Slow Learning

Abstract:

1 Introduction

The founders of the rational expectations (RE) revolution thought that, to be useful, rational expectations equilibria (REE) must be the outcome of learning equilibria.¹ While convergence to an REE may be a necessary condition for rational expectations to be a useful model of expectations, it is not sufficient. As Vives, 1993 stresses, in a changing world, for all practical purposes, “‘slow’ convergence may mean no convergence.” The critical issue is whether the speed of convergence is fast enough to render RE a useful guide for normative and positive analyses. In this paper, we characterize, for a large class of models, the speed of convergence of a learning equilibrium to an REE.

We depart from the RE framework and assume that people learn about their environment by forming beliefs about future economic outcomes and updating those beliefs as data arrive. We analytically characterize the speed of convergence in a broad class of non-stochastic learning models. The models that we consider have two important characteristics. First, people learn using either standard Bayesian methods or least-squares. Second, people’s beliefs about model outcomes are central determinants of equilibrium outcomes. We demonstrate that our results can be applied to a broad class of macroeconomic models that are solved using linearization methods. This class includes medium-scale dynamic stochastic general equilibrium (DSGE) models with shocks, like Christiano et al., 2005 (hereafter, CEE).

An important contribution of this paper is to provide a formal characterization of the asymptotic rate of convergence for a broad class of nonlinear, non-stochastic multivariate learning models studied in Evans and Honkapohja, 2000. Proposition 2 of this paper shows that a particular scalar statistic, \(b\), of the system determines the asymptotic speed of convergence to an REE. That parameter, which also determines the E-stability of an REE (\(b<1\)), can be calculated from the model solution (see Evans and Honkapohja, 2000 for a discussion of E-stability). A model exhibits slow learning when \(b\) is less than but close to unity. We investigate the economic determinants of \(b\), i.e., whether learning is fast or slow.

Our central finding is that, in empirically plausible new-Keynesian (NK) models, learning equilibria converge slowly to REE. Indeed, learning can be extraordinarily slow. We develop an asymptotically valid formula that depends only on \(b\) which approximates the time required to close a given fraction of the gap between initial and RE beliefs. Using simulations, we show that our formula is quite accurate for all the models considered. In the simple NK model, when wages and prices are sticky, progress to the REE is measured in decades.² An even stronger result holds for the benchmark DSGE model developed in Christiano et al., 2005: meaningful convergence to the REE is measured in centuries. Even more dramatically, in the simple NK model, when the zero lower bound (ZLB) is binding, progress is measured in millennia. We argue that stickiness in wages and the parameters of the monetary policy are the critical determinants of the speed of convergence.

The intuition behind our results is that people’s beliefs influence actual economic outcomes. For example, suppose people revise their expectations of future inflation upward. Wage setters, recognizing they would like higher nominal wages in the future, have an incentive to front-load those increases when wage stickiness is present. By increasing nominal marginal cost, this rise in the current wage exerts upward pressure on current inflation. In effect, sticky wages make inflation expectations at least partially self-fulfilling, much like in the wage-price spiral literature (see Blanchard, 1986). The more self-fulfilling expectations are, the slower people are to abandon initial priors (the higher is \(b\)) and the longer it takes for expectations to converge. Monetary policy also plays an important role. The more (less) aggressive the monetary authority’s response to inflation (the output gap), the less likely it is that inflation expectations will be self-fulfilling.

Sections 5 and 6 analyze the robustness of our key conclusions to a variety of perturbations. First, as noted above, those conclusions hold in empirically plausible DSGE models like the model of Christiano et al., 2005.³ Second, we consider the speed of learning when nominal rigidities arise from Rotemberg rather than Calvo-style nominal rigidities. Under RE, Calvo-style and Rotemberg-style nominal rigidities are equivalent in a first-order approximation to the model.⁴ However, they need not be equivalent in a learning equilibrium. As it turns out, our key results about the speed of learning are robust across the two different ways of modeling nominal rigidities. Third, our results are robust to whether or not the zero lower bound (ZLB) on the nominal interest rate is binding. Fourth, we show that our results are robust to whether people solve their problems using a version of Kreps, 1998’s anticipated utility (our benchmark analysis) or internally rational learning. In the anticipated utility framework, people update their beliefs at each period as new data comes in. But when they make their decisions, they proceed as though their beliefs will never be revised.⁵ Under internally rational learning, people fully integrate the fact that they are learning when they solve their problems, i.e., households and firms are internally rational in the sense defined by Adam and Marcet, 2011. Implementing internal rationality in the model’s nonlinear solution is computationally challenging.⁶ The associated computational burden explains why much of the learning literature works with a version of Kreps, 1998’s anticipated utility approach. Because of computational challenges, we investigate the implications of the two different learning models using the full nonlinear solution to the simple NK model when the ZLB is binding. Fifth, we demonstrate that our results are robust to whether we work with the fully nonlinear or linear approximations to model solutions. Again, for computational reasons it is difficult to work with a fully nonlinear version of the DSGE model in Christiano et al., 2005. So we use the simple NK model when the ZLB is binding. In the linear and nonlinear versions of that model, the speed of convergence to the REE is very slow.

Section 7 analyzes the impact of learning on how fiscal and monetary policies impact the economy. We show that the speed of convergence plays a crucial role in assessing the efficacy of government policies, such as increases in government purchases. These results are most stark in the simple NK model when the ZLB is binding.⁷ We make this case using a nonlinear version of the NK model in which only prices are sticky. This model has been widely used in the literature that analyzes the effect government purchases when the ZLB is binding. In the model we consider, the effects of a rise in government purchases are much smaller under learning than under RE. Under RE, the government spending multiplier is very large when the ZLB binds because an increase in government purchases raises expected inflation (see Christiano et al., 2011). Because the nominal interest rate is fixed, this rise generates a fall in the real interest rate, a rise in consumption, and a multiplier substantially larger than unity. In the learning model, expected inflation is partially backward-looking and changes little after an increase in government purchases. So, the real interest rate does not fall by very much, the key driver of the large REE multiplier is effectively eliminated, and the multiplier is close to unity.

We also consider the effects of monetary policy in the form of forward guidance when the ZLB is binding. To convey intuition as transparently as possible, we consider a simple form of forward guidance: the monetary authority commits to keeping the nominal interest rate at zero for one period after the shock that makes the ZLB binding returns to its steady-state level. Consistent with the existing literature (for example, Farhi and Werning, 2019, Del Negro et al., 2023, and Woodford, 2012), we find that even this simple form of forward guidance is powerful under RE. As is well known, the power of forward guidance under RE reflects its strong effect on expected inflation. Under learning, expectations are partially backward-looking, and forward guidance is not very powerful. So, as with fiscal policy, an REE-based analysis of monetary policy can be very misleading.

The models we work with have multiple REEs. Benhabib et al., 2001b document that NK models have two REE steady states. In one steady state the ZLB is binding, in the other it is not. Mertens and Ravn, 2014 and Bilbiie, 2022 discuss ZLB REEs that are caused by nonfundamental shocks. Throughout our paper, when we use linear approximations of the model, we focus on learning equilibria that are local to E-stable REEs.⁸ We do so for three reasons. First, E-stable REEs are the most-commonly studied in the related literature. Second, a full treatment of equilibrium multiplicity in the NK model is beyond the scope of this paper. Third, we find that learning can be very slow, even in learning equilibria that are local to E-stable REEs. This finding poses a challenge to policy analyses based on E-stable REEs that are the focus of the literature. We discuss equilibrium multiplicity in Appendix G.

The remainder of this paper is organized as follows. Section 2 discusses the relationship of our paper to the related literature. Section 3 presents our theoretical results regarding the speed of convergence to an REE. Section 4 analyzes the speed of convergence in the simple NK model when the ZLB is not binding. Section 5 analyzes the speed of convergence in the CEE model when the ZLB is not binding. Section 6 discusses the robustness of our analysis, including linear and nonlinear versions of the simple NK model when the ZLB is binding, and versions of the model under anticipated utility and internally rational learning. Section 7 considers whether learning matters for policy analysis. Finally, section 8 contains concluding remarks.

2 Related Literature

Our paper is related to several literatures. First, a large literature studies the properties of recursive estimators in learning models with decreasing gain.⁹ Ljung, 1977 establishes almost sure convergence. Marcet and Sargent Marcet and Sargent, 1989b; Marcet and Sargent, 1989a, Woodford, 1990, Evans and Honkapohja Evans and Honkapohja, 2000; Evans and Honkapohja, 2001, and others build on Ljung, 1977 to study the conditions under which learning equilibria converge to an REE. In contrast, we study the speed at which learning occurs. Marcet and Sargent, 1995 study the speed of convergence using numerical methods. In contrast, we provide analytic results for a large class of models. In addition, our numerical results pertain to a class of empirically plausible DSGE models.

We work in a nonstochastic environment in which it is natural to focus on the rate of convergence of deterministic sequences. The stochastic approximation literature focuses on convergence in distribution. Benveniste et al., 1990 show that \(t^{1/2}\theta_{t}\) converges to a Normal distribution with mean zero for \(b<1/2\).¹⁰ Christopeit and Massmann, 2018 provide a derivation of the speed of convergence in distribution of a learning equilibrium for \(b\in[1/2,1)\). They do so for a scalar linear regression model. Their theoretical results do not, in general, hold for nonlinear, multivariate environments. We provide multivariate, nonstochastic results on the speed of convergence of \(\theta_{t}\) for \(b<1\) and apply them to the nonlinear NK model.

A third literature examines the speed of convergence in the simple NK model (see, e.g., Ferrero, 2007). We study rates of convergence in a range of models, including an empirically plausible medium-sized DSGE model. Additionally, Ferrero, 2007 uses the Euler equation approach to learning.¹¹ We consider that approach to learning as well as the anticipated utility and internally rational approaches. In contrast to Ferrero, 2007, we also study convergence rates in linear and nonlinear versions of the NK model when the ZLB is binding.

Heemeijer et al., 2009 and Hommes, 2011 study positive and negative feedback loops from expectations to outcomes using laboratory experiments and univariate models with constant gain. Their results provide experimental support for the class of learning models considered in this paper.

A different literature assumes that people have RE and studies how people form beliefs about exogenous variables when they have imperfect information. For example, Vives, 1993 asks: how quickly do people’s beliefs about an exogenous cost parameter converge? Other authors like Erceg and Levin, 2003, Gust et al., 2018, and Farmer et al., 2024 study how people draw inferences about the movements in the unobserved transitory and persistent components of an exogenous variable. In contrast, we study convergence rates for beliefs about objects whose values depend on those beliefs. Other papers that study the dynamic properties of Bayesian learning with incomplete or noisy information include Collin-Dufresne et al., 2016 and Angeletos and La\'o, 2020. In this paper, we consider learning equilibria along the lines of those studied in, for example, Eusepi and Preston, 2018, and characterize the rate of convergence of beliefs.

Another literature studies the policy implications of learning when the ZLB is binding. Evans et al., 2008 study the effects of fiscal policy and alternative money supply rules for Euler-equation learning equilibria when the ZLB is binding. Eusepi, 2010 shows that central bank communication can have an important impact on Euler-equation learning equilibria when the ZLB is binding. Benhabib et al., 2014 and Evans et al., 2022 study convergence in the nonlinear simple NK model with anticipated-utility learning. Eusepi et al., 2021 discuss forward guidance when the ZLB is binding in the simple NK model with anticipated utility. Preston, 2005 and Eusepi et al., 2022 use the anticipated utility approach to study the effects of monetary policies in linearized NK models under learning. In contrast to our analysis, these papers do not characterize rates of convergence or consider internally rational learning.

3 Learning in a Nonlinear Environment

In this section, we study the speed of convergence of learning in a non-stochastic environment. Let \(\theta_{t}\) be a \(k-\)dimensional vector of variables that summarizes people’s period-\(t\) beliefs about the values of the parameters governing a perceived data-generating process. We interpret \(\theta_{t}\) as a deviation from a particular fixed point of beliefs in our learning algorithm.¹² As in Evans and Honkapohja, 2000, the vector \(\theta_{t}\) evolves according to \[\begin{equation} \theta_{t}=\theta_{t-1}+\gamma_{t}\left[M\left(\theta_{t-1},\gamma_{t}\right)-\theta_{t-1}\right]\tag{1}\end{equation}\] for \(t=1,2,3,\dots\) where \(M\left(\theta_{t-1},\gamma_{t}\right)\) is a potentially non-linear function. Here, \(\theta_{0}\) is given and \(\gamma_{t}\) the learning gain. Let the \(k\times k\) matrix \(D_{1}M\) denote the derivative of \(M\) with respect to the vector \(\theta_{t-1}\) and let \(b\) be the largest real part of the eigenvalues of \(D_{1}M\left(0,0\right)\). Similar to Evans and Honkapohja, 2000, we adopt the following two assumptions:

Our environment is the same as Evans and Honkapohja, 2000’s, except that we have a more-restrictive specification of \(\gamma_{t}\).¹³ They establish that for \(b<1\) and \(\theta_{0}\) sufficiently close to \(0,\) \(\left\Vert \theta_{t}\right\Vert\) converges to zero. We study the rate at which \(\left\Vert \theta_{t}\right\Vert\) converges to zero.

We adopt the following definition of the asymptotic rate of convergence of a variable in a nonstochastic environment. Our definition enables us to establish the precise sense in which the parameter \(b\) determines the rate of convergence.

Here, \(\left\Vert \cdot\right\Vert\) denotes a norm on \(\mathbf{R}^{k}\). The first part of Definition 1 says that for any small, positive \(\delta,\) \(\left\Vert x_{t}\right\Vert\) asymptotically converges no slower than \(t^{b-1+\delta}\). The second part says that \(\left\Vert x_{t}\right\Vert\) asymptotically converges no faster than \(t^{b-1-\delta}\). In this sense, \(b\) characterizes the power rate of convergence.

Christopeit and Massmann, 2018 report results which imply that when \(M\) is linear and \(\theta_{t}\) is a scalar then \(t^{1-b}\left\Vert \theta_{t}\right\Vert \rightarrow\kappa\) for \(0<\kappa<\infty\). Our concept of convergence in Definition 1 is weaker than convergence to a finite, nonzero constant limit. We work with our definition because \(t^{1-b}\left\Vert \theta_{t}\right\Vert\) may not converge to a finite, nonzero constant in nonlinear or multivariate cases.¹⁴

Conditions (i) and (ii) in Definition 1 describe a space of sequences. It is easy to verify that it includes \(\theta_{t}=\kappa t^{b-1},\) \(0<\kappa<\infty\). It also includes sequences that cycle around \(t^{b-1}\) with decreasing amplitude as \(t\) increases, e.g. \(\theta_{t}=\left[2+\sin\left(t\right)\right]t^{b-1}\). In addition, it includes sequences that deviate from \(t^{b-1}\) at a slow enough rate, such as \(\theta_{t}=\log\left(t\right)t^{b-1}.\) Definition 1, in effect, defines a set of sequences that exhibit a power rate of convergence, up to slowly varying terms. Consistent with this definition, all of the elements in the set of sequences identified by Definition 1 converge to zero at a rate slower than a geometric rate (i.e., the sequence \(\lambda^{t}\), \(t=1,2,...,\) for \(0<\lambda<1).\)

Let \(U\) denote a neighborhood of the origin that has the property that every \(\theta_{0}\in U\) generates a sequence \(\left\Vert \theta_{t}\right\Vert\) that converges to zero (Proposition 1 in Evans and Honkapohja, 2000 establishes that such a \(U\) exists). We now state our proposition, which is proved in Appendix A.

For intuition, it is useful to consider the following special case: If \(M\left(\theta_{t-1},\gamma_{t}\right)=M\theta_{t}\), where \(M\) is a matrix and all the eigenvalues of \(M\) are real and distinct, then \(t^{1-b}\left\Vert \theta_{t}\right\Vert\) converges to a non-zero limit for \(\theta_{0}\) in a dense subset of \(U\).

Proposition 2 leaves open the possibility that there are values of \(\theta_{0}\in U\) that do not have the property, \(\theta_{t}\simeq t^{b-1}\). An obvious, but uninteresting, example is \(\theta_{0}=0\) which implies \(\theta_{t}=0\) for all \(t\). A more informative example is given by: \[\theta_{t}=\theta_{t-1}+\gamma_{t}\left(M-I\right)\theta_{t-1}, M=\left[\begin{array}{cc} b & 0\\ 0 & a \end{array}\right],\] where \(a<b<1\). Suppose that \(\theta_{0}=\left[0,1\right]^{\prime},\) so that \(\theta_{t}=\left[\begin{array}{cc} 0, & \prod_{j=1}^{t}\left(1+\gamma_{j}\left(a-1\right)\right)\end{array}\right]^{\prime}.\) It is straightforward to show that \(\theta_{t}\simeq t^{a-1}\), so that the speed of convergence is faster than \(t^{b-1}\).¹⁵ However, any nonzero perturbation to the first element of \(\theta_{0}\), no matter how small, gives rise to a \(\theta_{t}\) sequence that has the property, \(\theta_{t}\simeq t^{b-1}\). This example, in which there are special initial beliefs that converge faster than \(t^{b-1}\), generalizes to any system in which \(M\) is linear, satisfies Assumption 1, and has an eigenvalue with real part less than \(b\). In any such system, there are initial beliefs that are arbitrarily close to any \(\theta_{0}\) that give rise to a sequence \(\theta_{t}\simeq t^{b-1}\).

In our analysis, the function, \(M\), in equation (1) embeds the equilibrium map from beliefs, \(\theta_{t-1},\) and the gain, \(\gamma_{t}\), to equilibrium outcomes under learning. It can be convenient to approximate equation (1) around the limit points of \(\theta_{t}\) and \(\gamma_{t}\) using perturbation methods.¹⁶ The first-order expansion of equation (1) implies \(\theta_{t}=\theta_{t-1}\), so that there is no learning in a neighborhood of the limit points. This property is a harbinger of the slow-learning results in this paper. A second-order expansion of equation (1) simply replaces \(M\left(\theta_{t-1},\gamma_{t}\right)\) in equation (1) with its first-order expansion, \(D_{1}M\left(0,0\right)\)\(\theta_{t-1}\).¹⁷ This result provides an interpretation of a standard procedure that uses a linear approximation of \(M\) when solving learning model.

4 The Speed of Learning in the Simple NK Model

In this section, we analyze the speed of learning in a simple textbook version of the NK model that incorporates a version of Kreps, 1998 anticipated utility.¹⁸ Our key finding is that the degree of wage stickiness, as well as the parameters of the monetary policy rule, play a central role in the speed of learning.

4.1 The Simple NK Model with Sticky Nominal Wages and Prices

In this subsection, we present a variant of the simple NK model in which both nominal wages and prices are sticky. We model nominal price rigidities as in Calvo, 1983. In the case of wage rigidities, we work with a variant of Erceg et al., 2000.

4.1.1 Firms’ Problems

A final homogeneous good, \(Y_{t}\), is produced by a continuum of competitive, identical firms using the technology \(Y_{t}=\left(\int_{0}^{1}Y_{f,t}^{\frac{\varepsilon-1}{\varepsilon}}df\right)^{\frac{\varepsilon}{\varepsilon-1}}\), where \(Y_{f,t}\) denotes differentiated inputs and \(\varepsilon>1.\) The representative homogeneous-good firm chooses inputs to maximize profits, \(Y_{t}P_{t}-\int_{0}^{1}Y_{f,t}P_{f,t}df\), where \(P_{t}\) denotes the time-\(t\) aggregate price level and \(P_{f,t}\) denotes the time \(t\) price of the \(f^{th}\) good. The representative firm’s first-order condition for the \(f^{th}\) input is: \[\begin{equation} Y_{f,t}=\left(\frac{P_{f,t}}{P_{t}}\right)^{-\varepsilon}Y_{t}.\tag{2}\end{equation}\]

A monopolist produces the \(f^{th}\) intermediate good with production technology \(Y_{f,t}=N_{f,t}\), where \(N_{f,t}\) denotes labor hired by firm \(f\). With probability \(\xi_{p}\) the monopolist sets its time-\(t\) price equal to \(P_{f,t-1}\). With probability \(1-\xi_{P}\), the monopolist resets its price optimally to \(\tilde{P}_{f,t}\). It chooses \(\tilde{P}_{f,t}\) to maximize the expected discounted value of profits: \[\begin{equation} \mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\xi_{P}^{j}C_{t+j}^{-1}\left(\frac{\tilde{P}_{f,t}}{P_{t+j}}-\left(1-\tau_{f}\right)s_{t+j}\right)\left(\frac{\tilde{P}_{f,t}}{P_{t+j}}\right)^{-\varepsilon}Y_{t+j}.\tag{3}\end{equation}\] Here, \(\mathbf{E}_{t}\) denotes the expectation operator under monopolists’ subjective beliefs at time \(t\), \(s_{t+j}\) denotes real marginal cost in period \(t+j\), and \(C_{t+j}^{-1}\) denotes the marginal value of profits in units of the consumption good in period \(t+j\). Finally, \(\tau_{f}\) is a government tax subsidy designed to eliminate the effect of monopoly distortions in the steady state.¹⁹

4.1.2 Households’ Consumption Problem

There is a continuum of identical households, \(h\in\left(0,1\right)\). Each household enters a period with a stock of bonds, \(b_{h,t-1}=B_{h,t-1}/P_{t-1}.\) Here, \(B_{h,t-1}\) denotes the nominal payoff at the beginning of period \(t\) to nominal bonds purchased in the previous period. In period \(t\), the household chooses consumption, \(C_{h,t}\), and end-of-period bond holdings, \(b_{h,t}\), to maximize: \[\begin{align} \mathbf{E}_{t}\sum_{j=0}^{\infty} & \beta^{j}\left\{ \log\left(C_{h,t+j}\right)-\frac{\chi}{2}\int_{0}^{1}N_{i,h,t+j}^{2}di\right\}\tag{4} \end{align}\] subject to the budget constraints, for \(j\geq0,\) \[\begin{align} C_{h,t+j}+\frac{b_{h,t+j}}{R_{t+j}} & \leq\frac{b_{h,t+j-1}}{\pi_{t+j}}+\int_{0}^{1}\left(1+\tau_{w}\right)w_{i,t+j}N_{i,h,t+j}di-\tau_{t+j}.\tag{5}\end{align}\] Here, \(\pi_{t}=P_{t}/P_{t-1}\), \(\tau_{t}\) denotes lump-sum taxes net of profits from the firms, and \(R_{t}\) denotes the nominal rate of interest. The \(h^{th}\) household has a variety of members, indexed by \(i\in\left(0,1\right)\), with imperfectly substitutable types of labor, \(N_{i,h,t}.\) The household takes \(w_{i,t}\) and \(N_{i,h,t}\) as given, for \(i\in\left(0,1\right)\). We assume households have the same expectation operator, \(\mathbf{E}_{t}\), as monopolists. For reasons similar to those discussed in Adam and Marcet, 2011 we place an upper and lower bound on \(b_{h,t}\). When we work with the linearized version of the model we mimic the effects of these bounds by adding the term, \(\frac{\vartheta}{2}b_{h,t+j}^{2},\) to the left side of the household budget constraint, equation (5). In practice, we set \(\vartheta\) to be a very small number and no costs are actually paid (even in learning equilibria) because \(b_{h,t}\) must be zero for bond markets to clear.

