Optimal consumption of the finite time horizon Ramsey problem
type: Item
instance of: scholarly article
title: Optimal consumption of the finite time horizon Ramsey problem
zbMATH Open document ID: 1168.65005
author: Takashi Adachi
author: Hiroaki Morimoto
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-07-09T00:00:00Z
review text: The economic growth model mathematically is presented by the Ramsey problem. In the Introduction the concept of the stochastic Ramsey problem with a finite horizon time \(T > 0\) is considered. The aim of this consideration is to choose a consumption rate processes \(c = \{c(t)\}\) maximizing the discounted expected utilities with a constant discount rate \(\alpha \geq 0.\) The associated Hamilton-Jacobi-Bellman (HJB) equation is given by a parabolic partial differential equation. By changing variables the HJB equation is transformed to a partial differential equation with respect to the viscosity \(V.\) In Section 2 the existence of the viscosity solution \(V\) with \(T = \infty\) is shown. The stochastic control problem \(V(t,x)\) is considered. The auxiliary control problems \(V_{n}(t,x)\) are introduced and in Proposition 2.3 for each \(t,x \geq 0\) the convergence of \(V_{n}(t,x)\) to \(V(t,x)\) as \(n \to \infty\) is proved. In Theorem 2.5 the dynamic programming principle for \(V_{n}\) is established. Theorem 2.8 shows that \(V\) is a viscosity solution to the associated HJB equation with given boundary conditions. In Section 3 the \(C^{1,2}-\)regularity of the continuous viscosity solution \(V\) is studied. A synthesis of the optimal consumption \(c^{*}\) for the stochastic Ramsey problem is presented. The function \(u(t,x) = V(T-t,x)\) is defined and in Theorem 3.2 it is shown that this function is a solution of the associated HJB equation. In Section 4 an application of the obtained results to the infinite time horizon problem is made. The concept of the discounted expected utilities with a discount rate \(\alpha \geq 0\) is developed. A function \(\widehat{V}(x),\) \(x \geq 0\) is defined. Theorem 4.1 shows that the function \(\widehat{V}(x)\) is a solution of a equation of type of HJB equation. Theorem 4.2 shows the form of the optimal policy for the discounted expected utilities problem. Two examples are considered.
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 65C20
Mathematics Subject Classification ID: 65C35
Mathematics Subject Classification ID: 68U20
Mathematics Subject Classification ID: 82C80
Mathematics Subject Classification ID: 91B62
zbMATH DE Number: 5577092
zbMATH Keywords: Ramsey problem
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
zbMATH Keywords: viscosity solutions
zbMATH Keywords: economic growth
reviewed by: Vassil Grozdanov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.041
OpenAlex ID: W1988607162
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DOI: 10.1016/J.JMAA.2009.04.041
description: scientific article; zbMATH DE number 5577092