3.1 Determining an equilibrium path

Suppose $K \geq 2$ perfect foresight solutions are found using our Algorithm. To resolve the indeterminacy problem (incompleteness), we consider a “sunspot” approach rather than trying to “purge” solutions based on a selection criterion. We assume the researcher or policymaker has some “prior probabilities” $p_1,\ldots, p_K \in [0,1]$ , $\sum _{k=1}^K p_k = 1$ , over the different perfect foresight solutions, where $p_k$ is the probability that agents will coordinate expectations on equilibrium $k$ at date 1 (the first period in which expectations are formed).

The simple first step is to draw a sunspot at date 1 that represents this coordination and will thereby ensure that a particular solution (i.e. perfect foresight path) is realized. We index the $K$ perfect foresight solutions by $(x_t^k)_{t=1}^\infty$ for $k=1,\ldots, K$ , and let $u_1 \sim \mathcal {U}_{(0,1)}$ be a sunspot that is drawn from the uniform distribution on the interval $(0,1)$ at date $t=1$ . Given all this, we have the following rule for determining which equilibrium is realized given the sunspot $u_1$ , which is analogous to how the values of categorical random variables, such as realizations of $K$ -state Markov processes or outcomes of rolling a die, are determined.

Remark 1. Suppose there are $K \geq 2$ perfect foresight solutions. Given probabilities $p_1,\ldots, p_K$ and a random draw $u_1 \sim \mathcal {U}_{(0,1)}$ , the realized perfect foresight path at date 1 is

(9) \begin{equation} (x_t)_{t=1}^\infty = \begin{cases} (x_t^1)_{t=1}^\infty & \text {if } u_1 \in (0,p_1] \\ (x_t^2)_{t=1}^\infty & \text {if } u_1 \in (p_1,p_1+p_2] \\ \vdots \\ (x_t^K)_{t=1}^\infty & \text {if } u_1 \gt p_1 + \ldots + p_{K-1} \\ \end{cases} \end{equation}

i.e. for $u_1 \in (\sum _{k=0}^{k^*-1} p_{k},\sum _{k=0}^{k^*}p_{k}]$ , where $k^* \in \{1,\ldots, K\}$ and $p_0 \,:\!=0$ , the unique (realized) perfect foresight solution is $(x_t)_{t=1}^\infty = (x_t^{k^*})_{t=1}^\infty$ .

Remark 1 simply gives a way to choose a realized perfect foresight path when there are multiple candidates. It applies to any finite set of perfect foresight solutions and is flexible due to the free specification of probabilities. For instance, if the researcher thinks some solution(s) somewhat “unrealistic” they may attach low probability to those solution paths. On the other hand, a perfectly agnostic researcher would use “flat priors” $p_k = 1/K$ for all $k$ .

As noted, our Algorithm stores all (found) perfect foresight solutions. Given some specified probabilities $p_1,\ldots, p_K$ , Remark 1 will then choose one solution as the realized equilibrium, akin to a lottery among perfect foresight paths (we give an example in Section 3.3). With this addition, our algorithm can be used to automate equilibrium determination (to some extent) while taking account of prior beliefs and their economic implications.Footnote ¹⁷

Further, we will show that combining Remark 1 with some additional timing assumptions allows the computation of expected outcomes as a probability-weighted average of perfect foresight paths, such that expected welfare or other outcomes may be evaluated under the “veil of ignorance” – i.e. before the sunspot is realized. We do this in the next section.

3.2 Expected outcomes

Suppose now that before expectations are formed in period 1, there is some initial state, period 0, in which the initial conditions $x_0, (e_t)_{t=1}^\infty$ are known, but the sunspot $u_1$ and thus the expectations $x_1,x_2,\ldots, $ are not. To motivate this, note that if the sunspot $u_1$ were determined at date 0, multiplicity could not arise since expectations would already have coordinated on a particular equilibrium. Viewed this way, the sunspot $u_1$ coordinates agents’ expectations on a particular equilibrium in period 1 and so “stands in” for some psychological process or “animal spirits” that are extraneous to the economy.Footnote ¹⁸ By contrast, the news shocks $(e_t)_{t=1}^\infty$ are structural and we take them as a predetermined aspect of the economic environment.

Thus, we think of date 0 as an initial position that is subject to the “veil of ignorance” in that the expectations of agents—an aspect of their behavior—are unknown and not pinned down. Given this view, an expected path can be defined as follows.

