is because the unique set of central bank credibility consistent with rational expectations and perfect information is γ T +i = 0 ∀i ≥ 0. 9 If a policy deviation occurs in period t, the central bank credibility is lost completely and permanently, i.e. γ t+i = 0 ∀i ≥ 1. Plug the guess forp t (equation 16) and the guess for π d t into the re-expressed Phillips curve (equation 14) and solve recursively for a t ∈ (0, 1).
Realize that a deviation from the announced policy path in t requiresp t−1 = 0 (cf. equation 13). From the guess forp t (equation 16) we know thatp d t = c t u t . Furthermore, by definition,p d t = π d t . Because I assume π d t =ĥ t u t it must be thatĥ t = c t ∀t. Solve recursively for c t ∀t ∈ {0, . . . , T }, using {a t } T +1 t=0 from above.
As a result of the classic policy trade-off, the central bank optimally commits to a conditional (future) deflation in response to a positive cost-push shock. Crucially, to decrease E t π t+1 sufficiently, the central bank must announce a more pronounced deflation, the shorter the horizon over which the central bank is expected to implement the policy commitment. In other words, the central bank must implement a more persistent (negative) output gap, the sooner the central bank is expected to return to a discretionary mode. Formally, the lower T and/or the lower the values in {γ j } T j=t , the higher a t . The degree to which the household's expectations adjust to policy commitments determines the severity of the policy trade-off between the output gap and the inflation rate in period t. More specifically, if the (representative) household expect the policy commitment to be implemented over a shorter horizon, the policy trade-off becomes more severe (ĥ t rises).
To illustrate, assume that the central bank's optimal policy is a commitment to a (conditional) future deflation. Since the inflation rate can be expressed as a (positive) function of discounted future expected output gaps, E t π t+1 is ceteris paribus higher, the lower T and/or the lower the values in {γ j } T j=t . From above, we know that this off-equilibrium increase in E t π t+1 induces the central bank to commit to a higher persistence in the output gap. However, households discount future expected inflation rates and incur convex losses from x t and π t . For this reason, the (off-equilibrium) rise in E t π t+1 cannot be offset completely by the central bank's optimal commitment to a higher persistence in the output gap. It is for that reason that the current period policy trade-off accentuates. Formally, the lower T and/or the lower the values in {γ j } T j=t , the higher the impact coefficient (ĥ t ).

Strategic Deviations in Optimal Monetary Policy
In contrast to previous work on limited commitment in optimal monetary policy, I allow the central bank to take strategic policy decisions. More specifically, the central bank can either honor past policy commitments or deviate. It delivers on past policy commitments if and only if the value of doing is strictly greater than the value associated to a policy deviation.
Introducing strategic policy deviations is important for two reasons. First, it shows under which circumstances a central bank is more or less likely to deviate from the announced policy path. Debortoli et al. (2014) were the first who adressed this question: They report the potential welfare gains of a policy deviation over the horizon of the impulse response function to a cost-push shock. My work complements their analysis by showing that the temptation to deviate is not only time-dependent, but also state-dependent.
Second, the introduction of strategic policy deviations provides an endogenous criterion based on which we can assess whether or not the central bank would deviate from past policy commitments. This debate seemed to be resolved because the static perspective suggests that it is always weakly preferable to implement the discretionary solution. My work shows that there are dynamic considerations which induce the central bank to implement past policy commitments. More specifically, if for some reason it is optimal to implement policy commitments in the future, it may also become optimal to honor policy commitments today.
The strategic policy problem is a recursive representation of the central bank optimization problem (equation 9) that takes into account that the continuation values differ depending on the central bank's policy choice. Importantly, it is assumed that the central bank looses all its credibility forever as soon as it first deviates from the announced policy path. Proposition 2 proves that the complete and permanent loss of credibility implies policy deviations in every future period. In contrast, if the central bank honors past policy commitments in t, it can (again) take a strategic policy decision in t + 1. Formally, where V d t denotes the value associated to a policy deviation in period t, V h t the value associated to honored commitments in period t, and U i t the period objective function of the central bank evaluated at the optimal x t,i and π t,i , i.e. evaluated at the x t,i and π t,i which satisfy the optimal monetary policy rule under limited credibility (equation 20 and equation 150). x t,i and π t,i depend on the central bank's policy choice i ∈ {d, h} where d stands for a policy deviation and h stands for the implementation of past policy commitments.

The Driving Process and Model Calibration
The driving force is a cost-push shock u j t which evolves according to a 2-state Markov process where the (discrete) magnitude of u j t is indexed by j ∈ {L, H}. Formally, where 1 − p L,L denotes the probability of transitioning from the low state L to the high state H (which is associated to and assume, for simplicity, that p H,H = p L,L = 0.5. The model is calibrated to quarterly data as suggested in Galí (2015, 67). In particular, β = 0.99 (implying a annualized steady state real interest rate of approximately 4%), = 9 (implying a steady state mark-up of 12.5%), α = 0 (reflecting a simplifying constant returns to scale assumption), σ = 1 (log-utility), ψ = 5 (implying a Frisch elasticity of the labor supply equal to 0.2), θ = 0.75 (implying an average duration of a price equal to four quarters), ρ u = 0 (where not stated otherwise), and u H t = 0.005.

