Expectation Formation and Endogenous Fluctuations in Aggregate Demand

It is apparent that the realized utility depends on two factors. First of all,
it depends on the state of nature, i.e., the true values of x_t+ι, p_{t +} ι and r_t+,
and on ν_{xt + 1} a particular signal received for x_t+,. However, when an agent
decides to purchase a signal she neither knows what signal she will receive
nor she knows the true state of nature. Nevertheless, she must decide on the
quality of a signal she wishes to receive before any uncertainty is resolved.
Observe that conditional on a given state of nature x_t+, the loss ε² resulting
from insufficient consumption smoothing is a function of a specific signal drawn
ν_{xt + 1}. Moreover, both the signal and the resulting loss are unknown ex ante.
Consequently economic agents must form expectations with the regard of the
potential loss ε². Let G {y_{xt + 1} ∣x_t+ι) be the conditional, also obtained by Bayes
rule, distribution function of a given value of a signal given that the state of
nature is x_t+,. Then the "expected" level of realized utility, again performing
for analytic convenience certainty equivalence calculation, conditional on a given
state of nature x_t+ι is given by

Observe that the level of realized utility depends on the true state of nature x_t+ι
and on the expected error resulting from suboptimal consumption smoothing.
Naturally, different realizations of x_t+1 will lead to different levels of realized
utility. Unfortunately, the value of x_t+1 is also ex ante unknown, therefore,
the unconditional "expected" realized utility can only be obtained by using the
underlying priors. A certainty equivalence evaluation leads to
where G (—) denotes the unconditional distribution of ν_xt+1 and the prior F_{xt + 1} is
used to determine E_t (l + x_t+∙∣)².

The specific form of the distribution of signals (26) implies that the condi-
tional expected value of x_t+ι can be expressed as

E(x.+i ∣Vχt + 1 ) — qtVxt + 1 + (l — qt) Etxt+1                  (29)

and the conditional variance takes the form

V (x_t+1 ∣Vχt + 1) — qt (l — qt)(νχt+1 — E.x.₊i)² + (l — qt) σX_t+1.     (30)

Therefore, the realized utility under the assumption of certainty equivalence
behavior before a specific signal of quality q_t is purchased is given by

^{u — A}t ^{+ lo}g^(Et^{(l + x}t+ι^{)2 — (l — q}t‰X_{t + 1} ⁾-

Naturally, the more informative a given signal is the higher the "expected"
realized utility. Similarly, the larger the ex ante uncertainty the smaller the
"expected" realized utility. Recall that signals can be obtained only at some

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