Based on Definition2, the expected path of the vector of endogenous variables $x_t$ is:

(10) \begin{equation} \left ( E_0[x_1], \hskip 0.2cm E_0[x_2], \hskip 0.2cm \ldots \hskip 0.2cm E_0[x_s], \hskip 0.2cm \ldots \hskip 0.2cm \right ) \end{equation}

where the expected values are

(11) \begin{equation} E_0[x_t] \,:\!= \sum _{k=1}^K p_k x_t^k \end{equation}

i.e. a linear combination of the date- $t$ points of the perfect foresight solutions $1,\ldots, K$ .

Similarly, one may compute expected welfare, for example, based on a quadratic loss function or some other approximation to a social welfare function. If $W_k \in \mathbb {R}$ is the welfare associated with solution $k$ , then analogous to (11) the expected welfare is

(12) \begin{equation} E_0[W] \,:\!= \sum _{k=1}^K p_k W_k. \end{equation}

It should be emphasized that expressions such as (11) and (12) also provide a basis for robustness-type analysis in the sense that the impact of different “prior beliefs” can easily be studied and best and worst-case scenarios identified. We now give a simple example.

Example 1. Consider again the Fisherian model with two solutions (Figure 1). A policymaker assigns probabilities $p_1$ to Solution 1 (slack) and $p_2 = 1-p_1$ to Solution 2 (bind), and has an ad hoc loss function that penalizes squared deviations from a zero inflation target:

\begin{equation*}L = \sum _{t=1}^\infty \beta ^{t-1} \pi _t^2.\end{equation*}

By (12) , the expected loss is $E_0[L] = p_1 L_1 + p_2 L_2$ , where $L_1 = \sum _{t=1}^\infty \beta ^{t-1} (\pi _t^1)^2 = \frac {\omega ^2}{1-\beta \omega ^2}\pi _0^2$ , $L_2 = \sum _{t=1}^\infty \beta ^{t-1} (\pi _t^2)^2 = \frac {(r/\omega )^2}{1-\beta \omega ^2}$ , and $\pi _t^1,\pi _t^2$ are the inflation solutions at date $t$ . Hence,

\begin{equation*} E_0[L] = \frac {p_1 \omega ^2 \pi _0^2 + (1-p_1) (r/\omega )^2}{1-\beta \omega ^2} \end{equation*}

which is linear in $p_1$ and quadratic in the initial inflation $\pi _0$ .

In Figure 2 (left panel), we plot the expected loss as the probability of Solution 1, $p_1$ , is increased from 0 to 1; we do this for several values of $\pi _0\gt 0$ , including the initial inflation $\pi _0^* = r/\omega ^2$ for which the losses are equal, i.e. $L_1=L_2$ .Footnote ¹⁹ For inflation rates above $\pi _0^*$ , the expected loss increases with $p_1$ , since inflation declines geometrically from its initial value $\pi _0$ under the “good”solution (see Figure 1). Going in the other direction, reducing initial inflation below $\pi _0^*$ makes inflation under Solution 1 “closer”to the zero inflation target, so loss $L_2$ is higher. For Solution 2, where the constraint binds in period 1, the loss $L_2$ is independent of the initial inflation $\pi _0$ , as is clear from Figure 1 (and the equation above).

Figure 2. Expected loss $E_0[L]$ as $p_1$ is increased for various $\pi _0$ (left panel) and the expected loss due to the zero lower bound $E_0[L]-L_1$ as $\pi _0$ is increased (right panel). The initial values in the left panel satisfy $\pi _0^h \gt \pi _0^m\gt \pi _0^*\gt \pi _0^l$ , where $\pi _0^*$ is the positive initial inflation that makes the loss under Solution 1, $L_1(\pi _0)$ , equal to the loss $L_2$ under Solution 2.

Figure 2 (right panel) plots the expected loss relative to Solution 1, $E_0[L]-L_1$ , which can be interpreted as the extra loss attributable to the lower bound friction, which makes Solution 2 (binding constraint) a possible equilibrium path. The expected loss attributed to the lower bound falls as initial inflation is increased, since this raises the Solution 1 loss $L_1$ by more than it raises the expected loss $E_0[L]$ . Increasing the probability of Solution 1 “flattens”the relationship between $E_0[L]-L_1$ and $\pi _0$ since the expected loss given the lower bound, $E_0[L_1]$ , gets closer to $L_1$ the higher the probability attached to Solution 1 (right panel).

The above exercise can be motivated by thinking of an asset like central bank digital currency that could potentially eliminate the lower bound. From a policy perspective, we would like to know ex ante whether it is worth providing such a currency (at some cost); this will depend partly on the expected benefit of eliminating the lower bound on nominal rates, shown in the right panel. While this is a simple example, the results are suggestive of the policy and robustness-type analysis possible using our expected outcomes approach.