A Simple Model With Strategic Policy Deviations
Suppose t = {0, 1, 2, 3}, with T = 2, and assume x −1 = 0. The central bank decides strategically whether or not to deviate from the optimal monetary policy rule in t = {0, 1, 2}. In period T + 1, the central bank implements the discretionary solution with certainty.
Consistency with rational expectations requires that the agents' beliefs about the number of states in which a policy deviation occurs coincide with the actual number of states in which a policy deviation occurs. For example, γ t = 1 is consistent with rational expectations and perfect information if and only if the central bank implements past policy commitments in period t + 1 independent of the realization of u t+1 . γ t = 0.5 is consistent if and only if past policy commitments are implemented in exactly one state in period t + 1 (provided that u t+1 can only take on two values), and γ t = 0 is consistent if the central bank reneges on past policy commitments in period t + 1 independent of the realization of u t+1 .
Future central bank credibility affects the optimal allocation in period t via two channels: Directly via the optimal future allocation (i.e. via E t π t+1 ) and indirectly via today's solution coefficients (a t and h t ). For this reason, we must assess the consistency of each γ i conditional on an entire sequence of {γ t } T t=i rather than conditional on γ i only. A consistent sequence of {γ t } T t=0 is a sequence in which every individual γ t is consistent.

Results
In response to a positive cost-push shock, the central bank cannot simultaneously stabilize the inflation rate and the output gap. Because the central bank objective function is convex in π t and x t , it is moreover suboptimal to either stabilize the inflation rate or the output gap. Consequently, the optimal response to a positive cost-push shock consists of a positive inflation rate and a negative output gap. The implementation of a negative output gap exerts negative pressure on the inflation rate and partly offsets the (off-equilibrium) rise in the inflation rate caused by the positive cost-push shock. Furthermore, the optimal policy path involves a commitment to a prolonged (conditional) recession which is accompanied by a negative inflation rate.

Future Central Bank Credibility and Current Period Allocation
The optimal monetary policy rule with strategic policy deviations shows that the persistence in the output gap (a t ) as well as the severity of the policy trade-off (ĥ t ) depend on current and future central bank credibilities (γ t+i ∀i ≥ 0). These central bank credibilities may be time-varying. My model is hence flexible enough to study the effect of future (time-varying) central bank credibilities on the current optimal monetary policy rule. By that, it advances on the research of Galí (2015) To illustrate the dependence of the optimal monetary policy rule on future central bank credibilities, consider two simulations (indexed by j) with a deterministic sequence of central bank credibilities Policy commitments are assumed to be honored in t ∈ {0, 1, 2} but not in t = 3. 10 How does future central bank credibility affect the optimal monetary policy rule? Let us first investigate how the optimal persistence in the output gap 10 γ t = 0 does not per se imply a policy deviation in period t. It only reflects the central bank's inability to credibly commit to a future policy path. In other words, γ t = 0 does not, per se, refrain the central bank from implementing past policy commitments. Similarly, γ t−1 = 1 does not per se imply honored commitments in period t because γ t−1 solely reflects the probability with which the agents expect future central bank commitments to be implemented. is affected by future central bank credibilities. Suppose, for the sake of the argument, that the impact coefficient of the cost-push shock (ĥ t ) is equal in both simulations. Under this assumption, the allocation in period t depends on future central bank credibilities only because future central bank credibilites affect the optimal persistence in the output gap.
Monetary policy commitments affect the agents' expectations about future output gaps. By that, they exert (positive or negative) pressure on the current inflation rate. To see this, re-express the Phillips curve as follows: In simulation a (γ 2 = 1), agents expect the central bank to honor policy commitments in period 3 while in simulation b (γ 2 = 0), agents expect a return to the feasible (x t , π t ) = (0, 0) allocation in period 3. Facing a positive costpush shock in t = 0, a naïve central banker may consider the same policy commitment independent of future central bank credibilities. Is this optimal, knowing that E 0 x a (where G denotes a guess) an optimal monetary policy response, even though E 0 x a 3 < 0 and E 0 x b 3 = 0? The answer is no, because optimality would require the allocation (x a 0 , π a 0 ) to deliver the same welfare as (x b,G 0 , π b,G 0 ). Strict convexity in the central bank objective function suggests, however, that the two allocations do not deliver the same welfare. This is because the output gap is the same in both allocations is partly offset by a commitment to a more severe recession in simulation b. Figure 1 displays the optimal monetary policy responses for both simulations under the assumptionĥ a t =ĥ b t ∀t. 11 The output gap and the inflation rate are the same in period 0 because x −1 = 0 and becauseĥ 0 is independent γ 2 by assumption. As of period 1, however, simulation b features a more negative optimal output gap than simulation a. This is because in simulation b, the central bank must optimally implement a more pronounced recession to partly offset the shorter horizon for which it can provide a credible policy commitment. The persistence parameter in the optimal monetary policy rule is hence higher in simulation b (a b t > a a t ∀t ≤ T ). Let us now analyze how future central bank credibilities affect the severity of the policy trade-off between the output gap and the inflation rate (which is formally captured byĥ t ). To isolate this channel, suppose that a t is left unaffected by the sequence of current and future central bank credibilities.
The severity of the policy trade-off is greater, the less flexible public expectations adjust to monetary policy commitments. As before, agents expect the policy commitment to be implemented for two (three) periods in simulation a (b). Applying the same logic as in the argument above, this implies that the optimal x b 0 must lie below x a 0 to partly offset the shorter horizon over which the public expects the central bank to implement the policy commitments in simulation b. Importantly, the optimal x b 0 is not so low as to have π b t = π a t . The policy trade-off is hence more pronounced in simulation b. Figure 2 illustrates the optimal monetary policy responses for both simulations under the assumption a a t = a b t ∀t. In t = {0, 1, 2}, the inflation gap and the output gap are both further off steady state than in simulation b. Figure 3 summarizes the evolution of the optimal persistence and the optimal impact coefficient over time for both simulations. Independent of t, both coefficients are more favorable (that is: lower) in simulation a, compared to simulation b. Also, both coefficients become less favorable (that is: rise) over time because the horizon over which the central bank can provide a credible policy commitment becomes shorter, the more time has passed.

