The optimal consumption function in a Brownian model of accumulation Part A: The consumption function as solution of a boundary value problem

Abstract

We consider a neo-classical model of optimal economic 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 (open-loop control) comprising a finite welfare condition, a martingale condition for shadow prices and a transversality condition as t→∞. We then replace these by conditions for optimality of a plan generated by a consumption function (closed-loop control), i.e. a function H( z) expressing log-consumption as a time-invariant, deterministic function of log-capital z . Making use of the exponential martingale formula we replace the martingale condition by a non-linear, non-autonomous second-order o.d.e. which an optimal consumption function must satisfy; this has the form H″( z)=F[H′( z), θ( z), z] , where θ( z)= exp{H( z)− z} . Economic considerations suggest certain limiting values which H′( z) and θ( z) should satisfy as z→±∞ , thus defining a two-point 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 are given in a separate paper (Part B).