3.3 Stochastic simulations

Lastly, we consider stochastic simulations with switching between multiple equilibria. For such simulations, we assume that the agents think they know all future shocks but are mistaken; as result, they have imperfect foresight and ignore risk, such that their expectations are not strictly rational, in contrast to the perfect foresight solutions studied thus far.

We use this approach to construct stochastic simulations in which agents form expectations under certainty equivalence. Both Dynare (Adjemian et al. Reference Adjemian, Bastani, Juillard, Mihoubi, Perendia, Ratto and Villemot2011) and OccBin (Guerrieri and Iacoviello, Reference Guerrieri and Iacoviello2015) have built-in options for such simulations using an extended path method, which is widely used, for example, at policy institutions such as central banks.Footnote ²⁰ While the extended path approach is typically applied to occasionally-binding constraint models with a unique solution (or assuming uniqueness), we provide an approach that allows simulations with multiple equilibria, which should speak to a wide audience.

Our approach draws on Remark 1, but it uses that approach to determine an equilibrium in every period $t$ in which the model is simulated; note that this is necessary because while the entire solution path is known as of date 1 under perfect foresight, this is not true here since the actual (realized) path will generally differ from the one that agents expected. Hence, equilibrium determination (via a sunspot) arises every period in response to the new (unanticipated) initial conditions $x_{t-1},e_t$ with which agents are confronted.

To make this concrete, suppose that the shock vector $e_t$ is drawn from some distribution. Then given $e_t$ , $x_{t-1}$ and agents’ beliefs about future shocks, we can find the solution(s) for these initial conditions using our Algorithm, and one can be selected by a sunspot (akin to Remark 1). The same procedure is then repeated in period $t+1$ , given $x_t, e_{t+1}$ and the agents’ expected future shocks. Provided a solution exists for all simulated $t$ , we can construct a stochastic simulation path of desired length as follows.

1. 1. Choose some systematic rule for assigning probabilities to different equilibria at each date, e.g. the “flat priors” approach, $p_k = \frac {1}{\text {no.\ of equilibria}}$ for each solution $k$ .Footnote ²¹

2. 2. Given $x_0,e_1$ and expected shocks $e_2^a,\ldots, e_T^a$ , use the Algorithm to find the solution paths $(x_t^k)_{t=1}^{T}$ for $k=1,\ldots, K_1$ , where $K_1$ is the number of solutions at date 1. Using the approach in Remark 1, select one of these solutions, $(x_t^{k_1^*})_{t=1}^{T}$ , where $k_1^* \in \{1,\ldots, K_1\}$ . Set $x_1 = x_1^{k_1^*}$ (our first simulated point) and move to period 2.

3. 3. Draw vector $e_2$ from some distribution. Given $x_1,e_2$ and expected shocks $e_3^a,\ldots, e_{T+1}^a$ , use the Algorithm to find the solution paths $(x_t^k)_{t=2}^{T+1}$ for $k=1,\ldots, K_2$ , where $K_2$ is the number of solutions in period 2. Use the approach in Remark 1 to select one of these solutions, $(x_t^{k_2^*})_{t=2}^{T+1}$ , where $k_2^* \in \{1,\ldots, K_2\}$ . Set $x_2 = x_2^{k_2^*}$ (second simulated point).

4. 4. Repeat in periods 3, 4 etc. to get a simulation path of the desired length, say $(x_t)_{t=1}^{T^*}$ .

We now show stochastic simulation in action using our running example (Fisherian model).

Example 2 (Ex. 1 cont’d). We continue the Fisherian example (Section 2.4 and Example 1 ) but we add a shock $e_t \in \mathbb {R}$ in the Taylor-type rule, so $i_t = \max \{ 0, r + \phi \pi _t - \psi \pi _{t-1} + e_t\}$ as in Holden (Reference Holden2023, Section 2.3). We set parameter values: $r=0.01$ , $\phi =2$ , $\psi =0.93$ , $\pi _0=0.02$ , along with date- $1$ anticipated shocks $e_1=e_2^a=-0.001$ . The probability of choosing Solution 1 (away from the zero bound) is set at $p_1 = 0.95$ , so there is a 5% probability of choosing the Solution 2 that hits the bound. At dates $t\gt 1$ we solve for $i_t,\pi _t$ conditional on the inherited state $\pi _{t-1}$ and fresh draws for the monetary policy shocks $e_t,e_{t+1}$ which are drawn from a normal distribution with a mean of zero and a standard deviation $\sigma _e$ (see Figure 3).

Figure 3. Five stochastic simulations: $p_1=0.95$ and initial values $\pi _0=0.02$ , $e_1,e_2=-0.001$ .

