Growing Through ChaoticIntervals

We consider a growth model proposed by Matsuyama [K. Matsuyama, Growing through cycles, Econometrica 67 (2) (1999) 335–347] which describes the interaction of two sources of economic growth: The mechanism of capital accumulation (Solow regime) and the process of technical change and innovation (Romer regime). In this model the dynamics often alternates between the two different regimes: There is a tradeoff between growth and innovation. Analytically the model is represented by a piecewise smooth one-dimensional unimodal map described by two different functions, each of which characterizes a different regime. The existence of regimes with attracting equilibria or 2-cycles was already known, but the transition to complex behavior never explained properly. This is the object of the present paper. We shown that no stable cycle can exist, except for a fixed point and a cycle of period two. The Necessary and Sufficient conditions for regular or chaotic regimes are formulated. The bifurcation structure of the two-dimensional parameter plane is completely explained. It is shown how the border-collision bifurcation may lead from the stable fixed point either to another equilibrium or to an attracting cycle of period two or directly to a pure chaotic regime (which consists either in 4-cyclical chaotic intervals, 2-cyclical chaotic intervals or in one chaotic interval). All the bifurcation parameters are analytically detected making use of the bifurcation curves of a piecewise linear map in canonical form, which can be determined analytically.