Foldes, Lucien (2017) The optimal consumption function in a Brownian model of accumulation. Part C: a dynamical system formulation. SRC Discussion Paper (No 68). Systemic Risk Centre, The London School of Economics and Political Science, London, UK.

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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 
Sets:  Research centres and groups > Systemic Risk Centre 
Date Deposited:  07 Nov 2017 13:56 
Last Modified:  07 Nov 2017 14:17 
Projects:  ES/K002309/1 
Funders:  Economic and Social Research Council 
URI:  http://eprints.lse.ac.uk/id/eprint/85121 
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