We found two solutions in all simulated periods; these are versions of the “high”and the “low”inflation solutions in Figure 1 . To resolve the indeterminacy of two solutions, at each date $t$ we drew a sunspot $u_t \sim \mathcal {U}_{(0,1)}$ that selects either Solution 1 (away from bound) or Solution 2 (hits bound) in period $t$ . Given our assumption that $p_1 = 0.95$ , Solution 1 (Solution 2) is selected at date $t$ if and only if $u_t \in (0,0.95]$ ( $u_t \gt 0.95$ ); see Remark 1 .

In the upper panel of Figure 3 , the standard deviation of the policy shock is very small to isolate the impact of the sunspot, i.e. switching between the two equilibria. Of five simulations, three hit the zero lower bound in some period (see dashed lines), giving strong deflation (cf. Figure 1), in contrast to the solution paths that always remain away from the bound.

In the lower panel, the shock variance is large enough to make each individual stochastic simulation path discernible, but the main variations in inflation and interest rates come from switching between equilibria rather than from disturbances to the policy rule. Note that due to switching between equilibria, the average simulated values of inflation and nominal rates, in a long simulation, may differ somewhat from those at the terminal solution steady-state.

4.1 A New Keynesian model

We consider the New Keynesian model studied in Brendon et al. (Reference Brendon, Paustian and Yates2013). Besides the zero lower bound, the only other departure from the benchmark model is a policy response to the change in the output gap, similar to the “speed limit” policies considered by Walsh (Reference Walsh2003):

(13) \begin{equation} i_t = \max \{\underline {i}, \hskip 0.1cm i_t^* \} \end{equation}

(14) \begin{equation} i_t^* = \rho _i i_{t-1}^* + (1-\rho _i)(\theta _\pi \pi _t + \theta _{\Delta y}(y_t - y_{t-1})) \end{equation}

(15) \begin{equation} y_t = E_t y_{t+1} - \frac {1}{\sigma }(i_t - E_t \pi _{t+1}) + e_t \end{equation}

(16) \begin{equation} \pi _t = \beta E_t \pi _{t+1} + \kappa y_t \end{equation}

where $\theta _\pi \gt 1$ , $\beta \in (0,1)$ , $\theta _{\Delta y}, \kappa, \sigma \gt 0$ , $\rho _i \in [0,1)$ , $\underline {i} = \beta -1$ and all values of $e_t$ are known.

We start by setting parameters at $\beta = 0.99$ , $\sigma =1$ , $\rho _i=0$ (no interest rate smoothing) and $\kappa = \frac {(1-0.85)(1-0.85\beta )}{0.85}(2+\sigma )$ as in Brendon et al. (Reference Brendon, Paustian and Yates2013); additionally, we set $\theta _\pi = 1.5$ and $\theta _{\Delta y}=1.6$ to initially replicate the exercise in Holden (Reference Holden2023, Appendix E). Starting at steady state, we consider a 1% demand shock at date 1 (i.e. $e_1=0.01$ , $e_t = 0$ for $t\gt 1$ ) and search for solutions to model (13)–(16) using our algorithm. We found two perfect foresight solutions as expected and the solutions match the ones reported by Holden (see Figure 4).

Figure 4. Multiple equilibria in the Brendon et al., model: $e_1=0.01$ and $i^*_0=y_0=\rho _i=0$ .

There are two perfect foresight solutions in Figure 4: one where the lower bound is never hit and inflation and the output gap rise only marginally above their steady-state values; and a second solution where interest rates are at the lower bound in the first two periods and there is strong and persistent deflation and large negative output gaps (Figure 4, all panels). This “bad” solution is clearly inferior in terms of stabilization of inflation and output gaps and arises due to pessimistic self-fulfilling expectations: if agents expect low inflation, then the rise in the real rate lowers the output gap and inflation, validating the expectations.

Having a growth-based interest rate rule—for which the shadow rate responds to inflation and the change in the output gap—is important for the multiplicity result, as there is a unique perfect foresight solution if the shadow rate follows a Taylor rule with a response to the output gap in levels (Holden, Reference Holden2023, Section 4.3). In addition, multiplicity occurs only when the response to the change in the output gap—or “speed limit”—is strong enough, as is clear from Figure 5 below. Such speed limit policies have been studied in the literature, with encouraging results from both theoretical and practical perspectives (Walsh, Reference Walsh2003; Orphanides, Reference Orphanides2003; Yetman, Reference Yetman2006) and some empirical works suggest that central bank behavior is consistent with a speed limit rule (e.g. Mehra, Reference Mehra2002).Footnote ²³

Given the potential negative consequences of the interest rate rule (14) in this model, we consider some alternative policy rules, to see if they can restore uniqueness by eliminating the “bad” solution. Before doing so, we first confirm that multiplicity is a robust feature of this model by studying the properties of the $M$ -matrix (discussed in Section 2.3 above).

