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.

1 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 as follows: 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?

Answering the research questions sheds light on the importance of central bank credibility in monetary policy making. Central bank credibility has become increasingly relevant. For example, the effectiveness of forward guidance depends crucially on central bank credibility. The reason is that forward guidance works through agents’ expectations (cf. (Angeletos and Lian 2018)). Likewise, new policy proposals like average inflation targeting draw their merits from the central bank’s ability to make a credible policy commitment (cf. (Nessén and Vestin 2005)).

My paper connects to various strands of the literature, most notably to the literature on optimal monetary policy and the literature on limited commitment in monetary policy. The current literature on optimal monetary policy (e.g., Galí (2015); Woodford (2005); Clarida et al. (1999)) is mainly concerned with the central bank’s policy trade-off in the presence of a cost-push shock: either the central bank 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 time-inconsistent Footnote 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 when the cost-push shock hits the economy.

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 (e.g., forward guidance): it seems implausible to assume that a central bank issues a statement regarding its future policy conduct without any intention to deliver. 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 (e.g., Debortoli et al. (2014); Debortoli and Lakdawala (2016); Schaumburg and Tambalotti (2007)). In these models, the central bank is exogenously selected to deviate from past policy commitments with a time-invariant, state-independent probability. My work suggests that a time-invariant limited commitment scenario is not fully compatible with strategic policy decisions: In fact, central bank credibility is either time-varying or equal to one of the two extreme cases, i.e., full credibility or zero credibility Footnote 2 . In addition, a state-independent probability for a policy deviation may provoke 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.

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 transitory loss of central bank credibility. After a policy deviation, the central bank may regain access to a commitment technology with a non-zero probability.

The 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. Furthermore, 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 discusses some additional literature. Section 3 derives the solution to the optimal monetary policy problem under limited credibility. Furthermore, it introduces the notion of strategic policy decisions. Section 4 presents the results of the analysis. Section 5 concludes.

2 Additional literature

In addition to the previously mentioned papers, my work connects to the literature on rules versus discretion and forward guidance. Kydland and Prescott (1977) and Barro and Gordon (1983a) were the first to analyze the relevance of rules versus discretion in monetary policy. In particular, their work studies the permanent temptation to deviate from a monetary policy rule that prescribes a state-independent, 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. Furthermore, they 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 (e.g., Galí (2015); Woodford (2005); Clarida et al. (1999)). Such a cost-push shock drives a temporary wedge between the natural output level and the efficient output level.

The literature on forward guidance studies limited commitments in monetary policy. 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. In their paper, the timing of a policy deviation is exogenous. Such a setup provokes outcomes which are sub-optimal. In particular, it may be that the central bank is forced into a policy deviation when the continuation of the policy plan would have been optimal. In my model, policy deviations occur strategically, i.e., only if a policy deviations delivers a higher welfare than the implementation of the pre-announced policy path.

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. Furthermore, in their model, the actual decision of whether or not to implement past policy commitments is simply 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.

3 Model

This section is organized as follows. First, it presents the core of the New Keynesian model as in Galí (2015). Second, it derives the model solution under limited commitment as in Debortoli et al. (2014). Third, it introduces the notion of strategic deviations. Forth, it presents the driving process of the model and the model calibration. Fifth, it describe the finite period version of the model which is used to study strategic deviations in optimal monetary policy.

3.1 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 model features a representative household which maximizes a utility function over consumption and leisure. In addition, there is a continuum of monopolistically competitive firms, producing differentiated intermediary output goods with a linear technology. There is no capital in the model. In each period, firms may re-optimize the price of their output goods with probability 1−θ, where θ ∈ (0,1), as in Calvo (1983). The log-linearized non-policy equilibrium of the model is given by the dynamic IS Eq. (1) and the New Keynesian Philipps curve (2)

l> &x_=\mathbb_ x_-\frac\left(i_-\mathbb_\pi_-\rho\right) \end $$ $$\beginl> &\pi_=\kappa x_+\beta\mathbb_\pi_+u_ \end $$

where x_t is the efficient output gap, π_t the inflation rate, i_t the nominal interest rate, u_t the cost-push shock, σ≥0 the constant relative risk aversion or, equivalently, the inverse intertemporal elasticity of substitution, ρ the steady state real interest rate, κ≡ξ(σ+φ) with \(\xi \equiv \frac <(1-\theta)(1-\beta \theta)>\) , and β ∈ (0,1) the discount factor of the household. φ is the inverse of the Frisch labor supply elasticity Footnote 3 .

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

$$\beginl> &\mathbb_ \sum_^ <\infty>\beta^\frac\left(\pi_^+\vartheta x_^\right) \end $$

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 \(\vartheta \equiv \frac <\xi >\left (\sigma +\varphi \right)\) , where ε ∈ (1,∞) is the elasticity of substitution between intermediary goods Footnote 4 .

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.

