Expectation Formation and Endogenous Fluctuations in Aggregate Demand

remains deterministic, as the law of large numbers allows to establish ʃ ν^tdi =
Qtxt+1 +(1 — Qt) E_tx_t+ι, and is given by

^kt+ι = 2 (^{1 -} (^Qt2^xt + i ⁺ (^{1 - q2}) ^Et^xt+i)) ∙               (32)

Moreover, in equilibrium the level of investment is proportional to the true
value³ of x_t+ι, i.e., x_t+ι = 2ψk_t+ι. Therefore, the equilibrium evolution of
variable x assumes the form

ψ ^ψ (^{1 -} Qt²) _τ.

^xt+1 = 1 + Ψq2    1 + Ψq2 ^Etxt+1^

Recall that all agents receive the same amount of income in the first period of
their lives. Therefore, even though they save different fractions of their incomes,
dependent on specific signals they draw, the average propensity to save in the
aggregate remains constant and is equal to the level of investment in physical
capital, i.e., β_t = k_t+ι. Therefore, the level of output in terms of the price of
the consumption good at time t is given by

^{(1 - φβ}t ⁾ _kα

(1 - βt)^{α kt} ∙

Clearly, the level of output changes when β_t changes. The latter may change for
either a change in the expected value of x_t+ι or for a change in the precision of
signals Q_t. Naturally, whenever economic agents are ex ante optimistic about the
future the smaller the fraction of income saved and the larger the equilibrium
output. Similarly, a change in the informativeness of signals triggers a change
in the level of savings and in the level of output. However, changes if they do
happen are a result of an endogenous decision of rational agents.

4.2 Dynamics

The description of the equilibrium is now complete for a given set of prior
distributions. Nevertheless, priors and the process of their formation have not
been specified. Unfortunately, economic theory remains silent on the issue of
the origin of priors. Therefore, there is no natural benchmark to turn to. The
paper takes a subjective approach and assumes that priors are formed on the

³Observe that the aggregate investment can be used for perfect identification of variable
æt+1 and hence economic agents by observing the level of investment ^t+1 can learn the true
value of ^t+1∙ Naturally, if agents learn the true value of ^t+1 then they can use it in the
process of expectation formation and they would behave as if they had perfect foresight ability.
This shortcoming can be addressed in either of the two ways. First of all, it can be assumed
that information on aggregate variables arrives with a lag and hence ^t+1 cannot be used
for identification of ^t+1∙ Alternatively, it could be assumed that agent i receives a fraction
ωi of economy wide profits where ʃ ωi di = 1 and ωi is unknown and random. Then ω^t+1
becomes the relevant variable and expectations must be formed with regard to ωi^t + 1 rather
than ^t+1 and then the overall level of investment stops being fully informative. The paper
does not follow this alternative approach in order not to expand the dimensionality of the
model.

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