The optimal consumption function in a Brownian model of accumulation. Part C: a dynamical system formulation

This Paper continues the study of the Optimal Consumption Function in a Brownian Model of Accumulation, see Part A [2001] and Part B [2014]; a further Part D, dealing with the effects of perturbations of the Brownian model, is in preparation. We begin here with a review of the o.d.e. system S which was used in Part B for the proof of the existence of an optimal consumption function. This system is non-linear, two dimensional but bilaterally asymptotically autonomous with limiting systems as log-capital tends to plus/minus infinity, each of which has a unique saddle point. An important part is played in the existence proof by the sets of forward/backward ‘special’ solutions, i.e. solutions of S converging to the asymptotic saddle points, and by their representing functions f and g. A ‘star’ solution, which is both a forward and a backward special solution, corresponds to an optimal consumption function. It is shown here that the sets of special solutions of S are C(1) sub-manifolds of R(3), hence that the functions f and g are continuously differentiable. The argument involves the construction of an imbedding of S in a 3-D autonomous dynamical system such that the asymptotic saddle points are mapped to saddle points of the 3-D system and the sets of forward/backward special solutions are mapped into stable/unstable manifolds. The usual Stable/Unstable Manifold Theorem for hyperbolic stationary points then yields the required C(1) properties locally (i.e. near saddle points), and these properties can be extended globally. A ‘star’ solution of S then corresponds to a saddle connection in the 3-D system. A stability result for the saddle connection is given for a special case.