Policy Deviations vs. Honored Commitments
When is it optimal for the central bank to implement past policy commitments? Proposition 1 shows that if each agent forms a correct belief about the central bank's credibility in period T (which is γ T = 0 because the central bank implements the discretionary solution in period T + 1 with certainty) full credibility in t < T is inconsistent with strategic policy decisions.
Proposition 1 Time-invariant full credibility (γ t = 1 ∀t < T ) is inconsistent with strategic policy deviations.
Proposition 1 can be proven directly: Time-invariant full credibility is inconsistent with strategic policy deviations if the value function under a policy deviation is weakly greater than the value function under honored commitments in at least one period. 12 where ϑ is strictly positive. Full credibility in period t < T is inconsistent with strategic policy deviations because condition 28 cannot hold with γ T = 0.
Proposition 2 establishes the pair result that zero credibility is consistent with strategic policy deviations if each agent forms a correct belief about the central bank's credibility in period T . 13 Proposition 2 Time-invariant zero credibility (γ t = 0 ∀t ≤ T ) is consistent with strategic policy deviations.
If the central bank cannot affect the agents' expectations in period T (because all of them hold the correct belief γ T = 0), it cannot do better than the discretionary solution. As a consequence, agents assign a zero credibility to any central bank commitment made in period T − 1 (γ T −1 = 0). This, in turn, makes it optimal for the central bank to implement the discretionary solution in T − 1, such that γ T −2 = 0, and so on.
Taken together, proposition 1 and 2 confirm the long established result of Kydland and Prescott (1977) and Barro and Gordon (1983a). Their work shows that the central bank is always tempted to deviate from past policy commitments. This time-inconsistency problem precludes any credible policy commitment in a model with rational and perfectly informed agents.
As a side result, proposition 3 shows that time-invariant limited credibility is inconsistent with strategic policy deviations.
The logic of the proof is similar to the proof of proposition 1 but with which cannot hold with γ T = 0. Time-invariant limited credibility is hence inconsistent with strategic policy deviations if each agent holds a correct belief about the central bank's credibility in period T . This result suggests that the time-invariant limited commitment case assumed in Debortoli et al. (2014), Debortoli and Lakdawala (2014), and Schaumburg and Tambalotti (2007) is not fully relevant under strategic policy deviations.
Are there any circumstances under which the central bank is willing to honor past policy commitments? The answer is yes. Proposition 4 proves that there is a thresholdγ for which γ T ≥γ ∈ [0, 1] induces the central bank to honor past policy commitments in t ≤ T .
Proposition 4 There exists aγ ∈ [0, 1] such that time-invariant full credibility is consistent with strategic policy deviations if γ T ≥γ.
Proposition 4 is true if the value of honoring past policy commitments is strictly greater than the value of a policy deviation for both t ∈ {1, 2} and for each potential shock sequence, given that γ T ≥γ ∈ [0, 1]. 15 In period 2, the central bank is willing to honor past policy commitment if and only where Σ ≡ a 1 (1 − a 0 ) is affected by a T (and hence γ T ). In order to find γ, guess γ G T and computeā 1 such that equation 30 holds with equality. Then, re-arrangeā 1 such thatγ The coefficients associated to a 2 in equation 30 (in particular: Σ) depend on the initial guess γ G T . γ G T is hence not necessarily equal toγ 1 (which is found to satisfy equation 30 with equality if Σ is formed with γ G T ). Thus we have to solve for the fixed point ofγ in equation 30 by continued iterations. 16 ũ) for each potential shock sequenceũ, γ t = 1 ∀t < T is consistent with strategic policy deviations. The full proof is provided in the appendix, section 5.6.
16 Proceed as follows: First, compute the difference between γ G T andγ I (whereγ I denotes γ after I iterations). Second, if the difference between γ G T andγ I is above some critical value, For reasonable parameterization, there is a γ T ≥γ ∈ [0, 1] such that V h 2 (ũ) > V d 2 (ũ) for each potential shock sequenceũ. Full central bank credibility is hence compatible with strategic policy deviations if a sufficient fraction of agents (inconsistently) believes that the central bank implements past policy commitments in period T + 1.
Somewhat less rigorously, if there is uncertainty about the exact end date of the announced policy path, the central bank may find it beneficial to implement policy commitment in t ≤ T . Though the lens of my model, this explains (at least in parts) why monetary policy commitments were often implemented in reality.

