Growth with technological progress in John Von Neumann's Model.

ABSTRACT

In this paper we analyse the effects of including Technological Progress in the original John Von Neumann's Model. First, we show that it is possible to eliminate the assumption of constant technology of the von Neumman's original mode/maintaining the conditions of "balanced growth". As result, we show that it is possible to obtain a rate with "balanced growth" for the economy expanding with Technological Progress. Also, we show that the expanding economy rate with Technological Progress has a minimum which is the maximum expanding economy rate without Technological Progress. This paper is the first step in a new line of research in multi-sector economic growth models, and where there are no restrictions about the functional form between input and output. These two are the most important differences between this model and the Solow's model After showing that the incorporation of Technological Progress in the John von Neuman's model is possible, the following steps must include the effects of endogenous growth or open economies, between others.

1. INTRODUCTION

John von Neumann is well known in the realm of Economics for his discoveries in two fundamental fields of Economic Analysis: Theory of Games and Growth. In fact, he is considered to be the creator of the Theory of Games, having published his first article on the theme in 1928. He later collaborated with Oskar Morgenstern in the publication of The Theory of Games and Economic Behaviour in 1944 and the mathematical formulations described in the book have greatly influenced economic analyses from then on. Reinhard Selten, who shared the 1994 Nobel Prize for Economics with John F. Nash and John C. Harsanyi for their advances in the analysis of equilibrium in the Theory of Games, acknowledges, in his autobiography, that his first steps in analysing the Theory of Games were based on John von Neumann and Oskar Morgenstern's book. At the beginning of the 1950's, John C. Harsanyi published a series of articles based on the utility functions in economic welfare and ethics described by these authors. Furthermore, Kenneth Arrow, winner of the Nobel Economics Prize for 1972, and Gerard Debreu, the winner in 1983, both based their works on Neumann's model of the Utility Theory to solve General Equilibrium problems.

In 1937, however, John von Neumann published another innovative article on economic growth in the context of general equilibrium. It was a multi-sector model that demonstrated that growth in equilibrium could exist under the supposition of the existence of different sectors in the economy. It facilitated the analysis of the consequences of the circular nature of the production process, or one in which a good could be produced from others or even from itself. He developed the theory of prices that are determined exclusively by the minimum cost of the goods obtained from other goods as well as the theory of an interest rate that is determined by the greatest possible expansion rate of the economic system. Furthermore, it was the first rigorous and formal modelling of the theory of non-aggregated capital.

The definition of "growth in equilibrium" (or balanced growth) implied the indispensable inclusion of certain highly restrictive assumptions about technology in the model. Its definition of equilibrium was quite similar to that of the steady state, in the sense that, in equilibrium, an uniform expansion of the entire system was possible. Prices remained constant, the relative intensities of the different production processes did not vary and the production of all the different goods maintained their same proportions, although an uniform geometric rate of growth of the entire system was possible. Furthermore, apart from the assumption of perfect long-term competition and the existence of constant returns in any given economic process, (i.e., each process can be carried out up to x times its given scale, without any increases or decreases in the cost per unit of "output"), it seemed to imply the need for technology to remain constant over time. As each output generated within a given period of time would be used as input for the following period, it would have to appear in the economic system in the same proportion and with the same intensity as it did during the previous period to allow a uniform and "balanced" expansion of the entire system. The definition of "balanced growth" and the need for the relative intensities of the production processes of the system to remain constant over time, implicitly suppose that the relative intensities of the inputs and, therefore, of the outputs, (which are to be the inputs for the following period), would remain constant. This implicit assumption of constant relative intensities of inputs for all time periods "forced" J. von Neumann to work with the assumption of constant technology, so that for all goods, the output/input ratio remains constant, not only in each period and for each good, but for all goods in all time periods as well. The "need" to include such an assumption in his model and, therefore, the "impossibility" of considering Technological Progress and its probable effects on the greatest possible rate of the system's balanced growth, attracted a lot of criticism (Champernowne, 1945-1946; Koopmans, 1964), which almost condemned this precursor to the multi-sector growth model, in the context of general equilibrium, to oblivion.

