Von Neumann Ray

Abstract

The von Neumann ray determines the proportions of maximal balanced growth in a von Neumann technology. The economic growth trajectory, which realizes the maximum possible growth rate, that the economy could with-stand for infinite time is located on this ray. Let us give a more formal discription of the problem being discussed. The trajectory x(0),…, x(t),… generated by this technology is called stationary if the proportions between goods in the state x(t) are independent of time t. A stationary trajectory can be written in the form rm x (t) = γ^tx where x = x(0) is the initial state. This trajectory is generated by a technologically feasible activity (x, y) under which γx ≤ γ₁. Usually a stationary trajectory is called the trajectory of balanced growth (although actual growth will take place only for γ > 1). The maximum number γ which enables a balanced growth is called the von Neumann (or technological) rate of growth for the technology Z. Thus the technological rate α is the solution of the following optimization problem find α = max γ subject to

$$ \left(x,y\right)\in Z,y\ge \gamma x. $$

If y ≥ αx, the process (x, y) is called the von Neumann activity (process), the corresponding vector x – the von Neumann vector and the ray passing through x – the von Neumann ray.