2.1 Framework

To illustrate the main mechanisms behind our results, we now use the simple new Keynesian model as an example [Clarida et al. (Reference Clarida, Galí and Gertler1999)]. The general results for a broad class of models will be derived in Section 3.

In every period $t=0,1,2,\ldots$ , the private sector’s behavior is summarized by:

(1) \begin{equation} \pi _t = \beta \mathbb E_t \pi _{t+1} + \kappa u_t + \xi _t, \end{equation}

(2) \begin{equation} \xi _{t+1} = \rho \xi _t + \varepsilon _{t+1}, \end{equation}

where $\beta \in (0,1)$ is a common discount factor, $\kappa \gt 0$ , $\rho \in (0,1)$ , and the $\varepsilon _t$ ’s are i.i.d. shocks that are drawn from a normal distribution with mean zero. $\pi _t$ is the inflation rate and $u_t$ is the output gap. Equation (1) is the new Keynesian Phillips curve and (2) describes the evolution of markup shocks. The initial value of $\xi _t$ , $\xi _0$ , is exogenously given. Taking (1) and (2) into account, the central bank minimizes the expected discounted sum of losses:

(3) \begin{equation} l(\pi _t,u_t) = \frac 1 2 \pi _t^2+ \frac a 2 u_t^2 \end{equation}

with $a\gt 0$ . In our setup, $u_t$ constitutes the central bank’s instrument. $\pi _t$ is a forward-looking or non-predetermined variable. $\xi _t$ is a predetermined variable.Footnote ¹³ The IS curve is omitted to simplify the exposition.

2.2 Optimal commitment

We first construct the optimal commitment solution, which enables us to show later how a discretionary policy-maker can achieve this solution. The optimal commitment solution can be obtained by setting up the Lagrangian:

(4) \begin{equation} \mathcal L_0 = \mathbb E_0 \sum _{t=0}^\infty \beta ^t \left [\frac 1 2 \pi _t^2+\frac a 2 u_t^2 + \lambda _{t+1}\left (\beta ^{-1} \pi _t - \pi _{t+1} - \kappa \beta ^{-1} u_t -\beta ^{-1} \xi _t \right )\right ], \end{equation}

where $\lambda _t$ ( $t=1,2,\ldots$ ) are the multipliers associated with the new Keynesian Phillips curve.

As it is well known [Clarida et al. (Reference Clarida, Galí and Gertler1999) and Woodford (Reference Woodford1999)], the commitment solution can be characterized by:

(5) \begin{equation} u_{t+1} - u_t = - \frac \kappa a \pi_{t+1} \qquad \textit{for t=0,1,2, ... } \end{equation}

(6) \begin{equation} u_0 = - \frac \kappa a \pi _0 \end{equation}

together with (1) and (2).

In line with Backus and Driffill (Reference Backus and Driffill1986) and Söderlind (Reference Söderlind1999), the commitment solution can be described by equations that specify the joint evolution of $\xi _t$ and $\lambda _t$ as well as by equations that state how the non-predetermined variable $\pi _t$ and the instrument $u_t$ depend on the current values of $\xi _t$ and $\lambda _t$ . With the help of the optimal commitment solution in Clarida et al. (Reference Clarida, Galí and Gertler1999), it is straightforward to show that, for the simple new Keynesian model under consideration, these equations are

(7) \begin{eqnarray} \begin{pmatrix} \xi _{t+1} \\[5pt] \lambda _{t+1} \end{pmatrix} = \begin{pmatrix} \rho\;\;\;\; & 0 \\[5pt] -\frac{\beta \delta }{1- \delta \beta \rho }\;\;\;\; & \delta \end{pmatrix} \begin{pmatrix} \xi _t \\[5pt] \lambda _t \end{pmatrix} + \begin{pmatrix} \varepsilon _{t+1} \\[5pt] 0 \end{pmatrix}, \end{eqnarray}

where

(8) \begin{equation} \delta = \frac{1+ \beta + \frac{\kappa ^2} a - \sqrt{(1-\beta )^2+ 2(1+\beta )\frac{\kappa ^2}a + \frac{\kappa ^4}{a^2}}}{2 \beta } \in (0,1), \end{equation}

as well as

(9) \begin{equation} \begin{pmatrix} \pi _t \\[5pt] u_t \end{pmatrix} = \begin{pmatrix} \frac \delta{1-\delta \beta \rho } & \frac{1-\delta } \beta \\[5pt] - \frac{\kappa \delta }{a (1-\delta \beta \rho )} & \frac{\kappa \delta }{a \beta } \end{pmatrix} \begin{pmatrix} \xi _t \\[5pt] \lambda _t \end{pmatrix}. \end{equation}

For exogenous values of $\xi _0$ and $\lambda _0=0$ , (7) and (9) describe the entire dynamics of the system. Loosely speaking, equation (7) can be interpreted as introducing a new predetermined state variable in addition to $\xi _t$ : the Lagrange multiplier on the forward-looking constraint, $\lambda _t$ .

2.3 Discretionary policy-making

Abandoning the restriction to Markov-perfect strategies for the time being, we now construct a discretionary equilibrium that implements the commitment solution. This equilibrium involves an additional state variable. A general law of motion for the additional state variable $s_t$ is

(10) \begin{equation} s_{t+1} = \phi _\xi \xi _t + \phi _s s_t + \phi _u u_t, \end{equation}

with a given initial value $s_0$ . It is clear that, in every period $t$ , $s_t$ can be written exclusively as a function of past shocks and past values of the instrument. Thus, decisions made by the policy-maker or other agents that depend on $s_t$ are influenced by past events that are not payoff-relevant. This is a typical feature of reputational equilibria such as those considered by Chari and Kehoe (Reference Chari and Kehoe1990, Reference Chari and Kehoe1993). In classic analyses of reputational equilibria in the inflation bias literature [Barro and Gordon (Reference Barro and Gordon1983)], inflation expectations are specified to be functions of past deviations of inflation from some optimal level.

