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Third Quarter 2021, 
Vol. 103, No. 3
Posted 2021-07-01

Convergence to Rational Expectations in Learning Models: A Note of Caution


We show in a simple monetary model that the learning dynamics do not converge to the rational expectations monetary steady state. We then show it is necessary to restrict the learning rule to obtain convergence. We derive an upper bound on the gain parameter in the learning rule, based on economic fundamentals in the monetary model, such that gain parameters above the upper bound would imply that the learning dynamics would diverge from the rational expectations monetary steady state.

YiLi Chien is a research officer and economist at the Federal Reserve Bank of St. Louis. In-Koo Cho is a professor of economics at Emory University and Hayang Univerity and a research fellow at the Federal Reserve Bank of St. Louis. B. Ravikumar is a senior vice president, the deputy director of research, and an economist at the Federal Reserve Bank of St. Louis. The authors are grateful for helpful conversations with James Bullard, George Evans, Thomas J. Sargent, and participants of "Learning Week" at the Federal Reserve Bank of St. Louis in 2017. Financial support from the National Science Foundation is gratefully acknowledged.


Macroeconomic variables, such as inflation and interest rates, have been important objects of investigation in economics. A common assumption in economic models is that agents (households, firms, and governments) have rational expectations (RE) about these variables and thus accurately forecast the dynamics of the variables. An alternative approach limits the forecasting ability of the agents: They learn from the history of the variables. The central question of a learning model is whether the agents learn to behave over time in such a manner that the economy converges to the RE equilibrium in the long run. Thus, the convergence property of learning dynamics has been a key issue. Early examples that examine convergence to a RE equilibrium include Bray (1982), Bray and Savin (1986), Lucas (1986), Marcet and Sargent (1989a), and Woodford (1990).

We revisit the convergence issue using a simple two-period overlapping generations model of inflation—Example 1 in Bullard (1994). Ours is an endowment economy where money is the only store of value and monetary policy follows a constant money growth rule. When we do not impose RE, agents forecast inflation (or, equivalently, the rate of return on money) using a learning rule that is a convex combination of past expected inflation and actual inflation (a constant-gain algorithm), so the data on inflation affects learning. Based on the forecast, they choose consumption and real balances. These choices, in turn, affect the path of prices and generate feedback from learning to actual inflation. Other examples of such feedback include Bray (1982) on prices, Marcet and Sargent (1989b) on hyperinflation, and Evans and Honkapohja (1995) on business cycles.

Our results are as follows. First, we show numerically that, for some parameter configurations and initial conditions, the dynamic system produces cycles, nonmonotonic convergence, and a nonzero forecast error in the limit; that is, agents never learn the actual inflation. These simulation results show that the learning dynamics do not necessarily converge to the RE monetary steady state. (Besides the economic fundamentals such as endowments, preferences, and money growth, our simulations require initial conditions and one parameter of the learning rule—the constant gain or the weight that the learning rule places on the difference between actual and expected inflations, i.e., the forecast error.)

Second, we show that the gain parameter affects the convergence properties of the learning rule. We demonstrate that the convergence region around the monetary steady state varies with the gain. So, for some values of the gain parameter, the dynamic system monotonically converges to the steady state, while for other values it displays cycles. We derive a necessary condition that the gain has to satisfy in order to guarantee convergence to the monetary steady state. This condition depends on economic fundamentals. Alternative learning rules with a different information set break the dependence of the convergence property on the gain parameter.

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