4.1.3 Employment and Wage Determination

Corresponding to each \(i\in\left(0,1\right)\) there is a single labor union. For every household, \(h,\) the \(i^{th}\) member belongs to that union. The union sets the nominal wage, \(W_{i,t}\), subject to Calvo frictions. With probability \(1-\xi_{W}\), union \(i\) can reset \(W_{i,t}\) and we denote its choice by \(\tilde{W}_{t}\). Otherwise the union must set \(W_{i,t}=W_{i,t-1}\). Union \(i\) provides all labor demanded in equal amounts from each household. Households must supply whatever amount of the \(i^{th}\) type of labor that union \(i\) requires.

A competitive, representative firm uses the labor, \(N_{i,t}=\int_{0}^{1}N_{i,h,t}dh\), supplied by union \(i\in\left(0,1\right),\) to produce a homogeneous labor service, \(N_{t},\) using the technology \(N_{t}=\left(\int_{0}^{1}N_{i,t}^{\frac{\epsilon-1}{\epsilon}}di\right)^{\frac{\epsilon}{\epsilon-1}}\), \(\epsilon>1\). The firm takes \(W_{i,t}\) as given and sells \(N_{t}\) at the price, \(W_{t}.\) In equilibrium, \(W_{t}=\left(\int_{0}^{1}W_{i,t}^{1-\epsilon}di\right)^{\frac{1}{1-\epsilon}}\). Profit maximization by the competitive, representative firm implies: \[\begin{equation} N_{i,t}=\left(\frac{W_{i,t}}{W_{t}}\right)^{-\epsilon}N_{t}.\tag{6}\end{equation}\]

The wage, \(\tilde{W}_{i,t}\), is chosen to maximize: \[\begin{equation} \mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\xi_{W}^{j}\left\{ \frac{C_{t+j}^{-1}}{P_{t+j}}\left(1+\tau_{w}\right)\tilde{W}_{i,t}\left(\frac{\tilde{W}_{i,t}}{W_{t+j}}\right)^{-\epsilon}N_{t+j}-\frac{\chi}{2}\left(\left(\frac{\tilde{W}_{i,t}}{W_{t+j}}\right)^{-\epsilon}N_{t+j}\right)^{2}\right\} .\tag{7}\end{equation}\] We assume unions have the same expectation operator, \(\mathbf{E}_{t}\), as households and monopolists. In an REE, maximizing equation (7) corresponds to maximizing the utility of the representative household.

4.1.4 Monetary, Fiscal Policy, and Aggregation

Aggregate output is given by: \[\begin{equation} Y_{t}=C_{t}+G_{t}.\tag{8}\end{equation}\] Monetary policy sets the gross nominal interest rate, \(R_{t}\), according to: \[\begin{equation} R_{t}=\beta^{-1}+\alpha_{\pi}\left(\pi_{t}-1\right)+\alpha_{Y}\frac{Y_{t}-Y}{Y}.\tag{9}\end{equation}\] Here, \(Y\) is the value of \(Y_{t}\) in the target-inflation (\(\pi_{t}=1\)) REE steady state. The government finances subsidies to households and firms with lump-sum taxes and balances its budget each period. The government also purchases final goods, \(G_{t}=G\).

The firm production function implies that gross output is given by: \[\begin{equation} Y_{t}=p_{t}^{\ast}N_{t},\tag{10}\end{equation}\] where \(p_{t}^{\ast}\) is a measure of price dispersion given by \(\left(p_{t}^{\ast}\right)^{-1}=\int_{0}^{1}\left(\frac{P_{f,t}}{P_{t}}\right)^{-\varepsilon}df\) (see Yun, 1996).

4.1.5 The Evolution of Beliefs Under Anticipated Utility

For the period-\(t\) problems of households, unions, and firms to be well-defined, they must have beliefs about the aggregate variables that determine their payoffs: \[\begin{equation} a_{t+j}=\left[\begin{array}{ccccccccc} C_{t+j}, & \pi_{t+j}, & R_{t+j}, & Y_{t+j}, & N_{t+j}, & w_{t+j}, & \tau_{t+j}, & p_{t+j}^{\ast}, & G_{t+j}\end{array},s_{t+j}\right]^{\prime},\tag{11}\end{equation}\] where \(w_{t+j}=W_{t+j}/P_{t+j}\), for \(j=0,1,2,....\) . We describe a recursive mechanism that characterizes people’s beliefs about \(\left\{ a_{t+j}\right\} _{j=0}^{\infty}\).

To focus attention on learning about the data generating process of key variables that are determined by intertemporal decisions, we work with a smaller set of variables, \(x_{t+j}\): \[\begin{align} x_{t+j}=\left[\begin{array}{ccc} C_{t+j}, & \pi_{t+j}, & w_{t+j}\end{array}\right]^{\prime}.\tag{12}\end{align}\] We assume people know seven aggregate equilibrium conditions of the model which map from \(a_{t+j-1}\) and \(x_{t+j}\) to the non-\(x_{t+j}\) elements of \(a_{t+j}\).²⁰ We denote this mapping, together with the relevant identities, by \(F\): \[\begin{equation} a_{t+j}=F\left(x_{t+j},a_{t+j-1}\right).\tag{13}\end{equation}\]

Before people observe \(x_{t}\), their beliefs about \(x_{t+j}\) are summarized by a density function, \(D,\) parameterized by a finite dimensional vector \(\Theta_{t-1}\): \[\begin{equation} x_{t+j}\sim D\left(a_{t+j-1};\Theta_{t-1}\right),\tag{14}\end{equation}\] for \(j=0,1,...\) . The presence of \(\Theta_{t-1}\) in equation (14) for each \(j\geq0\) reflects the anticipated utility assumption. Under this assumption, people making time-\(t\) decisions act as if their beliefs, \(\Theta_{t-1},\) will remain unchanged in the future, regardless of the data they see.

Equations (13) and (14) provide a recursive representation of people’s beliefs about \(\left\{ a_{t+j}\right\} _{j=0}^{\infty}\) in period \(t\) before observing \(a_{t}\). These beliefs are conditional on \(a_{t-1}\) and \(\Theta_{t-1}\).

After time-\(t\) decisions are made and markets clear, people use the observed \(x_{t}\) to update their beliefs, \(\Theta_{t-1},\) as follows: \[\begin{equation} \Theta_{t}=L\left(\Theta_{t-1},x_{t},a_{t-1};\gamma_{t}\right).\tag{15}\end{equation}\] Under anticipated utility, people making their decisions in period \(t\) do not internalize that they will update their beliefs at every future date according to equation (15). The form of \(L\) depends on the assumed model of learning and decision making. The variable \(\gamma_{t}\) is a decreasing gain and, in the models that we consider, takes the form \(\gamma_{t}=\left(\lambda+t\right)^{-1}>0\).

4.1.6 Equilibrium

At the beginning of period \(t\) people observe the aggregate state, \(a_{t-1}\). People make their decisions, conditional on their individual beliefs, \(\Theta_{t-1}\), based on observing \(x_{t}\), which they interpret as a draw from \(D\) in equation (14) for \(j=0\). They compute the remaining current period aggregates, \(a_{t},\) using equation (13). When people solve their period-\(t\) problems, beliefs about the distribution of future aggregates, \(\left\{ a_{t+j}\right\} _{j=1}^{\infty}\), are constructed using equations (13) and (14). When aggregated, all these decisions give rise to an actual set of period-\(t\) aggregates. A period \(t\) equilibrium is a fixed point of this mapping so that the observed \(x_{t}\) is consistent with the decisions people make. Formally,

The seven aggregate equilibrium conditions used to construct \(F\) in equation (13) and the three market-clearing conditions in (ii) determine the values of the ten period-\(t\) equilibrium variables in \(a_{t}.\)²¹

4.2 Solving the Model and Specification of Beliefs

We analyze the model’s properties using the second-order approximation of equation (1) discussed in Section 3. That approximation works with the linearized map from period \(t-1\) beliefs to the period-\(t\) equilibrium.²²

Let \(\hat{z}_{t}\) denote the log deviation of the variable, \(z_{t},\) from the zero-inflation REE steady-state value, \(z\).²³ In case \(z=0,\) then \(\hat{z}_{t}=z_{t}\).

We assume that at the time people make their time-\(t\) decisions, they believe the aggregate variables, \(\hat{x}_{t+j}\), have the following law of motion:²⁴ \[\begin{align} \widehat{x}_{t+j}=\mu+\Omega\widehat{a}_{t+j-1}+\nu_{t+j},\:j=0,1,....\tag{16}\end{align}\] Here, \(\mu\) is a \(3\times1\) vector. People believe that the \(3\times1\) vector, \(\nu_{t}\), is \(i.i.d.\) over time and normally distributed with mean 0 and diagonal variance-covariance matrix. In Appendix B we show that the REE has the form of equation (16) with \(\mu=0\), a specific matrix \(\Omega,\) and \(\nu_{t}\equiv0\). The presence of \(\widehat{a}_{t-1}\) in equation (16) reflects that the measure of price dispersion, \(p_{t-1}^{*}\), and the lagged real wage, \(w_{t-1}\) are part of the state in a period-\(t\) equilibrium. Because we work with a linear approximation, people’s belief about the variance of \(\nu_{t}\) does not affect their decisions. People’s believed value of \(\mu\) is given by \(m_{t-1}\), so that \(m_{t-1}\) corresponds to the vector \(\Theta_{t-1}\). For tractability, we assume that people believe that \(\Omega\) is equal to its REE value in a first-order approximation to the model. At the time people make their time-\(t\) decision, they are perfectly certain in their beliefs about \(\mu\) and \(\Omega\) in the law of motion of \(\widehat{x}_{t+j}\) for \(j\geq0\). The value of \(\nu_{t}\) is then determined by the requirement that time-\(t\) markets clear (see Appendix B for further discussion). Note that this means people’s assumption that \(\nu_{t}\) is an independent random variable is incorrect.

After seeing time-\(t\) aggregate outcomes, people update beliefs using Bayes’ rule:²⁵ \[\begin{equation} m_{t}=m_{t-1}+\gamma_{t}\left(\widehat{x}_{t}-\Omega\widehat{a}_{t-1}-m_{t-1}\right)\tag{17}\end{equation}\] where \(\gamma_{t}=\left(\lambda+t\right)^{-1}\).²⁶

Appendix B shows that \(\widehat{x}_{t}-\Omega\widehat{a}_{t-1}=Hm_{t-1}\) in a first-order approximation to the period-\(t\) equilibrium. So, a second-order approximation to equation (17) is given by \[\begin{align} m_{t} & =m_{t-1}+\gamma_{t}\left(Hm_{t-1}-m_{t-1}\right)\tag{18}\end{align}\] where \(\gamma_{t}=\gamma_{t-1}\left(1-\gamma_{t-1}\right)\), \(0<\gamma_{1}<1\). A surprising feature of equation (18) is that only the variables \(m_{t-1}\) and \(\gamma_{t}\) (and not the variables \(\hat{x}_{t}\) and \(\hat{a}_{t}\)) appear on the right-hand-side. This feature of the solution exploits our assumption that people are only uncertain about intercept terms and not slope terms in equation (16).

Note that equation (18) is of the form of equation (1) with \(M\left(m_{t-1},\gamma_{t}\right)=Hm_{t-1}\). So, we can apply our Proposition to this environment under Assumptions 1 and 2.

4.3 Speed of Convergence

In this subsection, we analyze the asymptotic speed of convergence using a calibrated version of the simple NK model. We assume the following benchmark parameter values: \[\beta=0.995,\varepsilon=\epsilon=4,\xi_{P}=\xi_{W}=0.85,\chi=1.25,\alpha_{\pi}=1.5,\alpha_{Y}=0.25,G=0.2Y,\vartheta=10^{-6}.\] We choose \(\beta\) so that the steady state real interest rate is about 2 percent at an annual rate. We set \(\varepsilon=\epsilon=4\) implying a steady state markup of 33 percent in product and labor markets. These values are within the range of values considered in the related literature. We set \(\xi_{P}=0.85,\alpha_{\pi}=1.5,\alpha_{Y}=0.25\), which are also within the range of estimates considered in the literature. For symmetry, we set \(\xi_{W}=\xi_{P}\). The value of \(\chi\) is chosen so that in the target-inflation (\(\pi=1\)) steady state labor supply is unity.

Table 1 reports the value of \(b\) in the benchmark model as well as in the following variants of the model: (i) wages are flexible and the labor market is competitive; (ii) prices are flexible; (iii) the coefficient \(\alpha_{\pi}\) in the Taylor rule is increased to \(2.5\); (iv) the coefficient \(\alpha_{\pi}\) in the Taylor rule is reduced \(1.01\); and (v) the coefficient \(\alpha_{Y}\) is reduced to \(0.125\).

We use our asymptotic results to develop intuition for the speed of convergence in the benchmark NK model with sticky prices and wages. We exploit a result for an important special case that applies to the models that we consider. For arbitrary \(m_{t-1}\), the evolution of beliefs is determined by the full matrix \(H\) (see equation (18)). However, if the largest real part of the eigenvalues of \(H\), \(b,\) is real and not repeated, then \(Hm_{t-1}=bm_{t-1},\) if \(m_{t-1}\) is proportional to the eigenvector, \(q,\) associated with \(b\). Moreover, apart from special exceptions, for any initial beliefs, \(m_{0},\) the non-\(b\) eigenvalues play a vanishingly small role in the dynamics of \(m_{t}\) for large \(t\) and eventually \(m_{t}\) is very nearly proportional to \(q\).²⁷ At that point, the period \(t\) equilibrium is in effect completely characterized by the scalar, \(b\), and equation (18) approaches: \[\begin{equation} m_{t}=\left(1+\gamma_{t}\left(b-1\right)\right)m_{t-1},\tag{19}\end{equation}\] where \(m_{t-1}\) is proportional to \(q\). In the spirit of these asymptotic results, we consider initial conditions, \(m_{0}\), which are proportional to \(q\). We scale \(m_{0}\) so that \(\left\Vert m_{0}\right\Vert =1\) and normalize \(m_{0}\) so that the value of the intercept of the inflation process is positive.

Table 2 reports the values of \(m_{0}\) for each variant of the model that we consider. The vector of initial beliefs has different lengths depending on the nature of nominal rigidities. When both prices and wages are sticky, people have to form beliefs about aggregate inflation, consumption, and the real wage. When only wages or prices are sticky, people only have to form beliefs about aggregate inflation and consumption.

Table 1 reports that the value of \(b\) in the benchmark model (sticky wages and prices) is \(0.84\). This value implies that asymptotic convergence to REE beliefs is very slow, with convergence measured in hundreds of years. We simulate the model to find the number of periods, \(\hat{T}_{2/3}\), needed to close 2/3 of the gap, \(\left\Vert m_{0}\right\Vert ,\) by setting \(\gamma_{1}=0.5\) and iterating on equation (19).²⁸ We provide an alternative approximation that assumes \(\left\Vert m_{t}\right\Vert\) is asymptotically proportional to \(t^{b-1}\) (see the discussion after Proposition 2). Under this approximation, the amount of time needed to close \(2/3\) of an initial gap in beliefs from the REE is \(T_{2/3}=\left(1/3\right)^{\frac{1}{b-1}}.\) For the benchmark model, \(\hat{T}_{2/3}\) and \(T_{2/3}\) are 2343 and 881 quarters or, more than 500 and 200 years, respectively.

Below, we provide intuition for the forces governing the asymptotic speed of convergence to an REE. We do so by considering the impact on \(b\) of several perturbations in model parameters.

Sticky Prices and Flexible Wages

It is useful to begin by considering the case where prices are sticky and wages are flexible. According to the value of \(m_{0}\) in Table 1, in period 1, people believe that the intercept for consumption is lower and inflation is higher than in the REE. Table 1 reports that expectations are not very self-fulfilling, \(b=0.26\), so that beliefs converge quickly to their REE values: \(T_{2/3}(\hat{T}_{2/3})=5(6)\) quarters.

Consider the effect of high expected future inflation. Firms understand that high future inflation implies high future nominal wages, raising their desired future prices. In the presence of sticky prices, firms front-load price increases, thereby inducing upward pressure on period-1 inflation.

Next, consider the effect of low expected future consumption. Low expected aggregate consumption implies that households will have a high marginal utility of consumption and a high labor supply in the future. Other things equal, firms expect low nominal wages in the future. This expectation induces them to lower prices in period \(1\). Low expected aggregate consumption means households expect low future labor income and profits. So, consistent with firms’ expectations, households increase their labor supply and lower consumption. Together, these labor-market effects exert downward pressure on period-1 inflation.

Because of the competing forces at work, the deviation of period 1 inflation from its REE value could be higher or lower than its expected value, \(m_{\pi,0}\). Suppose, as is the case for our parameterized model and our specification of \(m_{0}\), that the actual deviation in inflation is lower. Since period 1 inflation above its REE value and \(\alpha_{\pi}>0,\) the nominal interest rate rises above its REE value. The magnitude of that rise is moderated by the fact that \(\alpha_{Y}>0\) and the fact that consumption is below its REE value. On net, the period 1 equilibrium value of inflation is above its REE value but substantially below the level implied by \(m_{\pi,0}\). Actual consumption is lower than its REE value, but substantially above the level implied by \(m_{C,0}\).

At the end of period 1, people update their expectations based on period 1 market outcomes. The muted change in period-1 consumption and inflation, relative to initial expectations, prompts people to substantially revise their expectations about future consumption and inflation. This process continues in future periods. The net result is a positive, relatively small value of \(b\) and quick convergence to the REE.

Sticky Wages and Flexible Prices

Consider next the case when wages are sticky and prices are flexible. Table 1 reports that convergence occurs much more slowly than the sticky-price only case price: \(b=0.81\) and \(T_{2/3}\:(\hat{T}_{2/3})\approx\) 100 (250) years. The intuition for this slow convergence is as follows. Initial beliefs, \(m_{0},\) are similar to their values in the sticky-price-only case. High expected inflation in subsequent periods leads unions to set higher nominal wages in the initial period to avoid a low relative wage in the future. Absent Calvo price frictions, firms set their nominal price as a markup over the nominal marginal cost which is equal to the nominal wage rate. Because nominal marginal cost is higher than in the sticky-price-only case, inflation is higher and more self-confirming. Other things equal, the higher inflation in the sticky-wage-only case results, through monetary policy, in a higher real interest rate. But that effect reinforces the household’s desire to reduce consumption. Again, there are conflicting forces acting on the realized inflation rate. But, on net, the equilibrium rate of inflation is higher, and consumption falls by more than in the pure sticky price case. So, the data is more self-confirming, \(b\) is higher, and the speed of learning is lower.

Sticky Wages and Sticky Wages (benchmark model)

Table 1 shows that the benchmark model has a value of \(b\) that is similar to the sticky wage, flexible price model. Evidently, the dynamics of sticky wages are the dominant factor determining the large value of \(b\) and the extent to which beliefs are self-fulfilling.

Alternative Values of \(\alpha_{\pi}\)

According to Table 1, learning is faster when \(\alpha_{\pi}\) is higher. As indicated by \(m_{0,\pi}\), when \(\alpha_{\pi}=2.5,\) initial beliefs about inflation are above their REE value. As discussed above, the positive value of \(m_{\pi,0}\) leads firms to raise period 1 prices. When \(\alpha_{\pi}\) is large, monetary policy aggressively raises the interest rate in response to higher inflation. Other things equal, this rise leads to a fall in aggregate consumption. That fall induces an increase in labor supply and, a fall in the aggregate wage. The resulting decline in marginal cost moderates the rise in inflation. All these effects are increasing in \(\alpha_{\pi}\). So, other things equal, a higher value of \(\alpha_{\pi}\) makes inflation beliefs less self-fulfilling and leads to a smaller value of \(b\).

Table 1 also reports that when \(\alpha_{\pi}=2.5\), initial beliefs about aggregate consumption are below their REE value. As a consequence, individual households believe their income will be lower than its REE value. So, they cut back on their spending and increase their labor supply. The latter effect leads to a fall in the wage, marginal cost, and prices. When \(\alpha_{\pi}\) is large, monetary policy cuts the interest rate aggressively, thus moderating the fall in consumption and inflation.

The net effect of the compound experiment involving \(m_{\pi,0}\) and \(m_{C,0}\) is that period 1 realized inflation and consumption are closer in period 1 to their REE values when \(\alpha_{\pi}\) is equal 2.5 rather than 1.5. Consequently, when \(\alpha_{\pi}\) is larger people revise their expectations by more, \(b\) is smaller, and convergence to the REE is quicker.

Consistent with the previous intuition, Table 1 reports that when \(\alpha_{\pi}=1.01\), \(b\) is higher than its benchmark value and convergence to the REE is extraordinarily slow.

The Impact of \(\alpha_{Y}\)

Finally, consider the case in which \(\alpha_{Y}=0.125\) so monetary policy reacts less to output than in the benchmark case. Table 1 reports that a smaller value of \(\alpha_{Y}\) reduces the value of \(b\). As a result, learning is faster, with T\(_{2/3}\) and \(\hat{T}_{2/3}\) falling to about 10 and 25 years, respectively. The intuition is that a lower value of \(\alpha_{Y}\) acts like a higher value of \(\alpha_{\pi}\), which accelerates the rate of convergence. In the polar case of \(\alpha_{Y}=0\), \(b\) falls to 0.64, and \(T_{2/3}\) and \(\hat{T}_{2/3}\) fall to about 5 and 0.5 years, respectively.

5 Speed of Learning in an Empirically Plausible DSGE Model

In this section, we show that our results about the speed of convergence in the simple NK model hold in an empirically plausible, medium-sized DSGE model. The model we analyze is similar to the one in CEE and has the following features: sticky prices and wages, internal habit formation in consumption, endogenous capital utilization rates, and adjustment costs on investment. Monetary policy is governed by a Taylor rule with interest rate smoothing. In Appendix [subsec:CEE-(2005)-model], we provide a detailed specification of this model. We set all non-policy parameters are to the point estimates reported by CEE. For convenience, these values are reported in Appendix [subsec:CEE-(2005)-model]. For the monetary policy rule, we set the coefficients on the inflation rate and the output gap to the values we used in the benchmark simple NK model analysis. We set the interest rate smoothing parameter to \(0.8\), as in CEE.

We approximate the learning equilibrium using the same method as in the simple NK model. As in the latter model, we assume that people know how aggregate prices and quantities react to their lagged values. However, they must learn about the intercepts of those processes. As above, we calculate the speed of convergence of those beliefs to their REE values.

