The Optimal Consumption Function in a Brownian Model of Accumulation, Part B: Existence of Solutions of Boundary Value Problems

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 log-consumption as a time-invariant, deterministic function H(z) of log-capital z (both variables being measured in 'intensive units). Writing h(z) = H'(z), ?(z) = exp{H(z)-z}, the conditions require that the pair (h,?) satisfy a certain non-linear, non-autonomous (but asymptotically autonomous) system of o.d.e.s (F,G) of the form h'(z)= F(h,?,z), ?' = G(h,?) = (h-l)? for z ? ?, and that (h(z) and ?(z) converge to certain limiting values (depending on parameters) as z ? ? ?. The present paper, which is self-contained mathematically, analyses this system and shows that the resulting two-point boundary value problem has a (unique) solution for each range of parameter values considered. This solution may be characterised by the connection between saddle points of the autonomous systems (F-?,G) and (F +?,G), where F??(h,?) = F(h,?,??).