A ‘funnel’ Turnpike Theorem for Optimal Growth Problems with Discounting

A particularly interesting question in mathematical economics is the ‘turnpike’ problem. There have been many papers written on the subject, bearing out Nikaido’s (1964) prediction that ‘it will continue to bewitch many people’. In this paper we present an extension to the classical turnpike theory for the discounted optimal control problem, developed in the context of Implicit Programming theory. The analytic background and development of turnpike theory may be found in the papers of Morishima (1961), Radner (1961), Atsumi (1963), Nikaido (1964) Inada (1964), Samuelson (1965), Cass (1966), Gale (1967), McKenzie (1968,1976), Liviatan and Samuelson (1969), Rockafellar (1973,1976), and Haurie (1976). The original description of the phenomenon was given by Dorfman, .Samuelson and Solow (1958, p. 331). This is only a partial list of publications, but these particular papers form an interesting sequence that share a similar set of assumptions and analyze a similar model. The earlier papers are in discrete time, with the exception of Cass’ and Samuelson’s. Haurie extends McKenzie’s (1968) results to continuous time. It is also important to note that only Cass, Liviatan-Samuelson, and Rockafellar (1976) deal with discounted problems.