Table 2 reports the values of \(b\), \(T_{2/3}\), and \(\widehat{T}_{2/3}\) for nine versions of the model. The speed of convergence to the REE beliefs in the benchmark model, labeled “CEE (2005),” is very slow. Measured by \(T_{2/3}\) and \(\widehat{T}_{2/3}\), it takes roughly 800 years to close 2/3 of the gap between the initial beliefs and the REE beliefs. So, moving from the simple NK model to a more empirically plausible model increases the time it takes to converge to the REE. The intuition for the role of sticky prices and the monetary policy parameters, \(\alpha_{\pi}\) and \(\alpha_{Y}\), is qualitatively similar to the intuition provided in the context of the simple NK model.

As in the simple NK model, the parameters of the monetary policy rule and sticky wages are key determinants of the speed of learning. Allowing wages to be flexible dramatically reduces \(T_{2/3}\) from more than 33,000 to 22 (roughly 5 years). Table 2 reports that, conditional on the monetary policy rule, removing interest rate smoothing, habit formation, and adjustment costs on investment has little effect on the speed of convergence. Reducing \(\alpha_{Y}\) to 0.125 also does not signifcantly reduce the speed of learning. But, in the polar case of \(\alpha_{Y}=0,\) \(b=0.27\) and learning is rapid (\(T_{2/3}\) is about one year). This extreme result depends on habit formation. If we remove this feature from the model, \(b\) is equal to 0.55, essentially the same value as in the simple NK model with \(\alpha_{Y}=0\) and sticky wages and prices.

Table 2: \(b\) in variants of the CEE (2005) model with Calvo-style nominal rigidities

Note: \(b\) corresponds to the largest real part of the eigenvalues of the learning model, as described in section 3, \(T_{2/3}\) is the approximate time it takes to close \(\frac{2}{3}\) of the initial gap in beliefs using \(t^{b-1}\) as an approximation for \(\left\Vert m_{t}\right\Vert\), \(\hat{T}_{2,3}\) is the amount of time it takes to close \(\frac{2}{3}\) of the initial gap in beliefs in a simulation of the model. To implement no sticky wages, no sticky prices, no interest rate smoothing, no habit formation, and and no adjustment costs on investment, we divide the benchmark values of the relevant parameters by 100. To calculate \(\hat{T}_{2/3}\), we set \(\gamma_{1}=0.5\) and simulate the model using \(\gamma_{t}=\gamma_{t-1}\left(1-\gamma_{t-1}\right)\). The latter is the second-order approximation to \(\gamma_{t}=\frac{1}{1+t}\).

Source: Authors’ calculations.

6 Robustness of the Analysis

In this section, we assess the robustness of our conclusions to various perturbations of the benchmark analysis. The key perturbations involve model economies in which the ZLB is binding. The ZLB is of particular interest because it allows us to analyze the role of the Taylor principle in governing the speed of learning. For computational reasons, we adopt Rotemberg, 1982-style rather than Calvo, 1983-style nominal rigidities. Subsection 6.1 assesses the robustness of our results for the benchmark NK model to this change. In subsection 6.2 we analyze the speed of learning in the simple NK model when the ZLB is binding. In subsection [subsec:zlblin] we compute \(b\) using a linear approximation to the model. In subsection 6.2.2 we redo the analysis of subsection [subsec:zlblin] in the fully nonlinear version of the simple NK model. Subsection 6.2.3 assesses the robustness of the results of section 6.2.2 to assuming that learning is internally consistent in the sense of Adam and Marcet, 2011. We find that our key conclusions about the speed of learning are robust to the perturbations we consider.

6.1 Rotemberg-Style Nominal Rigidities

Under RE, Calvo-style and Rotemberg-style nominal rigidities are equivalent to a first-order approximation (see, for example, Born and Pfeifer, 2020). However, they need not be equivalent in a learning equilibrium (see Definition 4). We choose adjustment cost parameters so that the model with Rotemberg-style nominal rigidities is equivalent to a first-order approximation, in an REE, to the model with Calvo-style nominal rigidities. Wherever possible, we maintain the notation used in Sections 4 and 5.

For the simple NK model with Rotemberg-style nominal rigidities, each household is a monopoly provider of a differentiated labor input, \(N_{h,t}\), that is aggregated by competitive firms using the technology \(N_{t}=\left(\int_{0}^{1}N_{h,t}^{\frac{\epsilon-1}{\epsilon}}dh\right)^{\frac{\epsilon}{\epsilon-1}}.\) Households choose \(C_{h,t}\), \(b_{h,t}\), and \(w_{h,t}\) to maximize \[\mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\left\{ \log\left(C_{h,t+j}\right)-\frac{\chi}{1+\zeta}N_{h,t+j}^{1+\zeta}-\frac{\Phi_{W}}{2}\left(\frac{w_{h,t}}{w_{h,t-1}}\pi_{t}-1\right)^{2}\right\}\] subject to their budget constraints \[C_{h,t}+\frac{b_{h,t}}{R_{t}}\leq\frac{b_{h,t-1}}{\pi_{t}}+\left(1+\tau_{w}\right)w_{h,t}N_{h,t}-\tau_{t}\] and demand curves \(N_{h,t}=\left(\frac{W_{h,t}}{W_{t}}\right)^{-\epsilon}N_{t}\). Because households are identical across \(h\), they all pick the same values for \(C_{h,t}\), \(b_{h,t}\), and \(w_{h,t}\).²⁹

The intermediate goods firms set their prices, \(\tilde{P}_{f,t}\), to maximize \[\mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\Lambda_{t+j}\left\{ \left(\frac{P_{f,t+j}}{P_{t+j}}-\left(1-\tau_{f}\right)\frac{W_{t+j}}{P_{t+j}}\right)\left(\frac{P_{f,t}}{P_{t+j}}\right)^{-\varepsilon}Y_{t+j}-\frac{\Phi_{P}}{2}\left(\frac{P_{f,t}}{P_{f,t-1}}-1\right)^{2}\left(C_{t+j}+G_{t+j}\right)\right\} .\] Because firms are identical across \(f\), they all pick the same value of \(\tilde{P}_{f,t}\). Goods market clearing implies \(Y_{t}=\left(C_{t}+G_{t}\right)\left(1+\frac{\Phi_{P}}{2}\left(\pi_{t}-1\right)^{2}\right)\). The rest of the model is the same as the one in section 4.1 with \(p_{t}^{\ast}=1\) and price adjustment costs subtracted from firm profits. Beliefs are updated as in section [sec:solving], with \(\Theta_{t-1}=m_{t-1}=\left[\begin{array}{ccc} m_{C,t-1}, & m_{\pi,t-1}, & m_{w,t-1}\end{array}\right]^{\prime}.\)

We implement Rotemberg-style adjustment costs in the CEE model in a similar way. Table 3 reports the values of \(b\) in variants of the simple NK model and CEE model with Calvo-style nominal rigidities and in the model with Rotemberg-style nominal rigidities. The value of \(b\) for the two different ways of modeling nominal rigidities is very similar.

6.2 The ZLB

In this subsection, we analyze the speed of learning when the ZLB is binding. Up to the assumption of RE, the environment is the same as the one considered in Eggertsson and Woodford, 2003. To facilitate comparison with most of the related literature, we focus on the case of sticky prices and flexible wages, \(\Phi_{W}=0\). We begin by analyzing the properties of a linear approximation to the model’s solution at the ZLB.

Monetary policy sets the gross nominal interest rate, \(R_{t}\), according to \[\begin{equation} R_{t}=\max\left\{ 1,\beta^{-1}+\alpha_{\pi}\left(\pi_{t}-1\right)\right\} ,\tag{20}\end{equation}\] where \(\alpha_{\pi}\beta>1\), and the max operator reflects the ZLB constraint.

In period \(t\), households and firms discount next period’s utility by \(1/\left(1+r_{t}\right)\). In steady state, \(r_{t}=r^{ss}>0\). We assume that, until period 0, the economy has been in the non-stochastic steady state REE in which the nominal interest rate is positive. At the end of period 0, unexpectedly, people know that everyone’s discount rate has dropped, \(r_{1}=r^{\ell}<r^{ss}\). People correctly understand that \(r_{t+1}\) is drawn from a two-state Markov chain: \[\begin{align} \Pr\left[r_{t+1}=r^{\ell}|r_{t}=r^{\ell}\right] & =p,\text{ }\Pr\left[r_{t+1}=r^{ss}|r_{t}=r^{\ell}\right]=1-p,\tag{21}\\ \text{ }\Pr\left[r_{t+1}=r^{\ell}|r_{t}=r^{ss}\right] & =0,\nonumber \end{align}\] for \(t=1,2,....\) . We assume that people know that when \(r_{t}\) reverts to \(r^{ss}\), the economy returns to the steady state REE.³⁰ This assumption allows us to focus attention on learning while the ZLB is binding.

We assume that, when people are confronted with the unprecedented drop in \(r_{t}\), they become very uncertain about the distribution of aggregate variables. Arguably, it is reasonable to assume that the priors of their means are centered on the historical values of those variables (i.e., \(\pi^{ss}\) and \(C^{ss}\)). While prior variances were degenerate before the shock to \(r_{t}\), they are large when people become aware of the drop in \(r\). With this specification of initial beliefs, we in effect focus on the real-time evolution of beliefs instead of only focusing on asymptotics.

In the applications below, we use the same parameter values as in section 6.1. In addition, we specify the two new parameters, \(p\) and \(r^{\ell}\), to be \(p=0.8\) and \(r^{\ell}=-0.0015\).³¹ In the \(R>1\) steady-state REE, \(C^{ss}=0.8\), \(\pi^{ss}=1\), \(N^{ss}=1\). We assume that the initial precision of the priors is \(\lambda=1\). Appendix E describes the details of how we solve the learning model using a second-order approximation method, which involves a first-order approximation to the solution of the nonlinear NK model.

6.2.1 Speed of Convergence Using the Second-Order Approximation Solution Method

Under RE, the ZLB equilibrium is characterized by constant values of consumption, \(C^{\ell}\), and inflation, \(\pi^{\ell}\). The model has multiple equilibria when \(r_{t}=r^{\ell}\). Here, we focus on the unique learnable ZLB equilibrium and linearize the model around that equilibrium.³²

Our assumption about the mean of people’s priors, \(m_{0},\) at the end of period 0 implies that \[m_{0}=[m_{0,C},m_{0,\pi}]=\left[\log\left(C^{ss}/C^{\ell}\right),\log\left(\pi^{ss}/\pi^{\ell}\right)\right].\] Our parameterized model implies that \(b=0.92\). It turns that \(m_{0}\) is roughly proportional to the eigenvector associated with \(b\).

Our key result is that the learning economy converges very slowly when the ZLB is binding: \(T_{2/3}=1.7\) million and \(\widehat{T}_{2/3}=2.6\) million. The reason is that, absent a policy response to inflation in the ZLB, beliefs about expected inflation are very self-confirming. To understand why, suppose people’s prior at the end of \(t-1\) is that inflation will be low in period \(t\), causing firms to want to lower prices in period \(t\). When the ZLB binds, the nominal interest rate can’t adjust in the current period to prevent the fall in inflation. Consequently, expectations are more self-fulfilling, and people only adjust their beliefs very slowly.

6.2.2 Speed of Convergence in a Fully Nonlinear Model

In this subsection, we assess the robustness of our results to using a fully non-linear solution of the simple NK model. In the model, macroeconomic aggregates move by large amounts when the ZLB is binding. This fact makes the ZLB environment particularly useful for assessing the importance of nonlinearities.

As above, households and firms form beliefs about \(x_{t}=\left[\log\left(C_{t}\right),\log\left(\pi_{t}\right)\right]'\). While \(r_{t}=r^{\ell},\) people believe that \[\begin{equation} \left[\begin{array}{c} \log\left(C_{t}\right)\\ \log\left(\pi_{t}\right) \end{array}\right]=\left[\begin{array}{c} \mu_{C}\\ \mu_{\pi} \end{array}\right]+\left[\begin{array}{c} \nu_{C,t}\\ \nu_{\pi,t} \end{array}\right]\tag{22}\end{equation}\] where \(\nu_{C,t}\sim N\left(0,\sigma_{C}^{2}\right)\), and \(\nu_{\pi,t}\sim N\left(0,\sigma_{\pi}^{2}\right)\). These distributions are independent across time. The above expression is a special case of equation (14).

Our baseline assumption is that when people are confronted with an unprecedented observation, they become very uncertain about how market-determined variables will evolve. Their prior about \(\mu_{i}\) conditional on \(\sigma_{i}^{2}\) is Normal, parameterized with a mean, \(m_{i}\), and variance, \(\sigma_{i}^{2}/\lambda_{i}\), where \(\lambda_{i}\) characterizes the precision of the prior about \(\mu_{i}\). The marginal density of their prior for \(\sigma_{i}^{2}\) is proportional to an inverse-gamma distribution, with shape and scale parameters, \(\alpha_{i}\) and \(\left(\psi_{i}^{2}\left(\alpha_{i}+\frac{1}{2}\right)\right)\), respectively (see Appendix F2.1 for further details). The prior for \(\sigma_{i}^{2}\) is not exactly an inverse-gamma distribution because we truncate the support of \(\sigma_{i}^{2}\) so that \(\mathbf{E}_{t}\left[C_{t+1}\right]\) and \(\mathbf{E}_{t}\left[\pi_{t+1}\right]\) have finite values.³³ We find it convenient to express the scale parameter in this way because \(\psi_{i}\) is a consistent estimator for \(\sigma_{i}\). The joint density of \(\mu_{i},\sigma_{i}^{2}\) is proportional to the Normal inverse-gamma distribution. We collect the prior parameters in the vector \(\Theta_{0}\).

Under anticipated utility, at time \(t\) people set the values of \(\mu_{C}\) and \(\mu_{\pi}\) equal to the elements of \(m_{t-1}=\left[m_{C,t-1},m_{\pi,t-1}\right]'.\) Similarly, they set \(\sigma_{C}^{2}\) and \(\sigma_{\pi}^{2}\) to \(\psi_{C,t-1}^{2}\), and \(\psi_{\pi,t-1}^{2},\)respectively. In making their time-\(t\) decisions, people behave as though they know with certainty that those parameters will remain unchanged in the future.

We set the initial value of \(\Theta_{0}\) as follows: \[\begin{align*} \Theta_{0}=\left(\begin{array}{cccccccc} m_{C,0}, & m_{\pi,0}, & 1/\lambda_{C}, & 1/\lambda_{\pi}, & \psi_{C}, & \psi_{\pi}, & \alpha_{C}, & \alpha_{\pi}\end{array}\right)^{\prime}\\ =\left(\begin{array}{cccccccc} \log\left(C^{ss}\right), & \log\left(\pi^{ss}\right), & 1, & 1, & 0.02, & 0.02, & 1/2, & 1/2\end{array}\right)^{\prime} \end{align*}\] These priors are diffuse in two senses. First, if \(\psi_{C},\psi_{\pi}\) were the standard deviation of \(\log C_{t}\) and \(\log\pi_{t},\) then the distribution of \(\log C_{t}\) and \(\log\pi_{t}\) would have a quarterly standard deviation of two percent. Second, the assumed values \(\left(\begin{array}{cccc} 1/\lambda_{C} & 1/\lambda_{\pi} & 1/\alpha_{C} & 1/\alpha_{\pi}\end{array}\right)\) imply low precision in people’s beliefs about both the mean and standard deviation of \(\log\left(C_{t}\right)\) and \(\log\left(\pi_{t}\right)\), in the sense that people’s beliefs are relatively responsive to new data. The posterior distribution is also proportional to the Normal inverse-gamma distribution, and the function, \(L\), in equation (15) can be constructed using standard updating formulas (see Appendix F for a detailed description of our computational approach to solving the nonlinear model).

The solid and dashed lines in Figure 1a display the evolution of inflation, consumption, and the real interest rate after the drop in \(r\) under REE and learning, respectively. In this figure, we assume that \(r_{t}=r^{\ell}\) throughout the period shown. Of course, after a few quarters, this is a very low probability event. Nevertheless, it is useful to consider how the economy would evolve in this event to analyze the speed of convergence at the ZLB. Throughout the simulation, people behave as though \(r_{t}\) will revert to \(r^{ss}\) with probability \(1-p\).

Two key features of Figure 1a are worth noting. First, the outcomes for consumption and inflation are very different in the REE and the learning equilibria. In the REE, there is a very large drop in inflation and consumption, and the real interest rate rises sharply. The fall in inflation and consumption and the rise in the real rate are much smaller under learning. Second, the learning equilibrium converges very slowly to the REE. People initially change their views somewhat quickly because the small values of \(\lambda_{i}\) and \(\alpha_{i}\) for \(i\in\left\{ C,\pi\right\}\) imply that \(\gamma_{1}\) in equation (15) is relatively large. The rate at which they change their views slows dramatically because \(\gamma_{t}\) becomes smaller as time passes (see Figure 1b).

Figure 1: Learning equilibrium in non-linear model
(a) Outcomes
(b) Beliefs
Note: Sub-figure (a) shows outcomes for inflation, consumption, and the real interest rate under learning and in an REE when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities. Sub-figure (b) shows the values of \(\mathbf{E}_{t}\log\left(\pi_{t+1}\right)\) and \(\mathbf{E}_{t}\log\left(C_{t+1}/C^{ss}\right)\) over the course of the simulation.
Source: Authors’ calculations.

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The dot labeled \(t=10,000\) in figure 1b displays people’s views about the variables after 10,000 quarters. Given our value, \(p=0.8\), \(r\) is expected to be low for only about 5 quarters. The probability of staying at the ZLB for \(10,000\) quarters is \(p^{10,000}\approx0\). So, the fact that the system converges to the REE is essentially irrelevant because the ZLB is likely to be over well before that time. The crucial point is that in a typical ZLB episode, the associated REE is very different from the learning equilibrium for an enormous amount of time. That is, our results about slow convergence when the ZLB is binding are consistent with those that we obtained with the hybrid log-linear approximation to the model’s solution in section 6.2.

6.2.3 Speed of Convergence with Internalized Learning in a Fully Nonlinear Model

In this subsection, we examine the robustness of our result to the assumption that people make decisions based on internalized learning. For comparability with the results in the previous subsection, we again work with the simple NK model when the ZLB is binding.

Recall that anticipated utility differs from internalized learning in three ways. First, under anticipated utility, people do not update their beliefs based on current-period data until after they have made their decisions in the current period. Second, under anticipated utility, people ignore the fact that in the future they will update their views about the distribution of unknown parameters as new data are observed. Third, under anticipated utility, people ignore their uncertainty about the mean and variance of unknown parameters. In internalized learning, people observe aggregate outcomes, then update their beliefs and make their choices. In solving their problems, they fully internalize that they will update their beliefs in response to new data. See Appendix F for a description of our computational approach to solving the model under internalized learning.

Figure 2: Anticipated utility versus internalized learning


Note: The figure shows outcomes for inflation, consumption, and the real interest rate under anticipated utility and under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities. Notice that the dashed black lines, labeled “Anticipated utility,” in this figure are the same as the dashed black lines in Figure 1(a).
Source: Authors’ calculations.

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Figure 2 compares the evolution of the learning equilibrium under anticipated utility (dashed black line) and internalized learning (dash-dot blue line). The figure also includes the REE outcomes. The key takeaway is that we obtain the same slow-learning result regardless of which approach we take to decision-making. However, consumption and inflation initially fall somewhat more under internalized learning. The reason is that under anticipated utility people do not update their beliefs in the initial period in response to low consumption and inflation until after they have made their decisions. In internalized learning, people update their beliefs before making decisions, leading them to expect lower future inflation and consumption. Because the ZLB is binding, the real interest rate is higher, leading to even lower inflation and consumption. In any event, consumption, inflation, and the real interest rate are persistently very different than the REE outcomes. So, our results on slow convergence when the ZLB is binding are robust to the use of internalized learning.

7 Does Learning Matter for Policy?

In this section, we discuss how slow learning affects the implications of monetary and fiscal policy. It is well known that the policy implications of the ZLB are dramatic in REE. For example, Eggertsson, 2010 and Christiano et al., 2011 argued that the size of the government purchases multiplier is much larger when the ZLB is binding. Additionally, Del Negro et al., 2023 and others have argued that forward guidance about the nominal interest rate is particularly effective when the ZLB is binding. They also emphasize the forward guidance puzzle, according to which a promise to change the nominal interest rate far in the future can have nearly the same effect as changing the nominal interest rate in the current period. Because of these dramatic policy implications, we use the non-linear solution to the simple NK model when the ZLB is binding, and people use internalize learning in their decision-making. As it turns out, similar conclusions obtain if we work with anticipated utility.

7.1 The Government Purchases Multiplier

In this subsection, we compare the efficacy of fiscal policy under RE and internalized learning. We compute the government purchases multiplier by considering the effect on GDP, \(C_{t}+G_{t},\) of a 5 percent rise in government purchases relative to its steady-state level while \(r_{t}=r^{\ell}\). We define the multiplier in the ZLB at time \(t+j\) as \[\begin{equation} \frac{\varDelta C_{t+j}+\varDelta G_{t+j}}{\varDelta G_{t+j}}.\tag{23}\end{equation}\] Here, \(\Delta G_{t+j}=0.05\times G\) and \(\Delta C_{t+j}\) is the difference between consumption in the equilibrium when \(G_{t+j}\) is high relative to the steady state value of \(G\).

In the REE, the multiplier in the ZLB for all \(j\geq0\) is equal to 3.95 (see figure 3). The multiplier is large when the ZLB is binding because a rise in G raises inflation and expected inflation. Because \(R_{t}\) is constrained by the ZLB, this rise leads to a fall in the real interest rate and a rise in \(C_{t}\). So, in this case, the multiplier is bigger than unity.

Figure 3: Equilibria with and without an increase in G_t
(a) Outcomes during ZLB
(b) G multiplier during ZLB
Note: Sub-figure (a) shows outcomes for inflation, consumption, and the real interest rate under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)) and \(G_{t}\) is either high or at it steady state value while \(r_{t}=r^{\ell}\). Sub-figure (b) shows the value of the multiplier defined in equation (23). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities.
Source: Authors’ calculations.

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Under learning, expected inflation is partially backward-looking and does not move much with a rise in \(G_{t}\). As a result, the real interest rate falls only slightly, and the response in consumption is small. Figure 3b displays the value of the multiplier over time in the REE and under learning in the ZLB. Because learning is slow, the multiplier is close to \(1\) over the 20 quarters displayed.³⁴

7.2 Forward Guidance About \(R_{t}\)

Under forward guidance, the monetary authority commits to keeping the nominal interest rate at the ZLB for \(J\) periods after \(r_{t}\) has returned to its steady-state level. To make our point as simply as possible, we consider the case \(J=1\).

Figure 4: Forward guidance under learning and in the REE
Note: The figure shows inflation, consumption, and the real interest rate paths under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)) with and without forward guidance about \(R_{t}\). Under forward guidance, the monetary authority credibly promising to set \(R_{t}=1\) for one period after \(r_{t}=r^{\ell}\). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities.
Source: Authors’ calculations.