Sufficient Central Bank Credibility
What central bank credibility (γ) sustains honored commitments in t ≤ T ? Understanding the determinants ofγ requires the understanding of two mechanisms: First, the higher γ T , the higher the relative cost of a policy deviation in period T (the reason is that the central bank benefits from a less severe policy trade-off, the higher its credibility). Second, the central bank is more willing to deviate, the greater the (potentially: counterfactual) inflation gap that is supposed to be implemented if past policy commitments are honored.
Taken together, these two forces suggest that a higher inflation gap under honored commitments (which increases the temptation to deviate) requires a higherγ (which decreases the temptation to deviate). Figure 4 presents the difference V d 2 (ũ) − V h 2 (ũ), conditional on γ 2 and the realized shock sequenceũ. A positive difference means that the central bank prefers to deviate from past policy commitments. More specifically, blueish colors indicate values of γ 2 which support γ 0 = γ 1 = 1. In contrast, yellowish colors indicates values of γ 2 which are too low to incentivize the implementation of past policy commitments. The black solid line is the thresholdγ which ensures honored commitments in t ≤ T .
To illustrate, for the shock sequence {H, H, H}, γ 2 ≥ 0.49 suffices to ensure the implementation of past policy commitments in t ≤ T . In contrast, for the shock sequence {H, H, L}, γ 2 ≥ 0.72 is necessary to avoid γ 1 = 0. The threshold forγ hence carries information about the central bank's temptation to deviate from the announced policy path.
Why is it thatγ has to be higher for {H, H, L} than for {H, H, H}? The reason is that the central bank commits to a conditional future deflation (inrepeat the computation ofā I (and the correspondingγ I ) with γ I−1 as an input. Repeat until γ I is sufficiently close toγ I and reportγ =γ I . The conditional future deflation is amplified if the cost-push shock changes its sign between the current and the future period. More specifically, after two consecutive positive cost-push shocks, the expected inflation rate for period 2 under honored commitments is negative. A negative cost-push shock in period 2 amplifies the (conditional) commitment to a deflation, i.e. the change in the sign of the cost-push shock between period 1 and 2 makes the inflation gap under honored commitments greater (compared to a shock sequence in which three consecutive positive cost-push shocks materialize  Of course, for γ 0 = γ 1 = 1 to be consistent with rational expectations, the central bank must already decide to implement past policy commitments in period 1 with probability 1. Moreover, γ 2 must be such that the central bank is known to be willing to implement policy commitments in period 2 with certainty, i.e. even if the realized u 2 is such that the temptation to renege in period 2 is maximized. The central bank is willing to honor past policy commitments under any realization of the shock sequence if γ 2 ≥ 0.72. 18

