Expectation Formation and Endogenous Fluctuations in Aggregate Demand

Finally, combining the above relationships it is possible to show that the
equilibrium level of the price of the final consumption good is given by

1      vint

* = ~<-T->““t.                   (1»)

where φ =(1 — γ (1 — α)).

It is natural to define the level of output in this model as the sum of incomes
of young and old agents, i.e.,

GDPt = yι,t + y2,t∙

The income of the young is simply equal to the wage, yι_jt = w_t, and the income
of the old comes from two sources: earnings received from physical capital rental
and profits, i.e., y^ _t = r_tk_t + π_t. Moreover, the demand for the consumption
good comes from two sources: the old devote all of their wealth towards pur-
chases of the consumption good and the young devote a fraction 1 — β_t of their
income towards consumption, hence

^Dt = ^2,t ^{+ (1 — β}t⁾ yι,t∙                          ⁽20⁾

Therefore, the level of output is given by

GDP_t = ¹ - ^φβt w_t.
^t 1 — φ ^t

Finally, the level of output expressed in terms of the price of the consumption
good is given by

= ^{(1 — φβ}t⁾ _kα                             (₂₁)

^yt =(1 — β_t)^{α kt ,                                (21)}

for completeness recall that the employment in the intermediate goods sectors
equals l^l_t^nt = 1 — β_t.

Equation (21) completes the description of the intratemporal equilibrium.
Moreover, the last equation illustrates that the aggregate output not only de-
pends on the supply side characteristics, i.e., the technology and the resources
available, the amount of physical capital k_t, but also on the marginal propensity
to save β_t. In other words, not only the supply side determines output, but also
the composition of aggregate demand influences the aggregate output. The last
finding stems from fact that the sectors of intermediate goods are monopolisti-
cally competitive whereas the investment good sector is perfectly competitive.
Such a structure leads to the existence of aggregate demand externality, which
in turn influences aggregate variables. Young agents by changing the amount
of income saved, affect the demand for the consumption good. However, by
affecting the demand they affect profits in the intermediate goods sectors. A
change in the profits in the intermediate goods sectors affects the income of the
old agents, which in turn affects the demand for the consumption good, which

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