Figure 5. Regions in which $M$ is not a $P$ -matrix (black) when $T = 16$ (Case: $\rho _i=0$ ).

In Figure 5, we show some parameter regions for which the $M$ matrix of impulse responses is a $P$ matrix and is not a $P$ matrix. Recall that if the $M$ matrix is a $P$ -matrix, then there is a unique solution for all initial conditions. We set $T=16$ and plot the regions for which the $M$ matrix is a $P$ -matrix (determinacy region, white), and is not a $P$ -matrix (black region). We consider different combinations of the response coefficients $\theta _\pi$ , $\theta _{\Delta y}$ in the interest rate rule and in each panel we vary the inverse elasticity of intertemporal substitution, $\sigma$ .

Figure 5 shows there is a unique solution if the response to the change in the output gap, $\theta _{\Delta y}$ , is not too strong relative to the inflation response $\theta _\pi$ (see the white regions). In the first panel, which uses the baseline value of $\sigma =1$ , we see that $M$ is a $P$ -matrix only if the response to the change in the output gap is smaller than the response coefficient on inflation. Note that the parameter combination used in Figure 4 ( $\theta _\pi =1.5, \theta _{\Delta y} = 1.6$ ), where we see two solutions, lies in the black region as expected. In fact, Brendon et al. (Reference Brendon, Paustian and Yates2013, Proposition 1) show that the model (13)–(16) has multiple perfect foresight equilibria if and only if $\theta _{\Delta y} \gt \sigma \theta _\pi$ , and Figure 5 is consistent with this conclusion.

In summary, multiplicity is a robust outcome and this raises the question of whether alternative monetary policies could restore uniqueness by eliminating the bad solution. We investigate this below while retaining the “speed limit” aspect of the policy rule, which may have theoretical and practical advantages as argued by Walsh (Reference Walsh2003) and others. We start with interest rate smoothing before turning to unconventional monetary policy rules.

4.2 Interest rate smoothing

We first ask whether policymakers could achieve a better outcome by smoothing the shadow interest rate in (14) by setting $\rho _i \in (0,1)$ . We started out by checking the regions where the $M$ matrix is a $P$ -matrix for $T=16$ , similar to Figure 5 but with three $\rho _i$ values and $\sigma =1$ .

Figure 6. Regions in which $M$ is not a $P$ -matrix (black) for $T = 16$ and various $\rho _i$ .

The $P$ -matrix regions in Figure 6 indicate that the determinacy region expands once the interest rate smoothing parameter $\rho _i$ is large enough. For $\rho _i=0.8$ , the parameter combination $\theta _\pi = 1.5$ , $\theta _{\Delta y}=1.6$ is now in the determinacy region (white), so there is a unique solution at these parameter values (plotted in Figure 7 below). However, indeterminacy remains if the response to the speed limit is strong enough (Figure 6, right) and as noted by Holden (Reference Holden2023, Appendix E), the $P$ -matrix regions under interest rate smoothing tend to those in the model without any smoothing as $T$ is increased, such that multiplicity remains a widespread problem and there are both “good” and “bad” equilibria.Footnote ²⁴

In Figure 7, we present a numerical simulation. We set $\theta _\pi =1.5$ , $\theta _{\Delta y} = 1.6$ and $e_1 = 0.01$ as in the baseline simulation in Figure 4; the only difference is that the interest rate smoothing parameter is set at either $\rho _i = 0.40$ (weak smoothing) or a high value $\rho _i = 0.80$ (strong smoothing). With moderate interest rate smoothing ( $\rho _i=0.40$ ) there are two solutions and the “bad” solution is exacerbated relative to Figure 4: inflation and the output gap fall by around 5 times as much initially and interest rates spend 7 periods at the lower bound, rather than 2. Intuitively, it is easier to induce a lengthy spell at the lower bound when the shadow rate is persistent, provided $\rho _i$ is not large enough to eliminate the bad solution. In the case of $\rho _i=0.8$ , there is a unique solution that is away from the bound in all periods (dashed gray line)—though as Figure 6 shows, uniqueness is not a general result.

Figure 7. Perfect foresight solutions with interest rate smoothing when $e_1=0.01$ , $i^*_0=y_0=0$ , $\sigma =1$ and $\theta _\pi = 1.5$ , $\theta _{\Delta y}=1.6$ : two different values of $\rho _i$ ( $\rho _i=0.4$ , solid; $\rho _i=0.8$ , dashed).