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Forward guidance takes the following form. In period \(I\), \(r_{t}\) reverts to \(r^{ss}\) (\(I\) is stochastic) and \(R_{t}=1\). In all earlier periods, when \(r_{t}=r^{\ell}\), people know that in period \(I\), \(C_{t}\) and \(\pi_{t}\) are equal to their REE values, \(C^{I}\) and \(\pi^{I}\). People also know that, in period \(I+1\), the economy reverts to the target-inflation REE steady state. In Appendix G we show that the number of REE proliferates under forward guidance. We study the learning equilibrium during the period when \(r_{t}=r^{\ell}\) using the same initial values for \(\Theta_{0}\) as in section 6.2.2.³⁵

Figure 4 shows that the learning equilibrium under forward guidance is very similar to the learning equilibrium without forward guidance. The reason is that forward guidance has small effects on inflation and consumption in the period \(I\) (the period when \(r_{t}\) reverts to \(r^{ss}\)). Those small effects have little impact on the learning equilibrium in which beliefs are partially backward-looking. Clearly, there is no forward guidance puzzle under learning.³⁶ That puzzle emerges under RE because of the strong effect of forward guidance on expected inflation. Under learning, expectations are backward-looking, and forward guidance has little influence on expected inflation. Moreover, the small effects of forward guidance persist for many quarters because learning is slow.

8 Conclusion

In this paper, we consider the speed with which people learn about their environment. To characterize the speed of convergence of people’s beliefs, we analytically extend results in the literature to characterize the asymptotic convergence rate of nonstochastic, multivariate systems. We then argue that the slow convergence result emerges naturally in empirically plausible NK models, including canonical DSGE models similar to CEE (2005). When learning is slow, analyses of fiscal and monetary policies under RE can be very misleading.

We leave several promising and related topics for future research, including: connecting our results to the literature on Bayesian REE, investigating how heterogeneity may affect the speed of learning, and analyzing how survey data might be used to inform estimates of model parameters by empirically disciplining the speed of learning.

A Proof of Proposition 1

Proposition 1 is an immediate result of two lemmas that we prove in this appendix. We assume that \(\theta_{t}\) evolves according to equation (1), and we maintain Assumptions 1 and 2 in the text. Recall that Assumption 1 implies that the largest real part of the eigenvalues of \(D_{1}M\left(0,0\right)\), which we denote by \(b\), is strictly less than unity. We also state the lemmas in terms of any neighborhood of the origin, \(U\), that has the property that every \(\theta_{0}\in U\) generates a sequence \(\left\Vert \theta_{t}\right\Vert\) that converges to zero. Proposition 1 in Evans and Honkapohja, 2000 establishes that such a \(U\) exists. Our first lemma establishes that for every \(\theta_{0}\in U\) the implied sequence, \(\theta_{t},\) satisfies part (1) of Definition 1. Our second lemma establishes that there exist uncountably many \(\theta_{0}\in U\) that imply a sequence, \(\theta_{t},\) that satisfies part (2) of Definition 1. The general strategy of the proofs of Lemmas 1 and 2 is similar to the strategy of the proofs of Propositions 1 and 2, respectively, in Evans and Honkapohja, 2000.

A1 Lemma 1

A2 Lemma 2

B Linearization Under Anticipated Utility and Learning About Intercepts

Here we describe our strategy for linearizing models under anticipated utility and learning. This strategy applies to our simple NK model and to our variant of the CEE (2005) model. To fix notation, if a variable, \(x_{t}\) has a steady state value \(x\) that is different from zero, then we define \(\widehat{x}_{t}=\log(x_{t}/x)\). Otherwise, we define \(\widehat{x}_{t}=x_{t}\).

For a given model, we assume that there is a bounded linear REE in which \[\begin{align} \widehat{a}_{t+j}=\Theta_{1,RE}\widehat{a}_{t+j-1}+\Theta_{\epsilon,RE}\epsilon_{t+j}\tag{33}\end{align}\] for all \(j\geq0\). Here, \(\widehat{a}_{t+j}\) is a vector (length \(n_{a}\)) of aggregate variables in (log-)deviation from steady state and \(\epsilon_{t+j}\) is a vector (length \(n_{\epsilon}\), possibly empty) of standard Normal structural shocks. In equation (33), and throughout this appendix, we use notation similar to that used in Sims, 2001. All matrices are assumed to be conformable.

In our learning model, people have beliefs about a vector (length \(n_{x}\)) of aggregate variables, \(x_{t+j}\subseteq a_{t+j}\). We assume that those beliefs are of the form \[\begin{align} \widehat{x}_{t+j}=c_{x}+\Gamma_{1,x}\widehat{a}_{t+j-1}+\Psi_{x,\epsilon}\epsilon_{t+j}+\Psi_{x,\nu}\nu_{t+j}.\tag{34}\end{align}\] People are uncertain about \(c_{x}\) and \(\Psi_{x,\nu}\). We assume that people know \(\Gamma_{1,x}\) and \(\Psi_{x,\epsilon}\) to be their REE values, consistent with equation (33). Agents also think that there is another vector (length \(n_{x}\)) of shocks, \(\nu_{t+j}\), that could affect \(\widehat{x}_{t+j}\). Agents believe that \(\nu_{t+j}\) is a vector of independent standard Normal random variables. We assume that the matrix \(\Psi_{x,\nu}\) is diagonal, reflecting the belief that the elements of \(\nu_{t+j}\) only affect the associated element of \(\widehat{x}_{t+j}\). In an REE each element \(c_{x}\) and \(\Psi_{x,\nu}\) is zero. Note that the elements of \(\nu_{t+j}\) do not appear in the dynamics of \(\widehat{a}_{t+j}\) in equation (33).

Under the anticipated utility assumption, agents believe with certainty that \[\begin{align} \widehat{x}_{t+j}=c_{x,t-1}+\Gamma_{1,x}\widehat{a}_{t+j-1}+\Psi_{x,\epsilon}\epsilon_{t+j}+\Psi_{x,\nu,t-1}\nu_{t+j}.\tag{35}\end{align}\] for some vector \(c_{x,t-1}\) (length \(n_{x}\)) and matrix \(\Psi_{x,\nu,t-1}\) (size \(n_{x}\times n_{x}\)). Additionally, we assume that agents know \(n_{a}-n_{x}\) relationships that allow them to uniquely determine \(a_{t+j}\) given \(x_{t+j}\) through the relationship \(a_{t+j}=F(x_{t+j},\epsilon_{t+j},a_{t+j-1})\), and that these relationships must hold in each period. Here, \(F\) embeds the associated identities for the elements of \(x_{t+j}\). The (log-)linear version of this mapping is given by \[\begin{align} \widehat{a}_{t+j}=\widehat{F}\left[\begin{array}{ccc} \hat{x}_{t+j}^{\prime}, & \epsilon_{t+j}^{\prime}, & \hat{a}_{t+j-1}^{\prime}\end{array}\right]^{\prime},\tag{36}\end{align}\] where \(\hat{F}\) is a conformable matrix.

In the model agents make a number of choices conditional on aggregate states and shocks. For example, in the simple NK model, households choose household-specific bond holdings that they contemplate being different from zero. We gather the agent-specific variables into the vector \(h_{t+j}\) (length \(n_{h}\)). We specify \(h_{t+j}\) so that it includes the versions of each element of \(x_{t+j}\) that would be implied by the agent-specific choices. An example of an element of \(h_{t+j}\) corresponding to an aggregate quantity in \(a_{t+j}\) in the simple NK model is household-specific consumption (in \(h_{t+j}\)) corresponding to aggregate consumption (in \(x_{t+j}\)).

After (log-)linearization, we assume agent-specific optimality conditions and budget constraints can be written as \(n_{h}\) equations of the form \[\begin{align} \Gamma_{0,h}\widehat{h}_{t+j}+\Gamma_{0,e}\widehat{e}_{t+j}+\Gamma_{0,a}\widehat{a}_{t+j}= & \Gamma_{1,h}\widehat{h}_{t+j-1}+\Gamma_{1,ha}\widehat{a}_{t+j-1}+\Gamma_{1,h\epsilon}\epsilon_{t+j}.\tag{37}\end{align}\] Here, the vector \(\widehat{e}_{t+j}=[\mathbf{E}_{t+j}\widehat{h}_{t+j+1}^{\prime},\mathbf{E}_{t+j}\widehat{a}_{t+j+1}^{\prime}]^{\prime}\) and \(\mathbf{E}_{t+j}\) is the expectation operator conditional on agents’ beliefs and the information they will have at time \(t+j\). Those beliefs are given by equation (35). Note that we assume that the model does not contain any agent-specific shocks in the sense that all decisions makers are always solving the same problem (though they do not know that). We emphasize that all of the matrices in equation (37) take the values they would in an REE. Deviations from rational expectations only appear through the coefficients in equation (35). Defining the vector \(\eta_{t+j}=[(\widehat{h}_{t+j}-\mathbf{E}_{t+j-1}\widehat{h}_{t+j})^{\prime},(\widehat{a}_{t+j}-\mathbf{E}_{t+j-1}\widehat{a}_{t+j})^{\prime}]^{\prime}\), we also need to impose the identity \[\begin{align} \eta_{t+j}=\begin{bmatrix}\widehat{h}_{t+j}, & \widehat{a}_{t+j}\end{bmatrix}^{\prime}-\widehat{e}_{t+j-1}\tag{38}\end{align}\]

Letting \(\widehat{y}_{t+j}=[\widehat{h}_{t+j}^{\prime},\widehat{a}_{t+j}^{\prime},\widehat{e}_{t+j}^{\prime}]^{\prime}\) (which has length \(2\left(n_{h}+n_{a}\right)\)), the \(2\left(n_{h}+n_{a}\right)\) equations in (35)-(38) can be written as \[\begin{align} \Gamma_{0}\widehat{y}_{t+j}=\Gamma_{1}\widehat{y}_{t+j-1}+Cc_{x,t-1}+\Psi_{\epsilon}\epsilon_{t+j}+\Psi_{\nu}\Psi_{x,\nu,t-1}\nu_{t+j}+\Pi\eta_{t+j}.\tag{39}\end{align}\] where the matrices without subscripts are independent of the values of \(c_{x,t-1}\) and \(\Psi_{x,\nu,t-1}\) and are equal to their values in an REE. Letting \(z_{t+j}=[\epsilon_{t+j}^{\prime},\nu_{t+j}^{\prime}]^{\prime}\), this is exactly the form of the system studied by Sims, 2001. The solution to this system described in Sims, 2001 ensures that \(\mathbf{E}_{t+j}\eta_{t+j+1}=0\). The analysis in Sims, 2001 is focused on RE models. It is worth stressing that the methodology we outline here could be used to compute decision rules for \(h_{t+j}\) in an REE by setting \(c_{x,t-1}\) and \(\Psi_{x,\nu,t-1}\) to be their REE values.

Under the conditions discussed in Sims, 2001, there is a unique solution to the system of equations in (39) that can be expressed as \[\begin{align} \widehat{y}_{t+j}=\Theta_{1}\widehat{y}_{t+j-1}+\Theta_{C}c_{x,t-1}+\Theta_{\epsilon}\epsilon_{t+j}+\Theta_{\nu}\Psi_{\nu}\Psi_{x,\nu,t-1}\nu_{t+j}.\tag{40}\end{align}\] This system represents how people believe they will act conditional on their beliefs. The time \(j=0\) version of this equation represents how they will act in period \(t\).

Importantly, equation (40) contains the mapping \[\begin{align} \widehat{h}_{t}=\Theta_{1,h}\widehat{y}_{t-1}+\Theta_{C,h}c_{x,t-1}+\Theta_{h,\epsilon}\epsilon_{t}+\Theta_{h,\nu}\Psi_{\nu}\Psi_{x,\nu,t-1}\nu_{t}\tag{41}\end{align}\] that defines decision rules for \(\widehat{h}_{t}\) that are optimal from the perspective of the agents in the model, given \(\widehat{y}_{t-1}\), \(\epsilon_{t}\), and \(z_{t}\). It follows from equation (43)-(45) in Sims, 2001 that because \(\Gamma_{0}\), \(\Gamma_{1}\), \(\Psi_{\epsilon}\), and \(\Pi\) in equation (39), are equal to their values used when computing an REE it must be that \(\Theta_{1,h}\), \(\Theta_{C,h}\), and \(\Theta_{h,\epsilon}\) are also equal to their values in an REE. So, we do not include a time subscript on these matrices.³⁹ Unlike in an REE, the system contains perceived shocks, \(\nu_{t}\). In the period equilibrium, the values of \(\nu_{t}\) are determined by the requirement that markets clear, which we discuss below.

In a learning equilibrium, we first note that the symmetry of the agents’ problems implies that any agent-specific values up to period \(t-1\) are equal to their aggregate quantities. So, we can write equation (41) as \[\begin{align} \widehat{h}_{t}=\Theta_{1,ha}\widehat{a}_{t-1}+\Theta_{C,h}c_{x,t-1}+\Theta_{h,\epsilon}\epsilon_{t}+\Theta_{h,\nu}\Psi_{\nu}\Psi_{x,\nu,t-1}\nu_{t}.\tag{42}\end{align}\] The vector \(\widehat{e}_{t-1}\) does not appear on the right-hand side of equation (42) because in the solution to the linear model its terms are are a linear combination of \(\nu_{t}\) and \(\epsilon_{t}\).

To construct a period equilibrium, we impose the restrictions of market clearing. Those restrictions take the form that the elements of \(\widehat{x}_{t}\) must be equal to corresponding elements of \(\widehat{h}_{t}\). An example of such a restriction in the simple NK model is that household-specific consumption must equal aggregate consumption. We have \(n_{x}\) such restriction, which is the length of \(\nu_{t}\). We write these restrictions as \[\begin{align} \mathcal{R}\widehat{h}_{t}= & \widehat{x}_{t}.\tag{43}\end{align}\] Here, \(\mathcal{R}\) is a matrix of ones and zeros. In a period equilibrium, equation (43) will hold. From the perspective of the agents, we can use equations (35), (42), and (43) to obtain \[\begin{align} \left(\mathcal{R}\Theta_{h,\nu}\Psi_{\nu}-I\right)\Psi_{x,\nu,t-1}\nu_{t}= & c_{x,t-1}-\mathcal{R}\Theta_{C,h}c_{x,t-1}.\tag{44}\end{align}\] If \(\left(\mathcal{R}\Theta_{h,\nu}\Psi_{\nu}-I\right)\) is invertible, then the equilibrium values of \(\Psi_{x,\nu,t-1}\nu_{t}\) can be determined uniquely. In practice we have not encountered an example in which invertibility fails. The values \(\Psi_{x,\nu,t-1}\nu_{t}\) induce agents to make decisions that lead to market clearing.

In a learning equilibrium, after agents observe \(\widehat{a}_{t}\) and \(\epsilon_{t}\) and make their decisions in the period equilibrium, they update their beliefs using Bayes rule. Recall that agents are only learning about \(c_{x}\) and \(\Psi_{x,\nu}\) in equation (34) and that the elements of \(\nu_{t}\) are perceived to be independent of one another and over time. We assume that agents beliefs are proportional to the Normal-inverse-gamma conjugate priors on each element of \(c_{x}\) and the associated element of \(\Psi_{x,\nu}\) and that they have the same precision in these priors.⁴⁰ Then \[\begin{align} c_{x,t}=c_{x,t-1}+\gamma_{t}\left(\widehat{x}_{t}-\Gamma_{1}\widehat{a}_{t-1}-\Psi_{x,\epsilon}\epsilon_{t}-c_{x,t-1}\right)=c_{x,t-1}+\gamma_{t}\Psi_{x,\nu,t-1}\nu_{t}.\tag{45}\end{align}\] Here, \(\gamma_{t}=1/(\lambda_{0}+t)\) where \(\lambda_{0}\) determines the precision of the time \(1\) prior on each element of \(c_{x}\). From equation (44), the values of \(\Psi_{x,\nu,t-1}\nu_{t}\) are do not depend on \(\widehat{a}_{t-1}\), \(\epsilon_{t}\), or \(\Psi_{x,\nu,t-1}\). Therefore, in a learning equilibrium in which \(\Gamma_{1,x}\) and \(\Psi_{x,\epsilon}\) are believed with certainty to be their REE values, the mean of agents’ beliefs about \(c_{x}\) evolves in a non-stochastic manner and \(c_{x,t}\) does not depend on \(\widehat{a}_{t-1}\). We can write the evolution of \(c_{x,t}\) as \[\begin{align} c_{x,t}=c_{x,t-1}+\gamma_{t}\left(H-I\right)c_{x,t-1}.\tag{46}\end{align}\] Equation (46) is in the form of equation (1) in the main text. Because \(M\left(\cdot,\cdot\right)\) is linear in \(c_{x,t-1}\) the smoothness assumptions stated in the text related to \(M\left(\cdot,\cdot\right)\) hold.

C A Comparison to Euler-Equation Learning

An alternative approach to learning is to start with aggregate Euler equations from the REE. This approach has been widely used.⁴¹ The Euler-equation approach to learning stands in contrast to the anticipated utility approach of Preston, 2005 in which households and firms solve their infinite horizon problems given beliefs.

C1 The Linear Approximation to the Equilibrium in the Simple NK Model with Only Sticky Prices Under Euler-Equation Learning

Consider the simple NK model with only sticky prices. The aggregate Euler equations from the REE are given by \[\begin{align*} \widehat{C}_{t} & =-\beta\alpha_{\pi}\widehat{\pi}_{t}+\mathbf{E}_{t}\left[\widehat{C}_{t+1}+\widehat{\pi}_{t+1}\right]\\ \widehat{\pi}_{t} & =\frac{2\left(\varepsilon-1\right)}{\Phi}\widehat{C}_{t}+\beta\mathbf{E}_{t}\widehat{\pi}_{t+1}. \end{align*}\] These equations hold under either Calvo-style or Rotemberg-style adjustment costs so long and \(\Phi\) is chosen accordingly and we do not treat \(p_{t}^{\ast}\) as a state variable in the Calvo model because it is always equal to 1 in a first-order approximation around \(\pi=1\). Under the assumption that people are learning about steady states (or means), the Euler equation learning approach proceeds by assuming that \(\mathbf{E}_{t}\widehat{C}_{t+1}=m_{C,t-1}\) and \(\mathbf{E}_{t}\widehat{\pi}_{t+1}=m_{\pi,t-1}\) so that \[\begin{align*} \widehat{C}_{t} & =-\beta\alpha_{\pi}\widehat{\pi}_{t}+m_{C,t-1}+m_{\pi,t-1}\\ \widehat{\pi}_{t} & =\frac{2\left(\varepsilon-1\right)}{\Phi}\widehat{C}_{t}+\beta m_{\pi,t-1}. \end{align*}\] The values of \(\widehat{\pi}_{t}\) and \(\widehat{C}_{t}\) are calculated from these equations. Beliefs are updated so that \[m_{x,t}=m_{x,t-1}+\frac{1}{\lambda_{0}+t}\left(\hat{x}_{t}-m_{x,t-1}\right).\] This updating equation (in vector form) has the same form as equation (1) in the main text of our paper. When we compute the value of \(b\) under the Euler equation learning approach, we get \(b=0.95\) and \(T_{2/3}\approx1.6\) billion. Under the learning approach we take in the text, and as reported in Table 1, when we assume households and firms solve their infinite horizon problems given beliefs we get \(b=0.26\) and \(T_{2/3}=5\). We conclude that modeling agents infinite-horizon decision problems, rather than using aggregate Euler equations from the REE, can have important implications for conclusions regarding the speed of learning.

C2 Euler-Equation Learning and the ZLB

Consider the simple NK model with only sticky prices and Rotember-style nominal rigidities when the ZLB is biding (as in Section 6.2). Let \(\tilde{X}_{t}\) denote the value of \(X_{t}\) in period \(t\) when \(r_{t}=r^{\ell}\) and the ZLB is binding. The aggregate version of the household intertemporal Euler equation is \[\frac{1}{\tilde{C}_{t}}=\frac{1}{1+r^{\ell}}\mathbf{E}_{t}\frac{p}{\tilde{\pi}_{t+1}\tilde{C}_{t+1}}+\frac{1}{1+r^{\ell}}\mathbf{E}_{t}\frac{1-p}{\pi^{ss}C^{ss}}.\] The aggregate version of the firm optimality condition is

\[\begin{align*} \left(\tilde{\pi}_{t}-1\right)\tilde{\pi}_{t}= & \frac{\varepsilon-1}{\Phi_{P}}\left[\chi\tilde{C}_{t}\left(\tilde{C}_{t}+\tilde{G}_{t}\right)\left(1+\frac{\Phi_{P}}{2}\left(\tilde{\pi}_{t}-1\right)^{2}\right)-1\right]\left(1+\frac{\Phi_{P}}{2}\left(\tilde{\pi}_{t}-1\right)^{2}\right)\\ & +\frac{p}{1+r^{\ell}}\mathbf{E}_{t}\left(\tilde{\pi}_{t+1}-1\right)\tilde{\pi}_{t+1}\left(\frac{\tilde{C}_{t+1}+\tilde{G}_{t+1}}{\tilde{C}_{t}+\tilde{G}_{t}}\right)\frac{\tilde{C}_{t}}{\tilde{C}_{t+1}}. \end{align*}\] The REE at the ZLB has no state variables. So, we can linearize these equations around the values of aggregate variables taken in the REE and use the Euler equation approach to learning. When we compute the value of \(b\) under the Euler equation learning approach, we get \(b=0.98\). This is a larger value for \(b\) than under anticipated utility. So, learning is slow using either approach when we have decreasing gain as in equation (1).

D The CEE (2005) Model with Calvo-Style Nominal Rigidities

In this appendix we detail the version of the CEE (2005) model that we use in the main text. The version of the model in this appendix has Calvo-style nominal rigidities. The simple NK model and the models with Rotemberg-style nominal rigidities involve minor modifications to these equations and use the parameter values discussed in the main text.

