MODELING THE EVOLUTION OF EXPECTATIONS AND UNCERTAINTY IN GENERAL EQUILIBRIUM

• Published date: 01 May 2016
• DOI: http://doi.org/10.1111/iere.12174
• Date: 01 May 2016
• Author: Leonardo Melosi,Francesco Bianchi

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 2, May 2016

MODELING THE EVOLUTION OF EXPECTATIONS AND UNCERTAINTY

IN GENERAL EQUILIBRIUM∗

BYFRANCESCO BIANCHI AND LEONARDO MELOSI1

Cornell University, Duke University, CPER, and NBER, U.S.A.; Federal Reserve Bank of

Chicago, U.S.A.

We develop methods to solve general equilibrium models in which forward-looking agents are subject to waves of

pessimism, optimism, and uncertainty that turn out to critically affect macroeconomic outcomes. Agents in the model

are fully rational and conduct Bayesian learning, and they know that they do not know. Therefore, agents take into

account that their beliefs will evolve according to what they will observe. This framework accommodates both gradual

and abrupt changes in beliefs and allows for an analytical characterization of uncertainty. We use a prototypical Real

Business Cycle model to illustrate the methods.

1. INTRODUCTION

A centerpiece of the rational expectations revolution is that economic outcomes critically

depend on agents’ beliefs about future events. Most general equilibrium models are solved

assuming that agents have perfect knowledge about the stochastic properties of all the real-

ized events. These are certainly strong restrictions imposed upon the dynamics of beliefs. For

instance, the private sector is likely to have limited information about the future path of pol-

icymakers’ decisions, the dynamics of dividend payments, or the likely duration of observed

changes in the returns to labor and capital. These assumptions, in turn, influence the expec-

tations formation mechanism and hence the predictions we draw from rational expectations

models.

In this article we develop methods to solve dynamic general equilibrium models in which

forward-looking and fully rational agents learn about the stochastic properties of realized events.

This modeling framework captures waves of pessimism, optimism, and uncertainty that turn

out to critically affect macroeconomic outcomes. Such outbursts of pessimism, optimism, and

uncertainty may happen abruptly or may gradually unfold over a long period of time in response

to the behavior of other agents or to the realizations of economic outcomes. Furthermore, this

framework is well suited to study the effects of shocks to beliefs and agents’ uncertainty in

Dynamic Stochastic General Equilibrium (DSGE) models. All results are derived within a

modeling framework suitable for structural estimation that will allow researchers to bring the

models to the data.

∗Manuscript received September 2014; revised December 2014.

1We thank the editor, Jesus Fernandez-Villaverde, and two anonymous referees for excellent comments and sugges-

tions. We also thank Fernando Alvarez, Gadi Barlevy, Jeff Campbell, Martin Eichenbaum, Eric Leeper, Alessandro

Pavan, Monika Piazzesi, Martin Schneider, Dan Waggoner, and Tao Zha for very useful comments and discussions.

We also wish to thank participants at Macro seminar of the Federal Reserve Bank of Chicago, Purdue University, the

SED 2013 in South Korea, and the Second Conference on Rational Inattention and Related Theories held at Oxford

University. Todd Messer provided excellent research assistance. Francesco Bianchi gratefully acknowledges financial

support from the National Science Foundation through grant SES-1227397. The views in this article are solely the

responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of

Chicago or any other person associated with the Federal Reserve System. Please address correspondence to: Francesco

Bianchi, Department of Economics, Cornell University, 464 Uris Hall, Ithaca, NY 14853. Phone: (607) 255-5030. E-mail:

francesco.bianchi@cornell.edu.

717

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(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

718 BIANCHI AND MELOSI

The evolution of agents’ beliefs is modeled assuming the existence of different states of the

world or regimes that differ according to the statistical properties of the exogenous shocks or

based on the behavior of some of the agents in the model. Such regimes follow a Markov-

switching process, which may be correlated with other aspects of the model. For example, the

government could be more likely to inflate debt away when the level of spending is high. Agents

are assumed to observe economic outcomes, but not the regimes themselves. Agents will then

adopt Bayesian learning to infer which regime is in place. This will determine the evolution of

agents’ beliefs about future economic outcomes.

Our modeling framework does not rely on the assumption of anticipated utility that is often

used in models characterized by a learning process. Such an assumption implies that agents

forecast future events assuming that their beliefs will never change in the future. Instead, agents

in our models know that they do not know. Therefore, when forming expectations, they take

into account that their beliefs will evolve according to what they observe in the future. In our

context, it is possible to go beyond the anticipated utility assumption because there are only a

finite number of relevant beliefs, and they are strictly linked to observable outcomes through

the learning mechanism in a way that we can keep track of their evolution. It should also be

noted that the proposed approach is based on agents being fully rational and hence their beliefs

always being consistent in equilibrium. Rationality in our approach is essential in that it puts

discipline on beliefs so as to make it possible to draw precise predictions from economic models.

The proposed model framework is flexible enough to encompass both abrupt and gradual

changes in beliefs. For example, augmenting the modeling framework with signals about the

regime in place allows one to capture the sharp effect of news on the evolution of the economy

or to study the macroeconomic implications of changes in animal spirits about future events. At

the same time, through the learning process, we can model situations in which agents’ beliefs

gradually change in response to the behavior of other agents or the realizations of stochastic

events. This sluggish adjustment of public expectations is hard to reproduce through rational

expectations models in which the functioning of the whole economy is common knowledge

among agents. Furthermore, the methods introduced in this article can be combined with tech-

niques developed by Bianchi (2016) to obtain an analytical characterization of the evolution of

uncertainty. Bianchi (2016) shows how to compute measures of expectations and uncertainty in

Markov-switching models with perfect information. The combination of these methods endows

researchers with a convenient toolkit to take dynamic general equilibrium rational expectations

models to the growing amount of data on macroeconomic uncertainty.