Concluding Remarks
The empirical motivation for this paper is the observation that central banks have recently used more or less explicit policy commitments to manage public expectations. Woodford (2012) and Andrade et al. (2016) argue that these commitments are subject to potential revisions. Current and future credibility therefore plays a crucial role in determining the effectiveness of monetary policy commitments. In particular, if the central bank's credibility is high, agents are more willing to adjust their expectations. The responsiveness 18 Because the agents know the first two realizations of the shock sequence in t = 1, the consistent γ 1 may depend on the state of the economy. In particular, after two consecutive, equally signed cost-push shocks, γ 1 = 0.5 is consistent with honored commitments in period 1 in some range of γ T , while γ 1 = 1 is consistent in the same range of γ T if the sign of the cost-push shock differs between period 0 and 1 (this is true for γ T ∈ (0.49, 0.72)). Central bank credibility in period 1 may be 0.5 because the inflation rate in period 2 under honored commitments, and with it the temptation to deviate, differs with the first two realizations of the cost-push shock. I constrain the attention to cases in which γ 2 ≥γ implies γ 1 = 1 for each potential shock sequence to avoid dealing with path dependent sequences of central bank credibilities.γ = 0.72 is hence the most conservative measure of the thresholdγ that is necessary to defer policy deviations in t ≤ T . of the agents' expectations feeds back into the optimal behavior of the central bank, i.e. affects the optimal monetary policy rule.
Most papers in the recent literature on limited commitment in monetary policy, e.g. Debortoli et al. (2014) and Bodenstein et al. (2012), make two critical assumption: First, central bank credibility is time-invariant and exogenous. Second, policy deviations occur randomly. My paper allows for time-variation in the central bank's credibility and endogenizes the monetary policy decision. It asks two questions: First, how does future central bank credibility affect the optimal monetary policy rule? Second, under which circumstances is it optimal for the central bank to implement past policy commitments?
To address this question, I use a simple New Keynesian dynamic stochastic general equilibrium (DSGE) model in which the central bank decides strategically if it wants to honor past policy commitments. Policy deviation come with a permanent and complete loss of central bank credibility, i.e. higher macroeconomic volatility in the future. The central bank decides to renege on past policy commitments if and only if a deviation delivers a higher welfare than the implementation of past policy commitments.
My results show that higher future expected central bank credibility attenuates the current period policy trade-off between a stable inflation rate and a stable output gap. Moreover, it decreases the optimal persistence in the output gap. Both is because agents are more willing to adjust their expectations if policy commitment are expected to be honored with a higher probability and/or over a longer horizon. The higher the future expected central bank credibility, the lower the cost of implementing policy commitments today.
The less costly implementation of the optimal monetary policy rule under high credibility, together with the loss of credibility after a policy deviation, provides the central bank with a incentive to implement past policy commitments. My main result shows that the central bank is willing to implement past policy commitments if a sufficient fraction of agents is not aware of the exact end date of the policy commitment. This finding challenges the timeinconsistency argument against monetary policy commitments and provides a potential explanation for the repeated implementation of monetary policy commitments in reality.
Further research is desirable along multiple dimensions of the model. First, the rather conservative assumption about the complete and permanent loss of credibility after a policy deviation could be relaxed. In particular, the central bank should, in some way, be enabled to restore its credibility after a policy deviation. Second, the model could be generalized to study path dependent evolutions of central bank credibilities. Third, one could introduce evolution-ary stable preferences to analyze whether a central bank would abstain from policy deviations under complete information.

Appendix
The appendix provides details on the derivation of the New-Keynesian dynamic stochastic general equilibrium (DSGE) model. Also, it contains the full proofs the propositions presented in the main text. The non-policy equilibrium is based on Galí (2015) and Canetg (2015), the second order approximation to the household's utility function on Galí (2015) and Woodford (2005)

Non-Policy Equilibrium
This section derives the non-policy equilibrium of the simple New Keynesian model.
Households: The (representative) household maximizes its expected discounted utility subject to a sequence of dynamic budget constraints and a No-Ponzi condition.
The intertemporal utility function is additively separable in a consumption bundle (C t ) and leisure (1 − N t ). β ∈ (0, 1) is the discount factor of the household, σ ≥ 0 the constant relative risk aversion or, equivalently, the inverse intertemporal elasticity of substitution, and ϕ ≥ 0 the inverse of the Frisch labor supply elasticity, i.e. the inverse wage elasticity of hours worked (N t ). The (sequence of) dynamic budget constraints are associated to the Lagrange multiplier(s) η t . P t denotes the price level, Q t the period t price of a risk-free security B t+1 which pays one unit in period t + 1, W t the nominal wage, and D t the (aggregated) nominal firm profit. The Lagrangian is given by The No-Ponzi condition prevents excessive debt, i.e. ensures that the household can repay its debt in every period. 19 Λ T,t ≡ β T −t U C,T U C,t is the stochastic discount factor, cf. Galí (2015,56,84).
The household takes prices (P t , Q t , W t ) and nominal firm profits (D t ) as given when solving its maximization problem. The first order conditions with respect to C t , N t , and B t+1 are The transversality condition is combining the first order conditions yields Equation 38 constitutes a household labor market optimum [LM], i.e. an optimal labor supply. It shows that hours worked react positively to the real wage W t /P t and more strongly to a change in the (real and nominal) wage, the lower ϕ (the lower the inverse wage elasticity of hours worked).
Equation 39 is the Euler equation [EE]. It postulates equality between the marginal utility of consuming an additional Q t units today and the discounted marginal utility of consuming one additional unit tomorrow. The intuition is as follows. The household can either consume or save an additional Q t units today. If the household consumes an additional Q t units today, it gets Q t C −σ t /P t utils from it. If the household saves an additional Q t units today, it has one additional unit to consume tomorrow. Because the household is impatient, getting C −σ t+1 /P t+1 utils tomorrow is only worth βC −σ t+1 /P t+1 utils today. If the marginal utilities of the two options (consuming an additional Q t units today and saving an additional Q t units today) were not the same, the household could increase its utility by re-allocating consumption (e.g. consume more today and less tomorrow). An optimal intertemporal allocation of consumption thus requires the marginal utilities of the two options to be equal.
Bundler: Suppose that there is an intermediary firm (the bundler) which bundles the intermediary consumption goods C it into a single good C t (output). I assume ∈ (1, ∞), i.e. some, but imperfect substitutability among intermediary consumption goods ("a taste for variety"). 20 More specifically, I impose the Dixit-Stiglitz (Dixit and Stiglitz (1977)) aggregator, assuming a continuum of goods on the interval [0, 1].
The (static) nominal profit maximization problem of the bundler is The optimal demand for the bundler's input good C it is i.e. a decreasing function of its price P ti and an increasing function of P t and C t .
Firms: There is a continuum of monopolistically competitive firms on the interval [0, 1], each producing a single differentiated intermediary output good Y it with a linear technology Y it = N it (with N it being the labor demand of firm i). As in Calvo (1983), in every period a firm decides on a new price P ti of its output good with probability 1 − θ. If selected to change its price, firm i maximizes the period t market value of current and discounted future expected real profits over P * t . 21 Future profits are weighted by the probability of seeing P * t in effect in that period (θ i ). As in Galí (2015, 84), the optimal price setting problem takes the following form.
Y t+k|t is the output of a firm that last reset its price in period t and C t+k (Y t+k|t ) is the associated nominal cost function. Use the constant elasticity (the nominal marginal costs of a firm that last reset its price in period t), and the definition of the steady state mark-up M ≡ −1 to write the firm's first order condition as By the definition of the log-expressions for the variables, we have that A first order Taylor expansion around a zero inflation steady state (characterized by p * t − p t+k = 0, Λ t,t+k = β i , Y t+k|t = Y and, consequently, Because the marginal product of labor is one, we have Ψ t+k|t = W t+k , i.e. Ψ t+k|t is independent of when the price of the intermediary good was last changed. Hence, we can write ψ t+k|t = ψ t+k ≡ p t+k − µ M C t+k . The above expression for p * t can be written recursively as From the definition of the aggregated price level (cf. Galí (2015, 55)), it follows that Combining the two equations yields Equilibrium: The following system of equations determines the nonpolicy equilibrium.

