The Von Neumann-Gale Growth Model and its Stochastic Generalization

This paper deals with the deterministic and stochastic versions of the von Neumann-Gale model. Von Neumann's (1937) original concern was to determine a balanced path growing at a maximal rate for a linear and stationary technology and a price system supporting that path.Such a pair (a path and a price system) was called a von Neumann equilibrium. Gale (1956) proposed a version of the model assuming a general, not necessarily linear technology. This generalization led to a rich and interesting theory aimed at the analysis of efficient growth paths and based on the concepts introduced by von Neumann. The theory was developed in the 1950s and 1960s by Rockafellar, Radner, McKenzie, Nikaido, Morishima, and others. Its extension to the stochastic case was an open problem for about three decades, and a substantial progress in this direction was made only in the 1990s. The main results obtained are concerned with stochastic generalizations of a von Neumann equilibrium and efficient paths. Fundamental questions about their existence, uniqueness and stability (turnpike properties) are answered. The chapter gives an account of the achievements in the field and outlines new applications of the von Neumann-Gale model in finance related to asset pricing and hedging in securities markets