A discretionary equilibrium for the economy under consideration can be defined as follows:

Definition 1. Consider a central bank with instrument $u_t$ , loss function (3), and discount factor $\beta$ , which faces the constraints (1), (2), as well as the law of motion (10) for the payoff-irrelevant state variable $s_t$ , where $\phi _u$ , $\phi _\xi$ , $\phi _s$ , and $s_0$ are given. Then

(11) \begin{equation} \begin{pmatrix} \pi _t \\[5pt] u_t \end{pmatrix} = M \begin{pmatrix} \xi _t \\[5pt] s_t \end{pmatrix} \end{equation}

with $2 \times 2$ -dimensional matrix $M$ describes a discretionary equilibrium if the following properties hold:

1. 1. The path of $\{\pi _t, u_t, \xi _t\}_{t=0}^\infty$ implied by (2), (10), and (11) satisfies (1).

2. 2. In each period $t$ , no profitable deviation exists for the central bank, that is, the choice of $u_t$ implied by (11) is optimal given (1), where the central bank takes the process by which rational inflation expectations $\mathbb E_t \pi _{t+1}$ are formed as given.

3. 3. Rational expectations $\mathbb E_t \pi _{t+1}$ are formed in line with (2), (10), and (11).

This definition is identical to the one used in the literature [see e.g., Backus and Driffill (Reference Backus and Driffill1986)] with the only modification that the additional state variable $s_t$ , which is not payoff-relevant, is introduced.Footnote ¹⁴

For payoff-relevant state variables, the coefficients in the law of motion are typically exogenously given. By contrast, $s_t$ is an auxiliary variable that is used to describe how decisions and expectations are affected by payoff-irrelevant past events. Note that in our paper as in other analyses of reputational mechanisms, there are multiple equilibria.Footnote ¹⁵ Thus, the $\phi$ ’s in the law of motion for $s_t$ , (10), are coefficients that will be pinned down in order to obtain the equilibrium of interest, namely the one implementing the commitment outcome. Other choices of the coefficients would lead to other reputational equilibria or may not result in reputational equilibria at all. The flexibility with which the additional state variable can be introduced may be reminiscent of sunspots.Footnote ¹⁶ A major difference between the two concepts is the fact that sunspot variables are exogenous stochastic processes, whereas the state variable that we introduce is endogenous and can be influenced by the policy-maker, in particular.

We make use of the flexibility with which $s_t$ can be introduced and set $s_0=0$ as well as

(12) \begin{equation} \phi _u = -\frac \kappa{1-\delta }, \end{equation}

(13) \begin{equation} \phi _\xi = -\frac 1{1- \delta \beta \rho }, \end{equation}

(14) \begin{equation} \phi _s = \frac 1 \beta. \end{equation}

It may be instructive to give an intuition for how the parameters for the law of motion for $s_t$ are chosen and why the central bank cannot profitably deviate in the discretionary equilibrium implementing the commitment solution that we are constructing.

In the Markov-perfect equilibrium of the standard new Keynesian model where the central bank minimizes social losses, there is no endogenous state variable, and inflation in all periods is only a function of the current markup shock. Hence, inflation expectations $\mathbb E_t \pi _{t+1}$ cannot be influenced by the central bank’s choice of $u_t$ , as the future markup shock $\xi _{t+1}$ is exogenous to monetary policy. Together with this observation, the Phillips curve (1) implies that an increase in $u_t$ affects current inflation only via the traditional marginal-cost channel, where an increase of $u_t$ by $\Delta u$ entails an increase in inflation by $\kappa \Delta u$ .

In a discretionary equilibrium that violates the Markov property, changes in the instrument $u_t$ may lead to changes in inflation expectations because changes in $u_t$ influence $s_{t+1}$ , which affects expectations about inflation in the future. These inflation expectations, in turn, influence current inflation, according to the new Keynesian Phillips curve. Thus, the central bank can affect current inflation $\pi _t$ not only via the traditional marginal-cost channel, which we have described above, but also via an expectations channel. As will be shown formally later, the particular choice of $\phi _u$ in (12) guarantees that the effects of a change in $u_t$ on inflation via the marginal-cost channel and the expectations channel exactly offset each other. As a consequence, inflation $\pi _t$ effectively becomes a predetermined variable, which cannot be influenced by the central bank’s choice in period $t$ .

This observation is key to understanding how the discretionary equilibrium we are constructing can implement the commitment solution. The central bank can choose its current instrument in a way such that inflation is expected to correspond to the value compatible with the commitment solution in the next period because, in the next period, it will not be possible for the central bank to influence inflation in that period. By contrast, a central bank in the discretionary equilibrium without an additional payoff-irrelevant state variable can always influence the current value of inflation and thereby implement a policy such that inflation differs from the value implied by the commitment solution.Footnote ¹⁷

Finally, we discuss how the remaining parameters are set. Obviously, $s_0=0$ ensures $s_0=\lambda _0$ .Footnote ¹⁸ The parameter choices for $\phi _\xi$ and $\phi _s$ were made in a way such that, for the value of $\phi _u$ given in (12) and $s_0=0$ , the evolution of the variables $\pi _t$ , $u_t$ , and $s_t$ equals the evolution of the corresponding variables $\pi _t$ , $u_t$ , and $\lambda _t$ in the commitment solution. Why the specific value of $\phi _s$ is required to implement the commitment solution is discussed in more detail in Section 2.4. In Appendix A, we comment on a potential concern that, together with (10), $\phi _s\gt 1$ could involve explosive dynamics for $s_t$ .

The following proposition shows that the commitment solution of the standard new Keynesian model can be implemented via a discretionary equilibrium:

Proposition 1. Consider a central bank with instrument $u_t$ , loss function (3), and discount factor $\beta$ , which faces the constraints (1), (2), as well as the following law of motion for the payoff-irrelevant state variable $s_t$ :Footnote ¹⁹

(15) \begin{equation} s_{t+1} = \frac 1 \beta s_t -\frac 1{1- \delta \beta \rho } \xi _t - \frac \kappa{1-\delta } u_t, \qquad \textit{with $s_0=0$}. \end{equation}

Then

(16) \begin{equation} \begin{pmatrix} \pi _t \\[5pt] u_t \end{pmatrix} = \begin{pmatrix} \frac \delta{1-\delta \beta \rho } & \frac{1-\delta } \beta \\[5pt] - \frac{\kappa \delta }{a (1-\delta \beta \rho )} & \frac{\kappa \delta }{a \beta } \end{pmatrix} \begin{pmatrix} \xi _t \\[5pt] s_t \end{pmatrix} \end{equation}

describes a discretionary equilibrium. This equilibrium implements the commitment solution.