As we demonstrate in this paper, however, under certain assumptions it is possible to include Technological Progress in von Neumann's model without altering the other necessary conditions and obtain a balanced growth rate (according to Von Neumann's definition of "equilibrium"). The supposition of constant relative intensities of inputs over time does not "impose" the supposition of constant technology. To facilitate the verification of this assumption, we resort to matrix calculus, so that both the conversion of Von Neumann's original model into matrix terms, as well as the construction of the output matrix that guarantees growth in equilibrium in all periods, would be the fundamental instruments that afford a clear insight into the true meaning of balanced growth and its implications. Identifying the key features of balanced growth, both with and without Technological Progress, would then allow us to identify the differences between the two different types of balanced growth. It would also afford the possibility of obtaining the maximum rate of balanced growth that depends on the capacity that Technical Progress has to increase the output. Furthermore, it would also be possible to demonstrate that the system's rate of balanced growth in the existence of such Technological Progress will have a minimum, which will be precisely the minimum rate of the system's balanced growth without the existence of technological progress.

The aim of this study is to reinitiate the line of research begun by von Neumann, considering multi-sector growth models that do not impose any restrictions on the sort of functional form that relates input to output. As such, some of the restrictions imposed by the traditional growth models are eliminated, such as the Cobb-Douglas functions, and two production factors (capital and work), among others.

2. TECHNOLOGY IN VON NEUMANN'S GENERAL EQUILIBRIUM MODEL

The aim of this section is to present, develop and amplify the original von Neumann model, with the aim of determining the implications of his definition of "balanced growth". In contrast to von Neumann's version of the model, in this study we present the model in its matrix form. The construction of the input and output matrixes, and especially that of the "output" required to maintain the same level of production in the following period, will be the basic instrument for achieving a clear vision of the implications that the original model has for technology.

Two of the main features of von Neumann's growth model are the existence of m production factors and n goods in the economic system in each period of time, instead of the two usual ones (capital and work), and, secondly, the circular nature of the production process in which each good to be produced requires a given amount of the other goods, and/or even of the same good. From this we deduce that the good [X.sub.1] is produced with the input of the good [X.sub.2] and in the same period, the good [X.sub.2] is in turn produced with the input of the good [X.sub.1].

To do so, yon Neumann supposes that there are n goods [X.sub.1], ..., [X.sub.n] that can be produced by m production processes [F.sub.1], ..., [F.sub.m], of which each production process [F.sub.i], i = 1,K,m, is defined as:

[1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.ij] is the number of units of the good j (input) required in the process i, and [b.sub.ij] is the coefficient of the good j (output) of the process i.

If we consider this economic system in terms of matrix, we could express the economic processes of input and output as follows:

[2] F (AX)=BX,

where:

[3] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We must point out that the j-th column of the matrix A represents the total number of productive units of good j required by each of the production processes, j = 1,K, [n.sup.()], and that the i-th row represents the total number of units of each of the goods required for production process i, i = 1,K ,[m.sup.()]. Likewise, the j-th column of the matrix B represents the total units of the good j, produced, j = 1,K,n, in coefficient terms, that are generated by all the production processes, and the i-th row represents the total units of all the goods produced, expressed in the coefficient that each production process generates i, i = 1,K,m.

In function of the total amount of goods that exist in the economy in each period of time, each of the production processes [F.sub.i] will be employed in at a given intensity [q.sub.i], i = 1,K ,m, so that the total production achieved in each period of the economy will be equal to:

[4] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where:

Q = ([q.sub.1], [q.sub.2], K, [q.sub.m]), with

[5] [q.sub.1] [greater than or equal to] [0.sup.()],

[6] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and Y is the total amount of goods in each period of time.

Given [2] and [4] we can define:

[7] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] : total requirements for the good j, j = 1,K,n

(since it represents the product of the matrix Q for each column of the matrix A) and

[8] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] : total production of the good j, j = 1,K, n

(since it represents the product of the matrix Q for each column of the matrix B).

So that if we denote the total number of units of the goods that exist at the beginning of the first period by the sub-indexes 0 and 1 respectively, once all the productive processes have been carried out we obtain:

[9] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The description of the economic system reflected in Expressions [1] to [9] implies the assumption of many suppositions, implicit, and even explicit, in von Neumann's model:

(i) Capital goods figure on both sides of the application, that is to say, there is both an input and an output to the production process.

(ii) Each process lasts one unit of time, so that if there are production processes that require more time, the processes are divided over time as many times as necessary to obtain the end product, and each of the goods obtained from each sub-process is considered an intermediate input, required for the production of the final good.

(iii) Each sub-process of Supposition (ii) is considered to be a process in itself, so that the inputs required for these new processes will be "natural production factors", or final goods obtained from the other productive processes carried out during the previous period, or intermediate goods, and the output obtained will be, in turn, a new intermediate good or the end product.

