Economic Dynamics under Heterogeneous Adaptive Learning: Alternative Conditions for Stability

Abstract

We provide criteria and sufficient and necessary conditions for stability of a linear structurally heterogeneous economy under heterogeneous learning of agents, thus extending the results of Honkapohja and Mitra (2006), Kolyuzhnov (2011), Bogomolova and Kolyuzhnov (2019, 2020). Using the alternative definition of the Db-stability approach, alternative to criteria and sufficient conditions written in terms of the corresponding Jacobian matrices in Bogomolova and Kolyuzhnov (2019), we provide criteria for strong δ-stability under mixed RLS/SG learning and sufficient conditions for weak δ-stability under heterogeneous RLS learning for four classes of models: models without lags and with lags of the endogenous variable and with t- or t-1-dating of expectations. For the case of the diagonal structure of the shocks process, this approach also allows us to provide criteria for strong δ-stability for univariate models without lags of the endogenous variable under mixed RLS/SG learning in economically meaningful terms, such as „same sign“ conditions and the E-stability of suitably defined associated economies, their subeconomies and average economies, and to provide quite weak sufficient conditions for weak δ-stability under heterogeneous RLS learning for univariate models with a lag of the endogenous variable using the same economic terms. We also provide sufficient conditions for strong δ-stability for bivariate models without lags of the endogenous variable under mixed RLS/SG learning for the case of the diagonal structure of the shocks process using the same economic terms. Using the characteristic equation approach, we provide economically tractable necessary conditions that can be used as an easy quick test for non-δ-stability. The fundamental nature of the approach adopted in the paper allows one to apply its results to a vast majority of the existing and prospective linear and linearized economic models (including estimated DSGE models used by central banks in most countries of the world) with the adaptive learning of agents.