Expectation Formation and Endogenous Fluctuations in Aggregate Demand

basis of past experience. Specifically, it is assumed that economic agents observe
data up to T periods into the past and based on these observations design a
predictor. Then they use the predictor on past data and calculate prediction
errors. Finally priors are formed from the predictor and the prediction errors.

Observe that in systems that converge to an equilibrium currently observed
values become, by definition, with time better and better predictors of future
variables. Therefore, it is natural to start with priors formed from a very simple
predictor that approximates future value with currently observed values. In par-
ticular, this simple predictor approximates the value of x_t+1 with x_t. Therefore,
the expected value of x_t+-∖ is simply equal to x_t, i.e., E_tx_{t +} -∖ = x_t. The variance
is equal to the variance of errors, ε_t-t = x_t-t₊ι — x_t-t, made by the predictor on
past data. Under these assumptions of priors formation the dynamic equation
of x_t takes the form

ψ ^ψ (1 ^- Qt²)
xt+1 = ₁ , _{; 2}--₁ , , ₂ xt,

1 + ΨQt²     1 + ΨQt²

as E_tx_t+1 = x_t. The optimal quality of signal satisfies (31). Dynamic properties
depend on the magnitude of ψ. If ψ is smaller than 1 in absolute value then the
level of capital stock and x_t converge to an equilibrium, whereas if ψ is greater
than 1 in absolute value then the level of capital stock and x_t follow explosive
paths. The paper analyses the two cases in turn.

Case 1 Let the parameters of the model be such that

(34)

In this case x and consequently the level of capital stock follow a path that
converges to an equilibrium irrespective of a given choice of q2. In other words,
even if agents choose, along the equilibrium path, not to learn and form ex-
pectations in an adaptive manner the economy converges to an equilibrium.
Moreover, convergence implies by definition that with time the distance be-
tween x_t+ι and x_t approaches zero, i.e., ε_t-t → θ and σ²(_{t + 1} → 0. This means
that as time progresses it pays more and more to form expectations in an adap-
tive manner, i.e., q_t² → 0 as t → ∞. Therefore, as long as condition (34) is met
x and consequently the level of capital stock converge to an equilibrium and the
economy is stable in the long run. The long run level of capital stock and the
long run level of output are given by

1

2(1+ ψ)

(1 — Φβ*) _k*α

(1 — β* )^{α    ,}

where β* = k*. Figure (3) depicts the evolution of the level of output in an
economy when condition (34) is met.

It is clear that in this case the economy settles into a steady state in the
long run and only exogenous shocks can cause fluctuations in macroeconomic

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