D1 Households

There is a continuum of households indexed by \(h\in\left(0,1\right)\) and a continuum of household members indexed by \(i\in\left(0,1\right)\). There is perfect consumption insurance within the household. Household member \(i\) provides labor of type \(i\). The wage rate and labor supply is set by a union, described below. Households maximize \[\begin{equation} \mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\left[\frac{\left(C_{h,t+j}-\gamma C_{h,t+j-1}\right)^{1-\sigma}}{1-\sigma}-\frac{\chi}{1+\zeta}\int_{0}^{1}N_{h,i,t+j}^{1+\zeta}di\right]\tag{47}\end{equation}\] by choosing \(C_{h,t}\), \(b_{h,t}\), \(u_{h,t}\), \(I_{h,t}\), and \(K_{h,t+1}\) subject to the budget constraints, for \(j\geq0\), \[\begin{align} R_{t+j}^{-1}b_{h,t+j}+\frac{\vartheta}{2}b_{h,t+j}^{2}+C_{h,t+j}+I_{h,t+j}+a\left(u_{h,t+j}\right)K_{h,t+j} & \leq\nonumber \\ \int_{0}^{1}\left(1+\tau_{w}\right)w_{i,t+j}N_{h,i,t+j}di & +r_{t+j}^{K}u_{h,t+j}K_{h,t+j}-\tau_{t+j}+\frac{b_{h,t+j-1}}{\pi_{t+j}}\tag{48}\end{align}\] and the capital accumulation equations \[\begin{equation} K_{h,t+j+1}=\left(1-\delta\right)K_{h,t+j}+\left(1-\frac{\phi_{I}}{2}\left(\frac{I_{h,t+j}}{I_{h,t+j-1}}-1\right)^{2}\right)I_{h,t+j}.\tag{49}\end{equation}\] If \(\sigma=1\), it is understood that the utility function for consumption takes the log form. Here, \(C_{h,t}\) is household-specific consumption, \(b_{h,t}\) are household-specific end-of-period real bond holdings, \(u_{h,t}\) is the household-specific capital utilization rate, \(I_{h,t}\) is household-specific investment, and \(K_{h,t}\) is the begining-of-period household-specific capital stock. \(R_{t}\) is the nominal interest rate on one-period nominal bonds, \(w_{i,t}\) is the wage rate paid to labor-type \(i\), \(r_{t}^{K}\) is the rental rate of capital, \(\tau_{t}\) are taxes net profits, and \(\pi_{t}\equiv P_{t}/P_{t-1}\). Note that capital and utilization provide capital services, \(u_{h,t}K_{h,t}\). The term \(a\left(u_{h,t}\right)K_{h,t}\) represents costs from utilization deviating from \(1\). In steady state, the form of \(a\left(\cdot\right)\) ensures \(u_{h,t}=1\). The first-order conditions of household \(h\) are \[\begin{align} \Lambda_{h,t}-\left(C_{h,t}-\gamma C_{h,t-1}\right)^{-\sigma}+\gamma\mathbf{E}_{t}\beta\left(C_{h,t+1}-\gamma C_{h,t}\right)^{-\sigma} & =0\tag{50}\\ \Lambda_{h,t}+\vartheta R_{t}b_{h,t}\Lambda_{h,t}-R_{t}\beta\mathbf{E}_{h,t}\frac{1}{\pi_{t+1}}\Lambda_{h,t+1} & =0\tag{51}\\ a^{\prime}\left(u_{h,t}\right)-r_{t}^{K} & =0\tag{52}\\ \Lambda_{h,t}-\left(1-\frac{\phi_{I}}{2}\left(\frac{I_{h,t}}{I_{h,t-1}}-1\right)^{2}\right)\eta_{h,t}+\phi_{I}\left(\frac{I_{h,t}}{I_{h,t-1}}-1\right)\frac{I_{h,t}}{I_{h,t-1}}\eta_{h,t}\nonumber \\ -\beta\mathbf{E}_{h,t}\phi_{I}\left(\frac{I_{h,t+1}}{I_{h,t}}-1\right)\left(\frac{I_{h,t+1}}{I_{h,t}}\right)^{2}\eta_{h,t+1} & =0\tag{53}\\ \eta_{h,t}+\beta\mathbf{E}_{t}a\left(u_{h,t+1}\right)\Lambda_{h,t+1}-\beta\mathbf{E}_{t}r_{t+1}^{K}u_{h,t+1}\Lambda_{h,t+1}-\beta\mathbf{E}_{t}\eta_{h,t+1}\left(1-\delta\right) & =0\tag{54}\end{align}\] Here, \(\Lambda_{h,t}\) is the Lagrange multiplier on the budget constraint and \(\eta_{h,t}\) is the Lagrange multiplier on the capital accumulation equation.

D2 Monopolists

Monopolist \(f\) produces a differentiated good, \(Y_{f,t}\). Goods are aggregated by competitive firms using the technology \(Y_{t}=\left(\int_{0}^{1}Y_{f,t}^{\frac{\varepsilon-1}{\varepsilon}}df\right)^{\frac{\varepsilon}{\varepsilon-1}}\). The price of \(Y_{t}\) is \(P_{t}=\left(\int_{0}^{1}P_{f,t}^{1-\varepsilon}df\right)^{\frac{1}{1-\varepsilon}}.\) Monopolists face demand curves of the form given in equation (2). Production is done so that \[\begin{equation} Y_{f,t}=N_{f,t}^{\upsilon}\left(\tilde{K}_{f,t}\right)^{1-\upsilon}\tag{55}\end{equation}\] where \(\tilde{K}_{f,t}\) are capital services purchased by firm \(f\). Notice that monopolists will solve a cost minimization problem that implies \[\begin{align} \frac{w_{t}}{r_{t}^{K}} & =\frac{\upsilon\tilde{K}_{f,t}}{\left(1-\upsilon\right)N_{f,t}}\tag{56}\\ s_{t} & =\frac{w_{t}^{\upsilon}\left(r_{t}^{K}\right)^{1-\upsilon}}{\upsilon^{\upsilon}\left(1-\upsilon\right)^{1-\upsilon}}.\tag{57}\end{align}\] Monopolists update their price with probability \(1-\xi_{P}\). When they update their price, they choose \(\tilde{P}_{f,t}\) to maximize the present-discount value of profits

\[\begin{equation} \mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\xi_{P}^{j}\Lambda_{t+j}\left(\frac{\tilde{P}_{f,t}}{P_{t+j}}-\left(1-\tau_{f}\right)s_{t+j}\right)\left(\frac{\tilde{P}_{t}}{P_{t+j}}\right)^{-\varepsilon}Y_{t+j}.\tag{58}\end{equation}\] Here \[\begin{equation} \Lambda_{t}-\left(C_{t}-\gamma C_{t-1}\right)^{-\sigma}+\gamma\mathbf{E}_{t}\beta\left(C_{t+1}-\gamma C_{t}\right)^{-\sigma}=0\tag{59}\end{equation}\] is the firm’s perception of how they should discount flows to the household, \(s_{t+j}\) is real marginal cost in period \(t+j\), and profits in period \(t+j\) are discounted by an aggregate version of the household marginal utility of consumption. Assuming \(\left(1-\tau_{f}\right)\frac{\varepsilon}{\varepsilon-1}=1\), the first-order condition is \[\begin{equation} \sum_{j=0}^{\infty}\beta^{j}\xi_{P}^{j}\mathbf{E}_{t}\Lambda_{t+j}\left(s_{t+j}-\tilde{p}_{f,t}\frac{P_{t}}{P_{t+j}}\right)\left(\frac{P_{t}}{P_{t+j}}\right)^{-\varepsilon}Y_{t+j}=0\tag{60}\end{equation}\] where \(\tilde{p}_{f,t}\equiv\tilde{P}_{f,t}/P_{t}\). Let \[\begin{align} F_{P,t} & =\sum_{j=0}^{\infty}\beta^{j}\xi_{P}^{j}\mathbf{E}_{t}\Lambda_{t+j}\left(\frac{P_{t}}{P_{t+j}}\right)^{1-\varepsilon}Y_{t+j}=\Lambda_{t}Y_{t}+\beta\xi_{P}\mathbf{E}_{t}\pi_{t+1}^{\varepsilon-1}F_{P,t+1}\tag{61}\end{align}\] Notice that we have assumed that the expectations of the firms satisfy the law of iterated expectations. Similarly, let \[\begin{align} Z_{P,t} & =\sum_{j=0}^{\infty}\beta^{j}\xi_{P}^{j}\mathbf{E}_{t}\Lambda_{t+j}\left(\frac{P_{t}}{P_{t+j}}\right)^{-\varepsilon}Y_{t+j}s_{t+j}=\Lambda_{t}Y_{t}s_{t}+\beta\xi_{P}\mathbf{E}_{t}\pi_{t+1}^{\varepsilon}Z_{P,t+1}\tag{62}\end{align}\] Then the first-order condition can be written as \[\begin{equation} \tilde{p}_{f,t}=\frac{Z_{P,t}}{F_{P,t}}.\tag{63}\end{equation}\]

D3 Unions

We assume that each household has a continuum of members indexed by \(i\in\left(0,1\right)\). Each member consumes the same amount in every period (perfect consumption insurance). The index \(i\) indicates the labor type provided by the household member. Members of type \(i\) from all of the households belong to a union that sets the wage for type-\(i\) labor. With probability \(1-\xi_{W}\) a union is able to update the wage for its type. Each union provides all labor demanded at the posted wage and demands equal amounts of labor from each union member.

Labor is combined into a productive labor unit using the technology \[N_{t}=\left(\int_{0}^{1}\left(\int_{0}^{1}N_{h,i,t}dh\right)^{\frac{\epsilon-1}{\epsilon}}di\right)^{\frac{\epsilon}{\epsilon-1}}\] . A unit of \(N_{t}\) costs \(W_{t}=\left(\int_{0}^{1}W_{i,t}^{1-\epsilon}di\right)^{\frac{1}{1-\epsilon}}\). Demand for \(N_{h,i,t}\) is given by equation (6). When a union sets its wage it does so by choosing \(\tilde{W}_{i,t}\) to maximize equation (7).

We assume unions have the same expectation operator as households and firms. Additionally, unions take the demand curve for their labor type as given. Assuming \(\frac{\epsilon-1}{\epsilon}\left(1+\tau_{w}\right)=1\), and defining \(\tilde{w}_{i,t}\equiv\tilde{W}_{i,t}/P_{t}\) and \(w_{t}\equiv W_{t}/P_{t}\), union \(i\)’s first-order condition is

\[\begin{equation} \mathbf{E}_{t}\sum_{j=0}^{\infty}\beta^{j}\xi_{W}^{j}\left[\frac{\Lambda_{t+j}P_{t}w_{t}}{P_{t+j}}\left(\frac{\tilde{w}_{i,t}}{w_{t}}\right)^{1+\epsilon\zeta}-\chi\left(\frac{w_{t}}{w_{t+j}}\frac{P_{t}}{P_{t+j}}\right)^{-\epsilon\zeta}N_{t+j}^{\zeta}\right]\left(\frac{w_{t}}{w_{t+j}}\frac{P_{t}}{P_{t+j}}\right)^{-\epsilon}N_{t+j}=0\tag{64}\end{equation}\] Define \[\begin{align} F_{W,t} & =\Lambda_{t}N_{t}w_{t}+\beta\xi_{W}\mathbf{E}_{t}\pi_{t+1}^{\epsilon-1}\left(\frac{w_{t}}{w_{t+1}}\right)^{1-\epsilon}F_{W,t+1}\tag{65}\\ Z_{W,t} & =\chi N_{t}^{1+\zeta}+\beta\xi_{W}\mathbf{E}_{t}\pi_{t+1}^{\epsilon\left(1+\zeta\right)}\left(\frac{w_{t}}{w_{t+1}}\right)^{-\epsilon\left(1+\zeta\right)}Z_{W,t+1}.\tag{66}\end{align}\] Because the law of iterated expectations holds for the subjective beliefs of unions, the first-order condition can be written as \[\begin{equation} F_{W,t}\left(\frac{\tilde{w}_{i,t}}{w_{t}}\right)^{1+\epsilon\zeta}=Z_{W,t}.\tag{67}\end{equation}\]

D4 Aggregation and Government Policy

Aggregate output is given by \[\begin{equation} Y_{t}=C_{t}+I_{t}+G_{t}+\int_{0}^{1}a\left(u_{h,t}\right)K_{h,t}dh.\tag{68}\end{equation}\] Integrating over demand curves for intermediate goods and firm-specific production functions yields \[\begin{equation} \int_{0}^{1}N_{f,t}^{\upsilon}\left(\tilde{K}_{f,t}\right)^{1-\upsilon}df=\int_{0}^{1}\left(\frac{P_{f,t}}{P_{t}}\right)^{-\varepsilon}dfY_{t}.\tag{69}\end{equation}\] Aggregate capital services are given by \[\tilde{K}_{t}=\int_{0}^{1}u_{h,t}K_{h,t}dh.\] All firms pick the same capital-to-labor ratio, so \[\begin{equation} p_{t}^{\ast}N_{t}^{\upsilon}\left(\tilde{K}_{t}\right)^{1-\upsilon}=Y_{t}.\tag{70}\end{equation}\] From the definitions of \(p_{t}^{\ast}\) and \(P_{t}\), \[\begin{equation} \left(p_{t}^{\ast}\right)^{-1}=\left(1-\xi_{P}\right)\left(\frac{1-\xi_{P}\pi_{t}^{\varepsilon-1}}{1-\xi_{P}}\right)^{-\frac{\varepsilon}{1-\varepsilon}}+\xi_{P}\pi_{t}^{\varepsilon}\left(p_{t-1}^{\ast}\right)^{-1}.\tag{71}\end{equation}\] Aggregate profits are given by \[\begin{align} \int_{0}^{1}\left(\frac{P_{f,t}}{P_{t}}-\left(1-\tau_{f}\right)s_{t}\right)\left(\frac{P_{f,t}}{P_{t}}\right)^{-\varepsilon}Y_{t}df & =\left(1-\left(1-\tau_{f}\right)\frac{s_{t}}{p_{t}^{\ast}}\right)Y_{t}.\tag{72}\end{align}\] We assume the government runs a balanced budget period by period. So, \[\begin{equation} \tau_{t}=G_{t}+\tau_{w}w_{t}N_{t}-\left(1-\frac{s_{t}}{p_{t}^{\ast}}\right)Y_{t}.\tag{73}\end{equation}\] We assume \[\begin{equation} G_{t}=G\tag{74}\end{equation}\] and \[\begin{equation} R_{t}=\left(1-\rho_{R}\right)\left(\beta^{-1}+\alpha_{\pi}\left(\pi_{t}-1\right)+\alpha_{Y}\frac{Y_{t}-Y}{Y}\right)+\rho_{R}R_{t-1}.\tag{75}\end{equation}\]

D5 REE Steady State

It is either the case that \(R=\beta^{-1}\) and \(\pi=1\) or it is the case that \(R=1\) and \(\pi=\beta\). We choose \(\chi\) so that \(N=1\) when \(\pi=1\). We also set \(G=0.2Y\) where \(Y\) is the steady state value when \(\pi=1\). The form of the function \(a\left(\cdot\right)\) ensures that \(u=1\) in steady state. See CEE (2005) for further discussion about \(a\left(\cdot\right)\).

D6 Beliefs and the Period-Equilibrium

Define \[\begin{align*} x_{t} & =\left[C_{t},\pi_{t},w_{t},I_{t}\right]^{\prime},a_{t}=\left[C_{t},\pi_{t},w_{t},I_{t},p_{t}^{\ast},R_{t},Y_{t},u_{t},s_{t},r_{t}^{K},N_{t},G_{t},K_{t+1},\tau_{t}\right]^{\prime}\\ h_{t} & =\left[C_{h,t},b_{h,t},u_{h,t},I_{h,t},K_{h,t+1},\Lambda_{h,t},\eta_{h,t},\tilde{p}_{f,t},\pi_{f,t},\tilde{w}_{i,t},w_{i,t},F_{P,t},Z_{P,t},F_{W,t},Z_{W,t},\Lambda_{t}\right]^{\prime} \end{align*}\] We include \(\Lambda_{t}\) in \(h_{t}\) because it represents firms’ and unions’ perceptions of how to discount flow to the household. We assume that agents form beliefs as in equation (35). So that the bond market clears, we impose that it must be that \(\hat{C}_{h,t}=\hat{C}_{t}\). So that the goods market clears, we to impose that firm pricing decisions are consistent with aggregate inflation. To do that, we find it useful to define \(\pi_{f,t}\) and \(p_{f,t}^{\ast}\) so that \[\begin{align} 1 & =\left(1-\xi_{P}\right)\tilde{p}_{f,t}^{1-\varepsilon}+\xi_{P}\pi_{f,t}^{\varepsilon-1}.\tag{76}\end{align}\] We also need to impose that union wage-setting decisions are consistent with the aggregate real wage. To do that, we find it useful to define \[\begin{equation} w_{i,t}^{1-\epsilon}=\left(1-\xi_{W}\right)\tilde{w}_{i,t}^{1-\epsilon}+\xi_{W}\pi_{t}^{\epsilon-1}w_{t-1}^{1-\epsilon},\tag{77}\end{equation}\] which follows from the definition of \(W_{t}\). Finally, we need to impose that \(I_{h,t}=I_{t}\) so that investment is consistent with overall aggregate output.

We write the restrictions we impose on the period equilibrium as \(C_{t}=C_{h,t}\), \(\pi_{t}=\pi_{f,t}\), \(w_{t}=w_{i,t}\), \(I_{t}=I_{h,t}\). It is now straightforward to put the model in the form discussed in Appendix B. Agents believe with certainty that the static equations  (68), (70), (71), (73), (74), and (75), hold in each period. They also believe that the following equations hold with certainty \(a^{\prime}\left(u_{t}\right)=r_{t}^{K}\), \(\frac{w_{t}}{r_{t}^{K}}=\frac{\upsilon u_{t}K_{t}}{\left(1-\upsilon\right)N_{t}}\), \(s_{t}=\frac{w_{t}^{\upsilon}\left(r_{t}^{K}\right)^{1-\upsilon}}{\upsilon^{\upsilon}\left(1-\upsilon\right)^{1-\upsilon}}\), and \(K_{t+1}=\left(1-\delta\right)K_{t}+\left(1-\frac{\phi_{I}}{2}\left(\frac{I_{t}}{I_{t-1}}-1\right)^{2}\right)I_{t}\). Here, \(K_{t}=\int_{0}^{1}K_{h,t}dh\). These equations are aggregate versions of equations that hold in every period. These equation, along with the identities from \(x_{t}\) to \(a_{t}\), define the mapping \(a_{t}=F(x_{t},a_{t-1})\) and their (log-)linear form can be expressed as in equation (36).

To construct equation (37), we use (log-)linear versions of equations (48), (49), (50), (51), (52), (53), (54), (59), (61), (62), (63), (65), (66), (67), (76), and (77).

D7 Parameter Values

Table 4: Parameter values for the CEE (2005) model

We use the following parameter values for our version of the CEE (2005) model with Calvo-style nominal rigidities.

E Linearization When the ZLB is Binding

Here, we describe how we construct the period equilibrium in the simple NK model with only sticky prices and Rotemberg-style adjustment costs when the ZLB is binding. We maintain the assumption that households and firms know the REE in the steady state. As a result, households and firms know the decision rules they will use in the event that the economy reverts to steady state.

To fix notation, we let \(\tilde{X}_{t}\) be value of the variable \(X_{t}\) when \(r_{t}=r^{\ell}\) and let \(\tilde{X}\) be its value in an REE. Let \(\widehat{\tilde{X}}_{t}=\log(\tilde{X}_{t}/\tilde{X})\). Note that we continue to use \(\widehat{b}_{h,t}\) and \(\widehat{p}_{f,t}\) at the ZLB, because the REE values of those variables are equal to the same values at the ZLB and in steady state. Households and firms form beliefs about \(\tilde{C}_{t}\) and \(\tilde{\pi}_{t}\) using \[\begin{align} \widehat{\tilde{C}}_{t}= & \tilde{m}_{C,t-1}+\tilde{\sigma}_{\nu,C,t-1}\tilde{\nu}_{C,t}\tag{78}\\ \widehat{\tilde{\pi}}_{t}= & \tilde{m}_{\pi,t-1}+\tilde{\sigma}_{\nu,\pi,t-1}\tilde{\nu}_{\pi,t}.\tag{79}\end{align}\]

Households and firms know that equations the following equations hold in every period: \(Y_{t}=N_{t}\), \(Y_{t}=\left(C_{t}+G_{t}\right)(1+\frac{\Phi}{2}(\pi_{t}-1)^{2})\), \(w_{t}=\chi N_{t}C_{t}\), \(\tau_{t}=G_{t}-(1-w_{t})Y_{t}+\frac{\Phi}{2}(\pi_{t}-1)^{2}(C_{t}+G_{t})\), \(G_{t}=G\), \(R_{t}=\max\left\{ 1,\beta^{-1}+\alpha_{\pi}\left(\pi_{t}-1\right)+\alpha_{Y}\frac{Y_{t}-Y}{Y}\right\}\). The Euler equation is \[\frac{1}{\tilde{C}_{h,t}\tilde{R}_{t}}+\vartheta\tilde{b}_{h,t}=\frac{1}{1+r_{\ell}}\mathbf{E}_{t}\frac{p}{\tilde{\pi}_{t+1}\tilde{C}_{h,t+1}}+\frac{1}{1+r_{\ell}}\mathbf{E}_{t}\frac{1-p}{\pi C_{h,t+1}\left(\tilde{b}_{h,t}\right)}.\] We emphasize that in the case that \(r_{t+1}=r^{ss}\), we will have \(\pi_{t+1}=\pi^{ss}\) and \(C_{h,t+1}\) is a known function of household bond holdings. The firm’s pricing optimality condition becomes \[\begin{align*} \left(\frac{\tilde{p}_{f,t}}{\tilde{p}_{f,t-1}}\tilde{\pi}_{t}-1\right)\tilde{\pi}_{t}\frac{\tilde{p}_{f,t}}{\tilde{p}_{f,t-1}}= & \frac{\varepsilon-1}{\Phi_{P}}\left[\tilde{w}_{t}-\tilde{p}_{f,t}\right]\tilde{p}_{f,t}^{-\varepsilon}\frac{\tilde{Y}_{t}}{\tilde{C}_{t}+\tilde{G}_{t}}\\ & +\frac{p}{1+r_{\ell}}\mathbf{E}_{t}\left(\frac{\tilde{p}_{f,t+1}}{\tilde{p}_{f,t}}\tilde{\pi}_{t+1}-1\right)\frac{\tilde{p}_{f,t+1}}{\tilde{p}_{f,t}}\tilde{\pi}_{t+1}\left(\frac{\tilde{C}_{t+1}+\tilde{G}_{t+1}}{\tilde{C}_{t}+\tilde{G}_{t}}\right)\frac{\tilde{C}_{t}}{\tilde{C}_{t+1}}\\ & +\frac{1-p}{1+r_{\ell}}\mathbf{E}_{t}\left(\frac{p_{f,t+1}}{\tilde{p}_{f,t}}\pi-1\right)\frac{p_{f,t+1}}{\tilde{p}_{f,t}}\pi\left(\frac{C+G}{\tilde{C}_{t}+\tilde{G}_{t}}\right)\frac{\tilde{C}_{t}}{C}. \end{align*}\] We can now proceed as in Appendix B to solve for the log-linear approximation to the period equilibrium. Note that we log-linearize \(\tilde{X}_{t}\) around \(\tilde{X}\). We need an approximation in which households and firms contemplate variables at the ZLB that grow slower than \(\tilde{R}^{-1}\tilde{\pi}\tilde{\beta}^{-1}p^{-1}\). Because of the probability of switching from the ZLB to the steady state, the growth condition ensures that the unconditional expectation of each variable in the system converges to zero as the horizon of the expectation goes to infinity. The growth condition is satisfied by only one solution to the linear system, and we study that solution. Note that in any learning equilibrium, \(\tilde{b}_{h,t}=b_{h,t}=0\) and \(\tilde{p}_{f,t}=p_{f,t}=1\) in every period.