While it is easy to confirm the statement of the proposition for specific parameter combinations with standard software routines that can be used to compute discretionary equilibria (Söderlind, Reference Söderlind1999), it is instructive to also consider a formal proof.

Proof

As a preliminary step, we note that the dynamics of $\pi _t$ and $u_t$ equal those in the commitment solution. This can be seen by comparing (15) and (16) to (7) and (9). In particular, plugging the expression for $u_t$ given in (16) into (15) yields an expression for $s_{t+1}$ as a function of $\xi _t$ and $s_t$ . With the help of (8) as well as $s_0=\lambda _0=0$ , it is straightforward to show that this equation for $s_{t+1}$ yields dynamics for the additional state variable $s_t$ that are identical to the dynamics for the Lagrange multiplier in the commitment solution, which are described by (7).

To prove that (16) describes a discretionary equilibrium, we show that, first, together with (2) and (15), it is compatible with the Phillips curve (1) and that, second, the central bank’s choice of $u_t$ in every period $t$ is optimal subject to the new Keynesian Phillips curve in this period $t$ , given the rational inflation expectations $\mathbb E_t \pi _{t+1}$ that are formed in line with (2), (15), and (16).

The candidate equilibrium is compatible with the new Keynesian Phillips curve, because (i) the economy evolves as in the commitment solution and (ii) the commitment solution satisfies the new Keynesian Phillips curve. To prove that the central bank behaves optimally requires a few steps. First, we calculate inflation expectations, as they enter the new Keynesian Phillips curve, which represents a constraint for the central bank. Using (2), (15), and (16), we obtain the following expression:

(17) \begin{equation} \mathbb E_t \pi _{t+1} = \frac 1{1- \delta \beta \rho } \left ( \delta \rho - \frac{1-\delta } \beta \right ) \xi _t + \frac{1-\delta }{\beta ^2} s_t - \frac \kappa \beta u_t. \end{equation}

As we have argued before, it is crucial that inflation expectations are a function of $u_t$ . This property stems from the fact that the central bank’s choice of $u_t$ affects $s_{t+1}$ , which in turn affects inflation in period $t+1$ .

As a next step, we plug the inflation expectations (17) into the new Keynesian Phillips curve (1) and obtain

(18) \begin{equation} \pi _t = \frac \delta{1- \delta \beta \rho } \xi _t + \frac{1-\delta } \beta s_t. \end{equation}

It is noteworthy that the central bank’s choice of $u_t$ has no influence on inflation in the same period. As has been explained before, this is a property of the particular equilibrium under consideration and is a consequence of the specific choice of $\phi _u$ . Section 2.5 studies a case where, due to small deviations from rational expectations, the influence of $u_t$ on $\pi _t$ is not zero but small. In this case, outcomes close to the commitment outcome can still be implemented.

Consider optimal central bank behavior in a particular period $t$ . For this purpose, we set up the corresponding Bellman equation:

(19) \begin{equation} \begin{split} W(\xi _t,s_t) =& \min _{u_t} \left \{\frac 1 2 \pi _t^2 + \frac a 2 u_t^2 + \beta \mathbb E_t W(\xi _{t+1},s_{t+1})\right \} \\[5pt] &\qquad \textit{subject to (2), (15), (18).} \end{split} \end{equation}

We obtain the following first-order condition as well as a condition that results from the envelope theorem:

(20) \begin{equation} 0 = a u_t -\frac \kappa{1-\delta } \beta \mathbb E_t W_s(\xi _{t+1}, s_{t+1}), \end{equation}

(21) \begin{equation} W_s(\xi _t, s_t) = \frac{1-\delta } \beta \pi _t + \mathbb E_t W_s(\xi _{t+1}, s_{t+1}), \end{equation}

where the subscript $s$ stands for the respective partial derivative. Equations (20) and (21) can be combined to

(22) \begin{equation} \mathbb E_t u_{t+1} - u_t = - \frac \kappa a \mathbb E_t \pi _{t+1} \qquad \textit{for t=0,1,2,...} \end{equation}

This condition characterizes optimal central bank behavior in the candidate discretionary equilibrium. As the paths of $\pi _t$ and $u_t$ satisfy the condition for optimal central bank behavior under commitment, (5), they also satisfy (22). Q.E.D.

The additional state variable $s_t$ can be interpreted as the burden of past promises. To make this more precise, note that (16) implies that the central bank’s instrument $u_t$ is an increasing function of $s_t$ in equilibrium. Thus, a positive value of $s_t$ represents a promise to choose a rather expansionary monetary policy, while a negative value represents a promise to choose a comparably contractionary policy. As (15) entails that $s_{t+1}$ is a decreasing function of $u_t$ , current expansionary monetary policy tends to entail the promise of contractionary monetary policy in the future. As a consequence, the central bank behaves as in the commitment solution. In the commitment solution, a negative markup shock calls for expansionary policy in the same period but contractionary policy in the next period.

2.4 Dynamic response to policy errors

We continue our discussion of the standard new Keynesian model by illustrating the dynamics of the economy if the central bank deviates from its optimal choice of $u_t$ in period $0$ but not in future periods. We set the parameters as follows: $\kappa =0.3$ , $\beta =0.99$ , $\rho =0.9$ , $a=0.05$ , and $\xi _0=0$ , in addition to $s_0=0$ . Moreover, assume that the central bank chooses $u_0=1$ rather than $u_0=0$ , which would be optimal.

Figure 1 illustrates the dynamics of the system in the absence of markup shocks $\xi _t$ , that is, for $\varepsilon _t=0$ in $t=1,2,3,\ldots$ . The deviation of the central bank cannot influence inflation (displayed as a solid line) in period $0$ , as the discretionary equilibrium under consideration involves that inflation is effectively predetermined.Footnote ²⁰ According to the dash-dotted line, which displays $s_t$ , the suboptimally high level of $u_0$ drives $s_1$ below zero (compare (10) and (12)). Because $s_{t+1}$ depends positively on $s_t$ on the equilibrium path ( $s_{t+1}=\delta s_t$ ), $s_t$ stays negative in the consecutive periods $t=2,3,..$ . As inflation and output are increasing functions of $s_t$ (compare (9) for $s_t=\lambda _t$ ), both output and inflation are negative from period $1$ onward.

