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Theory of Economic Growth$

Michio Morishima

Print publication date: 1969

Print ISBN-13: 9780198281641

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198281641.001.0001

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Existence of the Von Neumann Equilibrium

Existence of the Von Neumann Equilibrium

Source:
Theory of Economic Growth
Publisher:
Oxford University Press

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

α xC ( h ) xB , x X ,
(1)
where C(h) represents A + hLe and X is the set of all nonnegative x which are normalized such that the sum of the elements of each x is unity.

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:

α = α ( h ) , or g = α 1 = g ( h ) .
This traces out the curve which was called the Optimum Transformation Frontier by M. Bruno.1 Since h and g reflect the level of consumption and the level of investment, respectively, the curve gives the frontier of consumption and investment.

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

β C ( ω ) y By , y Y .
(2)
Regarding ω as given and y as variable, we minimize β subject to (2). The minimum value is a function of ω,
β = β ( ω ) , or r = β 1 = r ( ω ) ,
which gives the outer envelope of the factor‐price frontiers.

Let us now assume:

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

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

  3. Assumption 3. B + C(ω) > 0 for all ω 0 .

Assumptions 1 and 2 were introduced by Kemeny, Morgenstern, and Thompson, whilst Assumption 3 was made by von Neumann himself for assuring the indecomposability of the system. By Assumption 1, xC(h) ≠ 0 for all xX, so that α (h), the maximum of α, is finite. Next, let y > 0. By Assumption 1, C(ω)y > 0, so that (2) holds for some finite β for (p.310) y > 0. Therefore, β (ω), the minimum of β, should be finite. Moreover, it can be shown that α (h) and β (ω) are monotonic decreasing functions, because as Le 0 , an increase in ω gives rise to a monotonic increase of C(ω).

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

α ( ω ) x ( ω ) C ( ω ) x ( ω ) B ,
(1′)
β ( ω ) C ( ω ) y ( ω ) By ( ω ) .
(2′)
In view of the nonnegativity of x(ω) and y(ω), we have from these
α ( ω ) x ( ω ) C ( ω ) y ( ω ) x ( ω ) By ( ω ) β ( ω ) x ( ω ) C ( ω ) y ( ω ) .
Furthermore, by virtue of Lemma 2 to be proved in Section 2 below, α (ω) is at least as large as β (ω); therefore, the extreme right‐hand side of the above expression does not exceed its extreme left‐hand side. This implies that the above expression must hold with equality; hence
α ( ω ) x ( ω ) C ( ω ) y ( ω ) = x ( ω ) By ( ω ) ,
(3)
β ( ω ) x ( ω ) C ( ω ) y ( ω ) = x ( ω ) By ( ω ) .
(4)

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,

x ( ω ) By ( ω ) > 0 .
(5)

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 ω 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 yY such that Zy 0 , then xZ > 0 for some xX.

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

Z ( λ ) = [ z 1 , . . . , z n 2 , λ z n 1 + ( 1 λ ) z n ] .
Then for any λ0 in the interval [0, 1] there is no nonnegative, non‐zero, (n − 1)‐dimensional vector η such that Z ( λ 0 ) η 0 , because, by hypothesis, there is no yY such that Zy 0 . As the lemma is true for n − 1, there is an xX such that xZ0) > 0.

Let ζ (λ0) = x0 z n−1 + (1 − λ0)z n] be maximized at x = x0) subject to xZ ( λ 0 ) 0 and xX. Of course, max ζ (λ0) > 0, so that either x0)z n−1 or x0)z n (or both) is positive. When λ0 = 0, we have max ζ (0) = x(0)z n > 0; when λ0 = 1, x(1)z n−1 > 0.

Suppose now x ( 0 ) z n 1 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

x ( λ * ) z n 1 > 0 and x ( λ * ) z n > 0 .
(6)
Otherwise, we would have x ( 1 ) z n 1 0 , a contradiction.

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

x ( λ * ) z i 0 , i = 1 , . . . , n 2 .
But we have seen that xZ*) > 0 for some xX. Hence, for a sufficiently small positive number μ, we obtain [μ x + (1 − μ)x*)]Z > 0. Clearly, μ x + (1 − μ)x*) ∈ X. Q.E.D.

Lemma 2. α ( ω ) β ( ω ) .

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 yY such that Zy 0 . Hence, by Lemma 1, there is an x such that

xZ = x [ B α ( ω ) C ( ω ) ] > 0 ,
which implies that α (ω) is not the maximum of α subject to (1), a contradiction. Q.E.D.

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.