Equation
Name Source Households: Euler equation Households: Labor supply equation Firms: Optimal price setting y t = c t [MCC] Market clearing condition dY t+k|t and the fact that the marginal product of labor is 1) and [LM,PF] to get Furthermore, with flexible prices, θ = 0 (cf. Galí (2015, 62)) such that the firm's first order condition collapses to µ M C = p * t +ψ t . Combined with ψ t = w t (from Ψ t+k|t ≡ dC t+k (Y t+k|t ) dY t+k|t and the fact that the marginal product of labor is 1) and using [LM,PF] yields 68) i.e. a constant natural output level (which in turn implies y n t = y in every period). Consequentlyμ and Equation

Name
Descriptioñ New Keynesian IS-Curve π t = βE t π t+1 + κỹ t [PK] New Keynesian Phillips Curve Table 3: Non-Policy Equilibrium, in terms ofỹ t withỹ t ≡ y t − y n t and κ ≡ ξ(σ + ϕ+α 1−α ). Acknowledge that µ M C = 0 defines the efficient output level such that E t y e t+1 = y e t = 0. Define x t ≡ y t − y e t and u t ≡ κ(y e t − y n t ) (as in Galí (2015, 128)), and the non-policy equilibrium follows. 24

Steady State
This section derives the steady state of the New Keynesian model.

1)
C t = C t+i No driving force is growing over time 2) P t = P t+i By the definition of a zero inflation steady state 3) Λ t,t+i = β i Follows from 1) and 2) 4) P * t = P t−1 Follows from 2) and the definition of the aggregated price level 5) Y t+i|t = Y Follows from 1) and 4), no price dispersion in steady state 6) Ψ t+i|t = Ψ t Follows from 1) and 4), no price dispersion in steady state 7) P t = MΨ t Follows from 3), 4), 5), 6) and the firm's first order condition The steady state log-output level is determined by the household's labor market optimality condition, the market clearing condition, the production function, and the steady state log real wage The steady state log-consumption level is determined by the market clearing condition and the steady state log-output level The steady state log hours worked is determined by the production function and the steady state log-output level The steady state inflation rate is determined by the definition of the zero inflation steady state The steady state nominal interest rate is determined by the Euler equation [EE], the fact that in steady state C t = C t+i , and the steady state inflation rate. Use −log(Q t ) ≡ i t and −log(β) ≡ ρ.
The steady state real interest rate is determined by the Fisher equation and the steady state inflation rate