Figure 1.

Impulse responses of inflation (solid line), the output gap (dashed line), and the additional state variable $s_t$ (dash-dotted line) in response to a one-time deviation of the output gap. Markup shocks have been set to zero in all periods.

The figure shows that deviations from the equilibrium behavior in a particular period lead to changes in the additional state variable $s$ in future periods and thereby to increases in losses in these periods. It may be interesting to contrast the consequences of a deviation in this paper and the respective consequences in papers that use trigger strategies to overcome time-inconsistency problems [see, e.g., Loisel (Reference Loisel2008) and Levine et al. (Reference Levine, McAdam and Pearlman2008)].Footnote ²¹ Compared to equilibria with trigger strategies, a small deviation of the central bank only has small, transitory consequences for the economy. As can be seen easily, the response of the economy after a deviation is always proportional to the magnitude of the deviation.Footnote ²²

It may also be instructive to consider the situation where $s_t$ differs from zero in one period $t$ , there are no markup shocks, that is $\xi _t=0$ , and the central bank ignores the burden of past “promises,” $s_t$ , by setting $u_t=0$ in this period. In this case, $s_{t+1}=\beta ^{-1} s_t$ would hold. Thus, the burden of past “promises” would grow by a factor $\beta ^{-1}$ . This mild increase in $s_t$ deters deviations in the first place. It might also be noteworthy that the factor $\beta ^{-1}$ decreases with $\beta$ . This is plausible, as larger discount factors require smaller punishments in the future.

2.5 Deviations from rational expectations

In a sense, the discretionary equilibrium implementing the commitment solution represents a knife-edge case in which the effects of changes in the central bank’s instrument on inflation via the marginal-cost channel and the expectations channel cancel each other exactly. This may raise questions about the robustness of our approach.

To study this issue, we consider small deviations from rational expectations. In particular, we assume that inflation expectations are not given by (17) but by

(23) \begin{equation} \mathcal E_t \pi _{t+1} = \frac 1{1- \delta \beta \rho } \left ( \delta \rho - \frac{1-\delta } \beta \right ) \xi _t + \frac{1-\delta }{\beta ^2} s_t - \left ( \frac \kappa \beta + \epsilon \right ) u_t, \end{equation}

where we use $\mathcal E_t$ to denote expectations that may not be fully rational, and $\epsilon$ is a parameter that describes the magnitude of deviations from rationality. Obviously, for $\epsilon =0$ , (23) is identical to (17). Under the assumption that private sector expectations are governed by (23), the Phillips curve (1) implies that inflation is given by:

(24) \begin{equation} \pi _t = \frac \delta{1- \delta \beta \rho } \xi _t + \frac{1-\delta } \beta s_t - \beta \epsilon u_t, \end{equation}

which is a generalization to (18). Importantly, for $\epsilon \ne 0$ , it is no longer true that the central bank cannot influence current inflation by changing its instrument $u_t$ . However, for small values of $\epsilon$ , the effect of changes in the instrument $u_t$ on $\pi _t$ is small.

As a next step, we analyze the dynamic consequences of this departure from rational expectations. To complete the description of the variant of the model, we assume that the central bank knows that expectations are formed in line with (23) and that $s_t$ still evolves according to (15). Figure 2 shows the impulse responses of inflation (left panel) and the output gap (right panel) to a markup shock. The solid lines stand for $\epsilon =0$ , that is, rational expectations, and the dashed lines for small deviations from rational expectations where we have set $\epsilon =0.01$ .

Figure 2.

Impulse responses of inflation (left panel) and the output gap (right) panel for a markup shock with $\varepsilon _0=1$ . The solid lines represent the impulse responses when inflation expectations are perfectly rational ( $\epsilon =0$ ). The dashed lines show the impulse responses when inflation expectations not perfectly rational, that is, they are given by (23) with $\epsilon =0.01$ .

The figure shows that small deviations from rational expectations only have a small effect on the dynamics of the economy. The reason for this finding is that, while changes in the instrument affect current inflation for small but nonzero values of $\epsilon$ and thus inflation cannot be interpreted as an effectively predetermined variable, non-negligible changes in inflation require very large changes in the instrument, which tend to be costly for the central bank. Hence, small deviations from rational expectations allow the implementation of an equilibrium with social losses that are close to the losses under commitment.

While we have only studied a very specific deviation from the equilibrium implementing the commitment outcome, the conclusions from this analysis hold more generally: Small deviations from the equilibrium implementing the commitment solution lead to equilibria that involve social losses very close to the ones implied by the commitment solution. Consider, for example, a discretionary equilibrium where the central bank responds to the state variable $s_t$ but where $s_t$ follows a law of motion (10) with coefficients $\phi _u$ , $\phi _\xi$ , and $\phi _s$ slightly different from the ones specified in (12)–(14). Note that inflation and output in equilibrium can always be expressed as functions of current and lagged markup shocks and that the coefficients in front of these realizations of markup shocks are continuous functions of the $\phi$ ’s. As a consequence, small changes in the $\phi$ ’s only have small effects on the dynamic response of the economy to shocks and therefore only small effects on social losses.

2.6 Implementation through delegation

As a next step, we show that the commitment solution can also be implemented by a standard discretionary Markov equilibrium when monetary policy is delegated to a central bank with a loss function that is different from the social loss function.Footnote ²³ Our approach is related to the result by Woodford (Reference Woodford2003) that the delegation of monetary policy to a central bank with an interest-rate smoothing objective can be socially desirable even if interest-rate smoothing is not socially desirable per se. It extends this finding in two dimensions. First, our approach implements the commitment solution and thus allows the central bank to reap all of the welfare gains that are theoretically possible. By contrast, interest-rate smoothing typically allows the central bank to achieve only a part of the possible welfare gains. Second, our approach is more general as it can be applied to the broad class of linear models considered in Section 3 and is not restricted to the standard new Keynesian model. Optimal interest-rate targets that facilitate the implementation of the commitment solution in the new Keynesian model are derived in Section 3.4.