F Solution Algorithm for Nonlinear NK Model When the ZLB is Binding

In this appendix we detail our solution strategy for the non-linear NK model we consider in our paper. We exploit the model’s structure to simplify its solution. In particular, because the steady state is an absorbing state for the REE and the learning equilibria that we consider, we can solve the steady state decision rules without reference to the period when \(r_{t}=r^{\ell}\). Here, we maintain the assumption that households and firms know the REE in the steady state. With this solution in hand, we then turn to the period when \(r_{t}=r^{\ell}\), which is where we consider learning. We find it useful to represent the model recursively in this appendix. This appendix explains how to solve the model under internalized learning. At the end of the appendix, we explain how to solve the model using that solution to generate results for anticipated utility.

F1 Steady State

In the steady state, there is no uncertainty. However, households still face a bond-holding choice and firms still face a relative-price choice. In an REE, households will choose to hold zero bonds and firms will choose to set their price to the aggregate price level.

F1.1 Household Problem

In the steady state, the household value function is given by \[V_{h}^{ss}\left(b_{h,t-1}\right)=\max_{C_{h,t},N_{h,t},b_{h,t}}\left\{ \log\left(C_{h,t}\right)-\frac{\chi}{2}\left(N_{h,t}\right)^{2}+\beta V_{h}^{ss}\left(b_{h,t}\right)\right\}\] subject to \[C_{h,t}+\frac{b_{h,t}}{R^{ss}}\leq\frac{b_{h,t-1}}{\pi^{ss}}+w^{ss}N_{h,t}-\tau^{ss}.\] Here, \(b_{h,t-1}\) and \(b_{h,t}\) are household \(h\)’s real bond holdings chosen in the previous and current period, respectively. The variables \(C_{h,t}\) and \(N_{h,t}\) are household \(h\)’s consumption and labor supply. The aggregate variable \(R^{ss}\), \(\pi^{ss}\), \(w^{ss}\), and \(\tau^{ss}\) are the gross nominal interest rate, the gross inflation rate, the real wage, and taxes net of transfers and profits. The values of these aggregate variables are known to the household. We constrain households so that \(b_{h,t}\in\left[\underline{b},\overline{b}\right]\). Implicitly, we have functions \(C_{h,t}=C_{h}^{ss}\left(b_{h,t-1}\right)\), \(N_{h,t}=N_{h}^{ss}\left(b_{h,t-1}\right)\), and \(b_{h,t}=b_{h}^{ss}\left(b_{h,t-1}\right)\). Assuming the constraint on \(b_{h,t}\) is not binding, household maximization implies \[\begin{align} \frac{1}{C_{h}^{ss}\left(b_{h,t-1}\right)} & =\beta R^{ss}\frac{1}{C_{h}^{ss}\left(b_{h,t}\right)\pi^{ss}}\tag{80}\\ \chi C_{h}^{ss}\left(b_{h,t-1}\right)N_{h}^{ss}\left(b_{h,t-1}\right) & =w^{ss}\tag{81}\\ C_{h}^{ss}\left(b_{h,t-1}\right)+\frac{b_{h}^{ss}\left(b_{h,t-1}\right)}{R^{ss}} & =\frac{b_{h,t-1}}{\pi^{ss}}+w^{ss}N_{h}^{ss}\left(b_{h,t-1}\right)-\tau^{ss}\tag{82}\end{align}\] We define a grid over \(\left[\underline{b},\overline{b}\right]\) and approximate the functions \(C_{h}^{ss}\left(b_{h,t-1}\right)\), \(N_{h}^{ss}\left(b_{h,t-1}\right)\), and \(b_{h}^{ss}\left(b_{h,t-1}\right)\) on that grid in the following way.⁴²

1. We conjecture a value for \(b_{h}^{ss}\left(b_{h,t-1}\right)\) at each grid point, which we denote by \(b_{h}^{ss,i}\left(b_{h,t-1}\right)\).

2. Note that equations (81) and (82) can be written as \[\chi C_{h}^{ss}\left(b_{h,t-1}\right)\left(C_{h}^{ss}\left(b_{h,t-1}\right)+\frac{b_{h}^{ss,i}\left(b_{h,t-1}\right)}{R^{ss}}-\frac{b_{h,t-1}}{\pi^{ss}}+\tau^{ss}\right)=\left(w^{ss}\right)^{2}.\] The left-hand-side is increasing in \(C_{h}^{ss}\left(b_{h,t-1}\right)\geq0\). For every \(b_{h,t-1}\), we solve for the value of \(C_{h}^{ss}\left(b_{h,t-1}\right)\) that makes this hold with equality. We call this the conjectured value for \(C_{h}^{ss}\left(b_{h,t-1}\right)\) and denote it by \(C_{h}^{ss,i}\left(b_{h,t-1}\right)\). Note that with \(C_{h}^{ss,i}\left(b_{h,t-1}\right)\), we can back out \(N_{h}^{ss,i}\left(b_{h,t-1}\right)\) from equation (81).

3. For each grid point, \(b_{h,t-1}\), find \(b_{h,t}\) that solves the following version of equation (80) \[C_{h,t}\beta R^{ss}\frac{1}{C_{h}^{ss,i}\left(b_{h,t}\right)}-1=0\] where \(C_{h,t}\geq0\) solves \[\chi C_{h,t}\left(C_{h,t}+\frac{b_{h,t}}{R^{ss}}-\frac{b_{h,t-1}}{\pi^{ss}}+\tau^{ss}\right)=\left(w^{ss}\right)^{2}.\] We use linear interpolation to compute \(C_{h}^{ss,i}\left(b_{h,t}\right)\) for values of \(b_{h,t}\) that fall between grid points. If the procedure would set \(b_{h}^{\prime}>\overline{b}\) or \(b_{h}^{\prime}<\underline{b}\), we set \(b_{h,t}\)to the respective endpoint of the grid. We record the value of \(b_{h,t}\) by updating the conjectured function \(b_{h}^{ss}\left(\cdot\right)\) using \(b_{h}^{ss,i+1}\left(b_{h,t-1}\right)=b_{h,t}\).

4. Having computed \(b_{h}^{ss,i+1}\left(b_{h,t-1}\right)\) for every grid point, we check to see if \[\left|b_{h}^{ss,i+1}\left(b_{h,t-1}\right)-b_{h}^{ss,i}\left(b_{h,t-1}\right)\right|<\epsilon\] at every grid point for some small \(\epsilon\). If yes, we say that we have solved the household problem in steady state. If no, we set \(b_{h}^{ss,i}\left(b_{h,t-1}\right)=b_{h}^{ss,i+1}\left(b_{h,t-1}\right)\) at each grid point and repeat steps (ii), (iii), and (iv).

Because \(\beta\frac{R^{ss}}{\pi^{ss}}=1\), it is not surprising that we find that \(b_{h}^{ss}\left(b_{h,t-1}\right)=b_{h,t-1}.\)

F1.2 Firm Problem

In the steady state, the firm value function is given by \[\begin{align*} V_{f}^{ss}\left(p_{f,t-1}\right)=\max_{p_{f,t}} & \left\{ \frac{1}{C^{ss}}\left(p_{f,t}-\left(1-\tau_{f}\right)w^{ss}\right)\left(p_{f,t}\right)^{-\varepsilon}Y^{ss}\right.\\ & \left.-\frac{1}{C^{ss}}\frac{\Phi_{P}}{2}\left(\frac{p_{f,t}}{p_{f,t-1}}\pi^{ss}-1\right)^{2}\left(C^{ss}+G^{ss}\right)+\beta V_{f}^{ss}\left(p_{f,t}\right)\right\} . \end{align*}\] Here, \(p_{f,t-1}\) and \(p_{f,t}\) are the ratio of firm \(f\)’s price to the aggregate price level in the previous and current period, respectively. The aggregate values \(\pi^{ss}\), \(w^{ss}\), \(C^{ss}\), \(G^{ss}\), and \(Y^{ss}\) are known to the firm. We constrain firms so that \(\log\left(p_{f,t}\right)\in\left[\underline{p},\overline{p}\right]\).⁴³ Implicitly, we have a function \(p_{f}^{ss}\left(p_{f,t-1}\right)\). Assuming the constraint on \(\log\left(p_{f}\right)\) is not binding, firm optimality implies \[\begin{align} \Phi_{P} & \left(\frac{p_{f}^{ss}\left(p_{f,t-1}\right)}{p_{f,t-1}}\pi^{ss}-1\right)\frac{1}{p_{f}}\pi^{ss}\left(C^{ss}+G^{ss}\right)=\left(\varepsilon-1\right)\left(\frac{w^{ss}}{p_{f}^{ss}\left(p_{f,t-1}\right)}-1\right)\left(p_{f}^{ss}\left(p_{f,t-1}\right)\right)^{-\varepsilon}Y^{ss}\nonumber \\ & +\beta\Phi_{P}\left(\frac{p_{f}^{ss}\left(p_{f}^{ss}\left(p_{f,t-1}\right)\right)}{p_{f}^{ss}\left(p_{f,t-1}\right)}\pi^{ss}-1\right)\frac{p_{f}^{ss}\left(p_{f}^{ss}\left(p_{f,t-1}\right)\right)}{\left(p_{f}^{ss}\left(p_{f,t-1}\right)\right)^{2}}\pi^{ss}\left(C^{ss}+G^{ss}\right)\tag{83}\end{align}\] We define a grid over \(\left[\underline{p},\overline{p}\right]\) and approximate the function \(p_{f}^{ss}\left(p_{f,t-1}\right)\) on that grid in the following way.

1. We conjecture a value for \(p_{f}^{ss}\left(p_{f,t-1}\right)\) at each grid point. Call the conjectured value \(p_{f}^{ss,i}\left(p_{f,t-1}\right)\).

2. For each grid point, \(p_{f,t-1}\), find \(p_{f,t}\) that solves the following version of equation (83) \[\begin{align*} \Phi_{P} & \left(\frac{p_{f,t}}{p_{f,t-1}}\pi^{ss}-1\right)\frac{1}{p_{f,t-1}}\pi^{ss}\left(C^{ss}+G^{ss}\right)=\left(\varepsilon-1\right)\left(\frac{w^{ss}}{p_{f,t}}-1\right)\left(p_{f,t}\right)^{-\varepsilon}Y^{ss}\\ & +\beta\Phi_{P}\left(\frac{p_{f}^{ss,i}\left(p_{f,t}\right)}{p_{f,t}}\pi^{ss}-1\right)\frac{p_{f}^{ss,i}\left(p_{f,t}\right)}{\left(p_{f,t}\right)^{2}}\pi^{ss}\left(C^{ss}+G^{ss}\right) \end{align*}\] We use linear interpolation over \(\log\left(p_{f,t}\right)\) to compute \(p_{f}^{ss,i}\left(p_{f,t}\right)\) for values of \(\log\left(p_{f,t}\right)\) that fall between grid points. If the procedure would set \(\log\left(p_{f,t}\right)>\overline{p}\) or \(\log\left(p_{f,t}\right)<\underline{p}\), we set \(p_{f,t}\) to the respective endpoint of the grid. We record the value of \(p_{f,t}\) by updating the conjectured rule for \(p_{f}^{ss}\left(p_{f,t-1}\right)\) using \(p_{f}^{ss,i+1}\left(p_{f,t-1}\right)=p_{f,t}\).

3. Having computed \(p_{f}^{ss,i+1}\left(p_{f,t-1}\right)\) for every grid point, we check to see if \[\left|p_{f}^{ss,i+1}\left(p_{f,t-1}\right)-p_{f}^{ss,i}\left(p_{f,t-1}\right)\right|<\epsilon\] at every grid point for some small \(\epsilon\). If yes, we say that we have solved the firm problem in steady state. If no, we set \(p_{f}^{ss,i}\left(p_{f,t-1}\right)=p_{f}^{ss,i+1}\left(p_{f,t-1}\right)\) at each grid point and repeat steps (ii) and (iii).

F2 Solution when \(r_{t}=r^{\ell}\)

To address the case when \(r_{t}=r^{\ell}\), we assume that we have the steady state decision rules in hand and that households and firms know these decision rules with certainty.

F2.1 Beliefs

Before presenting the household and firm problems, some comments about beliefs are in order when \(r_{t}=r^{\ell}\). To simplify the model, we assume households and firms have the same beliefs (though they do not know that they have the same beliefs). Households and firms believe that so long as \(r_{t}=r^{\ell}\) , \(\log\left(C_{t}\right)\) and \(\log\left(\pi_{t}\right)\), have uncorrelated Normal distributions with unknown means and variances. That is, \(\log\left(\pi_{t}\right)\sim N\left(\mu_{\pi},\sigma_{\pi}^{2}\right)\) and \(\log\left(C_{t}\right)\sim N\left(\mu_{C},\sigma_{C}^{2}\right).\) We assume that households and firms have beliefs about the means and variances of the distributions for \(\log\left(C_{t}\right)\) and \(\log\left(\pi_{t}\right)\) that are characterized by density functions that are proportional to Normal-inverse-gamma distributions. (See Zellner, 1971 and Murphy, 2007 for a discussion of the Normal-inverse-gamma distribution). These beliefs are not exactly Normal-inverse-gamma distributions because the households and firms embed in their beliefs an upper bound on the variances. This upper bound is important because if variances were unbounded, \(\mathbf{E}\left[\pi_{t}\right]=\mathbf{E}\left[C_{t}\right]=\infty\), which would challenge the applicability of an expected utility framework (see Geweke, 2001 for related discussion). The distributions characterizing beliefs are independent across \(C_{t}\) and \(\pi_{t}\). That is, for \(i\in\left\{ \pi,C\right\}\), \(\mu_{i}\in\left(-\infty,\infty\right)\) and \(\sigma_{i}^{2}\in\left[0,\overline{\sigma}_{i}^{2}\right]\) we have \[\begin{align*} \Pr\left(\sigma_{i}^{2}|\alpha_{i,t-1},\beta_{i,t-1}\right) & =\frac{\frac{\beta_{i}^{\alpha_{i,t-1}}}{\Gamma\left(\alpha_{i,t-1}\right)}\left(\frac{1}{\sigma_{i}^{2}}\right)^{\alpha_{i,t-1}+1}\exp\left(-\frac{\beta_{i,t-1}}{\sigma_{i}^{2}}\right)}{\frac{\Gamma\left(\alpha_{i,t-1},\beta_{i,t-1}/\overline{\sigma}_{i}^{2}\right)}{\Gamma\left(\alpha_{i,t}\right)}},\\ \Pr\left(\mu_{i}|\sigma_{i}^{2},m_{i,t-1},\lambda_{i,t-1}\right) & =\frac{\sqrt{\lambda_{i,t-1}}}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left(-\frac{\lambda_{i,t-1}}{2\sigma_{i}^{2}}\left(\mu_{i}-m_{i,t-1}\right)^{2}\right). \end{align*}\] See Here, \(\Gamma\left(\cdot\right)\) is the gamma function and \(\Gamma\left(\cdot,\cdot\right)\) is the incomplete gamma function. Note that \(\Gamma\left(\cdot\right)=\Gamma\left(\cdot,0\right)\). Again, the advantage of truncating the prior of \(\sigma_{i}^{2}\) is that \(\mathbf{E}\left[\pi_{t}\right]<\infty\) if \(\overline{\sigma}_{\pi}^{2}<\infty\) and \(\mathbf{E}\left[C_{t}\right]<\infty\) if \(\overline{\sigma}_{C}^{2}<\infty\).

Even though we truncate the distributions for \(\sigma_{i}^{2}\), we maintain the usual recursive updating equations because the likelihoods associated with observations of \(\pi_{t}\) and \(C_{t}\) are not truncated. Beliefs about consumption and inflation are parameterized by four values, \(\alpha_{i,t-1}\), \(\beta_{i,t-1}\), \(m_{i,t-1}\), and \(\lambda_{i,t-1}\) for \(i\in\left\{ \pi,C\right\}\). So, we have 8 total parameters. The recursive updating formulas (see Murphy, 2007) for these variables are \(\lambda_{i,t}=\lambda_{i,t-1}+1\), \(\alpha_{i,t}=\alpha_{i,t-1}+1/2\), \[\begin{align*} m_{i,t} & =\frac{\lambda_{i,t-1}m_{i,t-1}+\log\left(i_{t}\right)}{\lambda_{i,t-1}+1}\text{, } \beta_{i,t}=\beta_{i,t-1}+\frac{\lambda_{i,t-1}\left(\log\left(i_{t}\right)-m_{i,t-1}\right)^{2}}{2\left(\lambda_{i,t-1}+1\right)}. \end{align*}\]

We need to include variables in \(\Theta_{t-1}\) that will fully capture the values \(\alpha_{i,t-1}\), \(\beta_{i,t-1}\), \(m_{i,t-1}\), and \(\lambda_{i,t-1}\) for \(i\in\left\{ \pi,C\right\}\). First, we keep \(\frac{1}{t_{\ell}}\) in \(\Theta_{t-1}\), which is the inverse of the 1 plus number of periods that \(r_{t}\) has been equal to \(r_{\ell}\) up to period \(t-1\). We keep the inverse because it is bounded between zero and one. From this value, we can trivially back out \(\lambda_{i,t-1}\) and \(\alpha_{i,t-1}\), given their values in the first period when \(r_{t}=r_{\ell}\). We set the initial value of \(\lambda_{i,t-1}=1\) and the initial value of \(\alpha_{i,t-1}=2\). We keep \(m_{C,t-1}\) and \(m_{\pi,t-1}\) in \(\Theta_{t-1}\). And we also keep \[\psi_{i,t}=\sqrt{\psi_{i,t-1}^{2}\frac{2\alpha_{i,t}}{2\alpha_{i,t}+1}+\frac{\lambda_{i,t-1}}{\left(\lambda_{i,t-1}+1\right)}\frac{1}{2\alpha_{i,t}+1}\left(\log\left(i_{t}\right)-m_{i,t-1}\right)^{2}}.\] Note that by setting \(\beta_{i,t}=\left(\psi_{i,t}\right)^{2}\alpha_{i,t+1}\) it is clear that we recover the exactly recursive structure of \(\beta_{i}\) (given above). An advantage of using \(\psi_{i,t}\) in \(\Theta_{t}\) rather than \(\beta_{i,t}\) is that \(\psi_{i,t}\) is a consistent estimator for the standard deviation, whereas \(\beta_{i,t}\) generally grows without bound (except when the standard deviation is zero). Keeping the values of \(\Theta_{t}\) within bounded grids will be important for the purposes of approximation. In total, \(\Theta_{t}=\left[\frac{1}{t^{\ell}},m_{\pi,t},m_{C,t},\psi_{\pi,t},\psi_{C,t}\right]\) has five elements (where \(t^{\ell}\) is the number of periods that the ZLB has persisted) and we have a mapping from \(\Theta_{t-1}\) to \(\alpha_{i,t}\), \(\beta_{i,t}\), \(m_{i,t}\), and \(\lambda_{i,t}\) for \(i\in\left\{ \pi,C\right\}\). We also have a law of motion for \(\Theta_{t}\) so that \(\Theta_{t}=L\left(\Theta_{t-1},\left[\pi_{t},C_{t}\right]^{\prime}\right).\)

An advantage of the Normal-inverse-gamma setup detailed above is that we can have analytic expressions for the distribution for the variables \(\log\left(\pi_{t}\right)\) and \(\log\left(C_{t}\right)\) conditional on \(\Theta_{t-1}\). (See Zellner, 1971 and Murphy, 2007 for related discussion. See Bakshi and Skoulakis, 2010 for a discussion of truncating the distribution of the variance.) In particular \[\begin{align} \Pr\left(\log\left(i_{t}\right)|\Theta_{t-1}\right)= & \left(\frac{\lambda_{i,t-1}\alpha_{i,t-1}}{\beta_{i,t-1}\left(\lambda_{i,t-1}+1\right)}\right)^{1/2}\frac{\Gamma\left(\alpha_{i,t-1}+1/2\right)}{\Gamma\left(\alpha_{i,t-1}\right)\sqrt{2\pi\alpha_{i,t-1}}}\nonumber \\ & \times\left(1+\frac{1}{2\alpha_{i,t-1}}\frac{\left(\log\left(i_{t}\right)-m_{i,t-1}\right)^{2}}{\left(\frac{\lambda_{i,t-1}\alpha_{i,t-1}}{\beta_{i,t-1}\left(\lambda_{i,t-1}+1\right)}\right)^{-1}}\right)^{-\alpha_{i,t-1}-1/2}\frac{\kappa_{i,t}}{\kappa_{i,t-1}},\tag{84}\end{align}\] where \(\kappa_{i,t}=\Gamma\left(\alpha_{i,t},\beta_{i,t}/\overline{\sigma}^{2}\right)\left[\Gamma\left(\alpha_{i,t}\right)\right]^{-1}.\) Notice that \(\kappa_{i,t}\) depends on the point of evaluation for \(\log\left(i_{t}\right)\). If we ignored the ratio \(\kappa_{i,t}/\kappa_{i,t-1}\), which would be correct in the case when \(\bar{\sigma}_{i}^{2}=\infty\), the pdf for \(\log\left(i_{t}\right)\) is a \(t\) distribution with location parameter \(m_{i,t-1}\), scale parameter \(\left(\frac{\lambda_{i,t-1}\alpha_{i,t-1}}{\beta_{i,t-1}\left(\lambda_{i,t-1}+1\right)}\right)^{-1/2}\), and \(2\alpha_{i,t-1}\) degrees of freedom. If \(\overline{\sigma}_{i}^{2}\) is large, \(\kappa_{i,t}/\kappa_{i,t-1}\neq1\) but is close to unity. For finite \(\bar{\sigma}_{i}^{2}\), the ratio \(\kappa_{i,t}/\kappa_{i,t-1}\) serves to thin the tails of the distribution of \(\log\left(i_{t}\right)\) by down-weighting the probability of extreme values for \(\log\left(i_{t}\right)\).⁴⁴ Because the density function of the \(t\) distribution is readily available and reliably computed in statistical software and because \(\kappa_{i,t-1}\) and \(\kappa_{i,t}\) are easily computed using readily available implementations of the gamma and incomplete gamma functions, we can use equation (84) for quadrature weighting. We use Gauss-Hermite quadrature with seven nodes to approximate integrals based on equation (84).