3.2 Optimal monetary policy under limited commitment

Building on the work of Debortoli and Lakdawala (2016) and Debortoli et al. (2014), who derive the welfare loss function under limited commitment, I additionally introduce time-variation in central bank credibility Footnote 5 .

$$\beginl> \mathbb_ \sum_^ <\infty>\beta^\prod_^\gamma_\frac\left(\pi_^+\vartheta x_^\right) \end $$

where γ_t denotes the central bank’s credibility in period t and \(\prod _^\gamma _=1\) . Note that γ₀ (rather than γ₁) is associated to (x₁,π₁) 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

$$\beginl> &\pi_=\kappa x_ +\beta \gamma_\mathbb_ \pi_+\beta(1-\gamma_)\mathbb_ \pi_^+u_ \end $$

where \(\mathbb _ \pi _\) is the inflation rate that is expected to prevail if commitments are honored in period t+1 and \(\mathbb _ \pi _^\) 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 \(\mathbb _ \pi _^=\mathbb _f_(u_)\) 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 Footnote 6 . Combining and iterating on the first order conditions with respect to π_t and x_t yields

$$\beginl> &x_=-\frac<\kappa>\left[\pi_+\pi_+\ldots+\pi_-\lambda_\right] \end $$

if γ_i>0 ∀ i ∈ t−1> Footnote 7 . By construction, deviations from the announced policy path in period s require λ_s−1=0, as in Debortoli et al. (2014). Setting λ_s−1=0 implies \(x_=-\frac <\kappa >\pi _\) which is 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_s−1=0 is akin to a policy deviation in period s Footnote 8 . Consequently, with t=0 being the initial period of the policy plan

where \(\hat

_\equiv \pi _+\hat

_\) and \(\hat

_=0\) Footnote 9 . 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 of \(\hat

_\) .

with \(\mu _\equiv \frac <\vartheta (1+\beta \gamma _)+\kappa ^>\) . Suppose u_t ∼ AR(1) with \(\mathbb (\varepsilon _^)=0\) and \(V(\varepsilon _^)=\sigma _<\varepsilon _^>^\) and guess the time-varying solution for \(\hat

_\) to be a linear function of \(\hat

_\) and u_t.

$$\beginl> &u_=\rho_ u_+\varepsilon_^ \end $$

Plug the guess for \(\hat

_\) (Eq. 11) and the guess for \(\pi _^\) into the re-expressed Phillips curve (Eq. 9) and solve recursively for a_t ∈ (0,1).

$$\beginl> &a_=\frac<\mu_><1-\mu_\beta\gamma_ a_>\hspace \forall\hspace t \end $$

Realize that a deviation from the announced policy path in t requires \(\hat

_=0\) (cf. Eq. 8). From the guess for \(\hat

_\) (Eq. 11), we know that \(\hat

_^=c_u_\) . Furthermore, by definition, \(\hat

_^=\pi _^\) . Because I assume \(\pi _^=\hat _ u_\) , it must be that \(\hat _=c_\hspace \forall t\) . Solve recursively for c_t ∀ t ∈ T>, using \(\_^\) from above.

$$\beginl> &c_=\frac<\mu_(1+\beta c_\rho_)><1-\mu_\beta\gamma_ a_>\hspace \forall t\in\ <0,\ldots,T\>\end $$

with \(c_=\frac <\kappa ^<2>+\vartheta (1-\beta \rho _)>\hspace \forall i\geq 1\) as in (Galí (2015), 130). The optimality condition (Eq. 8), together with the guess for \(\hat

_\) (Eq. 11) and the solution for the coefficients (in particular, \(c_=\hat _\) ) yields the time-varying model solution for γ_t ∈ (0,1)

$$\beginl> &x_=a_x_-\frac<\hat_\kappa>u_ \end $$

The output gap x_t depends on the entire sequence of current and future central bank credibilities \(\<\gamma _\>_^\) . More specifically, \(\<\gamma _\>_^\) determines the optimal persistence in the output gap as well as the severity of the policy trade-off.

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 \(\mathbb _\pi _\) 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 \(\<\gamma _\>_^\) , 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 ( \(\hat _\) 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, \(\mathbb _\pi _\) is ceteris paribus higher, the lower T and/or the lower the values in \(\<\gamma _\>_^\) . From above, we know that this off-equilibrium increase in \(\mathbb _\pi _\) 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 \(\mathbb _\pi _\) 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 \(\<\gamma _\>_^\) , the higher the impact coefficient ( \(\hat _\) ).

3.3 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.

The strategic policy problem is a recursive representation of the central bank optimization problem (Eq. 4) 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 its credibility for some time after deviating from the announced policy path. In each period after a policy deviation, including the period of the policy deviation, the central bank may regain access to a commitment technology with a non-zero probability. The central bank can take a strategic policy decision in period t+1 after it deviated in period t only if it regains access to a commitment technology in period t. By contrast, if the central bank honors past policy commitments in t, it can take a strategic policy decision in t+1 with certainty. Formally,