# Von Neumann Technology

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1377

## Abstract

The von Neumann technology is a convenient tool for the description and analysis of a wide variety of economic systems. It can be considered a special form of describing the production possibility set (i.e. the production process of the economic system, a form mostly designed for mathematical research of development dynamics).

The von Neumann technology is a convenient tool for the description and analysis of a wide variety of economic systems. It can be considered a special form of describing the production possibility set (i.e. the production process of the economic system, a form mostly designed for mathematical research of development dynamics).

The production process of this technology is determined by definition of input and output of goods corresponding to contiguous time intervals. An arbitrary production process can be described in this framework by introducing additional intermediate goods. We give a more formal description of the considered situation. Consider an economy with *n* goods, where we understand the term ‘goods’ in a very broad sense. Depending on the economic situation we can number among the goods not only goods in the usual sense of the world but also various types of capital, labour, natural resources as well as some conditional goods (e.g. the effect of consumption of some other goods).

A technology is a set *Z* consisting of technologically feasible processes (activities) *Z*. Every activity transforms a given set of goods (input vector) into another set (outout vector). Thus formally the activity is represented by a pair of vectors *Z* = (*x, y*), where *x* is the input vector and *y* the output vector, both of them being *n-*dimensional vectors with non-negative components.

Considering the technology we assume that all technologically admissible activities have the same duration (a unit time interval). This hypothesis is based on the assumption that a longrun process can be decomposed into several processes of unit length. As a result of this decomposition intermediate goods (e.g. capital vintages or unfinished products) can be introduced.

*Z*.

- (1)
Any activity can be used at any intensity: i.e. (

*x*,*y*) ∈*Z*,*λ*⩾ 0 implies*λ*(*x*,*y*) ∈*Z*. This property reflects the possibility of an unlimited use of resources. - (2)
Any two activities can be used jointly: \( \left(x,y\right)\in Z,\left(u,\upsilon \right)\in Z \) implies \( \left(x+u,y+u\right)\in Z \). Geometrically (1) and (2) mean that the von Neumann technology can be described by a convex cone.

- (3)
All goods can be produced. This means (together with (2) that there exists an activity (

*x, y*) such that all coordinates of the vector*y*are positive. - (4)
Non-zero output is impossible without input.

*Z*in the narrow sense (in another terminology: the von Neumann model, the model of an expanding economy) is defined through specification of m activities which are termed basic; it is the set of all input–output vectors which can be obtained by the joint use of the basic activities with arbitrary intensities. Geometrically,

*Z*is a polyhedral cone with activities as its extreme rays. Algebraically, it is convenient to define

*Z*by a pair of

*m*×

*n-*matrices: the input matrix

*A*and the output matrix

*B*. If (

*a, b*) is the

*i*th basic activity, the vector

*a*is the

*i*th column of the matrix

*A, b*is the

*i*th column of the matrix

*B*. Then

*u*is an

*m-*vector of intensities. The condition (3) (resp. 4) is equivalent to the absence of zero columns in the matrix

*A*(resp. to the absence of zero rows in the matrix

*B*). These properties were formulated by Kemeny, Morgenstern and Thompson in 1956. von Neumann in his fundamental paper (1937; English translation: 1946) assumed a stronger condition: in every activity every good is either consumed or produced.

The von Neumann technology in a broad sense (in another terminology: the Neumann–Gale model) is merely a closed (in the topological sense) set for which the conditions (1)–(4) are fulfilled. It was introduced by Gale in 1956. Such technologies arise, for example, in connection with the use of production functions.

The von Neumann technology is a formal mathematical object that can be used for modelling various economic situations. One such situation was considered by J. von Neumann. He studied a closed economic system (i.e. having no connections with the outer world). The production possibilities of the system are given by the input and output matrices. There is no outflow of consumption, the process of production includes the reproduction of labour force, the workers save nothing, all capitalists’ returns are invested. In other works, von Neumann abstracts from consumption and savings and concentrates solely on the process of production. A detailed analysis of the underlying economic assumptions is given in (Champernowne 1946).

Some deep generalizations of the von Neumann technology describing an open economy and explicitly taking into account consumption, labour and wages were studied by Morgenstern and Thompson and by J. Los and his pupils.

*F*(

*K, L*), the production function describing the performance of the firm where

*K*is the capital and

*L*the labour force. It is supposed that any part of the output

*F*(

*K, L*) obtained with the capital

*K*and the labour force

*L*can be turned into investment

*I*, the remaining part being used for purchasing the labour force

*l*. The wage rate ω and the capital deterioration rate μ re given. The set of states (

*k, l*) the firm can reach (in a unit time interval) from the state (

*K, L*) is described by a system of inequalities

*F*satisfies the traditional assumptions of concavity and homogeneity of degree 1 then the set of activities ((

*K, L*), (

*k, l*) satisfying (*)) is a von Neumann technology (in a broad sense).

The von Neumann technology is often used for representing the production part in various models of economic dynamics. Models with utility functions explicitly taking into account consumption, as well as dynamic Leontief models can be stated and analysed in this framework as well. We note furthermore than many other problems not connected with economic dynamics can be embedded into a von Neumann technology scheme, in particular, ‘bottleneck problems’.

