The optimal consumption function in a Brownian model of accumulation. Part C: a dynamical system formulation

This Paper continues the study of the Optimal Consumption Function in a Brownian Model of Accumulation, see Part A [2001] and Part B [2014]; a further Part D, dealing with the effects of perturbations of the Brownian model, is in preparation. We begin here with a review of the o.d.e. system S which was used in Part B for the proof of the existence of an optimal consumption function. This system is non-linear, two dimensional but bilaterally asymptotically autonomous with limiting systems as log-capital tends to plus/minus infinity, each of which has a unique saddle point. An important part is played in the existence proof by the sets of forward/backward ‘special’ solutions, i.e. solutions of S converging to the asymptotic saddle points, and by their representing functions f and g. A ‘star’ solution, which is both a forward and a backward special solution, corresponds to an optimal consumption function. It is shown here that the sets of special solutions of S are C(1) sub-manifolds of R(3), hence that the functions f and g are continuously differentiable. The argument involves the construction of an imbedding of S in a 3-D autonomous dynamical system such that the asymptotic saddle points are mapped to saddle points of the 3-D system and the sets of forward/backward special solutions are mapped into stable/unstable manifolds. The usual Stable/Unstable Manifold Theorem for hyperbolic stationary points then yields the required C(1) properties locally (i.e. near saddle points), and these properties can be extended globally. A ‘star’ solution of S then corresponds to a saddle connection in the 3-D system. A stability result for the saddle connection is given for a special case

The Optimal Consumption Function in a Brownian Model of Accumulation Part A: The Consumption Function as Solution of a Boundary Value Problem

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 of 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),?(z),z], where ?(z) = exp{H(z)-z}. Economic considerations suggest certain limiting values which H'(z) and ?(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 will be given in a separate paper (Part B).Consumption, capital accumution, Brownian motion, optimisation, orderinary differential equation, boundary value problems.

Martingale conditions for optimal saving: discrete time.

Necessary and sufficient conditions are derived for optimal saving in a stochastic neo-classical one-good world with discrete time. The usual technique of dynamic programming is replaced by classical variational and concavity arguments, modified to take account of conditions of measurability which represent the planner's information structure. Familiar conditions of optimality are thus extended to amit production risks represented by quite general random processes - no i.i.d.r.v.s., stationarity or Markov dependence are assumed - while utility and length of life also may be taken as random. It is found that the 'Euler' conditions may be interpreted as martingale properties of shadow prices.

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), J(z) = exp{H(z)-z}, the conditions require that the pair (h,J) 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,J,z), J'(z) = G(h,J) = (h-1)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 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 as the connection between saddle points of the autonomous systems obtained from (F,G) as z tends to + or - infinity

The effect of connection charges on the number of connections and on the prices and rents of houses

Comments on economic issues raised by the Monopolies Commission Report Connection Charges for Electricity and Gas, Cmnd. 5036, 1972