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.
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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 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.
| Item Type: | Monograph (Discussion Paper) |
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| Official URL: | http://www.systemicrisk.ac.uk/ |
| Additional Information: | © 2017 The Author |
| Divisions: | Systemic Risk Centre |
| Subjects: | H Social Sciences > HC Economic History and Conditions H Social Sciences > HG Finance |
| 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: | 11 Dec 2024 19:25 |
| Projects: | ES/K002309/1 |
| Funders: | Economic and Social Research Council |
| URI: | http://eprints.lse.ac.uk/id/eprint/85121 |
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