Learning and Endogenous Business Cycles in a Standard Growth Model

is equal to 0.2549. A strictly greater than zero Lyapunov exponent diagnostics the presence of sensitive
dependence on initial conditions in the dynamics. An attractor with this such property is said to be
chaotic and it is illustrated in the k_t - k_t+1 space by figure A17 which has been computed for a total of
40, 000 iterates where the first 100 were discarded.

The case where h =1

If the constant gain adaptive learning schemes (24), (25) do not include current input prices: h =1,
and G₁ = 1, then according to proposition 7, the model with boundedly rational households can only lose
its stability through a Hopf-Neimark bifurcation. Under the assumption that γ = 0.09, the corresponding
value for a is equal to 1.01375, and the Hopf-Neimark bifurcation occurs at ηb_H =0.Forγ = 0.09, and
η = 0.7, figure A18 illustrates in the kt - kt+1 spaces that the competitive equilibrium dynamics converges
to an invariant closed curve for a total of 40, 000 iterates where the first 100 were discarded.

6 Conclusion

In a standard decentralized production economy characterized by a saddle point dynamics under perfect
foresight, simple constant gain learning schemes may generate endogenous business cycles around the
non-trivial steady-state. The perfect foresight hypothesis has been relaxed by introducing boundedly
rational households forming one-period ahead input price forecasts from simple constant gain adaptive
learning schemes, and predicting two-period ahead physical capital holdings and future consumption
stream from expectation rules consistent with the economy’s growth path. Under these assumptions, the
representative households’ planning problem can be written as a succession of two-period constrained
optimization problems and analytical solutions can be found under a general class of preferences. For
different specifications of preferences and technology captured by the relative values of Gi and a, the com-
petitive equilibrium trajectories under constant gain adaptive learning schemes may exhibit distinct local
stability properties depending on the size of the gain parameter η and the properties of the information
sets. A given learning scheme can lead to opposite stability properties depending whether the underlying
information sets used in making forecasts accommodate or not current input prices. If expectations take
into account the latest announcement made by the ‘Walrasian auctioneer’: h =0, then endogenous busi-
ness cycles may occur through the Flip bifurcation theorem, or through the Hopf-Neimark bifurcation
theorem, and though the Hopf-Neimark bifurcation theorem only when information about current prices
is not used in making forecasts: h =1. In the former case, the competitive equilibrium trajectories con-
verges to the perfect foresight steady state for a broader range of constant gain parameters. Calibrated to
the U.S. economy, and for low values of the coefficient of relative risk aversion, the model with boundedly
rational households may exhibit limit cycles or chaotic competitive equilibrium trajectories that do not
exist under the perfect foresight hypothesis.

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