# Von Neumann Ray

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1866

## Abstract

*x*(0),…,

*x*(

*t*),… generated by this technology is called stationary if the proportions between goods in the state

*x*(

*t*) are independent of time

*t*. A stationary trajectory can be written in the form rm

*x*(

*t*) =

*γ*

^{t}

*x*where

*x*=

*x*(0) is the initial state. This trajectory is generated by a technologically feasible activity (

*x, y*) under which

*γx*≤

*γ*

_{1}. Usually a stationary trajectory is called the trajectory of balanced growth (although actual growth will take place only for

*γ*> 1). The maximum number γ which enables a balanced growth is called the von Neumann (or technological) rate of growth for the technology

*Z*. Thus the technological rate

*α*is the solution of the following optimization problem find

*α*= max

*γ*subject to

*y*≥

*αx*, the process (

*x, y*) is called the von Neumann activity (process), the corresponding vector

*x*– the von Neumann vector and the ray passing through

*x*– the von Neumann ray.

*x*(0),…,

*x*(

*t*),… generated by this technology is called stationary if the proportions between goods in the state

*x*(

*t*) are independent of time

*t*. A stationary trajectory can be written in the form rm

*x*(

*t*) =

*γ*

^{t}

*x*where

*x*=

*x*(0) is the initial state. This trajectory is generated by a technologically feasible activity (

*x, y*) under which

*γx*≤

*γ*

_{1}. Usually a stationary trajectory is called the trajectory of balanced growth (although actual growth will take place only for

*γ*> 1). The maximum number γ which enables a balanced growth is called the von Neumann (or technological) rate of growth for the technology

*Z*. Thus the technological rate

*α*is the solution of the following optimization problem find

*α*= max

*γ*subject to

*y*≥

*αx*, the process (

*x, y*) is called the von Neumann activity (process), the corresponding vector

*x*– the von Neumann vector and the ray passing through

*x*– the von Neumann ray.

J. von Neumann in his pioneering paper (1937) established that a stationary price trajectory corresponds to the growth rate *α*, i.e., there exists a sequence *p*(0), *p*(1),…, *p*(*t*),… of price vectors such that *p*(*t*) = *α*^{− t}*p* and *pw* ≤ *α pυ* for all (υ, *w*) ∈ *Z* (this means exactly that *p*(*t*) is a price trajectory). The vector *p* appearing in the definition of this trajectory is called the von Neumann price vector.

*Z*we can find the number

*α*which is the solution of the problem (1), and a technologically feasible activity (

*x, y*) and a price vector

*p*satisfying the relations

It can occur in degenerate cases that *p* = 0, i.e. all goods serving as inputs in a von Neumann process have zero prices. We shall exclude this (senseless from the economic point of view) situation and call (*α*, (*x, y*),*p*) a von Neumann equilibrium if it satisfies (2) (where *α* is the solution of the problem (1) and *px* > 0. The equilibrium has the following economic interpretation. If in the initial time period *t* = 0 the system is in the state *x*(0) = *X* then it can develop with the maximum possible rate of growth α (the same for all goods) maintaining the initial proportions between goods. This development is implemented by the activity (*x, y*). It is possible to choose time-constant prices in such a manner that the interest factor *pw*/*pv* (equal to 1 + the rate of return) for any technologically admissible activity (υ, *w*) does not exceed *α*. For the activity (*x, y*) this interest factor is maximal and equals α.

Using the notion of characteristic prices we can say that the stationary equilibrium trajectory of the economic system moving along the von Neumann ray with the rate *α* admits as a characteristic a stationary price trajectory with the same price decline rate *α*.

Now we consider a von Neumann technology in the narrow sense *Z*. Recall that it is defined by an input matrix *A* and an output matrix *B*. For this technology the conditions (2) reduce to the following inequality system *α Au* ≤ *Bu*, *pb* ≤ *α pA* where *u* is an *m*-vector of intensities.

*u, p*satisfy this system with

*p*is the vector of von Neumann prices,

*u*is the so-called vector of von Neumann intensities; it determines the equilibrium vector

*x*=

*Au*.

In terms of equilibrium it is possible to characterize goods for which growth at a rate exceeding the von Neumann growth rate *α* is technologically possible. Let (*x, y*) be an activity such that the output of good *i* is greater than its input multiplied by *α*. Then it can be easily seen that the equilibrium price of the good *i* is equal to zero; in other words, this good is free. In short, this property of the equilibrium can be stated as follows: if the growth rate for some good exceeds the technological growth rate, then this good is free.

Now we point out another property of equilibrium for a von Neumann technology, in the narrow sense defined by an input matrix *A* and an output matrix *B*. The pair (*a, b*), where *a* is the *i*th column of *A, b* is the *i*th column of *B*, defines the *i*th basic activity of this technology. To every basic activity we can associate its interest factor *pb/pa*. We can choose among the basic activities the most profitable ones, i.e. those for which the interest factor is maximal (equal to *pb/pa*). An important property of an equilibrium activity (*x, y*) is that it can be obtained by a joint use (with some intensities) only of the most profitable activities. If **u** is a von Neumann intensity vector then its components corresponding to the activities with non-maximal profitability are equal to zero.

