"The Optimal Consumption Function in a Brownian Model of Accumulation 
Part A: The Consumption Function as Solution of a Boundary Value Problem"
Lucien Foldes
London School of Economics
ABSTRACT
We consider a neoclassical model of optimal economics growth with c.r.r.a. utility in
which the traditional deterministic trends representing population growth, technological
progress, depreciation and impatience are replaced by Brownian motions with drift. When
transformed to 'intensive' units, this is equivalent to a stochastic model of optimal saving
with diminishing returns to capital. For the intensive model we give sufficient conditions
for optimality of a consumption plan (openloop control) comprising a finite welfare
condition, a martingale condition for shadow prices and a transversality condition as time tends
to infinity. We then replace these by conditions for optimality of a plan generated by a
consumption function (closedloop control), i.e., a function H(z) expressing logconsumption
as a timeinvariant, deterministic function of logcapital z. Making use of the exponential
martingale formula we replace the martingale condition by a nonlinear, nonautonomous second
order o.d.e. which an optimal consumption function must satisfy; this has the form
H"(z) = F[H'(z),J(z),z], where J(z) = exp{H(z)z}. Economic considerations suggest certain
limiting values which H'(z) and J(z) should satisfy as z tends to + or  infinity, thus defining
a twopoint boundary value problem (b.v.p.)  or rather a family of problems, depending on the
values of parameters. We prove two theorems showing that a consumption function which solves
the appropriate b.v.p. generates an optimal plan. Proofs that a unique solution of each b.v.p.
exists will be given in a separate paper (Part B).
Keywords: Consumption, capital accumulation, Brownian motion, optimisation, ordinary
differential equations, boundary value problems.
AMS (MOS) subject clasifications: 34B10, 49A05, 60J70, 90A16, 93E20.
JEL subject classifications: D81, D90, E13, O41.
Economics Department, London School of Economics, Houghton Street,
London WC2A 2AE, UK
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