For the parameters $\theta _\pi = 1.5$ , $\theta _{\Delta y}=1.6$ , the determinacy result seems to hold for smoothing of around $\rho _i=0.8$ or higher.Footnote ²⁵ Some intuition can be gained by scaling $\theta _{\pi }, \theta _{\Delta y}$ in Eq. (14) by $\frac {1}{1-\rho _i}$ and letting $\rho _i \rightarrow 1$ . In this case, the shadow interest rate tends to $\Delta i_t^* = \theta _\pi \pi _t + \theta _{\Delta y} \Delta y_t$ , which is consistent with any rule of the form $i_t^* = constant + \theta _\pi p_t + \theta _{\Delta y} y_t$ , where $p_t = \pi _t + p_{t-1}$ is the log price level. The latter is a price-level targeting rule without a speed limit term. Given that Holden (Reference Holden2023, Appendix E) finds that a levels rule restores determinacy, it is intuitive that sufficiently high values of $\rho _i$ lead to the same conclusion.

In short, interest rate smoothing does not, in general, prevent the occurrence of multiple equilibria, and when the “bad” solution is present we see that a smoothing rule worsens destabilization of inflation and the output gap somewhat. At the same time, however, we saw that highly inertial interest rate rules may eliminate the bad solution.

4.3 Forward guidance

Since interest rate smoothing is not a robust solution to indeterminacy and destabilization, we now consider forward guidance. We are motivated here by the observation that forward guidance promises an extended period of expansionary monetary policy, such that self-fulfilling pessimistic expectations of inflation and the output gap—as under the “bad” solution—might not be rational. This unconventional policy is not studied by Holden (Reference Holden2023).

To model forward guidance, we consider expansionary “news shocks” to the shadow interest rate, such that (14) becomes

(17) \begin{equation} i_t^* = \rho _i i_{t-1}^* + (1-\rho _i)(\theta _\pi \pi _t + \theta _{\Delta y}(y_t - y_{t-1})) + e_t^{FG} \end{equation}

where $e_t^{FG} \leq 0$ is a forward guidance “news shock” and $e_1^{FG} = 0$ .

We assume that $e_t^{FG}\lt 0$ only in periods when forward guidance is in place. Our question is whether such a policy eliminates the “bad” solution or better stabilizes inflation and the output gap. Table 1 records the percentage of initial conditions (out of 800) for which both a “good” and “bad” solution exist at different forward guidance horizons; here, different initial conditions refers to varying the size of the news shocks $e_t^{FG}$ within a range and we consider 800 such variations (i.e. cases) for each forward guidance horizon; see the Table 1 notes. By forward guidance horizon we mean the number of periods in which there are “expansionary” news shocks $e_t^{FG}\lt 0$ and we consider only consecutive periods of such shocks. All other parameters and initial conditions are kept fixed at the baseline values used in Figure 4.

Table 1. Determinacy at various forward guidance horizons (800 cases, $\rho _i=0$ )

Note: Forward guidance is modeled via news shocks $e_t^{FG} = -0.01 - Uniform(0,0.01)$ which start in period 2 and last to the stated date. Each row: 800 cases of different news shocks.

The results in Table 1 show that forward guidance is not effective in eliminating indeterminacy. Up to a 5-period forward guidance horizon (rows 1–5), all 800 sets of news shocks led to multiple solutions (i.e. 100% of cases).Footnote ²⁶ If the forward guidance horizon is 5 periods, the length of spells at the zero lower bound is no longer the same across all 800 cases of news shocks (see Table 1, final column) and there can be multiple spells at the lower bound (see Figure 8, bottom right for an example); however, neither a longer horizon nor interest rate smoothing seem to alter the conclusion of widespread multiplicity.

Figure 8 shows that the “good” solutions under forward guidance have inflation and output “overstimulated” (upper panel), while the “bad” solutions have very poor stabilization relative to the original policy given by the solid black line ("Baseline," lower panel). Importantly, stabilization under the “bad” solutions is worse when forward guidance is present and stabilization of inflation and the output gap deteriorates as forward guidance becomes more aggressive (i.e. when it delivers a similar “dose” for a larger number of periods). Hence, while some previous works suggest that forward guidance could be stabilizing at the zero lower bound (e.g. Eggertsson et al. Reference Eggertsson, Egiev, Lin, Platzer and Riva2021), our results point in the opposite direction.