The following proposition is shown in Appendix C:

Proposition 2. Suppose that monetary policy-making is delegated to a central bank whose instantaneous losses are

(25) \begin{equation} l_{CB}(\pi _t,u_t,s_t) = \frac 1 2 \pi _t^2+ \frac a 2 u_t^2 + \frac b 2 (u_t-u_t^*)^2, \end{equation}

where

(26) \begin{equation} u_t^* = -\frac{\kappa \delta }{a(1-\delta \beta \rho )} \xi _t + \frac{\kappa \delta }{a\beta } s_t \end{equation}

and $b$ is an arbitrary parameter with $b\gt 0$ . $s_t$ is given by (15). The central bank discounts its future losses with the same discount factor $\beta$ as society. Then the commitment solution associated with the social loss function (3) can be implemented by a discretionary Markov equilibrium.

Compared to the social loss function, the central bank loss function (25) includes the additional term $b (u_t-u_t^*)^2/2$ , which penalizes deviations of the central bank’s instrument from the time-varying target $u_t^*$ . $u_t^*$ mimics the evolution of $u_t$ in the discretionary equilibrium implementing the commitment solution (compare (16)). Thus, on the equilibrium path, $u_t^*$ equals the central bank’s choice of instrument for the commitment solution. In this sense, the central bank’s modified loss function punishes deviations from the commitment solution.Footnote ²⁴

While it is perhaps unsurprising that large costs of deviations from the commitment solution ensure that the central bank implements this solution, the proposition implies that even small incentives, that is, arbitrarily small values of $b$ , are sufficient. Loosely speaking, the reason why small values of $b$ are sufficient to ensure that the central bank behaves as under optimal commitment is that the term $b (u_t-u_t^*)^2/2$ does not alter the incentives of the central bank compared to the equilibrium discussed in Proposition 1, as the central bank already chooses $u_t=u_t^*$ in this equilibrium. The purpose of the additional term in the central bank’s loss function is to add an additional state variable to the central bank’s optimization problem.

It may also be noteworthy that it is possible to completely eliminate $s_t$ from the central bank’s optimization problem, as it is straightforward to formulate $u_t^*$ as a function of the current value of $\xi _t$ as well as lagged values of $u_t^*$ and $u_t$ . More specifically, we can write

(27) \begin{equation} u^*_t=\frac 1 \beta u_{t-1}^* -\frac{\kappa \delta }{a(1-\delta \beta \rho )} \xi _t -\frac{\kappa ^2 \delta }{a\beta (1-\delta )} u_{t-1}. \end{equation}

One might think that the coefficient $\frac 1 \beta$ implies explosive dynamics for the output-gap target $u_t^*$ . However, it is easy to see that this is not the case. For this purpose, we rewrite (27) as:

(28) \begin{equation} u^*_t=\frac 1 \beta \left [ 1- \frac{\kappa ^2\delta }{a(1-\delta )}\right ] u_{t-1}^* -\frac{\kappa \delta }{a(1-\delta \beta \rho )} \xi _t -\frac{\kappa ^2 \delta }{a\beta (1-\delta )} (u_{t-1}-u_{t-1}^*) \end{equation}

and note that, in equilibrium, $u_t=u_t^*$ holds for all periods and that the coefficient in front of $u_{t-1}^*$ is smaller than 1 (and positive) for all admissible parameter values.

For the parameter values, we have used before ( $\kappa =0.3$ , $\beta =0.99$ , $\rho =0.9$ , and $a=0.05$ ), we obtain

(29) \begin{equation} u^*_t=0.2851 u_{t-1}^* -2.2929 \xi _t -0.7250 (u_{t-1}-u_{t-1}^*). \end{equation}

The property that the target value $u_t^*$ depends on lagged values of the central bank’s instrument is crucial for the implementability of the commitment solution. In particular, it is responsible for the key feature of our approach that non-predetermined economic variables are effectively turned into predetermined ones.

Figure 3 illustrates the dynamics of the output-gap target $u_t^*$ for a markup shock with $\varepsilon _0=1$ and $\varepsilon _t=0$ for $t=1,2,3,\ldots$ .Footnote ²⁵ The evolution of the output gap under commitment is identical to the evolution of the output-gap target as well as the output gap chosen by a discretionary central bank under optimal delegation (solid line). In addition, we show the dynamics of the output-gap target (dotted line) if the central bank deviates in period $0$ and chooses only $50\%$ of the optimal value for the output gap but implements the optimal policy thereafter. As a consequence of this deviation, the output-gap target drops in period $1$ but quickly returns toward the equilibrium path. The output gap (dashed line) differs from its target in period $0$ by construction but corresponds to the target in all remaining periods. We have confirmed that, in line with our theoretical results, the value of $b$ , which describes the intensity of the additional incentives to align output with the additional target $u_t^*$ , does not affect the equilibrium outcomes under delegation.

Figure 3.

Impulse responses of the equilibrium output gap and the corresponding target in response to a markup shock with $\varepsilon _0=1$ (solid line); the output gap in response to the markup shock with a deviation in period 0 (dashed line), and the output-gap target for the deviation (dotted line).

At this point, it is instructive to discuss the relationship to the interest-rate smoothing objective considered by Woodford (Reference Woodford2003) in more detail. Woodford augments the central bank’s loss function by terms that are proportional to $(i_t)^2$ and $(i_t-i_{t-1})^2$ , where the central bank’s instrument $i_t$ is the deviation of the nominal interest rate from its long-run level. These additional terms in the loss function can be interpreted as introducing an additional objective for the central bank’s instrument, where the target value $i_t^*$ is proportional to $i_{t-1}$ . This is obviously related to our approach, where the target $u_t^*$ depends on lagged values of the instrument $u_t$ as well. By adding an IS curve to the new Keynesian model studied in this section, one can easily specify an optimal interest-rate target $i_t^*$ that allows for the commitment solution to be implemented. This approach is pursued in Section 3.4.

