In this paper, we investigate real-time behavior of constant-gain stochastic gradient (SG) learning, using the Phelps model of monetary policy as a testing ground. We find that whereas the self-confirming equilibrium is stable under the mean dynamics in a very large region, real-time learning diverges for all but the very smallest gain values. We employ a stochastic Lyapunov function approach to demonstrate that the SG mean dynamics is easily destabilized by the noise associated with real-time learning, because its Jacobian contains stable but very small eigenvalues. We also express caution on usage of perpetual learning algorithms with such small eigenvalues, as the real-time dynamics might diverge from the equilibrium that is stable under the mean dynamics.

We study the conditions that ensure rational expectations equilibrium (REE) determinacy and expectational stability (E-stability) in a standard sticky-price model augmented with the cost channel. We allow for varying degrees of pass-through of the policy rate to bank-lending rates. Strong cost-side effects limit the size of the policy rate response to inflation that is consistent with determinacy, so that inflation-targeting policies may not be capable of ensuring REE uniqueness. In this case it is advisable to combine policy rate responses to inflation with an appropriate reaction to the output gap and/or firm profitability. The negative reaction of real activity and asset prices to inflationary shocks adds a negative force to inflation responses that counteracts the borrowing cost effect and prevents expectations of higher inflation from becoming self-fulfilling.

We investigate the stability properties of Muth's model of price movements when agents choose a production level using replicator dynamic learning. It turns out that when there is a discrete set of possible production levels, possible stable states and stability conditions differ between adaptive learning and replicator dynamic learning.

We study adaptive learning in economic environments subject to recurring structural change. Stochastically evolving institutional and policymaking features can be described by regime-switching models with parameters that evolve according to finite state Markov processes. We demonstrate that in nonlinear models of this form, the presence of sunspot equilibria implies two natural schemes for learning the conditional means of endogenous variables: under mean value learning, agents condition on a sunspot variable that captures the self-fulfilling serial correlation in the equilibrium, whereas under vector autoregression learning (VAR learning), the self-fulfilling serial correlation must be learned. We show that an intuitive condition ensures convergence to a regime-switching rational expectations equilibrium. However, the stability of sunspot equilibria, when they exist, depends on whether agents adopt mean value or VAR learning: coordinating on sunspot equilibria via a VAR learning rule is not possible. To illustrate these phenomena, we develop results for an overlapping-generations model and a New Keynesian model.

This study examines the expectational stability of rational expectations equilibria (REE) under alternative Taylor rules when trend inflation is nonzero. We find that when trend inflation is high, the REE is likely to be expectationally unstable. This result holds true regardless of the nature of the data (such as contemporaneous, forecast, or lagged data) introduced in the Taylor rule. Our results suggest that high macroeconomic volatility during the period of high trend inflation can be explained well by introducing the concept of expectational stability.

This note studies fiscal–monetary policy interactions in an endogenous growth model with multiple assets. The “growth-rate Laffer curve” clarifies an important tension between economic growth and government revenue and reveals that higher economic growth does not always finance a larger budget deficit. There are two Pareto-ranked balanced-growth equilibria, which can both be E-stable. Although fiscal policy can eliminate the expectational indeterminacy, it rules out the equilibrium with a higher growth rate and higher welfare. Near the lower bound of the nominal interest rate, an arbitrarily small budget deficit will select the low-growth equilibrium to be the unique E-stable equilibrium.

By endowing his agents with simple forecasting models, or representations, M. Woodford (“Learning to Believe in Sunspots,” Econometrica 58, 277–307, 1990) found that finite state Markov sunspot equilibria may be stable under learning. We show that common factor representations generalize to all sunspot equilibria the representations used by Woodford. We find that if finite state Markov sunspots are stable under learning then all sunspots are stable under learning, provided common factor representations are used.

Bullard and Mitra [Journal of Monetary Economics 49 (2002), 1105–1130] find that, in a New Keynesian economy without capital and under four variants of the Taylor rule, the Taylor principle is sufficient to guarantee both determinacy and E-stability of equilibrium in most cases. Xiao [Macroeconomic Dynamics 12 (2008), 22–49] claims that with capital and mild increasing returns the Taylor principle cannot guarantee either determinacy or E-stability with any of the four rules. In this paper we show that in the Calvo-type sticky price models a second-order condition for profit maximization must be satisfied in firms' pricing decision problem, and we point out that the examples given in Xiao's paper to support his conclusion violate this condition. After imposing this condition, we find that increasing returns have little effect on determinacy and E-stability under two of the policy rules but significant effects under the other two. These results are obtained in models both with and without capital.

We introduce increasing returns to scale into an otherwise standard New Keynesian model with capital, and study the determinacy and E-stability of equilibrium under Taylor-type interest rate rules. With very mild increasing returns supported by empirical research, the conventional wisdom regarding the design of interest rate rules can be overturned. In particular, the “Taylor principle” no longer guarantees either determinacy or E-stability of the rational expectations equilibrium.

The concept of E-stability is widely used as a learnability criterion in studies of macroeconomic dynamics with adaptive learning. In this paper, it is demonstrated, via a counterexample, that E-stability generally does not imply learnability of rational expectations equilibria. The result indicates that E-stability may not be a robust device for equilibrium selection.