Adaptive expectation models are ways of predicting an agent’s behaviour based on their past experiences and past expectations for that same event. They are first used by Irving Fisher in his book “The Purchasing Power of Money”, 1911, and further developed in the 1940s and 1950s, especially by Phillip Cagan in his article “The Monetary Dynamics of Hyper-Inflation”, 1956 and, most famously, by Milton Friedman in 1957, in his book “A Theory of the Consumption Function”.
Models are usually based around the following formula:
Et xt+1 is our expectation (E) in year t for a variable x in the year t+1. It is based on our expectation from the year before (t-1) for variable x in our current year, and a weighted proportion of our past expectations. Remember that our expectations from last year were, in turn, based on those of the year before, so all our expectations from the first time we ever dared assume anything are contained within the equation.
λ will depend on how much we were off last year versus this year: if our predictions are proving to be volatile, we will be more likely to adjust them. It will also depend on how much groundbreaking new information we have received, rendering previous expectations useless. Basically, it depends on how much we feel we might have been off last year when we predicted things for next year.
The easiest way to know how adaptive expectations work, is to understand the expectations-augmented Phillips curve. Using also this same curve, it is also easy to understand how rational expectations work.