*Z*is ‘indecomposable’, i.e. for the production of some goods all goods are (directly or indirectly) used, then this growth rate admits an economic description. To demonstrate this consider stationary price trajectories, i.e. sequences of the form.

*q*is the price vector such that

*qw*\( \leq \)

*β q*υ for all technologically admissible activities (υ,

*w*). If

*q*is given then the minimal number

*β*for which the sequence (3) is a price trajectory coincides with

*β*(

*q*) = max}(

*q*(

*w*)/

*q*(

*υ*) : (

*υ*,

*w*) ∈

*Z*} which is the maximal (at prices

*q*) growth rate. The quantity

*β*(

*q*) − 1 is the maximal rate of return at prices

*q*,

*Z*is the minimal number

*β*for which a stationary price trajectory exists. If this trajectory is generated by a price vector

*p*, i.e. has the form

*p*(

*t*) =

*β*

^{−1}

*p*then the vector

*p*is such that the maximal rate of return

*β*(

*p*) − 1 defined by

*p*does not exceed the rate of return

*β*(

*q*) − 1 for any price vector

*q*.

It turns out that if the technology *Z* is indecomposable in the aforementioned sense then the economic growth rate *β* co-incides with the technological growth rate *α*, the prices *p* with the minimal rate of return *β* (*p*) − 1 being von Neumann prices. To clarify the situation, introduce the following definition. The number *α* for which there exist an activity (*x, y*) and a vector *p* satisfying (2) and the inequality *px* > 0 is called a growth rate. It turns out that for the indecomposable nondegenerate case the technology admits only one growth rate which is simultaneously the technological and the economic one. Thus if some number *α*, for some (*x, y*)) ∈ *Z* and *p* the inequalities (2) and *px* > 0 are satisfied, then *α* simultaneously solves the problems of maximizing the rate of reproduction and of minimizing the rate of return *β*(*p*) − 1.

In the decomposable case the situation is much more tangled: several growth rates can exist. Nevertheless their number does not exceed the number of goods.

Further we shall consider only indecomposable technologies. Let *x* = *x*(0) be a vector with non-negative components representing the endowments at the moment *t* = 0. Choosing in one way or another the activities we can form various trajectories of length *T* begining in *x*(0). Among those of special interest are trajectories which are optimal in terms of some price vector *q*. If the point *x*(0) belongs to the von Neumann ray and *q* coincides with the von Neumann price vector then optimal behaviour consists in moving with the maximum technologically possible rate *α* along the von Neumann ray. It turns out that for a sufficiently wide class of initial states *x*(0) and vectors *q* the optimal trajectories must grow with a rate which differ little from *α*.

Let us discuss this in more detail. Let *p* be the von Neumann price vector. If the trajectory *x*(0), …, *x*(*T*) of length *T* is such that for a sufficiently large number of moments *t* the inequality *px*(*t* + 1)/*px*(*t*) ≤ with *γ* < *α* holds then the mentioned trajectory cannot be optimal. This assertion can be elaborated in many ways. It has a very elegant and transparent geometrical interpretation.

Consider a von Neumann technology *Z* and choose among its activities the most profitable ones (i.e. those with the maximal rate of return according to von Neumann prices *p*).

These activities form a facet of the convex cone *Z* which is called a von Neumann facet. The further it is from the von Neumann facet the less profitable is any activity. Thus, an overwhelming majority of the activities taking part in the construction of the optimal trajectory lie near the von Neumann facet. Such assertions are usually caled turnpike theorems in the weak form. More precisely, the number of activities lying ‘far’ from the facet does not exceed some number independent of the length of the trajectory. Under some additional assumptions the activities essentially different from the facet can occur only at the beginning and the end of the trajectory (turnpike theorem in the strong form). Finally, some additional assumptions guarantee that the activities forming the trajectory simply belong to the facet (turnpike theorem in the strongest form).

Suppose that *Z* is a von Neumann technology in the narrow sense. Then the von Neumann facet has as its extreme rays the most profitable basic activities. We recall that every activity (υ, *w*) from *Z* is formed as a combination of basic activities with some intensities. The closeness of (υ, *w*) to the facet means that in its formation the most profitable activities are used with substantially greater intensities than the other activities. This activity belongs to the facet if only the most profitable activities are actually used.

We mention now the case when there is only one most profitable activity (*x, y*) (this case is typical for the technologies described by production functions). The von Neumann facet in this case coincides with the ray passing through the 2*n*-dimensional vector (*x, y*). Instead of deviation of the activities from this ray we can speak about the deviation of the trajectory itself (more precisely, of its state *x*(*t*)) from the von Neumann ray which in this case is spanned by the vector *x*. The fact that a point has a small deviation from the von Neumann ray means simply that the proportions between its coordinates differ insignificantly from the proportions on the ray. This permits us to interpret the turnpike theorems from another point of view, for example, the theorem in the strong form means that the proportions between products for the states of the optimal trajectory can differ substantially from those on the ray only at the beginning and the end of the trajectory. (The first is caused by the difference of the initial state *x*(0) from the von Neumann vector *x*, the second by the difference of the optimality criterion from the vector of von Neumann prices.)

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