Approximating the Household's Utility
This subsection presents the derivation of a second order approximation to the household's utility function. Using the approximated household's utility function simplifies the monetary policy problem because it expresses the welfare loss in terms of the welfare relevant output gap and the inflation rate. The derivation follows Galí (2015, 85, 117, 154) and Woodford (2005, 399, 694). A second order approximation to the household's felicity function ("current period utility function") around the steady state yields Up to a second order approximation Xt−X Further use σ ≡ − U CC U C C (the inverse intertemporal elasticity of substitution) and ϕ ≡ U N N U N N (the inverse of the Frisch labor supply elasticity) to write Next, realize that aggregated employment is the sum of employment across firms N t = 1 0 N ti di. Combine this with C ti = Y ti (the market clearing condition which holds on the firm level and in the aggregate), C ti = P ti Pt − C t , and the (strictly concave and total factor productivity augmented) production function. 25 Take logs and subtract the (log) production function evaluated at the steady state.n Approximating d t consists of three steps. First, from Galí (2015, 53), A second order approximation to the aggregated price level equation yields instead of Y ti = N ti is without loss of generality since the later is a special case of the former (with A t = 1 and α = 0).
which is, up to a second order approximation and using the insight from before (equation 92) equal to Third, E i (p 2 ti ) equals, up to a second order approximation, using x = p ti − p t (with f (x) = 0) and Plugging this into the insight from before (equation 97) and taking the log yields which equals, up to a second order approximation, using x = 1 2 1−α 1 Θ V i (p ti ) and x 0 = 0, and realizing that V i (p ti ) 2 is higher than second order, Back to the approximated felicity function: Combine equation 84 and 88 with the approximation for d t (equation 104), ignoring terms of third and higher order, and use that in an efficient steady state − U N U C = M P N with Galí (2015, 21, 23)).
Using the above result in equation 106 yields Let us now turn to V i (p ti ). Claim: Ignoring terms of third and higher order, V i (p ti ) = θV i (p t−1,i ) + θ 1−θ π 2 t . Proof, following Woodford (2005, 399, 694): Similarly, using the above result q.e.d. Use V i (p ti ) ≡ ∆ t and iterate the above equation backwards.
Back to the approximated felicity function, i.e. equation 112. Multiplied by (−1) and the expectation operator E 0 , summed over each (current and future) period, and appropriately discounted by β t Using the insight from before (equation 126) yields the expected welfare loss function.
Define W ≡ −E 0 ∞ t=0 β t Ut−U U C C as the sum of the discounted current and future consumption loss equivalents as a fraction of steady state consumption (for more details, see below). Finally, the expected welfare loss function can be represented in two ways θ Θ (as above), W = ξ W , and ϑ = ξ σ + ϕ+α 1−α . W is a rescaled welfare loss measure computed with a normalized weight of 1 on the inflation variability.

Objective Function Under Limited Commitment
This subsection proves that under limited commitment, minimizing the (rescaled) welfare loss function (equation 131) of the representative household amounts to minimizing a central bank objective function of the form as in Debortoli et al. (2014) (cf. equation 141). 26 In particular, it shows that future period losses are discounted by β times the probability that the central bank has not deviated from the announced policy path until that period. This is so because the central bank cannot control the expectations about the allocation prevailing off the path on which the agents expect commitments to be honored.  In our case, the planner's value function is the minimized (rescaled) welfare loss function of the representative household is the control variable depending on the history of events up to period t. (ω t ) and k t (ω t ) is the state variable.
The planner's value function can be expressed as The first line is the path on which commitments are honored in every period. The second line captures each path starting with a deviation from the announced policy plan in period 1. The third line captures each path starting with a deviation from the announced policy plan in period 2, and so on. For The term i k t (ω t R,i ) captures each path starting with a deviation from the announced policy plan in period i > 0. Put differently, it captures each path on which the central bank deviates from its policy plan.
Let the degree of commitment be time-invariant. Then p(ω t ) = γ t and which is equal to Expectations about allocations prevailing off the path on which the agents expect commitments to be honored, i.e.
∞ t=0 E 0 t+1 , are exogenous to the central bank's optimization problem. Hence The central bank objective function as in Debortoli et al. (2014) (140) where t.i.p denotes terms independent of policy. In sum, we proved that minimizing the (rescaled) welfare loss function (equation 131)