F2.2 Household Problem

When \(r_{t}=r^{\ell}\), the household value function is given by

\[\begin{align*} V_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)=\max_{C_{h,t},N_{h,t},b_{h,t}} & \left\{ \log\left(C_{h,t}\right)-\frac{\chi}{2}\left(N_{h,t}\right)^{2}\right.\\ & \left.+\frac{1}{1+r^{\ell}}\left[p\mathbf{E}_{t}V_{h}\left(b_{h,t},\Theta_{t},x_{t+1}\right)+\left(1-p\right)V_{h}^{ss}\left(b_{h,t}\right)\right]\right\} \end{align*}\] subject to \(C_{h,t}+\frac{b_{h,t}}{R_{t}}\leq\frac{b_{h,t-1}}{\pi_{t}}+w_{t}N_{h,t}-\tau_{t}.\) Here, \(x_{t}=\left[\pi_{t},C_{t}\right]^{\prime}\), \(V_{h}^{ss}\left(\cdot\right)\) is the steady state value function for the household, which is defined above, and \(\mathbf{E}_{t}\) denotes expectations of the household computed conditional on \(\Theta_{t-1}\) and \(x_{t}\). Given \(x_{t}\) and \(\Theta_{t-1}\), we have \(\Theta_{t}=L\left(\Theta_{t-1},x_{t}\right)\). So, the expectation of the household is taken with respect to \(x_{t+1}\), which is believed to be iid. We assume that households know the monetary and fiscal policy rules. We also assume that they correctly think that \(Y_{t}=\left(C_{t}+G_{t}\right)\left(1+\frac{\Phi_{P}}{2}\left(\pi_{t}-1\right)^{2}\right)\), \(N_{t}=Y_{t}\), and \(w_{t}=\chi C_{t}Y_{t}\). Given \(x_{t}\), with these assumptions \(R_{t}\), \(w_{t}\), and \(\tau_{t}\) can be computed. The steady state values of aggregate variables are known to the household. We constrain households so that \(b_{h,t}\in\left[\underline{b},\overline{b}\right]\). The household optimization problem gives us implicit functions for \(C_{h,t}=C_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\), \(N_{h,t}=N_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\), and \(b_{h,t}=b_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\). Considering interior solutions for \(b_{h,t}\), we have \[\begin{align} \frac{1}{C_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)} & =\frac{1}{1+r^{\ell}}R\left[p\mathbf{E}_{t}\left\{ \frac{1}{\pi_{t+1}C_{h}\left(b_{h,t},\Theta_{t},x_{t+1}\right)}\right\} \right.\nonumber \\ & \left.\phantom{\frac{1}{1+r_{\ell}}R[[[[}+\left(1-p\right)\frac{1}{\pi^{ss}C_{h}^{ss}\left(b_{h,t}\right)}\right]\tag{85}\\ w_{t} & =\chi C_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)N_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\tag{86}\\ C_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right) & =\frac{b_{h,t-1}}{\pi_{t}}+w_{t}N_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)-\tau_{t}-\frac{b_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)}{R_{t}}.\tag{87}\end{align}\] Instead of approximating \(C_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\), \(N_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\), and \(b_{h}\left(b_{h,t-1},\Theta_{t-1},x_{t}\right)\) directly, we approximate \[v_{h}\left(b_{h,t},\Theta_{t}\right)=\mathbf{E}_{t}\left\{ \frac{1}{\pi_{t+1}C_{h}\left(b_{h,t},\Theta_{t},x_{t}\right)}\right\} .\] We take this approach because it eliminates \(x_{t}\) as a state variable in the approximation. We define grids on the elements of \(\Theta_{t}\) and use the grid defined for \(b_{h,t}\) in the steady state. We then approximate \(v_{h}\left(b_{h,t},\Theta_{t}\right)\) in the following way.⁴⁵

1. We conjecture a value for \(v_{h}\left(b_{h,t-1},\Theta_{t-1}\right)\) at each grid point in the cross product of the grids over the elements of \(b_{h,t-1}\) and \(\Theta_{t-1}\).⁴⁶ Call the conjectured value \(v_{h}^{i}\left(b_{h,t-1},\Theta_{t-1}\right)\).

2. For a given grid point we use quadrature to get a value for \(\mathbf{E}_{t-1}\left\{ \left(\pi_{t}C_{h,t}\right)^{-1}\right\} .\)To solve for the expectation of interest, we need to solve for \(C_{h,t}\) given many different values for \(x_{t}\). Conditional on a value for \(x_{t}\), equations (86) and (87) can be written as \[\chi C_{h,t}\left(C_{h,t}+\frac{b_{h,t}}{R_{t}}-\frac{b_{h,t-1}}{\pi_{t}}+\tau_{t}\right)=w_{t}^{2}\] The left-hand-side is increasing in \(C_{h,t}\geq0\). For a given \(b_{h,t}\), we solve for the value of \(C_{h,t}\) that makes this hold with equality. We then search for the value of \(b_{h,t}\) that makes the following version of equation (85) hold with equality: \[\frac{1}{C_{h,t}}=\frac{1}{1+r^{\ell}}R\left[pv_{h}^{i}\left(b_{h,t},\Theta_{t}\right)+\left(1-p\right)\frac{1}{\pi^{ss}C_{h}^{ss}\left(b_{h,t+1}\right)}\right].\] We use linear interpolation to compute \(v_{h}^{i}\left(b_{h,t},\Theta_{t}\right)\) for values of \(b_{h,t}\) and \(\Theta_{t}\) that fall between grid points. If the procedure would set \(b_{h,t}>\overline{b}\) or \(b_{h,t}<\underline{b}\), we set \(b_{h,t}\)to the respective endpoint of the grid for \(b_{h,t}\). We record the associated value of \(C_{h,t}\) and use it in the quadrature to compute \(v_{h}^{i+1}\left(b_{h,t-1},\Theta_{t-1}\right)=\mathbf{E}_{t-1}\left\{ \left(\pi_{t}C_{h,t}\right)^{-1}\right\}\).

3. Having computed \(v_{h}^{i+1}\left(b_{h,t-1},\Theta_{t-1}\right)\) for every grid point, we check to see if \[\left|v_{h}^{i}\left(b_{h,t-1},\Theta_{t-1}\right)-v_{h}^{i+1}\left(b_{h,t-1},\Theta_{t-1}\right)\right|<\epsilon\] at every grid point for some small \(\epsilon\). If yes, we say that we have solved the household problem when \(r_{t}=r^{\ell}\). If no, we set \(v_{h}^{i}\left(b_{h,t-1},\Theta_{t-1}\right)=v_{h}^{i+1}\left(b_{h,t-1},\Theta_{t-1}\right)\), repeat steps (ii) and (iii).

The grid that we use on \(\frac{1}{t^{\ell}}\) is special. In particular, we let that grid be \(\left[0,\frac{1}{99},\frac{1}{98},\dots,1\right]\). The first element of the grid corresponds to the case when infinite time has past. In this case households think that they would update their beliefs so that \(\Theta_{t}=\Theta_{t-1}\) because \(m_{i,t-1}\) and \(\psi_{i,t-1}\) are consistent estimators for the means and variances. In our numerical computations, we utilize this fact to first approximate \(v_{h}\) in this case. We then approximate \(v_{h}\) in the case where \(t^{\ell}=99\). When \(t^{\ell}=99\), we need to interpolate between the solution to the case when \(t^{\ell}=\infty\) and the conjectured value of \(v_{h}^{i}\left(b_{h,t-1},\Theta_{t-1}\right)\) when \(t^{\ell}=99\) to evaluate \(v_{h}^{i}\left(b_{h,t},\Theta_{t}\right)\).⁴⁷ That is, when \(t^{\ell}=99\) we have to find a fixed point of this interpolation, which is computationally intense. When \(\left(t^{\ell}\right)^{-1}=98\), having approximated \(v_{h}\left(b_{h,t},\Theta_{t}\right)\) for \(\left(t^{\ell}\right)^{-1}=99\) means that can evaluate \(v_{h}\left(b_{h,t},\Theta_{t}\right)\) exactly at \(\left(t^{\ell}\right)^{-1}=99\) without reference to \(v_{h}^{i}\left(b_{h,t},\Theta_{t}\right)\). We approximate for \(v_{h}\left(b_{h,t},\Theta_{t}\right)\) when \(\left(t^{\ell}\right)^{-1}=98\) and work work back in this way to \(\left(t^{\ell}\right)^{-1}=1\). This strategy fits this into the structure of steps 1-3 because we know that the value of \(v_{h}\left(b_{h,t-1},\Theta_{t-1}\right)\) will not depend on its value at any any \(t^{\ell}\) that is smaller than implied by \(\Theta_{t-1}\). So, we have a block dependent structure to \(v_{h}\left(b_{h,t-1},\Theta_{t-1}\right).\) Additionally, we know that \(t^{\ell}\) will only take integer values.

F2.3 Firm Problem

When \(r_{t}=r^{\ell}\), the firm value function is given by

\[\begin{align*} V_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)=\max_{p_{f}^{\prime}} & \left\{ \frac{1}{C}\left(\left(p_{f,t}-\left(1-\tau_{f}\right)w_{t}\right)\left(p_{f,t}\right)^{-\varepsilon}Y_{t}-\frac{\Phi_{P}}{2}\left(\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}-1\right)^{2}\left(C_{t}+G_{t}\right)\right)\right.\\ & +\frac{1}{1+r^{\ell}}\left[p\mathbf{E}_{t}V_{f}\left(p_{f,t},\Theta_{t},x_{t+1}\right)+\left(1-p\right)V_{f}^{ss}\left(p_{f,t}\right)\right] \end{align*}\] Here, \(x_{t}=\left[\pi_{t},C_{t}\right]^{\prime}\), \(V_{f}^{ss}\left(\cdot\right)\) is the steady state value function for the firm, which is defined above, and \(\mathbf{E}_{t}\) denotes expectations of the firm conditional on \(\Theta_{t-1}\) and \(x_{t}\). Given \(x_{t}\) and \(\Theta_{t-1}\), we have \(\Theta_{t}=L\left(\Theta_{t-1},x_{t}\right)\). So, the expectation of the firm is taken with respect to \(x_{t+1}\), which is believed to be iid. We assume that firms know the monetary and fiscal policy rules. We also assume that they correctly think that \(Y_{t}=\left(C_{t}+G_{t}\right)\left(1+\frac{\Phi_{P}}{2}\left(\pi_{t}-1\right)^{2}\right)\), \(N_{t}=Y_{t}\), and \(w_{t}=\chi C_{t}Y_{t}\). Given \(x_{t}\), with these assumptions \(w_{t}\), \(G_{t}\), and \(Y_{t}\) can be computed. The steady state values of aggregate variables are known to the firm. We constrain firms so that \(\log\left(p_{f,t}\right)\in\left[\underline{p},\overline{p}\right]\). Implicitly, from firm optimization we have a function \(p_{f,t}=p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\). Considering interior solutions for \(p_{f,t}\), firm maximization implies \[\begin{align} \Phi_{P} & \left(\frac{p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)}{p_{f,t-1}}\pi_{t}-1\right)\frac{1}{p_{f,t-1}}\pi_{t}\left(C_{t}+G_{t}\right)=\nonumber \\ & \left(\varepsilon-1\right)\left(\frac{w_{t}}{p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)}-1\right)\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\right)^{-\varepsilon}Y_{t}\nonumber \\ & +\frac{1}{1+r_{\ell}}p\mathbf{E}_{t}\frac{C_{t}}{C_{t+1}}\Phi_{P}\left(\frac{p_{f}\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right),\Theta_{t},x_{t+1}\right)}{p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)}\pi_{t+1}-1\right)\frac{p_{f}\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right),\Theta_{t},x_{t+1}\right)}{\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\right)^{2}}\pi^{\prime}\left(C^{\prime}+G^{\prime}\right)\nonumber \\ & +\frac{1}{1+r_{\ell}}\frac{C_{t}}{C^{ss}}\left(1-p\right)\Phi_{P}\left(\frac{p_{f}^{ss}\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\right)}{p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)}\pi^{ss}-1\right)\frac{p_{f}^{ss}\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\right)}{\left(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\right)^{2}}\pi^{ss}\left(C^{ss}+G^{ss}\right).\tag{88}\end{align}\] Instead of approximating \(p_{f}\left(p_{f,t-1},\Theta_{t-1},x_{t}\right)\) directly, we approximate \[v_{f}\left(p_{f,t},\Theta_{t}\right)=\mathbf{E}_{t}\left\{ \frac{1}{C_{t+1}}\Phi_{P}\left(\frac{p_{f,t+1}}{p_{f,t}}\pi_{t+1}-1\right)\frac{p_{f,t+1}}{p_{f,t}}\pi_{t+1}\left(C_{t+1}+G_{t+1}\right)\right\} .\] We take this approach because we can eliminate \(x_{t}\) as a state variable in the approximation. We use the same grids on the elements of \(\Theta_{t-1}\) that we use for the household problem and the grid defined for \(\log\left(p_{f,t-1}\right)\) in the steady state and we approximate \(v_{f}\left(p_{f,t-1},\Theta_{t-1}\right)\) in the following way.

1. We conjecture a value for \(v_{f}\left(p_{f,t-1},\Theta_{t-1}\right)\) at each grid point in the cross product of the grids over the elements of \(p_{f,t-1}\) and \(\Theta_{t-1}\). Call the conjectured value \(v_{f}^{i}\left(p_{f,t-1},\Theta_{t-1}\right)\).

2. For a given grid point we use quadrature to get a value for \[\mathbf{E}_{t-1}\left\{ \frac{1}{C_{t}}\Phi_{P}\left(\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}-1\right)\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}\left(C_{t}+G_{t}\right)\right\} .\] To solve for the expectation of interest, we need to solve for \(p_{f,t}\) given many different values for \(x_{t}\). Conditional on a value for \(x_{t}\), we find a value of \(p_{f,t}\) that solves the following version of equation (88) \[\begin{align*} \Phi_{P} & \left(\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}-1\right)\frac{1}{p_{f,t-1}}\pi_{t}\left(C_{t}+G_{t}\right)=\\ & \left(\varepsilon-1\right)\left(\frac{w_{t}}{p_{f,t}}-1\right)\left(p_{f,t}\right)^{-\varepsilon}Y_{t}+\frac{1}{1+r_{\ell}}pv_{f}^{i}\left(p_{f,t},\Theta_{t}\right)\frac{C_{t}}{p_{f,t}}\\ & +\frac{1}{1+r_{\ell}}\frac{C_{t}}{C^{ss}}\left(1-p\right)\Phi_{P}\left(\frac{p_{f}^{ss}\left(p_{f,t}\right)}{p_{f,t}}\pi^{ss}-1\right)\frac{p_{f}^{ss}\left(p_{f,t}\right)}{\left(p_{f,t}\right)^{2}}\pi^{ss}\left(C^{ss}+G^{ss}\right). \end{align*}\] We use linear interpolation over \(\log\left(p_{f,t}\right)\) to compute \(v_{f}^{i}\left(p_{f,t},\Theta_{t}\right)\) for values of \(\log\left(p_{f,t}\right)\) and \(\Theta_{t}\) that fall between grid points. If the procedure would set \(\log\left(p_{f,t}\right)>\overline{p}\) or \(\log\left(p_{f,t}\right)<\underline{p}\), we set \(p_{f,t}\) to the respective endpoint of the grid for \(p_{f}\). We record the value of \(p_{f,t}\) and the associated aggregate variables so that the quadrature procedure can approximate \[v_{f}^{i+1}\left(p_{f,t-1},\Theta_{t-1}\right)=\mathbf{E}_{t-1}\left\{ \frac{1}{C_{t}}\Phi_{P}\left(\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}-1\right)\frac{p_{f,t}}{p_{f,t-1}}\pi_{t}\left(C_{t}+G_{t}\right)\right\} .\]

3. Having computed \(v_{f}^{i+1}\left(p_{f,t-1},\Theta_{t-1}\right)\) for every grid point, we check to see if \[\left|v_{f}^{i}\left(p_{f,t-1},\Theta_{t-1}\right)-v_{f}^{i+1}\left(p_{f,t-1},\Theta_{t-1}\right)\right|<\epsilon\] at every grid point for some small \(\epsilon\). If yes, we say that we have solved the household problem when \(r_{t}=r_{\ell}\). If no, we set \(v_{f}^{i}\left(p_{f,t-1},\Theta_{t-1}\right)=v_{f}^{i+1}\left(p_{f,t-1},\Theta_{t-1}\right)\), repeat steps (ii) and (iii).

Our use of the same grids as in the household problem allows us to exploit the same block dependent structure in \(\left(t^{\ell}\right)^{-1}\).

F3 Learning Equilibria

Here we detail how we construct learning equilibria, given the solutions to the household an firm problems—\(v_{h}\) and \(v_{f}\).

1. Set \(r_{t}=r^{\ell}\) and assume a value for \(\Theta_{t-1}\) for \(t=1\).

2. Conjecture a value for \(\pi_{t}\).

   1. Find the value of \(C_{t}\) that would make the following equation hold \[\frac{1}{C_{t}}=\frac{1}{1+r^{\ell}}R_{t}\left[pv_{h}\left(0,L\left(\Theta_{t-1},\left[\pi_{t},C_{t}\right]\right)\right)+\left(1-p\right)\frac{1}{\pi^{ss}C_{h}^{ss}\left(0\right)}\right].\]

   2. Check to see if the following equation holds \[\begin{align*} \Phi_{P} & \left(\pi_{t}-1\right)\pi_{t}\left(C_{t}+G_{t}\right)=\left(\varepsilon-1\right)\left(w_{t}-1\right)+\frac{1}{1+r_{\ell}}pv_{f}\left(1,L\left(\Theta_{t-1},\left[\pi_{t},C_{t}\right]\right)\right)C_{t}\\ & +\frac{1}{1+r_{\ell}}\frac{C_{t}}{C^{ss}}\left(1-p\right)\Phi_{P}\left(\pi^{ss}-1\right)\pi^{ss}\left(C^{ss}+G^{ss}\right). \end{align*}\] If yes, we have a period equilibrium for period \(t\) and we record \(\pi_{t}\) and \(C_{t}\). If no, conjecture a different value for \(\pi_{t}\).

3. Set \(\Theta_{t+1}=L\left(\Theta_{t},\left[\pi_{t},C_{t}\right]\right)\) and repeat step (ii).

When we consider “anticipated utility,” we define \(\tilde{\Theta_{t}}\) to be \(\Theta_{t}\), but with \(\frac{1}{t_{\ell}}=0\). We then perform step 2 with \(\tilde{\Theta}_{t}\) instead of \(\Theta_{t}\). However, in step 3 we continue to use \(\Theta_{t}\). The switch between \(\tilde{\Theta}_{t}\) and \(\Theta_{t}\) highlights the way in which “anticipated utility” is not internally rational. Because it is much less computationally demanding, the anticipated utility solution can also be calculated directly using \(m_{i,t-1}\) and \(\psi_{i,t-1}\) without relying on interpolation between grid points of those variables.

F4 Forward Guidance

When we implement forward guidance, we assume that households and firms know the aggregate values of consumption and inflation in period \(I\) (the period that \(r_{t}\) changes back to \(r^{ss}\) but \(R_{t}\) remains at the ZLB). We denote these quantities by \(C^{I}\) and \(\pi^{I}\). We also assume that households and firms know how they will act in period \(I\). However, they are unsure about the period when \(r_{t}=r^{\ell}\). They have to learn about the data generating process for \(C_{t}\) and \(\pi_{t}\) while \(r_{t}=r^{\ell}\). With these assumptions, the model can now be cast as in the previous three sub-sections, but with householding and firms thinking that with probability \(1-p\) they will be in period \(I\) and using the value functions and decision rules in period \(I\).

G REE Multiplicity

In this appendix, we consider REE multiplicity in the context of the simple NK model with only sticky prices that we study in the main text. Some of the material in this appendix appeared in earlier versions of this paper (see Christiano et al., 2018). We consider stability under learning (E-stability) of the different REEs. See Evans and Honkapohja, 2001 for discussion of E-stability and its use in equilibrium selection. See Boneva et al., 2016 for a discussion of multiple REE at the ZLB. The first two sub-sections of this appendix work with the model with Rotemberg-style nominal rigidities because that model has no state variable, which simplifies the analysis. Because the specification of adjustment costs with Rotemberg-style nominal rigidities is somewhat ad hoc, and because different specifications lead to the same first-order approximation to the model, we also consider multiplicity in the context of the model with Calvo-style nominal rigidities in the third sub-section. We consider that model more micro-founded and, as a result, perhaps a better model for analysis of equilibrium multiplicity. However, because of the similarities between the models with Rotemberg-style nominal rigities and Calvo-style nominal rigidities, it is unclear that either should be regarded as strictly better.

We consider equilibrium multiplicity in the fully nonlinear version of the simple NK model. Note that analyses that linearize around the target-inflation steady state while also maintaining the nonlinearity associated with the ZLB would not identify the multiplicity we highlight.

G1 Rotemberg-Style Nominal Rigidities

In this sub-section, we focus on the version of the model with Rotemberg-style adjustment costs. Evans et al., 2008 consider a version of this model when there is no shock to the natural rate of interest. They analyze the model’s two steady states (in one steady state, \(\pi=1\), in the other \(\pi=\beta\)).⁴⁸ They find that only the target-inflation steady state in which \(\pi=1\) is stable under learning, and they argue that learning dynamics are affected by the zero lower bound. Here, we focus on equilibria following a shock to the natural rate of interest, as in Eggertsson and Woodford, 2003. Specifically, we assume that \(r_{t}\) is given as in section 6.2.

With Rotemberg-style adjustment costs, the model has no state variable. We consider minimum-state-variable REEs (see McCallum, 1983 for related discussion). An REE is a set of values for output, employment, inflation, and consumption, \(Y^{\ell},N^{\ell},\pi^{\ell},C^{\ell}\), respectively, when \(r=r^{\ell}\). We assume that the economy reverts to the unique rational equilibrium steady state, \(Y^{ss},N^{ss},\pi^{ss},C^{ss}\), with \(R^{ss}>1\) when \(r=r^{ss}\).⁴⁹

The four equilibrium conditions associated with the four unknowns, \(\pi^{\ell},C^{\ell},R^{\ell},N^{\ell},\) are \[\begin{align} 1=\frac{1}{1+r^{\ell}} & \left[p\frac{1}{\pi^{\ell}}+\left(1-p\right)\frac{C^{\ell}}{C^{ss}}\right],\tag{89}\\ \left(\pi^{\ell}-1\right)\pi^{\ell}\left(C^{\ell}+G^{\ell}\right)= & \frac{\varepsilon-1}{\phi}\left(\chi N^{\ell}C^{\ell}-1\right)N^{\ell}\nonumber \\ & +\frac{1}{1+r^{\ell}}p\left(\pi^{\ell}-1\right)\pi^{\ell}\left(C^{\ell}+G_{\ell}\right),\tag{90}\\ N^{\ell}= & \left(C^{\ell}+G^{\ell}\right)\left(1+\frac{\phi}{2}\left(\pi^{\ell}-1\right)^{2}\right)\text{, and}\tag{91}\\ R^{\ell}= & \max\left\{ 1,1+r^{ss}+\alpha\left(\pi^{\ell}-1\right)\right\} .\tag{92} \end{align}\] Equations (89) and (90) take into account that \(\pi^{ss}=1\). In addition, we verify and use the fact that \(R^{\ell}=1\). We compute \(C^{ss}\) using the steady state of the model. Equation (90) can be expressed as one equation in the unknown, \(\pi^{\ell},\) after using equations (89) and (91), to express \(C^{\ell}\) and \(N^{\ell}\) as functions of \(\pi^{\ell}\). Then, we can find a candidate equilibrium by finding a value of \(\pi^{\ell}\) that sets a function, \(f\left(\pi^{\ell}\right)=0\). To verify that a candidate value of \(\pi^{\ell}\) is an equilibrium, we must verify that the implied aggregate quantities and firm values are non-negative.

