Abstract
, where 
is real
balance, 
is expected value of 
and 
is the
offer curve as in Grandmont (1985) in which agents use a simple
adaptive learning rule of the form 
. These two relations
yield a one-dimensional adaptive learning dynamics
and (ii) to compare the adaptive learning dynamics generated by
the equation (1) to a simple least squares learning scheme where
agents would know the law of the dynamics 
, and choose 
so as to minimize the sum of
the squares of past forecasting errors, that is,
First, we analyze bifurcations and chaos of the learning dynamics
(1), which would clearly distinguish between the influence of the
income effect and that of the error correction parameter . We demonstrate that (i) if the income effect is weak, then
the stationary monetary equlibria are globally stable when
is less that 2, and (ii) if the income effect is strong,
then cyclic and chaotic learning dynamics occur under 
. In other word, we demonstrate that the income effect and the
error correction parameter offers the key to an understanding of
the learning dynamics.
Then we illustrate that (iii) a least squares learning (2) for
the error correction parameter , is able to lead to the
stationary moneraty equilibrium, provided that the endogenous
fluctuations are caused by the strong overractionary expectations,
that is, 
, and on the contrary (iv) the least squares
learning cannot lead to the stationary monetary equilibrium,
provided that the endougenous fluctuations are caused by the
strong income effect.
Finally we show that (v) a stabilization policy can stabilize an
unstable learning dynamics when the least squares learning cannot
lead the economic system to the stationary monetary equilibrium.