Figure 8. "Good" and “bad” solutions with forward guidance: $e_1=0.01$ , $i^*_0=y_0=0$ , $\theta _\pi = 1.5$ , $\theta _{\Delta y}=1.6$ , $\rho _i=0$ , $\sigma = 1$ . Forward guidance news shocks $e_t^{FG} = -0.015$ for 2 periods (FG1), 4 periods (FG2), 5 periods (FG3). Start: $t=2$ and $e_1^{FG} =0$ in all cases. Baseline: no FG.

4.4 Price-level targeting

Lastly, we consider price-level targeting. A key motivation is work showing that price-level targeting interest rate rules can mitigate or resolve indeterminacy in New Keynesian models (Giannoni, Reference Giannoni2014; Holden, Reference Holden2023). We ask whether a response to the price level is sufficient to restore determinacy when retaining the “speed limit” term in the interest rate rule. We retain the speed limit because policymakers may find such policies attractive (Walsh, Reference Walsh2003), but inadvertently bring about multiplicity (Figures 4–8); this is a problem we want to solve.

Accordingly, we assume the shadow interest rate under price-level targeting is

(18) \begin{equation} i_t^* = \rho _i i_{t-1}^* + (1-\rho _i) \left (\theta _p p_t + \theta _{\Delta y}(y_t - y_{t-1}) \right ) \end{equation}

where $\theta _p\gt 0$ and $p_t\,:\!= \pi _t + p_{t-1}$ is the log price level.

Differently from the rule in Holden (Reference Holden2023, Appendix E), the shadow interest rate still responds to the change in output gap. We now consider implications for determinacy.

Figure 9. Regions in which $M$ is not a $P$ -matrix (black): price-level targeting when $T=16$ . Note that $\theta _p$ is the response coefficient on the log price level and $\rho _i=0$ .

Figure 9 plots the regions in which $M$ is a $P$ -matrix, again for $T=16$ as in Figure 5. The indeterminacy region shrinks substantially, but we see that indeterminacy is not ruled out when a strong response to the “speed limit” is combined with a relatively weak response to the price level (black regions). Thus, only a sufficiently strong interest rate response to the price level—a large enough $\theta _p$ in (18)—ensures determinacy. Intuitively, this says that a long-lasting, aggressive expansionary policy to restore the price level to target is sufficient to rule out the “bad” equilibrium based on pessimistic expectations. Perfect foresight simulations suggest that the “good solution” under price-level targeting (with the nominal rate away from the bound) has good performance in terms of stabilization, and allowing interest rate smoothing in the rule reinforces this conclusion (see Supplementary Appendix, Section 3.4).

Interestingly, if the reaction to the price level $\theta _p$ is small enough for both solutions to exist, then a very weak response to the price level works well from a stabilization perspective. In this case, the “bad” solution under price-level targeting looks somewhat “better” than with the inflation targeting, interest rate smoothing or forward guidance rules (see Figure 10). We see that inflation and the output gap drop far less initially (i.e. in period 1), and these variables and interest rates then oscillate around the “good” solution, but within quite a narrow range and deviations “die out” quite quickly. It should be noted, however, that not all “bad” solutions under price-level targeting are “tamer” than for the original rule.Footnote ²⁷

Notably, our conclusions on price-level targeting are less positive than in Holden (Reference Holden2023), where analytical and numerical results are used to show that a price-level targeting rule ensures determinacy in a range of New Keynesian models. What our results highlight is that the assumption of a response to the level of the output gap, rather than its first-difference, is crucial. In the present model, where the “speed limit” $y_t-y_{t-1}$ enters the interest rate rule, determinacy is not guaranteed but instead requires a sufficiently strong response to the price level relative to the response to the speed limit, as shown in Figure 9. A particular example of multiplicity when the response to the price level is “too weak” can be seen in Figure 10.

4.5 Welfare analysis

Finally, we consider some welfare implications of multiplicity, an issue not considered in Holden (Reference Holden2023). We make use here of the “expected outcomes” approach in Section 3.2. In the above model, the microfounded social loss function has the form:

(19) \begin{equation} L = \sum _{t=1}^\infty \beta ^{t-1}( \pi _t^2 +\lambda y_t^2 ) \end{equation}

where $\lambda \gt 0$ is the relative weight on output gap variations.Footnote ²⁸

As we have seen, there are two perfect foresight solutions for some parameter values. In Table 2, we report the losses under each policy rule based on (19), for both the “good” solution (Solution 1) and the “bad” solution (Solution 2, when it exists), i.e.

(20) \begin{equation} L_k = \sum _{t=1}^\infty \beta ^{t-1}( \pi _{k,t}^2 +\lambda y_{k,t}^2 ), \quad \quad k \in \{1,2\} \end{equation}

where $k=1$ denotes the good solution and $k=2$ denotes the bad solution.