Thus with the help of the von Neumann technology we can study a demographic model of population movement, based on the following hypothesis: the number of marriages between men and women under certain ages is proportional to the minimum of the numbers of unmarried men and women under these ages. Men and women under certain ages, and also their newly created families which are distinguished according to the terms of their existence, play the role of ‘products’ here.

The technological activities describe a shift of ‘products’ from one age group to another, and the processes of family increase and decrease.

As a rule the von Neumann technology is analysed from two viewpoints. First, equilibrium states of the economic system can be determined in these terms. J. von Neumann introduced it specially for this purpose. Second, this technology is a convenient tool for analysing development trajectories of the economic system. Both directions are closely interconnected. The concept of von Neumann equilibrium (geometrically: the von Neumann ray) is extremely important in these problems. Here we focus our attention on the trajectory concept.

*x*(0), …,

*x*(

*T*) is called a trajectory of length

*T*generated by a von Neumann technology

*Z*if the relations

*y*(

*t*). The trajectories which are optimal in the sense that, the output value

*p*(

*T*)

*x*(

*T*) at moment

*T*is greater than or equal to the output value for any other trajectory beginning at

*x*(0) are of special interest here (

*p*(

*T*) ⩾ 0) is the given price vector at the moment

*T, px*is the scalar product of the vectors

*p*and

*x*).

Sometimes efficient trajectories *x*(0), …, *x*(*T*) are considered. Efficiency means that from the point *x*(0) it is impossible to reach in *T* steps the point *λx*(*t*) with *λ* ⩾ 1; in other words trajectories of the form *x*(0), …, *λx*(*T*) do not exist. Under some natural assumptions on the technology the trajectory is efficient if and only if there exists a price vector *p*(*T*) for which the trajectory is optimal. One can consider infinite trajectories \( x(0),\dots, x(t),x\left(t+1\right),\dots \) as well. An infinite trajectory is called efficient if each of its segments *x*(0), …, *x*(*t*) is efficient for any *t* > 0. The interest is infinite efficient trajectories is not motivated solely by the desire to understand the system’s behaviour in the far future. Much more concretely, the fact that *x*(1) must belong to the infinite efficient trajectory beginning at *x*(0) is often a very restrictive assumption, which allows us to determine uniquely the output *x*(1) among all feasible outputs generated by the input *x*(0).

*Z*generates not only the trajectories of goods describing the material flows in the economy but the price trajectories describing the financial flows. It is supposed that the price vector

*q*⩾ 0 at the moment

*t*+ 1 (given the price vector

*p*at the moment

*t*) is chosen in such a manner that the value of any output

*y*(at moment

*t*+ 1) does not exceed the value of the input

*x*at moment

*t*). Thus we have the following definition: the sequence

*p*(0), …,

*p*(

*t*), … is a price trajectory if \( p\left(t+1\right)y\le p(t)x \) for all \( \left(x,y\right)\in Z,t=0,1\dots, \) If

*x*(0), …,

*x*(

*t*), … is a goods trajectory, and

*p*(0), …,

*p*(

*t*), … is a price trajectory, then the inequalities

Let us consider now the case of a von Neumann technology (in the narrow sense) given by an input matrix *A* and an output matrix *B*. The (goods) trajectory *x*(0), …, *x*(*t*) generated by the technology *Z* is determined by the sequence of intensity vectors *u*(*t*) such that \( Bu(t)\le Au\left(t+1\right) \). This sequence is called the intensity trajectory. In this case the price trajectory is a sequence *p*(*t*) such that \( p\left(t+1\right)B\le p(t)A \).

*x*(0), …,

*x*(

*t*), … generated by some von Neumann technology

*Z*can be characterized by a system of ‘shadow prices’

*p*(0), …,

*p*(

*t*), …. The corresponding result (often called the characteristic theorem) is in a sense analogous to the duality theorem of linear programming and can be interpreted in a similar manner. Under some natural additional assumptions it is: the trajectory

*x*(0), …,

*x*(

*t*), … is efficient if and only if there exists a price trajectory

*p*(0), …,

*p*(

*t*), … such that

*p*(

*t*) ≠ 0 for all

*t*and

All this can be fully carried over to the case when at every moment *t* a new technology *Z*(*t*) is used. The discussion of trajectory properties and, in particular, the characteristics theorem is contained in Makarov and Rubinov (1977).

## See Also

### Bibliography

- Champernowne, D.G. 1945–6. A note on J. von Neumann’s article on ‘A model of general economic equilibrium’.
*Review of Economic Studies*13: 10–18.Google Scholar - Gale, D. 1956. The closed linear model of production. In
*Linear inequalities and related systems*, Annals of Mathematics Studies, vol. 38, ed. H.W. Kuhn and A.W. Tucker. Princeton: Princeton University Press.Google Scholar - Kemeny, J., O. Morgenstern, and G. Thompson. 1956. A generalization of the von Neumann model of an expanding economy.
*Econometrica*24: 115–135.CrossRefGoogle Scholar - Los, J. 1978. Mathematical theory of von Neumann economic models. Report on recent results.
*Colloquia Mathematica*40(2): 327–346.CrossRefGoogle Scholar - Makarov, V.L., and A.M. Rubinov. 1977.
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