Existence of the Von Neumann Equilibrium
Existence of the Von Neumann Equilibrium
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, e _{j}, j = 1, . . . , n, and e denotes the n‐dimensional row vector (e _{1}, . . . , e _{n}). Suppose now each worker buys h baskets. In a state of balanced growth, α x L workers (p.309) are employed, so that after subtracting their consumption from the outputs, xB, having resulted from the activities in the previous period, there remain goods of the amounts, xB − α xhLe. The feasibility requires that these be not exceeded by the current inputs, α xA. Thus we have
Regarding α and x as variable but h as given, we consider a problem to maximize α subject to (1). It is evident that the maximum value of α depends on the value of h specified:
Next, let w be the wage rate which prevails when the prices y are normalized such that the sum of the elements of y equals unity. (We define Y as the set of all such nonnegative y's.) By spending w, one can buy ω baskets so that w = ω ey. The augmented‐input‐coefficient matrix is then defined as C(ω) = A + ω Le, which is a function of ω, a kind of the real wage rate. Obviously, C(ω)y = Ay + wL; so that the price‐cost inequalities can be written as
Let us now assume:

Assumption 1. For any $\omega \geqq 0$, every row of C(ω) has at least one positive entry.

Assumption 2. Every column of B has at least one positive entry.

Assumption 3. B + C(ω) > 0 for all $\omega \geqq 0$.
Now, let x(h) be an x at which α is maximized subject to (1), and let y(ω) be a y at which β is minimized subject to (2). In the original von Neumann model, because of the assumption that workers cannot save, every worker has to consume ω baskets, so that h = ω. Therefore, we obtain
Let us confine our examination to the case of indecomposability.^{1} From Assumption 3 we have x(ω)[B + C(ω)]y(ω) > 0, so that either x(ω)By(ω) or x(ω)C(ω)y(ω) (or both) must be positive. Suppose now x(ω)C(ω)y(ω) is positive and x(ω)By(ω) is zero. Then from (3), α (ω) should be zero. But this contradicts Assumption 2, because it implies that xB > 0 for all x > 0, so that the maximum, α (ω), has to be positive. Hence,
Obviously, (1′), (2′), (3), (4), (5) establish the von Neumann equilibrium. Therefore, α (ω), β (ω), x(ω), y(ω) are solutions. It follows from (3), (4) and (5) that α (ω) equals β (ω). This result is valid for any $\omega \geqq 0$, provided that the system is indecomposable. Thus we obtain the conclusion which Bruno called the fundamental duality theorem; that is to say, the optimum transformation frontier is mathematically identical with the outer envelope of the factor‐price frontiers. It must, however, be emphasized that the theorem is no longer true when the system is decomposable, as I have seen in the paper just mentioned.
(p.311) 2. Now we have to prove necessary lemmas.
Lemma 1. Let Z be an m × n matrix. If there is no y ∈ Y such that $\mathrm{Zy}\leqq 0$, then xZ > 0 for some x ∈ X.
Proof. The lemma is true when n = 1. Let us show that it is true for n if it is so for n − 1.
Let z _{i} be the i‐th column of Z. Let Z(λ) be the m × (n − 1) matrix defined as
Let ζ (λ_{0}) = x[λ_{0} z _{n−1} + (1 − λ_{0})z _{n}] be maximized at x = x(λ_{0}) subject to $\mathrm{xZ}({\lambda}_{0})\geqq 0$ and x ∈ X. Of course, max ζ (λ_{0}) > 0, so that either x(λ_{0})z _{n−1} or x(λ_{0})z _{n} (or both) is positive. When λ_{0} = 0, we have max ζ (0) = x(0)z _{n} > 0; when λ_{0} = 1, x(1)z _{n−1} > 0.
Suppose now $x(0){z}_{n1}\leqq 0$. Both x(λ)z _{n−1} and x(λ)z _{n} are continuous functions of λ, and the latter is positive so long as the former is non‐positive. Hence there is a λ, say λ^{*}, such that
On the other hand, if x(0)z _{n−1} > 0, then it is evident that we have (6) at λ^{*} = 0. Thus, in any case, we have a λ^{*} at which the two inequalities of (6) hold.
With respect to other elements we only have
Lemma 2. $\alpha (\omega )\geqq \beta (\omega )$.
Proof. Suppose the contrary, i.e., α (ω) < β (ω). Lemma 1 can be applied to Z = B − α (ω)C(ω). Since β (ω) is the minimum of β subject to (2) and α (ω) is less than β (ω), there is no y ∈ Y such that $\mathrm{Zy}\leqq 0$. Hence, by Lemma 1, there is an x such that
Notes:
(1) The original proof by von Neumann uses a fixed‐point theorem; the proof by Kakutani uses a more general (and more difficult) fixed‐point theorem; the proofs by Georgescu‐Roegen and Gale use the theorem of the separating hyperplane; the proof by Kemeny, Morgenstern and Thompson uses theorems of game theory; the proof by Howe uses a duality theorem derived by Tucker. See J. von Neumann, ‘A Model of General Economic Equilibrium,’ Review of Economic Studies, Vol. 13 (1945–46); S. Kakutani, ‘A Generalization of Brouwer's Fixed Point Theorem,’ Duke Mathematical Journal, Vol. 8 (1941); N. Georgescu‐Roegen, ‘The Aggregate Linear Production Function and its Applications to von Neumann's Economic Model,’ Activity Analysis of Production and Allocation, ed. by T. C. Koopmans (1951); D. Gale, ‘The Closed Linear Model of Production,’ Linear Inequalities and Related Systems, ed. by H. W. Kuhn and A. W. Tucker (1956); J. G. Kemeny, O. Morgenstern and G. L. Thompson, ‘A Generalization of the von Neumann Model of an Expanding Economy,’ Econometrica, Vol. 24 (1956); Charles W. Howe, ‘An Alternative Proof of the Existence of General Equilibrium in a von Neumann Model,’ Econometrica, Vol. 28 (1960). The following is a variation of elementary proof devised by Gale who proved the key lemma by mathematical induction. There is a similar elementary proof by Loomis. See D. Gale, The Theory of Linear Economic Models (McGraw‐Hill, 1960); L. H. Loomis, ‘On a Theorem of von Neumann,’ Proceedings of National Academy of Sciences, U.S.A., Vol. 32 (1946).
(1) M. Bruno, ‘Fundamental Duality Relations in the Pure Theory of Capital and Growth,’ Review of Economic Studies, Vol. XXXVI (1969).
(2) See pp. 93–94 above.
(1) My article, ‘Consumption‐Investment Frontier, Wage‐Profit Frontier and the von Neumann Growth Equilibrium,’ Zeitschrift für Nationalökonomie, 1971, deals with a more general case where Assumption 3 does not necessarily hold.