How could one ensure that the central bank cares about deviations from an additional output target, that is, that $b (u_t-u_t^*)^2/2$ enters its loss function? First, following Walsh (Reference Walsh1995), incentive contracts could be used and central banker’s pay could depend on deviations of their policies from the targets $u_t^*$ . Importantly, the monetary incentives can be arbitrarily small. Second, as has been discussed in the Introduction, it is plausible that a public announcement that the central bank intends to set its instrument in line with $u_t^*$ will entail small direct costs for the central bank if it deviates from this announcement at a later stage.Footnote ²⁶

3.1 Setup

In the following, we generalize the results for the simple new Keynesian model to a broad class of linear-quadratic models with rational expectations. There are $n_x$ predetermined variables, contained in the column vector $x_t$ , and $n_y$ non-predetermined variables, contained in the vector $y_t$ . Let $z_t$ be the $(n_x+n_y)$ -dimensional vector that contains all predetermined and all non-predetermined variables, where the predetermined variables come first. There is also a $k$ -dimensional vector of instruments $u_t$ .

Social losses are described by:

(30) \begin{equation} L = \mathbb E_0 \sum _{t=0}^\infty \beta ^t \left (z'_{t} Q z_t + 2 z'_{t} U u_t + u'_{t} R u_t\right ), \end{equation}

where $\beta \in (0,1)$ . $Q$ is an $(n_x+n_y)\times (n_x+n_y)$ -dimensional matrix, $U$ is an $(n_x+n_y)\times k$ -dimensional matrix, and $R$ is a $(k\times k)$ -dimensional matrix. Without loss of generality, we assume $Q$ and $R$ to be symmetric. For the time being, we assume that the policy-maker’s loss function is identical to the social loss function. In Section 3.4, we will examine the case where the policy-maker’s losses include an additional term that involves target values for the instruments $u_t$ .

The predetermined and non-predetermined variables evolve according to

(31) \begin{equation} x_{t+1} = A_{xx} x_t + A_{xy} y_t + B_x u_t + \varepsilon _{x,t+1}, \end{equation}

(32) \begin{equation} \mathbb E_t y_{t+1} = A_{yx} x_t + A_{yy} y_t + B_y u_t, \end{equation}

where $A_{xx}$ , $A_{xy}$ , $A_{yx}$ , $A_{yy}$ , $B_x$ , and $B_y$ are given matrices whose coefficients have been obtained from log-linearized equations describing the private sector equilibrium, for example. The $n_x$ components of $\varepsilon _{x,t+1}$ describe the innovations to the predetermined variables $x_{t+1}$ . They have zero mean and covariance matrix $\Sigma$ .

As is well known, models with lagged variables and expectations more than one period ahead can also be cast in the form considered here. The same is also be true for some models with lagged expectations of present and future variables [see Blanchard and Kahn (Reference Blanchard and Kahn1980)].

3.2 Commitment

The commitment solution can be obtained by setting up the Lagrangian [Backus and Driffill (Reference Backus and Driffill1986) and Söderlind (Reference Söderlind1999)]:

(33) \begin{equation} \mathcal L_0 = \mathbb E_0 \sum _{t=0}^\infty \beta ^t \left [z'_{t} Q z_t + 2 z'_{t} U u_t + u'_{t}R u_t + 2 \rho' _{t+1} \left (A z_t + B u_t -z_{t+1} \right ) \right ], \end{equation}

where

(34) \begin{equation} A = \begin{pmatrix} A_{xx}\;\;\;\; & A_{xy} \\[5pt] A_{yx}\;\;\;\; & A_{yy} \end{pmatrix}, \,B = \begin{pmatrix} B_x \\[5pt] B_y \end{pmatrix}, \end{equation}

and $\rho _t$ is an $(n_x+n_y)$ -dimensional vector of Lagrange multipliers ( $t=1,2,3,\ldots$ ). The first-order conditions with respect to $z_t$ and $u_t$ are

(35) \begin{equation} \beta A' \mathbb E_t \rho _{t+1} = -\beta Q z_t - \beta U u_t + \rho _t, \end{equation}

(36) \begin{equation} -B' \mathbb E_t \rho _{t+1} = U' z_t + R u_t. \end{equation}

We assume that the commitment solution exists and involves unique paths of $z_t$ and $u_t$ . As shown by Backus and Driffill (Reference Backus and Driffill1986), the commitment solution can then be described by an equation that specifies the evolution of the predetermined variables $x_t$ and the multipliers $\rho _{y,t}$ associated with the non-predetermined variables:

(37) \begin{equation} \begin{pmatrix} x_{t+1} \\[5pt] \rho _{y,t+1} \end{pmatrix} = H \begin{pmatrix} x_t \\[5pt] \rho _{y,t} \end{pmatrix} + \begin{pmatrix} \varepsilon _{t+1} \\[5pt] 0 \end{pmatrix}, \end{equation}

where $H$ is an $(n_x+n_y) \times (n_x + n_y)$ -dimensional matrix. The initial values of the predetermined variables $x_t$ are exogenously given. The initial values of the elements in $\rho _{y,t}$ are zero, because the non-predetermined variables can be chosen freely in $t=0$ . The non-predetermined variables $y_t$ , the Lagrange multipliers $\rho _{x,t}$ associated with the predetermined variables, and the policy-makers’ instruments $u_t$ can be expressed as functions of the current values of $x_t$ and $\rho _{y,t}$ . In particular, $y_t$ can be written as:

(38) \begin{equation} y_t = C \begin{pmatrix} x_t \\[5pt] \rho _{y,t} \end{pmatrix}. \end{equation}

$C$ is an $n_y \times (n_x + n_y)$ -dimensional matrix that can be decomposed as $C= \left (C_x,C_{\rho _y} \right )$ , where $C_x$ has dimensions $n_y\times n_x$ and $C_{\rho _y}$ has size $n_y\times n_y$ .

3.3 Discretionary policy-making

We are now in a position to formulate a generalization of Proposition 1 to a large class of models:Footnote ²⁷

Proposition 3. Consider the unique commitment solution of the model characterized by (31), (32) and loss function (30). Let $C=(C_x,C_{\rho _y})$ be the matrix that describes how $y_t$ depends on $x_t$ as well as $\rho _{y,t}$ (see (38)). Then $C_{\rho _y}$ is invertible. For the discretionary policy-maker, introduce an $n_y$ -dimensional vector of additional state variables, $s_t$ . In the initial period, set $s_0=0_{n_y}$ .Footnote ²⁸ For $t=0,1,2,\ldots$ , assume that $s_t$ evolves according to

(39) \begin{equation} s_{t+1} = A_{sx} x_t + A_{ss} s_t + B_s u_t, \end{equation}

where the matrices $B_s$ , $A_{sx}$ , and $A_{ss}$ are given by:

(40) \begin{equation} B_s = C_{\rho _y}^{-1} \left ( B_y - C_x B_x\right ), \end{equation}

(41) \begin{equation} A_{sx}= C_{\rho _y}^{-1} \left [\left ( A_{yy} - C_x A_{xy} \right ) C_x - C_x A_{xx} + A_{yx}\right ], \end{equation}

(42) \begin{equation} A_{ss} = C_{\rho _y}^{-1} \left ( A_{yy} - C_x A_{xy} \right ) C_{\rho _y}. \end{equation}

Then a discretionary equilibrium for (31), (32), (39), and loss function (30) exists that implements the commitment solution.