We show how to apply these methods using a prototypical Real Business Cycle (RBC) model.

In the model, total factor productivity (TFP) growth can assume two values: high or low. For

each value of TFP growth, we allow for a long-lasting and a short-lasting regime. Therefore,

although agents can observe the current TFP growth rate, they are uncertain about its future

values, because they do not know if the current value is likely to last for a short time or for

a long time. We consider a wide range of specifications, allowing for smooth transitions or

abrupt changes in agents’ optimism about future realizations of TFP growth. Each of these

different specifications can be easily captured with the appropriate transition matrix governing

the evolution of TFP growth. This has the important implication that the dynamics of pessimism,

optimism, and uncertainty are consistent in equilibrium. Whenever a short-lasting regime is in

fact realized, with the benefit of hindsight, agents’ beliefs turn out to overreact to the regime

change because agents always take into account the possibility that the economy entered a

long-lasting regime. However, if, in fact, the regime is long-lasting, it takes time for agents’

beliefs to line up with the actual realization. This implies that although agents are fully rational,

their beliefs are generally misaligned with respect to the actual state of the economy. Such a

misalignment is found to substantially influence consumption and capital allocation in the RBC

model.

We show that the assumption of anticipated utility is not innocuous in the class of models

studied in this article. This is because anticipated utility leads to periods of overpessimism

and overoptimism with respect to the case in which agents are fully rational. These mistakes

MODELING THE EVOLUTION OF BELIEFS 719

cumulate over time because of agents’ investment decisions with respect to physical capital

accumulation. These findings are different from Cogley and Sargent (2008), who argue that

there are only minor drawbacks stemming from the anticipated utility assumption as long as

precautionary motives are not strong. Two reasons explain this discordance of results. First,

the learning problem is different in the two papers. In Cogley and Sargent (2008) agents have

to learn the transition matrix governing the evolution of regime changes, although in our case

agents have to learn the regime that is in place. Second, they consider a model with no capital

accumulation. In our setting, sluggish changes in physical capital imply that mistakes that were

done in the past due to overpessimism or overoptimism accumulate over time and cannot be

immediately undone.

The methods developed in this article are based on the idea of expanding the number of

regimes to take into account the learning mechanism. The central insight consists of recognizing

that the evolution of agents’ beliefs can be captured by defining an expanded set of regimes

indexed with respect to agents’ beliefs themselves. Once this structure has been imposed, the

model can be recast as a Markov-switching dynamic stochastic general equilibrium (MS-DSGE)

model with perfect information. If regime changes enter additively, the model can be solved

with standard solution methods such as gensys (Sims, 2002) and Blanchard and Kahn (1980),

following the approach described in Schorfheide (2005) and Liu et al. (2011). If instead regime

changes enter multiplicatively, the model can be solved with any of the methods developed for

solving MS-DSGE models, such as Davig and Leeper (2007), Farmer et al. (2009), Cho (2015),

and Foerster et al. (2013).

In both cases, the resulting solution is suitable for likelihood-based estimation. This is be-

cause even if the final number of regimes is very large, there is a tight link between observable

outcomes and the evolution of agents’ beliefs. In other words, the transition matrix govern-

ing the joint dynamics of the economy and agents’ beliefs is highly restricted. For example,

Bianchi and Melosi (2014b) apply these methods and Bayesian techniques to estimate a model

in which agents are uncertain about the future stance of monetary policy. This article is therefore

related to a growing literature that models parameter instability to capture changes in the evo-

lution of the macroeconomy. This consists of two branches: Schorfheide (2005), Justiniano and

Primiceri (2008), Bianchi (2013), Davig and Doh (2014), and Fernandez-Villaverde and Rubio-

Ramirez (2008) introduce parameter instability in DSGE models, although Sims and Zha (2006),

Primiceri (2005), and Cogley and Sargent (2005) work with structural VARs. Finally, to the

extent that we can model situations in which agents’ beliefs evolve in response to policy makers’

behavior, our work is also linked to papers that study how inflation expectations respond to

policy decisions, such as Mankiw et al. (2004), Nimark (2008), Del Negro and Eusepi (2011),

and Melosi (2014a, 2014b).

Schorfheide (2005) pioneers a method to estimate general equilibrium models in which agents

learn the realization of a discrete Markov-switching process that affects the constants of the

model-implied laws of motion. Specifically, they have to learn if the current central bank target

for inflation is high or low. Our work sharply differs from this contribution. First, our framework

can accommodate situations in which agents learn about regime changes that affect not only

the constant terms of the model, but also its autoregressive component. For example, Bianchi

and Melosi (2014b) use the proposed framework to estimate a model in which agents have

to learn about future policymakers’ behavior. Second, in our framework agents always have

enough information to infer what the current state of the economy is or what other agents

are doing: high or low growth, Hawkish or Dovish monetary policy, etc. Nevertheless, agents

face uncertainty about the statistical properties of what they are observing. For example, agents

could be uncertain about the persistence and the destination of a particular state. As we shall

show, in a model in which agents are forward looking these sources of uncertainty have pervasive

effects on the law of motion of the economy. Third, as previously pointed out, in our framework

agents know that they do not know. In other words, when forming their expectations, they take

into account that their beliefs will evolve according to what they will observe in the future. In