Learning and Endogenous Business Cycles in a Standard Growth Model

I consider that the representative household derives his two-period ahead physical capital holdings and
future consumption stream from simple time invariant expectation rules consistent with the economy’s
balanced growth path:

x_t^j = (1 + e)x_t^j-1
ct^j = (1 + e)ct^j-1

where j ∈ Z+ - {1}. The lifetime utility function (1) defined over an infinite planned stream of consump-
tion satisfying forecasting rule (16) is denoted by υ: R²₊ → R, which can be written as:

∞

^υ (^{c0 c}D = u (^c0y + X ^βj u ((1 + e)^j-1 ^c1)

j=1

Proposition 2 Under a constant relative risk aversion instantaneous utility function u with a coefficient
of relative risk aversion γ ∈ R+ - {1}, and the assumption that β (1 + e)^1-γ < 1, the expected lifetime
utility function (17) can be written as: υ (c0, c1) = u (c⁰y + φu (c1) + ξ where φ = β/ [1 — β (1 + e)1 ^γ]
and ^ξ = ^-β2 (1 ^- (1 + e) ^γ´ / h(1 ^{- β}) (1 ^- γ) (1 ^{- β} (1 + e) ^γ´i.

c_t^{1 γ} (β(1 + e)^{1 γ}yj 1 — βj ¹ i / (1 — γ). For β (1 + e)^{1 γ} < 1, this geometric series converges
to βc1 ^γ/ £(1 — β(1 + e)^{1 γ}) (1 — γ)] — β/ [(1 — β) (1 — γ)]. Therefore, we can easily show that equation
(17) can be written as: υ (c0, c1) = u (c°) + φu (c1y + ξ, where φ = β/ [1 — β (1 + e)1 ^γ], and ξ =
^—β2 (1^— (1 +^{e)1 γ}´ / h(1^{— β}) (1^— γ) (1^{— β} (1 +^{e)1 γ}´i. ■

The expected budget constraint for the current period is given by equation (5) for j =0and w_t⁰ = wt,
r_t⁰ = rt . The next period’s expected budget constraint is obtained by plugging the forecasting rule (15)
into equation (5) for j =1:

c1= w1+ (R1 — δ — e — n — ne} x1 + π1                          (18)

Under bounded rationality, the representative household’s infinite horizon lifetime utility maximizing
problem consists of a succession of two-period constrained optimization problems in which planned con-
sumption plans for the current and the next period are derived at every time period t for t ∈ Z+ ∪ {0}
by maximizing (17) subject to (5) for j =0and w_t⁰ = wt , r_t⁰ = rt , (18), the zero profit conditions:
π_t⁰ = π_t¹ =0, along with the current and the next period’s expected non-arbitrage conditions: R_t =
r_t + δ, R_t¹ = r_t¹ + δ :

ν(x_t ,k_t ,B_t ) =ArgMax υ c_t⁰ ,c_t¹

ct⁰,ct¹

s.t. c_t⁰+(1 + n)x_t¹= wt +(1+rt ) xt                                (19)

c_t¹ = w_t¹ + r_t¹ — e — n — ne x_t¹

given xt , wt, rt , w_t¹, r_t¹

where v : R³₊ → R denotes the expected value function. Because there is no uncertainty about the present
once the equilibrium prices have been found by the ‘Walrasian auctioneer’, consumption plans for the
current period are always carried out: c_t⁰ = ct . Therefore: x_t¹ = xt+1 for t ∈ Z+ ∪ {0}, and the solution
to the household’s optimization problem (19) can be written as:

ct = c xt ,wt ,rt ,w_t¹,r_t¹                                          (20)

c_t¹ = c¹ xt ,wt ,rt ,w_t¹,r_t¹                                         (21)

xt+1 = x xt ,wt,rt ,w_t¹,r_t¹                                      (22)

where both the market clearing real wage and the market clearing real interest rate depend on the current
state of the economy according to equations (6) and (7). In equilibrium, the individual physical capital

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