Learning and Endogenous Business Cycles in a Standard Growth Model

γ =1, and a Cobb-Douglas production technology: F (Kt,EtNt)=K_t^α (EtNt)^1-α where α represents
the capital’s share with 0 <α<1. In order to make the balanced growth path of the model consistent
with the long-run characteristics for the U.S. Economy, I set the value for α to be equal to 0.4 which
corresponds to the standard labor-income share of 60%. The estimates for the yearly rate of growth for
the population and the labor augmenting technical progress over the sample period 1954-1992 are taken
from Cooley and Prescott (1995) and are equal to 0.012 and 0.0156 respectively. Following Cooley and
Prescott (1995), I use the law of motion for the capital stock and the first-order necessary condition for
consumption both evaluated at the permanent regime to pin down the values for δ and β. Since the law
of motion for the capital stock per capita in the steady state is given by: (1 + e) (1 + n) = (1 — δ) + i∕k,
and the yearly investment-capital ratio i∕k is equal to 0.076, then the resulting value for the yearly
depreciation rate for physical capital δ is equal to 0.048 under above specifications for n and e. The first-
order necessary condition for consumption per effective amount of labor evaluated in the steady state is:
(1 + e)^γ (1 + n) = β (1 — δ + αy∕k). For a yearly capital-output ratio k∕y equals to 3.32, and a coefficient
of relative risk aversion γ equals to 1, the optimality condition for consumption is satisfied in the steady
state for a yearly discount factor β of 0.958. If we consider instead relative risk aversion coefficients of
0.02, 0.09, we get yearly discount factors of 0.944, 0.945 respectively. Table 1 in the appendix summarizes
in quarterly terms all the parameter values considered for the simulations. For a CRRA specification
of instantaneous preferences, the constrained utility maximizing problem under perfect foresight (8)
cannot be solved analytically, and the competitive equilibrium trajectories of the model are derived by
log-linearizing the households’ first-order necessary condition around the permanent regime as in King,
Plosser and Rebelo (1988). Under the bounded rationality hypothesis, the constrained planning problem
(18) can be solved analytically for any t ∈ Z+ ∪ {0}, and the utility maximizing consumption plans per
effective amount of labor are given by:

e =------^^p1∖ ₁√7-1)/7 £(1 + ^e)w^γ1 + ρ1 (W^jt + (1 + rt) et)]                   (26)

1 + ^^-1/Y (p1)(^{γ )/Y}

e =-------_ιz 1 ₁ a(7-1)/7 £e + p1 (Wt + (1 + rt) et)]                      (27)

1 + v^-1h (p1)(^{γ )/Y}

Using the parameter specifications of table 1, the model has a unique non-trivial steady state level of
physical capital per effective amount of labor k equals to 75.4963, and the corresponding value for G1 is
equal to 1 for h ∈ {0, 1}.

The case where h =0

If the constant gain adaptive learning schemes (24), (25) include current input prices: h =0,and
for G1 = 1, then according to proposition 4, the model with boundedly rational households can only
lose its stability through a Flip bifurcation. Using a total of 518, 400 distinct ordered pairs (1∕γ, η) ∈
[1.1, 100] ® [0.1, 1], figure 14A illustrates in the parameter space 1∕γ — η the local dynamic properties of
the competitive equilibrium trajectories where 1∕γ denotes the intertemporal elasticity of substitution be-
tween current and expected future consumption. The color codes used in computing this two-dimensional
bifurcation diagram are defined in figure 15A. For each ordered pair (1∕γ, η) considered, figure 14A is
obtained by simulating the model for a maximum of 10, 000 iterates and deleting the first 1, 000. Under
the assumption that γ = 0.02, the corresponding value for a is equal to 4.56187, and the first Flip bi-
furcation occurs at ηb_F = 0.609583 according to proposition 4. For γ = 0.02, figure 16A illustrates the
local qualitative behavior of the model under 475 distinct values of the gain parameter η ranging from
0.05 to 1. In this one-dimensional bifurcation diagram, the competitive equilibrium tra jectory of physical
capital per capita has been computed for a total of 1, 000 iterates. The first 100 iterations have been
discarded, and the 900 remaining have been plotted vertically just above the corresponding value of η
represented along the horizontal axis. If η<ηb_F , the plotted horizontal segment reveals that the physical
capital per effective amount of labor converges to the unique non-trivial stationary state k.Ifη ≥ bη_F ,
the dynamic of physical capital per effective amount of labor alternates between cycles of every order or
irregular behavior. At η = 0.7, the largest Lyapunov exponent we get after 40, 000 iterates of the model

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