Table 2. Welfare losses and policy rules: good and bad solutions ( $\lambda =0.1$ )

Note: IT1 is the baseline case in Figure 4. IT2 adds interest rate smoothing with $\rho_i = 0.4$ (Figure 7). FG1 and FG2 shown and described in Figure 8. PLT1 (PLT2) sets $\theta_p=1.5$ ( $\theta_p=0.015$ , see Figure 10). Reported losses are computed as the ratio of the loss relative to the ‘good’ loss $L_1$ for rule IT1.

For each type of policy rule we consider two parameterizations and other parameters and initial conditions are set at their baseline values as in the exercises in Figures 4, 7, 8, 10. We see that the price-level targeting rule PLT1 ( $\theta _p=1.5$ ) gives the lowest loss among the rules shown and restores determinacy (penultimate row); it also outperforms an interest rate smoothing rule with $\rho _i =0.8$ (Figure 7) or higher.Footnote ²⁹ Even when the price level response is very small ( $\theta _p = 0.015$ , Figure 10)—such that multiple solutions arise—the loss under the good solution is much smaller than under the forward guidance rules (at 3.5 times the loss under the baseline “good solution”), while the bad solution imposes a relatively small welfare loss compared to the alternatives. These main conclusions are not sensitive to the relative weight $\lambda$ on output gap deviations in the loss function (which is set at 0.1 in Table 2).Footnote ³⁰

We now consider expected social welfare losses in the above model. To do so, we follow the “expected outcomes” approach of Section 3.2 that computes a weighted average in which the weights are the probabilities that expectations will coordinate on each solution. The latter can be viewed as an expected outcome from some initial position or state (period 0) in which it is not yet known which perfect foresight solution agents’ expectations will coordinate on. Therefore, we can think of this welfare exercise as an assessment made by policymakers (or researchers) who are viewing the economy from under the “veil of ignorance.”

Figure 10. “Good” and “bad” solutions under price-level targeting for $e_1=0.01$ , $i^*_0=y_0=0$ and a “weak” response to the price level $\theta _p = 0.015$ when $\sigma = 1$ , $\theta _{\Delta y}=1.6$ and $\rho _i=0$ .

We stick with the same policy rule (and parameter values) as in Table 2. Given (20) and a probability $p_1$ of Solution 1, the expected loss under a given rule is

(21) \begin{equation} E_0[L] = p_1 L_1 + (1-p_1) L_2 \end{equation}

where $L_1=L_2$ in the case of rules for which there is a unique solution.

In Figure 11, we plot the expected welfare losses under each rule. While the price-level targeting rule PLT1 ( $\theta _p=1.5$ ) performs best as expected, there is no clear ranking among the other rules because the expected losses are sensitive to the probability $p_1$ of Solution 1 (both panels). The “weak response” price-level targeting rule PLT2 ( $\theta _p=0.015$ ) is ranked second for all values of $p_1$ up to approx. 0.9998 (right panel), but is then outperformed by the inflation targeting rules IT1 (no smoothing), and as the probability $p_1$ approaches 1, the inflation targeting rule IT2 (smoothing, $\rho _i=0.4$ ) outperforms both the PLT2 rule and IT1, though this requires that the bad solution is a near-zero probability event.

Figure 11. Expected welfare losses as the probability of Solution 1, $p_1$ , is varied. The different interest rate rules are the same ones as in Table 2. The left panel uses a log scale.

Overall, the performance of price-level targeting is quite robust compared to the other policies for which losses are very sensitive to the probability that agents coordinate expectations on the “good” solution, 1. If the “bad equilibria” have non-trivial probability, price-level targeting looks highly attractive; however, if the such equilibria are considered extremely unlikely, then the potential benefits of price-level targeting look smaller and might not be convincing to policymakers. Thus, we see the potential value of assessing robustness of expected welfare to the probability of the good solution, as in Figure 11.

Finally, a word of caution is in order. It should be noted that since the “good” and “bad” solutions differ under each policy rule, the probabilities of each solution could also differ in these cases; for example, the “tame” bad solution under price-level targeting with $\theta _p=0.015$ (Figure 10) might be considered more plausible than the highly destabilizing bad solution under inflation targeting with moderate interest rate smoothing (Figure 7, $\rho _i = 0.4$ ). If so, then an analysis like that in Figure 11 is still useful, but with the caveat that the expected welfare losses under each rule should be computed conditional on the different probabilities of each solution that the researcher or policymaker has in mind.Footnote ³¹