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In John von Neumann's 1937 system we have "production of commodities by means of commodities" - akin to a Sraffa-Leontief classical system. But it also introduces activity analysis, duality, slack conditions, fixed-point theorems, disaggregate capital and other concepts profitably used in later Walrasian general equilibrium theory. Data
Activity Analysis Technology:
There are k technological processes indexed m = 1....k. The
"intensity of use" of the mth process denoted by coefficient 0
£zm£ 1. By intensity, we mean the
proportion of the total economy that uses that process. Thus:
Let there be n commodities indexed i = 1, ..., n.
zmbmi is the output of commodity i from the process m. We have n prices, pi from i = 1...n normalized in a unit simplex (i.e. åpi = 1)
Conditions with No Growth: (1) Quantity Side: Let an economy be "self-replacing" (produces outputs that it needs as inputs with the same set of intensities). Thus, total output of good i from all processes must be sufficient to meet total demand for good i as an input into all processes:
If total output exceeds total input demand (strict inequality holds) for any good i then pi = 0 (excess supply of i implies i is a free good). Letting z be an intensity vector and A the input matrix (typical element = ami) and B the output matrix (typical element = bmi), this system can be rewritten:
(2) Price Side: Assuming an independent budget for each process and perfect competition (no rents), then total revenue obtained from the sale of output of a process cannot exceed the costs of inputs in the process:
Conditions (with growth) We assume intensity grows at the balanced exponential rate (1+g). Thus, zm(t+1) = (1+g)zm(t). Thus, for the quantity side, total outputs in this period must meet total input demands in the next period, i.e.
or:
Similarly, we assume a uniform rate of profit (surplus) which gives us a rate of decay of price: pi(t+1) = pi(t)/(1+r). Thus the surplus is transformed into a simple uniform rate of interest for intertemporal discounting. We thus obtain:
Further restrictions are imposed by von Neumann (1937) including non-negativity of z and p and the condition that every good is part of every process either as input or as output. This last was loosened by Kemeny, Morgenstern and Thompson (1956) and replaced with:
(2) For every good i, there is some process m such that bmi > 0 (3) output value, pBz > 0. These are often known as the "KMT" conditions. An appropriate fixed point theorem proves existence of solution for quadruplet (z*, p*, g*, r*). Result There will be a pair (p*, z*) which solves our equation and gives us the Golden Rule, g* = r* for a maximal growth rate and a minimal interest rate. The Golden Rule can be easily shown. By the excess supply rule for free goods, then pre-multiplying the first equation by the price vector:
where the strict equality will hold by the free goods assumption. Similarly, by the excess cost rule, then post-multiplying the second equation by the intensity vector:
where the strict equality holds by the excess cost assumption. Then, obviously:
Thus the solution, the von Neumann "ray", will have maximum growth and minimal profit rate equal to each other (Golden Rule). Outline of Proof: John von Neumann used his 1928 game-theoretic "minimax" theorem of saddlepoint as contraction mapping:
i.e. if min and max are contractible for ¦(x0, y0), then ¦ has a saddle point. So, think of a game of "Producer" versus "Competition": Producer attempts to maximize growth, competition attempts to minimize profits. Recall that we want (z, g) from quantity side and (p, r) from price side. So game is as follows:
Competition (Price Side): for given p, take maximum r, i.e. (1+r) = maxm(pB/pA). Then choose p which yields the lowest of the maximum r, i.e. find p such that (1+r)* = minimaxm(pB/pA). Existence of minumum r* exists by KMT(2). Then construct mapping F (p, z) = pBz/pAz (which is positive by KMT(3)). von Neumann then shows that given a z = z0, then this function reaches a minimum for p whereas, given p = p0, then this function reaches a maximum for z. If there is a solution, then that solution is characterized as
so the solution to the system, F (p0, z0) = F(p*, z*), is the saddlepoint. Does it exist? Take a particular z0 and associated with this vector is a non-empty, convex, compact set of price vectors P(z0) each of which minimize the function F(p, z0). Equivalently, with some initial p0, we can associate a non-empty, convex, compact set of intensity vectors, Z(p0), each of which maximize the function F(p0, z). A "fixed point" is a pair of vectors (p*, z*) Î (P(z*), Z(p*)). von Neumann provides sufficient conditions for the existence of such a fixed point and proves that it is the saddlepoint. Go on to "Von Neumann System with Consumption" ________________________________________________________ Selected References: John von Neumann (1937) "Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes", 1937, in K. Menger, editor, Ergebnisse eines mathematischen Kolloquiums, 1935-36. [English 1945 trans. as "A Model of General Economic Equilibrium", Review of Economic Studies, Vol. 13 (1), p.1-9.]. J.G. Kemeny, O. Morgenstern and G.L. Thompson (1956) "A Generalization of the von Neumann Model of an Expanding Economy", Econometrica, Vol. 24 (2), p.115-35. M. Dore, S. Chaktravarty and R. Goodwin (1989), editors, John von Neumann and Modern Economics. Oxford: Clarendon Press.
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