Model Solution
This subsection derives the optimality condition of the optimal monetary policy problem under time-varying limited commitment. The derivation is based on the work of Debortoli et al. (2014), Galí (2015, and Woodford (2005). The central bank problem is to minimize the (rescaled) welfare loss function where γ t denotes the central bank's credibility in period t and −1 i=0 γ i = 1. 28 The policy problem is subject to the New Keynesian Phillips curve where E t π t+1 is the inflation rate that is expected to prevail if commitments are honored in period t + 1 and E t π d t+1 is the inflation rate that is expected to prevail if the central bank deviates from the announced policy path in period t + 1.
Let us express the optimization problem as a Lagrangian "subject to the constraint that [the optimal evolution of both choice variables from t = 0 onward] represents a possible rational expectation equilibrium" (Woodford (2005, 488)). Assume that the inflation rate which is expected to prevail if the central bank reneges on past policy commitment in t + 1 is an arbitrary (linear) function of the state variable(s) in t + 1. Formally, assume E t π d t+1 = E t f t+1 (u t+1 ) with the (time-varying) functional form of f t+1 unknown.
with λ t being the Lagrange multiplier associated to the New Keynesian Phillips curve in period t. The first order condition (FOC) with respect to π t , for each t and every potential state of the world is if γ i > 0 ∀i ∈ {0, . . . , t − 1}. The first order condition (FOC) with respect to x t , for each t and every potential state of the world is wherep t ≡ π t +p t−1 andp −1 = 0 as in Galí (2015, 135) if γ i > 0 ∀i ∈ {0, . . . , t − 1}. 29 By construction, deviations from the announced policy path in period t imply λ t−1 = 0, as in Debortoli et al. (2014). 30 31 Consequently, with t = 0 being the initial period of the policy plan wherep t = π t +p −1 andp t−1 = 0. For t > 0, the optimal output gap depends not only on the current inflation rate but also on lagged inflation rates. That is, there is a history dependence in the optimal output gap.
This finding previews the result that under (limited) commitment it is both possible and optimal to commit to future policy responses when facing a current period cost-push shock. The reason is that such a commitment affects the household's expectations which in turn affect current period variables (in particular, the inflation rate). Consequently, less of a current period variability in the output gap is necessary to achieve the optimal inflation rate. This is beneficial because the welfare loss function is strictly convex in the inflation rate and the output gap. 32 As an intermediary step, re-express the New Keynesian Phillips curve in terms ofp t . Use the definition ofp t , and replace x t with the (combined) first order condition of the optimal monetary policy problem (equation 154).
30 Suppose λ −1 = 0. Then, from equation 146 and 148 it follows that x 0 = − κ ϑ π 0 which is the the optimality condition for the period in which the policy plan is first implemented (cf. Galí (2015, 130, 135)). In other words, setting the non-physical λ −1 = 0 is akin to a deviation from the announced policy path in period 0. 31 In the period of the policy implementation (t = 0), the ratio between the inflation rate and the output gap is independent of central bank credibilities. However, the level of the inflation rate and the output gap changes with {γ t } T t=0 (cf. equation 173). 32 A more detailed discussion is provided in Galí (2015, 137).
Realize that, by definition, Apply the method of undetermined coefficients and guess the time-varying solution forp t to bep Further guess the time-varying (linear) solution π d t =ĥ t u t withĥ t unknown. Plug the two guesses into the re-expressed Phillips curve (equation 159). 33 Solve recursively for a t ∈ (0, 1).
if past policy commitments are honored (which is indicated by the subscript h). In terms ofũ = {u t } T +1 t=0 , the optimal {x t,h } 2 t=0 are In terms ofũ, the optimal {π t,h } 2 t=0 are π 0,h =ĥ 0 u 0 (179) π 2,h =ĥ 2 u 2 − (1 − a 2 ) ĥ 1 u 1 + a 1ĥ0 u 0 As in Debortoli et al. (2014), policy deviations in period s are associated to x s−1 = 0. Moreover, policy deviations are accompanied by a complete and permanent loss of central bank credibility, i.e. γ t = 0 ∀t ∈ {s, ..., 3}. In other words, if the central bank deviates from past policy commitment in period s, the optimal x t,d and π t,d are, for t = {s, ..., 3}, equal to the discretionary solution (policy deviations are indicated by the subscript d). 34 In period 3, the central bank implements the discretionary solution with certainty. 35 This implies a 3 = ϑ ϑ+κ 2 . Using equation 164, a 2 is given by 34 Note that γ t = 0 does not per se imply that the central bank behaves discretionary in period t. In equilibrium, however, γ t−1 = 0 always implies a policy deviation in period t. The proof is provided below. 35 The only consistent central bank credibility in T is therefore γ T = 0.
Claim: Time-invariant limited credibility is inconsistent with strategic policy deviations if γ T = 0. Proof: Suppose γ T = 0, γ t = 0.5 ∀t < T , V h 1 (u j 1 ) > V d 1 (u j 1 ) for exactly one j, and V h 2 (u k 2 ) > V d 2 (u k 2 ) for exactly one k. Logic of the proof: By contradiction: ∀k it is proven that γ t = 0.5 ∀t < T is inconsistent with strategic policy deviations.
Claim: Time-invariant full credibility is inconsistent with strategic policy Logic of the proof: By contradiction: for some k it is proven that γ t = 1 ∀t < T is inconsistent with strategic policy deviations.

is satisfied if and only if
(1 − a 2 ) 2 + κ 2 ϑ a 2 2 < 0 (212) Because ϑ > 0, the condition for V h 2 (u k 2 ) > V d 2 (u k 2 ) is never satisfied. This suggests γ 1 = 0, contradicting what was supposed to be true (namely that γ t = 1 ∀t < T is consistent with strategic policy deviations). Consequently, we have shown that full credibility in period t < T is inconsistent with strategic policy deviations if γ T = 0.
Claim: Time-invariant zero credibility is consistent with strategic policy deviations. Proof: Suppose γ t = 0 ∀t ≤ T . Then V h 1 (u j 1 ) ≤ V d 1 (u j 1 ) ∀j and V h 2 (u k 2 ) ≤ V d 2 (u k 2 ) ∀k. Logic of the proof: Direct proof: If V h 1 (u j 1 ) ≤ V d 1 (u j 1 ) ∀j and V h 2 (u k 2 ) ≤ V d 2 (u k 2 ) ∀k it is proven that γ t = 0 ∀t ≤ T is consistent with strategic policy deviations.
For reasonable parameterization, there is a γ T ≥γ ∈ [0, 1] such that V h 2 (ũ) > V d 2 (ũ) for each potential shock sequenceũ. As a consequence, the only consistent central bank credibility in T − 1 is γ T −1 = 1. This finding