Foldes, Lucien (2017) The optimal consumption function in a Brownian model of accumulation. Part C: a dynamical system formulation. Systemic Risk Centre Discussion Papers (68). Systemic Risk Centre, The London School of Economics and Political Science, London, UK.

Text
 Published Version
Download (1MB)  Preview 
Abstract
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 nonlinear, two dimensional but bilaterally asymptotically autonomous with limiting systems as logcapital 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) submanifolds 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 3D autonomous dynamical system such that the asymptotic saddle points are mapped to saddle points of the 3D 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 3D system. A stability result for the saddle connection is given for a special case.
Item Type:  Monograph (Discussion Paper) 

Official URL:  http://www.systemicrisk.ac.uk/ 
Additional Information:  © 2017 The Author 
Divisions:  Systemic Risk Centre 
Subjects:  H Social Sciences > HB Economic Theory 
JEL classification:  D  Microeconomics > D9  Intertemporal Choice and Growth > D90  General E  Macroeconomics and Monetary Economics > E1  General Aggregative Models > E13  Neoclassical O  Economic Development, Technological Change, and Growth > O4  Economic Growth and Aggregate Productivity > O41  One, Two, and Multisector Growth Models 
Date Deposited:  07 Nov 2017 13:56 
Last Modified:  14 Nov 2022 16:18 
Projects:  ES/K002309/1 
Funders:  Economic and Social Research Council 
URI:  http://eprints.lse.ac.uk/id/eprint/85121 
Actions (login required)
View Item 