Our baseline parameters are the same as in the main text. In the \(R>1\) steady-state REE, \(C^{ss}=0.8\), \(\pi^{ss}=1\), \(N^{ss}=1\). While \(r=r^{\ell}\), we set \(G^{\ell}=G(r^{\ell})=G^{ss}\). We also consider an alternative specification for government purchases, given by \[\begin{equation} G^{\ell}=G\left(r^{\ell}\right)=1.05\times G\left(r^{ss}\right).\tag{93}\end{equation}\]

Figure 5: f(π^ℓ) Corresponding to the Target-Inflation Steady-State Equilibrium
Note: The zeros of the function \(f\), which is defined in the text, identify REE values for \(\pi^{\ell}\) assuming that the economy goes to the target-inflation steady state after \(r_{t}=r^{ss}\). The dashed line is the case when \(G^{\ell}\) is high. Source: Authors’ calculations.

Accessible version

Figure (5) displays the function \(f\left(\pi^{\ell}\right)\) for a range of values of \(\pi^{\ell}\) in the baseline (solid blue line) and alternative (dashed blue line) cases. In each case, there are two values of \(\pi^{\ell}\) for which \(f\left(\pi^{\ell}\right)=0.\) Table 5 reports the values of \(C^{\ell},\,w^{\ell},\,N^{\ell},\,R^{\ell}\) and \(\pi^{\ell}\) at these zeros of \(f\). Each crossing corresponds to an interior equilibrium in which the ZLB binds.

Table 5: Equilibrium Values While \(r_{t}=r^{\ell}\), Returning to Target-Inflation Steady State

Note: This table reports \(\left\{ \pi^{\ell},R^{\ell},C^{\ell},N^{\ell},w^{\ell}\right\}\) for two equilibria indicated by \(A\) and \(B\) when \(G^{\ell}=G^{ss}\) (5a) and when \(G^{\ell}=1.05G^{ss}\) (5b). Each equilibrium returns to the target-inflation steady state as soon as \(r=r^{ss}\). The government purchases multiplier reported in the last line of panel is the change in GDP per unit increase in \(G\) within each of the type \(A\) and \(B\) equilibria.

Source: Authors’ calculations.

Panel A: \(G^{\ell}=G^{ss}\)

Panel B: \(G^{\ell}=1.05\times G^{ss}\)

With regard to the fiscal multiplier, we calculate the effect of an increase in \(G\) comparing \(A\) to \(A^{\prime}\) and \(B\) to \(B^{\prime}\)–that is, comparing two Bad-ZLB equilibria and two Good-ZLB equilibria (see Figure 5). Table 5 shows that the multiplier is very large in the latter case and very small in the former case. Consistent with this observation, expected deflation is much larger at \(A^{\prime}\) than at \(B^{\prime}\).

When we apply the methodology of section E, we find that the REE associated with \(B\) is E-stable (\(b<1\)) while the REE associated with \(A\) is not E-stable (\(b>1\)).

G2 Forward Guidance in the Model with Rotemberg-Style Nominal Rigidities

In this subsection, we consider multiplicity of equilibria in the context of forward guidance, as described in Section 7. We work with the simple NK model with only sticky prices and Rotemberg-style nominal rigidities. The model has a shocks to the natural rate of interest, as in the previous sub-section. The monetary authority commits to keeping the nominal interest rate at the ZLB for \(J\) periods after the discount rate has returned to its steady-state level. To make our point as simply as possible, we consider the case \(J=1\). In this sub-section we show that the number of REE proliferates under forward guidance.

In period \(I\), \(R=1\) even though \(r_{t}=r^{ss}.\) People know that the economy will transition to steady state in period \(I+1\). The equilibrium conditions in period \(I\) are equations (89) through (91) adjusted for forward guidance: \[\begin{align} 1 & =\frac{1}{1+r^{ss}}\frac{C^{I}}{\pi^{ss}C^{ss}}\tag{94}\\ 0 & =\left(\pi^{I}-1\right)\pi^{I}\left(C^{I}+G^{ss}\right)-\frac{\varepsilon-1}{\phi}\left(\chi N^{I}C^{I}-1\right)N^{I}\tag{95}\\ N^{I} & =\left(C^{I}+G^{I}\right)\left(1+\frac{\phi}{2}\left(\pi^{I}-1\right)^{2}\right).\tag{96}\end{align}\]

Equations (94) and (96) define functions mapping \(\pi^{I}\) to \(C^{I}\) and \(N^{I}.\) These functions allow us to express the left-hand side of equation (95) as a function of \(\pi^{I}\). We denote this function by \(f^{I}\left(\pi^{I}\right).\) A candidate continuation equilibrium in period \(I\) is a value of \(\pi^{I}\) such that \(f^{I}\left(\pi^{I}\right)=0\) along with the associated values of \(C^{I},N^{I},w^{I}\), and the present value of the intermediate good firm in period \(I\). The four variables must be non-negative for a candidate equilibrium to be an equilibrium. Figure 6 displays the \(f^{I}\) function for a range of values of \(\pi^{I}\). We find two continuation REE corresponding to the two zeros of \(f^{I}\) displayed in the figure (see points \(\mathcal{A}\) and \(\mathcal{B}\)).⁵⁰ The REE at \(\mathcal{B}\) features very high inflation, and price adjustment costs are very unrealistically high.

Figure 6: Equilibria in Period of Switch from r_t = r^ℓ to r_t = r^ss Under One-Period Forward Guidance


Notes: Graph of the function, \(f^{I}\left(\pi^{I}\right),\) discussed after equation (96). The two crossings with the zero line correspond to REE in period \(I,\) the date when \(r\) switches from \(r_{t}=r^{\ell}\) to \(r_{t}=r^{ss}\). Monetary policy in period \(I\) corresponds to one-period forward guidance–that is, the interest rate is held at zero in period \(I\) and then reverts to \(R^{ss}\). The red star indicates the level of inflation in period \(I\) in the absence of forward guidance.

Accessible version

We now compute the REE allocations in the periods before \(I\), which we denote by \(I_{-1}\), conditional on the continuation equilibrium starting in period \(I\). The period \(I_{-1}\) equilibrium conditions are the appropriate analog of equations (89) through (91): \[\begin{equation} 1=\frac{1}{1+r^{\ell}}\left[p\frac{C^{\ell}}{\pi^{\ell}C^{\ell}}+\left(1-p\right)\frac{C^{\ell}}{\pi^{I}C^{I}}\right]\tag{97}\end{equation}\] \[\begin{align} \left(\pi^{\ell}-1\right)\pi^{\ell}\left(C^{\ell}+G^{\ell}\right)-\frac{\varepsilon-1}{\phi}\left(\chi N^{\ell}C^{\ell}-1\right)N^{\ell}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\nonumber \\ -\frac{1}{1+r_{\ell}}\left[p\left(\pi^{\ell}-1\right)\pi^{\ell}\left(C^{\ell}+G^{\ell}\right)+\left(1-p\right)\left(\pi^{I}-1\right)\pi^{I}\frac{C^{\ell}}{C_{I}}\left(C^{I}+G^{ss}\right)\right]=0\tag{98}\end{align}\] \[\begin{equation} N^{\ell}=\left(C^{\ell}+G^{\ell}\right)\left(1+\frac{\phi}{2}\left(\pi^{\ell}-1\right)^{2}\right)\tag{99}\end{equation}\] Here, we impose the condition that \(R^{\ell}=1\), which we verify.

Given \(C^{I}\) and \(\pi^{I}\), equations (97) through (99) define a mapping from \(\pi^{\ell}\) to \(C^{\ell}\) and \(N^{\ell}\). Now, we can express the left-hand side of equation (98) as a function of \(\pi^{\ell}\). We denote this function by \(f^{I_{-1}}\left(\pi^{\ell};\pi^{I},C^{I}\right)\). There are two functions, \(f^{I_{-1}},\) conditional on the \(\pi^{I},C^{I}\) associated with the period \(I\) continuation equilibria, \(\mathcal{A}\) and \(\mathcal{B}\).

Figure 7 displays both \(f^{I_{-1}}\) functions for a range of values of \(\pi^{\ell}\); see the dotted and dot-dashed lines. We chose the range of \(\pi^{\ell}\) so that the graph only displays zeros of \(f^{I_{-1}}\) that correspond to equilibria. We find two equilibria corresponding to the \(f^{I_{-1}}\) associated with \(\mathcal{A}\) (see \(A\) and \(B\) in Figure 7) and one associated with \(\mathcal{B}\) (see \(C\) in Figure 7). So there are three REEs with forward guidance. The two REEs without forward guidance can be seen in the solid line in Figure 7 (we take this curve from Figure 5).

Figure 7: REE Equilibria at the ZLB with and without Forward Guidance
Notes: The solid line reproduces the solid line in Figure 5 and corresponds to the case of no forward guidance. The dashed and dot-dashed lines correspond to the case of forward guidance. The dashed line corresponds to the case in which the economy goes to point \(\mathcal{B}\) in the period of the switch in \(r_{t}\) to \(r^{ss}\) (period \(I\)). It crosses the zero line more than once, but the other crossing involves very high inflation and is not an equilibrium because the present value of intermediate goods monopolists is negative. The dot-dashed line corresponds to the case in which the economy goes to point \(\mathcal{A}\) in period \(I\) (see Figure 6).

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G3 Calvo-Style Nominal Rigidities

In this sub-section, we work with a model of Calvo-style nominal rigidities, but that is otherwise the same as in the sub-section [subsec:Rotemberg-REEs].⁵¹ The nonlinear equations defining an REE in this model are \(\frac{1}{C_{t}}=R_{t}\frac{1}{1+r_{t}}\mathbf{E}_{t}\frac{1}{\pi_{t+1}C_{t+1}}\), \(F_{P,t}=\frac{C_{t}+G_{t}}{C_{t}}+\frac{1}{1+r_{t}}\xi_{P}\mathbf{E}_{t}\pi_{t+1}^{\varepsilon}F_{P,t+1}\), \(Z_{P,t}=\chi N_{t}\left(C_{t}+G_{t}\right)+\frac{1}{1+r_{t}}\xi_{P}\mathbf{E}_{t}\pi_{t+1}^{\varepsilon-1}Z_{P,t+1}\), \(p_{t}^{\ast}N_{t}=C_{t}+G_{t}\), \(\tilde{p}_{t}=\frac{Z_{P,t}}{F_{P,t}}\), \(1=\left(1-\theta_{P}\right)\tilde{p}_{t}^{1-\varepsilon}+\xi_{P}\pi_{t}^{\varepsilon-1}\), \(\left(p_{t}^{\ast}\right)^{-1}=\left(1-\theta_{P}\right)\tilde{p}_{t}^{-\varepsilon}+\xi_{P}\pi_{t}^{\varepsilon}\left(p_{t-1}^{\ast}\right)^{-1}\), \(R_{t}=\max\left\{ 1,\beta^{-1}+\alpha_{\pi}\left(\pi_{t}-1\right)\right\} .\) When \(r_{t}=r^{ss}\), we assume that the system converges to the target-inflation steady state. In that solution, \(p_{t-1}^{\ast}\) is the only state variable. In that solution, let \(X_{t}=X^{ss}\left(p_{t}^{\ast}\right)\). Because \(r_{t}=r^{ss}\) is an absorbing state, we can solve for these decision rules without reference to the solution when \(r_{t}=r^{\ell}\).

When \(r_{t}=r^{\ell}\), the nonlinear equations defining an REE in this model are the same as in the previous paragraph, but we can write the first three as \[\begin{align*} \frac{1}{C_{t}} & =R_{t}\frac{1}{1+r^{\ell}}\frac{p}{\pi_{t+1}C_{t+1}}+R_{t}\frac{1}{1+r^{\ell}}\frac{1-p}{\pi^{ss}\left(p_{t}^{\ast}\right)C^{ss}\left(p_{t}^{\ast}\right)}\\ F_{P,t} & =\frac{C_{t}+G_{t}}{C_{t}}+\frac{1}{1+r^{\ell}}\xi_{P}\left(p\pi_{t+1}^{\varepsilon-1}F_{P,t+1}+\left(1-p\right)p\left[\pi^{ss}\right]^{\varepsilon-1}\left(p_{t}^{\ast}\right)F_{P}^{ss}\left(p_{t}^{\ast}\right)\right)\\ Z_{P,t} & =\chi N_{t}\left(C_{t}+G_{t}\right)+\frac{1}{1+r^{\ell}}\xi_{P}\left(p\pi_{t+1}^{\varepsilon}Z_{P,t+1}+\left(1-p\right)p\left[\pi^{ss}\right]^{\varepsilon}\left(p_{t}^{\ast}\right)Z_{P}^{ss}\left(p_{t}^{\ast}\right)\right). \end{align*}\] In the context of a nonfundamental shock, Mertens and Ravn, 2014 noted that this system of equations defines a nonlinear function of \(p_{t}^{\ast}\). Further, they noted that if the shock remained in this state for a long time then \(p_{t-1}^{\ast}\) would converge to a constant (and so would the other variables). Here, we have a fundamental shock, \(r_{t}\). However, the system still defines a nonlinear function of \(p_{t-1}^{\ast}\) if \(r_{t}=r^{\ell}\) for a long time then \(p_{t-1}^{\ast}\) would converge to a constant (and so would the other variables). If we consider only the point to which the system could converge if \(r_{t}=r^{\ell}\) for a long time, and if we only consider interior equilibria, these equations define a function \(f^{C}\left(\pi^{\ell}\right)\) that must be zero at any limit point of an REE.

Figure 8: f^C(π^ℓ) Corresponding to the Target-Inflation Steady-State Equilibrium
Note: The function, \(f_{C},\) is defined in the text. The dashed line is the case when \(G_{t}\) is high. Source: Authors’ calculations.

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Figure (8) displays the function \(f_{C}\left(\pi^{\ell}\right)\) for a range of values of \(\pi^{\ell}\). There are two values of \(\pi^{\ell}\) for which \(f_{C}\left(\pi^{\ell}\right)=0.\) We also consider the case when \(G^{\ell}=1.05\times G^{ss}\) (the dashed lines). As was the case in the model with Rotemberg-style nominal rigidities, there are two values of \(\pi^{\ell}\) for which \(f_{C}\left(\pi^{\ell}\right)=0.\) Additionally, only the crossings on the right are E-stable.

H The Learning Equilibrium Studied by Mertens and Ravn (2014)

In this appendix, we explore the findings of Mertens and Ravn, 2014 using the simple NK model with only sticky prices and parameter values that we study in our paper. Mertens and Ravn, 2014 also study a simple NK model with Calvo-style nominal rigidities and only sticky prices. They show that there exists a nonfundamental REE. In that equilibrium, the natural rate of interest is always equal to its steady state value. Similar to the stochastic process for \(r_{t}\) in section 6.2, the nonfundamental shock, \(S_{t}\), follows a stochastic process to that \[\begin{align} \Pr\left[S_{t+1}=1|S_{t}=1\right] & =p,\text{ }\Pr\left[S_{t+1}=0|S_{t}=1\right]=1-p,\tag{100}\\ \text{ }\Pr\left[S_{t+1}=1|S_{t}=0\right] & =0.\nonumber \end{align}\] Notice that \(S_{t}=0\) is an absorbing state. When \(S_{t}=1\), Mertens and Ravn, 2014 show that there is an REE where \(R_{t}=1\) while \(S_{t}=1\). In this section, we set \(p=0.9\).

H1 REE

When \(S_{t}=1\), the nonlinear equations defining an REE in this model are the same as in Appendix G3, but we can write the first three as \[\begin{align*} \frac{1}{C_{t}} & =R_{t}\beta\frac{p}{\pi_{t+1}C_{t+1}}+R_{t}\beta\frac{1-p}{\pi_{ss}\left(p_{t}^{\ast}\right)C_{ss}\left(p_{t}^{\ast}\right)}\\ F_{P,t} & =\frac{C_{t}+G_{t}}{C_{t}}+\beta\xi_{P}\left(p\pi_{t+1}^{\varepsilon-1}F_{P,t+1}+\left(1-p\right)p\left[\pi_{ss}\right]^{\varepsilon-1}\left(p_{t}^{\ast}\right)F_{P}^{ss}\left(p_{t}^{\ast}\right)\right)\\ Z_{P,t} & =\chi N_{t}\left(C_{t}+G_{t}\right)+\beta\xi_{P}\left(p\pi_{t+1}^{\varepsilon}Z_{P,t+1}+\left(1-p\right)p\left[\pi_{ss}\right]^{\varepsilon}\left(p_{t}^{\ast}\right)Z_{P}^{ss}\left(p_{t}^{\ast}\right)\right). \end{align*}\] Mertens and Ravn, 2014 noted that this system of equations defines a nonlinear function of \(p_{t}^{\ast}\). Further, they noted that if \(S_{t}=1\) for a long time then \(p_{t-1}^{\ast}\) would converge to a constant (and so would the other variables).

H2 Euler-Equation Learning

Mertens and Ravn, 2014 consider a learning equilibrium using the Euler equation approach to learning (see Appendix C for a discussion of Euler equation learning). To implement the Euler equation approach with the zero lower bound, Mertens and Ravn, 2014 log-linearize the equilibrium conditions around the target-inflation steady state but preserve the nonlinearity in the monetary policy rule. Mertens and Ravn, 2014 assume that agents know that \(\widehat{C}_{t}=\widehat{\pi}_{t}=0\) when \(S_{t}=0\). When \(S_{t}=1\), the system of equations an be written as \[\begin{align*} \widehat{C}_{t} & =p\mathbf{E}_{t}\widehat{C}_{t+1}+p\mathbf{E}_{t}\widehat{\pi}_{t+1}-\max\left\{ \beta\log\left(\beta\right),\beta\alpha_{\pi}\widehat{\pi}_{t}\right\} \\ \widehat{\pi}_{t} & =\frac{\left(1-\beta\xi_{P}\right)\left(1-\xi_{P}\right)}{\xi_{P}}\left[\left(1+C\right)\widehat{C}_{t}+G\widehat{G}_{t}\right]+\beta p\mathbf{E}_{t}\widehat{\pi}_{t+1} \end{align*}\] In a learning equilibrium, beliefs are set so that \(\mathbf{E}_{t}\widehat{C}_{t+1}=m_{C,t-1}\) and \(\mathbf{E}_{t}\widehat{\pi}_{t+1}=m_{\pi,t-1}\). These values are then updated according to \[m_{i,t}=m_{i,t-1}+\gamma_{t}\left(\widehat{i}_{t}-m_{i,t-1}\right).\] Mertens and Ravn, 2014 use constant gain learning (\(\gamma_{t}=0.1\)). As in the rest of our paper, we use decreasing gain and set \(\gamma_{t}=\frac{1}{t+1}\).

As in Mertens and Ravn, 2014 we define the government purchases multiplier to be \(\frac{\Delta C_{t}+\Delta G_{t}}{\Delta G_{t}}\) where \(\Delta\) indicates the difference relative to the case when \(G_{t}=0\). Mertens and Ravn, 2014 consider two experiements. In the first experiment, Mertens and Ravn, 2014 calculate the government purchases multiplier in the REE. In the second experiment, Mertens and Ravn, 2014 calculate the size of the government purchases multiplier under learning. To calculate the multiplier under learning, Mertens and Ravn, 2014 set \(\widehat{G}_{t}=0\) and initial beliefs about consumption and inflation to be somewhat above their REE values. They then calculate a path for \(C_{t}\) . Then Mertens and Ravn, 2014 set \(\widehat{G}_{t}=\widehat{G}>0\) and also change initial beliefs to be lower, similar to the way initial beliefs are lower in the REE when \(\widehat{G}_{t}=\widehat{G}>0\). The they calculate an alternative path for \(C_{t}\) and compute the multiplier.

We consider a third experiment in which \(\widehat{G}_{t}=\widehat{G}>0\) and initial beliefs are set to the same initial value as in the learning equilibrium when \(\widehat{G}_{t}=0\). Figure 9a displays the value of the government purchases multiplier in these three experiments under the assumption that \(S_{t}=1\) for the entire duration of the period shown. Note that the REE multipler is low. As in Mertens and Ravn, 2014, we find that the compound, second experiment in which beliefs and \(G_{t}\) change exhibits a low multiplier. In the third experiment, when we do not change initial beliefs for the higher value of \(G_{t}\), the multiplier is larger than unity. Figure 9a is also consistent with the findings of Mertens and Ravn, 2014 that Euler equation learning is slow. The value of \(b\) for the system of equations associated with Euler equation learning is \(1.03\), which is consistent with expectations diverging from the nonfundamental equilibrium (as reported by Mertens and Ravn, 2014), but at a slow rate.

Figure 9: Size of Government Purchases Multiplier
(a) Euler Equation Learning
(b) Anticipated Utility
Note: The figure shows the size of the government purchases multiplier under three experiments in our simple NK model with only sticky prices. Experiment 1 is the REE. Experiment 2 changes intial expectations when \(G_{t}\) changes, as in . Experiment 3 leave initial expectations unchanged when \(G_{t}\) changes. Panel (a) is constructed under Euler equation learning, as in . Panel (b) is constructed under anticipated utility as outline in Appendix E. The figure displays the value of the government purchases multiplier in learning equilibria and the REE under the assumption that \(S_{t}=1\) for the entire duration of the period shown. This is a low probability event that would be surprising.
Source: Authors’ calculations.

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H3 Anticipated Utility

Straightforward modifications of the approach outlined in Appendix E allow us to solve for the linear approximation to the learning equilibrium under anticipated utility using the methodology of Appendix B. Note that we solve for the limit point of \(p_{t}^{\ast}\) (and the other variables) while \(S_{t}=1\) linearize the equations around this limit point while \(S_{t}=1\).⁵² We also linearize the equations once \(S_{t}=0\) around the target inflation steady state.

We conduct the same three experiments that we described in the previous section. There are dynamics associated with the REE in Figure 9b because \(p_{t}^{\ast}\) is a state variable of the model and it is linearized around a value different from unity. The value of the multiplier in the REE is somewhat different from the value in Figure 9a because we are linearizing around the nonlinear REE, rather than around the target inflation steady state. As in the previous sub-section, we find that the compound, second experiment in which beliefs and \(G_{t}\) change exhibits a low multiplier but the multiplier is larger when we do not change initial beliefs (experiment 3).

It is notable that the dynamics in Figure 9b are also consistent with slow learning under anticipated utility in the compound second experiment. The value of \(b\) under anticipated utility is \(1.7\). Because \(b>1\), the system diverges from the sunspot REE that Mertens and Ravn, 2014 study, however, as discussed above, this happens slowly for beliefs that are close enough to the REE in the compound second experiment. For experiment 3 (when we do not change beliefs at the same time that we change \(G_{t}\) at the ZLB), divergence from the REE happens quickly because initial beliefs when \(\widehat{G}_{t}>0\) are not so close to the REE. In this case, we only show the first few quarters of the simulation because the ZLB would no longer be binding after that period in the simulation and the linearization of the model would then have to incorporate the interest rate response from the Taylor rule.

Footnotes