1. This appendix aims at facilitating a clearer comprehension of Chapters VI and VII. A large literature is already available on existence of the von Neumann balanced-growth equilibrium. Nevertheless, an additional exposition will be welcomed, because most of the previous writings are highly sophisticated and average economic students still find them difficult to read. 1
For the sake of simplicity we confine ourselves to examination of the original von Neumann system fulfilling, among other things, the following assumptions: (a) capitalists do not consume and automatically invest their whole income; (b) workers cannot save and are prohibited from consumer choice; and (c) the system is 'indecomposable' so that every good is involved, either as input (in the 'augmented' sense) or as output, in every process. As much as possible, we shall stick to the notation that was used in the text: A denotes the input-coefficient matrix, B the output-coefficient matrix, L the labour-input-coefficient vector, x the intensity vector, y the normalized price vector, α one plus the rate of growth, and β one plus the rate of profits.
In the market there is only one kind of 'basket' which workers can buy. Each basket contains commodities in the fixed amounts, ej, j = 1, . . . , n, and e denotes the n-dimensional row vector (e1, . . . , en). Suppose now each worker buys h baskets. In a state of balanced growth, α x L workers