Expected price is the previous expected price plus a fraction (a) of the previous error in expectation. Some algebra shows that expected price is also an exponentially decaying function of past prices. The adaptive expectations model corresponds to simple exponential smoothing of observed prices.
By a suitable transformation, output is a function of lagged price and lagged output. Similarly, output can be expressed as previous output plus a partial adjustment of the difference between anticipated and previous output, the technical rigidities model. The final result is that output is a function of price lagged one period and output lagged one and two periods.