The proof is given in Appendix D.

The main idea for the proof of the general result is similar to the one for the new Keynesian model. First, the fact that the commitment solution can be described by the joint dynamics of $x_t$ and $\rho _{y,t}$ suggests that $n_y$ additional state variables should be added to the discretionary policy-maker’s problem in addition to the payoff-relevant state variables $x_t$ . These additional state variables are the components of $s_t$ . Second, to ensure that the policy-maker does not renege on past promises, the dynamics of $s_t$ are specified in a way such that, in every period $t$ , the policy-maker cannot influence current non-predetermined variables (as opposed to future non-predetermined variables). This is achieved by (40), which is a straightforward generalization to (12). Third, (41) and (42) specify the matrices $A_{ss}$ and $A_{sx}$ in a manner such that the dynamics of $s_t$ on the equilibrium path of the discretionary equilibrium equal those of $\rho _{y,t}$ in the commitment solution. Finally, one needs to show that the discretionary policy-maker does not wish to deviate. At this point of the proof, it is helpful that the value function of the discretionary policy-maker can be constructed from the commitment solution under the assumption that both approaches lead to identical dynamics.Footnote ²⁹

Proposition 3 shows that policy-makers that lack a means of commitment can nevertheless implement socially desirable policies that could not be supported in a Markov-perfect equilibrium. Thus, our approach clarifies that it is possible to implement history-dependent strategies like average inflation targeting, which was discussed at the beginning of the Introduction.

It may also be interesting to relate Proposition 3 to a theorem in Backus and Driffill (Reference Backus and Driffill1986) that examines whether the commitment solution can be supported by trigger strategies.Footnote ³⁰ In particular, they consider the case where deviations from the commitment solution are punished by a grim trigger, that is, a permanent switch to the standard Markov-perfect discretionary equilibrium. According to their theorem, the commitment solution can be sustained (i) if the support of shocks is bounded and (ii) if the discount factor is sufficiently large. By contrast, the discretionary equilibria constructed in this paper always allow for the implementation of the commitment solution, irrespective of the magnitude of the discount factor and for an arbitrary support of shocks.Footnote ³¹ This result stems from the fact that, loosely speaking, the punishment for deviations from the commitment solution is not restricted to a switch to the standard discretionary equilibrium.

3.4 Implementation through delegation

Having shown that the commitment solution can be implemented by a discretionary non-Markovian equilibrium, we proceed by showing that the finding of Section 2.6 about optimal delegation and the implementation of the commitment solution by a discretionary Markov equilibrium can be generalized to the broad class of models specified in Section 3.1. For this purpose, we consider a policy-maker whose loss function is identical to the social loss function, except for an additional term that penalizes deviations of the instrument from an additional target for the instrument.

In Appendix E, we show:

Proposition 4. Suppose that policy-making is delegated to a policy-maker whose instantaneous loss is

(43) \begin{equation} L_{del} = z'_{t} Q z_t + 2 z'_{t} U u_t + u'_{t} R u_t + (u_t-u_t^*)' \mathcal B (u_t-u_t^*), \end{equation}

where $u_t^* = - F \begin{pmatrix} x_t \\ s_t \end{pmatrix}$ and $s_t$ is given by (39). $F$ is the matrix that describes how the instrument $u_t$ is set as a function of $x_t$ and $\rho _{y,t}$ in the commitment solution, that is, it satisfies $u_t = - F \begin{pmatrix} x_t \\ \rho _{y,t} \end{pmatrix}$ .Footnote ³² $\mathcal B$ is an arbitrary $k\times k$ -dimensional positive definite matrix. The policy-maker discounts its future losses with the same discount factor $\beta$ as society. Then the commitment solution associated with the social loss function can be implemented by a discretionary Markov equilibrium.

For the standard new Keynesian model, we have shown that the additional incentives for the policy-maker that are necessary to ensure the implementation of the commitment solution can be very small. Importantly, this finding extends to the general setup considered here. Formally, this is an implication of the fact that $\mathcal B$ is an arbitrary positive definite matrix. As a consequence, it is possible to choose a matrix $\mathcal B$ with very small positive eigenvalues.Footnote ³³

In the following, we explain in more detail how delegation of monetary policy to a central bank with an optimally designed interest-rate target can implement the commitment solution in the new Keynesian model. For this purpose, we augment the standard new Keynesian model discussed in Section 2 by an IS curve:

(44) \begin{equation} y_t = -\sigma ^{-1} \left (i_t - \mathbb E_t[\pi _{t+1}]\right ) + \mathbb E_t [y_{t+1}], \end{equation}

where $i_t$ , the central bank’s instrument, is the nominal interest rate net of the natural real rate of interest, $\sigma$ is a positive parameter, and we use $y_t$ rather than $u_t$ for the output gap, as the output gap does not correspond to the instrument in this variant of the model. Henceforth, we will call $i_t$ the nominal interest rate for simplicity.

We select $\sigma =1$ and set the remaining parameter values to the levels used before. As there is only one instrument, $\mathcal B$ corresponds to an arbitrary positive number, whose value does not affect our results. It is straightforward to describe the dynamics of the two auxiliary variables $s_t$ with the help of (39), where, to compute $A_{ss}$ , $A_{sx}$ , and $B_s$ , we have to use matrices $A$ and $B$ that take the IS curve into account. The additional interest-rate target is then given by $i_t^* = - F ( x_t' , s_t')'$ , where $F$ is defined as in Proposition 4. The loss function under optimal delegation incorporates the additional interest-rate target in the following way:

(45) \begin{equation} l_{CB,i}(\pi _t,u_t,i^*_t) = \frac 1 2 \pi _t^2+ \frac a 2 y_t^2 + \frac b 2 (i_t-i_t^*)^2, \end{equation}

where $b$ is an arbitrary positive number.

