On The Dynamical Complexities Of Heterogeneous Growth Rates Paths

Dating from the seminal contribution of Lucas (1988), it has been understood that generalised production set which make explicit account of spillover effects within competitive economies may allow for unbounded capital paths along which capital goods grow at different rates. But to the exception of Benhabib & Perli [1996] through the examination of a particular parameterised case and despite their theoretical interest, there has however been no attempt to circumscribe the bifurcations set of this enriched class of solutions. This contribution is an attempt to progress towards their understanding. The argument proceeds through the consideration of production set for which generic production technologies are augmented in a multiplicative way by external effects that stem for the capital inputs --- no fixed factor is considered throughout the analysis. These technologies are attached to the production of a pure consumption output and two pure capital inputs. Integrating the equilibrium value of the rates of returns on the capital goods then allows the analysis to rest upon an equilibrium production possibility frontier. It is proved that the latter is homogeneous and that this property is in its turn at the core of the possibility of defining a growth solution with heterogeneous growth rates for the capital goods. No overall constant returns to scale is required at a sectoral level but it must be recovered at the aggregate level. It is proved that the sign of the second order derivatives of this frontier rest upon augmented sectoral capital intensities which make simultaneous account of the private and external components of the production technologies. A careful examination of the set of equilibrium growth rates then allows for establishing the possibility of multiple long-run equilibria when a. the growth rates of the two capital goods actually differ, b. strictly increasing returns prevail in at least one capital goods industry. It is then proved that only half of these solutions recover a local uniqueness but that the others may be locally or even globally indetermiate. Some advanced material in the theory of three-dimensional dynamical systems then allows for establishing the occurrence of a Poincar’-Hopf bifurcation and the emergence of periodic orbits in the neighbourhoods of long-run solutions with heterogeneous growth rates for the capital goods.