"The Optimal Consumption Function in a Brownian Model of Accumulation 
Part B: Existence of Solutions of Boundary Value Problems"
Lucien Foldes
London School of Economics
ABSTRACT
In Part A of the present study, subtitled 'The Consumption Function as
Solution of a Boundary Value Problem' Discussion Paper No. TE/96/297, STICERD, London School
of Economics, we formulated a Brownian model of accumulation and derived sufficient conditions
for optimality of a plan generated by a logarithmic consumption function, i.e. a relation
expressing logconsumption as a timeinvariant, deterministic function H(z) of logcapital z
(both variables being measured in 'intensive' units). Writing h(z) = H'(z), J(z) = exp{H(z)z},
the conditions require that the pair (h,J) satisfy a certain nonlinear, nonautonomous (but
asymptotically autonomous) system of o.d.e.s (F,G) of the form h'(z) = F(h,J,z), J'(z) = G(h,J)
= (h1)J for real z, and that h(z) and J(z) converge to certain limiting values (depending on
parameters) as z tends to + or  infinity. The present paper, which is selfcontained
mathematically, analyses this system and shows that the resulting twopoint boundary value
problem has a (unique) solution for each range of parameter values considered. This solution
may be characterised as the connection between saddle points of the autonomous systems obtained
from (F,G) as z tends to + or  infinity.
Keywords: Consumption, capital accumulation, Brownian motion, optimisation, ordinary
differential equations, boundary value problems.
AMS (MOS) subject clasifications: 34B10, 49A05, 90A16, 93E20.
JEL subject classifications: D81, D90, E13, O41
Abbreviated Title: Optimal Consumption as a Boundary Value Problem, Part B.
Economics Department, London School of Economics, Houghton Street,
London WC2A 2AE, UK
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