Expectation Formation and Endogenous Fluctuations in Aggregate Demand

The Evolution of Output and the Precision of Signals in Time when 2                               ψ a(1)>1+ ψ

0.53

0.525

0.52

0.515

0.51

0.505

0.5

0                   20                  40                  60                  80                  100                  120

time

0                   20                  40                  60                  80                  100                  120

time

Figure 6: The Evolution of Output when 2ψα (1) > 1 + ψ.

In summary, the economy does not settle into a steady state. The path
it follows is oscillatory and is chosen in equilibrium. Specifically, the form of
expectation formation is the key determinant of the path. The quality of sig-
nals purchased determines the size of the intertemporal errors. Moreover, it
defines the characteristic of the underlying equilibrium dynamic equation (33).
If agents choose to be relatively well informed, q_t close to one, then the coeffi-
cient in front of E_tx_t+1 is less than one. This implies that the economy follows
a convergent, albeit oscillatory, path, which terminates in a stable, given a high
choice of q_t, steady state. However, if complete convergence were to occur priors
would become more informative. Specifically past realized values would have to
be close to the true realized value and the variance would become arbitrarily
small. Moreover, the quality of signals depends on the variance of the underlying
prior. Therefore, economic agents would stop learning as the priors would be
sufficient to predict the future nearly with certainty and learning is costly. This,
however, would change the nature of the law of motion of variable x . For q_t
low enough, when there is no learning, the law of motion describes an explosive
path. Therefore, convergence would lead economic agents to choose not to learn,
but that would push the economy out of the convergence regime and would put
it on an explosive path into a divergence regime. This would, however, ruin
the informativeness of priors, their variance would increase, and would again
make agents learn. However, learning would again lead to convergence and the
cycle would repeat itself indefinitely. In equilibrium the economy bounces back
and forth between two regimes. Economic agents either learn and this leads
to convergence or rely on past observations and this leads to divergence. In
equilibrium, the two regimes alternate.

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