Expectation Formation and Endogenous Fluctuations in Aggregate Demand

3.1 Consumer Problem

Consumers live for two periods. Young agents face a standard intertemporal
choice problem, i.e., young agents must decide how to allocate their life time
earnings between the two periods. Therefore, they maximize their utility

U (cι_jt,C2_ιt+ι) = log(c∙ι,_t) + log(c2_jt+ι)

subject to the budget constraints

PtC!₁t + s_t = yι₁t

к 7

St = P_tkt+1

Pt+1c2,t+1 = rt+1kt+1 +(1 - δ) P^k+1⅛t+1 + ∏t+ι,

where p_t, and p_t+1 denote the prices of the consumption good in period t and
period t + 1, respectively, p^k and p^k₊₁ are the prices of physical capital in the
two periods, and S_t denotes the amount saved in period t. Physical capital is
the only investment good hence in equilibrium the amount saved must be equal
to the purchases of physical capital, i.e., it must be p^kk_t+ι = S_t.

The calculations pertaining to the intertemporal considerations in this model
are by far more demanding than those in the original Diamond model. There-
fore, in order to simplify the exposition, to avoid a discussion on potential corner
solutions, the rate of depreciation of physical capital is assumed to be one, i.e.,
from now on it is assumed that δ = 1.The last assumption implies that physical
capital once purchased can only be rented out and cannot be resold.

Optimality along the intertemporal margin requires that

_ 1 ( p^kk ∏t+Λ                     _m

^S‘ = 2 C'' - —)'               ⁽⁸⁾

Therefore, the capital stock in the economy evolves according to the following
dynamic equation

k_t+1   ¹ f ^y-1t- - ^π+- ʌ .                      (9)

^t+1    2 <pk rt+ι)                           ⁽⁾

3.2 Intratemporal Allocation

Let k_t denote the capital stock available in period t (savings of the young in
period t — 1.) Moreover, let D_t denote the demand for the final consumption
good at time t and let D_t be the demand for the investment good, i.e., the
amount saved at time t. Finally, let β_t be the fraction of income saved at time
t.

The market for the investment good is perfectly competitive and the pro-
duction function for the investment good is linear hence the price of a unit of
physical capital is equal to the marginal cost, i.e., to the wage p^k = w_t. Recall
that the income of the young is equal to the wage and that the savings take

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