It is easy to show that the central bank’s optimization problem can be formulated without the auxiliary variables $s_t$ . More specifically, the interest-rate target $i^*_t$ can be written as $i^*_t = I^*_t + \Delta _t$ with the properties that $I_t^*=i_t$ and $\Delta _t=0$ on the equilibrium path and where, for our parameter values and an arbitrary positive value of $b$ , the dynamics of $I^*_t$ and $\Delta _t$ are governed by:

(46) \begin{equation} I_{t+1}^* = 0.353645\, \xi _{t+1} +0.490248\, \xi _t +0.285081\, I^*_t -1.443798\, (i_t-I^*_t), \end{equation}

(47) \begin{equation} \Delta _{t+1} = 0.584252\, (\Delta _t + I^*_t - i_t), \end{equation}

with initial values $I^*_0=0.353645\, \xi _0$ and $\Delta _0=0$ .

It is noteworthy that $\Delta _{t+1}=0$ whenever the central bank set its instrument in line with its overall target $i_t^*=I_t^* + \Delta _t$ in the previous period. Thus, optimal delegation involves that the central bank faces an interest-rate target $i_t^*$ that can be decomposed into a long-term component $I^*_t$ , which evolves recursively and is a decreasing function of the lagged instrument, as well as a short-term component $\Delta _t$ , which affects the central bank’s overall target only in periods following deviations of the central bank’s instrument $i_t$ from the overall target $i_t^*=I_t^* + \Delta _t$ . In particular, whenever the central bank’s choice of instrument $i_t$ is too high compared to the target $i_t^*$ , then $\Delta _{t+1}\lt 0$ , that is, the target for period $t+1$ is pushed below $I_{t+1}^*$ .

The dynamics of the economy after a markup shock are illustrated in Figure 4.Footnote ³⁴ The left panel shows inflation (black solid line) and output (gray solid line) under optimal delegation, which correspond to the respective values under commitment. The corresponding dashed lines show the dynamics of output and inflation after a deviation in period $0$ where the central bank selects only 50% of the optimal value in that period and implements the optimal policy in subsequent periods. As has been mentioned before, our construction turns non-predetermined variables (inflation and output in this case), into predetermined ones. This explains why output and inflation correspond to their optimal levels in the period of the deviation. From period $1$ onward, they differ from the values under optimal commitment. However, these differences become very small after a few periods.

Figure 4.

Dynamic responses to a markup shock under optimal delegation to a central bank with an interest-rate target. Left panel: inflation (solid black line) and output (solid gray line) under optimal delegation as well as optimal commitment; inflation (dashed black line) and output (dashed gray line) under delegation when the central bank chooses a level of $i_0$ that corresponds to only $50\%$ of the optimal level. Right panel: the interest-rate target $i_t^*=I_t^* + \Delta _t$ and the interest rate $i_t$ in equilibrium (solid line), the nominal interest rate under the deviation in the first period (dashed line), the component $I_t^*$ of the interest-rate target (dash-dotted line) and $\Delta _t$ , the second component of the interest-rate target (dotted line).

The right panel of the figure illustrates the dynamics of the interest rate in more detail. The solid line stands for the interest rate under optimal delegation and optimal commitment. The interest-rate path for the one-time deviation described above corresponds to the dashed line. $\Delta _t$ differs from zero only in period $1$ (dotted line). In particular, it is negative due to the fact that the interest-rate set by the central bank in period $0$ was too high under the deviation. The overall interest-rate target $i_t^*$ is the sum of $\Delta _t$ (dotted line) and $I_t^*$ (the dash-dotted line). One can see that, following the deviation in period $0$ , the interest rate and the overall interest-rate target $i_t^*$ tend to move back to their equilibrium values over time.

3.5 Optimal delegation in practice

In general, one may be concerned that the additional targets for the central bank may be complex while other approaches to optimal delegation involve objectives that may be easier to interpret. For example, in Woodford (Reference Woodford2003), the central bank is assigned four objectives: (1) an inflation stabilization objective, (2) an output stabilization objective, (3) an interest-rate stabilization objective, and (4) an objective to stabilize changes in interest rates (see equation 4.1 in his paper). While these objectives have straightforward interpretations, optimal delegation involves that one has to set the relative weights that the central bank attaches to these four targets to specific values.

In practice, hiring a central banker who attaches the right weights to these different objectives may be challenging. As a consequence, one may design incentive contracts that specify how central bankers’ pay depends on the degree to which the different targets are met. However, even such an approach may be problematic as central bankers are plausible to be intrinsically motivated as well and, as an additional complication, the intrinsic motivation may interact with the extrinsic incentives in non-obvious ways. To sum up, it appears challenging to ensure that, in practice, a central bank acts in line with a loss function with specific, optimally selected weights on different targets.

By contrast, the approach proposed in this paper does not involve this problem as the weight that the central bank attaches to the additional target compared to its other objectives is irrelevant. The specification of the additional target may be complex, but measuring this target and the central bank’s performance with respect to meeting this target is straightforward.

Even for a comparably rich model like the one by Smets and Wouters (Reference Smets and Wouters2007), the term $(u_t-u_t^*)' \mathcal B (u_t-u_t^*)$ in Proposition 4 can be condensed into a one-dimensional index that can be calculated in a straightforward manner. This index could be given an interpretation as a measure of “dynamic central bank performance” and could be published on a regular basis. This performance index would take a value of zero when the central bank acts as in the equilibrium implementing the commitment solution and would take higher values otherwise.

The use of quality indices, which condense complex information in order to make it easier to digest, is not uncommon in other areas. For example, in the EU, energy labels on an A-G scale are used to provide summary information on the energy efficiency of appliances such as fridges.Footnote ³⁵ Moreover, many central banks publish inflation forecasts, which also condense various pieces of information. These arguments suggest that the publication of an additional performance index, which can be obtained by calculating $(u_t-u_t^*)' \mathcal B (u_t-u_t^*)$ , may be a way of implementing our approach in practice.