Learning and Endogenous Business Cycles in a Standard Growth Model

expected consumption plans by solving a two-period approximation of his infinite lifetime planning prob-
lem he would be able to solve all at once if he was endowed with both perfect foresight and unlimited
computing skills. This model is a discrete time decentralized version with exogenous labor supply and
labor augmenting technical progress of the standard neoclassical growth framework developed by Cass
(1965), and Koopmans (1965). For different specifications of the underlying information sets, I consider
that each boundedly rational household uses both constant gain adaptive learning mechanisms to fore-
cast next period’s input prices, and simple forecasting rules consistent with the economy’s growth path
to predict his two-period ahead physical capital holdings and future consumption stream. As a result, his
lifetime planning problem corresponds to an infinite succession of two-period optimization problems in
which analytical solutions can be derived under large specifications of preferences. Such representation of
the households’ planning problems has been suggested by Leontief (1958), and analyzed in simple growth
settings by Day (1969), Day and Lin (1992) for particular specifications of preferences and expectation
functions. I consider in this paper a standard growth framework with general specifications of preferences
and expectation functions, and characterized the local dynamic properties around the perfect foresight
steady-state in term of the properties of the functional forms and the information sets. Then I calibrate
the model to the U.S. economy and present numerical illustrations. For particular values for the coef-
ficient of relative risk aversion, the model may generate complex attractors that do not exist under the
perfect foresight hypothesis.

The paper is organized as follows. In section 2, I describe the model. In section 3, I present the
household’s problem under perfect foresight as well as the local dynamic properties of the competitive
equilibrium trajectories. In section 4, I present the household’s problem under bounded rationality and
analyze the local competitive equilibrium trajectories. In section 5, I calibrate the model to the US
economy and provide numerical illustrations. In section 6, I conclude the paper.

2TheModel

Let us consider a perfectly competitive production economy populated at time t for t ∈ Z+ ∪ {0} by Nt
identical and infinitely lived households who own all firms and production factors. From time t to t +1,
each household rents to firms xt units of physical capital at the real rental rate Rt as well as 1 unit of
labor service at the real wage rate w_t , receives a fraction π_t of the firms’ profits, and allocates his resulting
total income between current consumption ct and investment it. By introducing superscripts to denote
the planning time, I consider that the representative household’s rental income and resource constraint
expected to be received in j period(s) from time t where both t, j ∈ Z+∪{0},aregivenby: y_t^j = w_t^j + R_t^j x_t^j
and c_t^j + i_t^j = y_t^j + π_t^j respectively. Planned individual investment, physical capital holdings, and lifetime
consumption stream, are assumed to be derived from a constrained utility maximizing problem based on
expected future prices and quantities. The household’s expected lifetime utility evaluated at time t is

represented by a function U defined over an infinite stream of planned consumption starting at time t:

I consider the usual time separable specification of expected lifetime preferences by assuming
that U is an infinite weighted sum of instantaneous utility functions u:

∞

U (c^0,ct^1,c2^,.J=X ^βj u(c^j )

j=0

where β denotes the discount factor with 0 <β<1.

Assumption 1 The instantaneous utility function u: R+ → R+ is strictly increasing, concave, twice
continuously differentiable with u⁰ (c) homogenous of degree p for -1 < p < 0, guarantees in each period
the normality of the consumption commodity, and satisfies Inada conditions.

The relationship between the aggregate level of output Y , the utilized aggregate level of physical
capital K , and the employed aggregate level of labor L is described by a constant returns to scale
aggregate production function F :

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