(iv) For each production process the relationship [a.sub.ij]/[a.sub.ij+1] remains constant, j = 1,K,n. This implies

that the intensities of the factors of each process i do not vary, i = 1,K,m. Variations in such intensities would imply a new process. This is one of the fundamental and most restrictive suppositions of the von Neumann model. Each production process requires a fixed combination of inputs that cannot be altered. Once a production process has been chosen, no variations are allowed in the relative intensities of its factors, so that it remains constant over time.

To now, the model reflects an economic system, within a given period of time, that allows the transformation of certain goods into others. To be able to speak of "economic growth", we must define the input-output relationships that should be maintained throughout the different periods of time. Furthermore, if what we seek is a "balanced growth" that could justify a continuous and permanent growth of the output, the input-output relationships that must be established are not only more complex but more restrictive as well. To achieve this, von Neumann, (in an effort to find the rate that permits the maximum possible growth of the economic system, but in such a way that, with what is produced in each period, the increase in production in the following period is guaranteed), includes the following suppositions either explicitly or implicitly:

(v) The input of any given period will be the output of the preceding period. This is the key supposition of his model. It considers a closed economy in which the production requirements for certain goods, in any given period, cannot come from outside the economy, but must proceed from the production of the previous period of the same economy. This implies that in each in time period there will have to be a series of production processes that are capable of generating the necessary output in terms of relative intensities and relative proportions which, when they are used as inputs in the following period, would guarantee the production of an output that maintains the same structure in terms of relative proportions and intensities as the output produced during the previous period. Only in such a case can we speak about "balanced growth".

(vi) The production factors can grow to an unlimited extent. This supposition, however, does not imply that in each period, the amount of each input required, or those generated by the different production systems will be unlimited. On the contrary, given that it is based on the existence of a limited number of goods at the beginning of the process, the amount of output that is obtained from each production process is limited. Since the output of one period is the input for the following period, in each period, the amount of input and output are assured.

(vii) The capitalists reinvest all of the output and the workers consume all of their income. This way, the reinvestment of any excess production of the workers' and employees' needs is assured.

(viii) There are constant returns in production, which implies that each process can be carried out at x times its given scale. The output/input relationship does not vary in function of the number of times that a process is repeated.

(ix) The ratio between the intensities of the use of the different processes [q.sub.i]/[q.sub.i+1] remains constant over time, although [q.sub.1],K,[q.sub.m] might change, which allows the use of any unit for measuring time, i = 1,K,m. This supposition, however, also impedes the entry of any new production process that would alter the intensities of the factors employed for each good.

Supposition (v) is a clear restriction. It prevents any guarantee on the future production of any good since, for each good, it is imperative that, as a minimum, the amount of each of the goods that is produced in each period be equal to the total input required by the different processes. If what is desired is the growth of the system, then an indispensable prerequisite will be that the output obtained from each of the goods must be greater than its respective input. Given [7] and [8], this implies that the total of each good produced must be higher that the total requirements of the good implied. That is to say:

[10] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[11] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, for the economic system it must be proven that:

[12] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the [[alpha].sub.j] can be different for each good, there will be certain goods that increase at a higher rate than others. Supposition (v), however, determines what will happen in such a case. As the input for any given period is necessarily the output from its preceding period, to guarantee that the production is assured in the future, the distribution of the total output at the end of a period must be proportional to the input required at the beginning of the period to be able to repeat the production of the good during the following periods, which implies that all the goods must grow at the same rate to be able to maintain the same distribution. In such a case, we could certainly speak of balanced growth. It implies that balanced growth exists when, for all of the goods, the output obtained, which serves as the input for the next period is proportionally equal to its respective input for the previous period. That is to say:

[13] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, furthermore, [q'.sub.i] and [b'.sub.ij] must fulfil the suppositions expressed in (iv) and (ix), with [q'.sub.i] being the intensity with which el production process i in the following period will be repeated, i = 1,K,m with [b'.sub.ij] being each of the output coefficients obtained in this period, but will be employed in the following period, j = 1,K,n.

If we express [13] in matrix, we can state that: balanced growth, according to J. von Neumann, exists when it is verified that:

[14] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[15] [[alpha].sup.*] > 1.

And this is true for

[16] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Expressions [13] to [16] require two additional explanations:

1. Conditions [13] to [16] are prerequisites to the balanced growth of production. Balanced growth requires the same [alpha] ([[alpha].sup.*]) for all the goods and, furthermore, that this [[alpha].sup.*] must be equal to the minimum of all the existing [[alpha].sub.j]. An [[alpha].sup.*] > [[alpha].sub.min] does not permit balanced growth, since there would be a shortage of output for the good j for which [[alpha].sub.j] < [[alpha].sup.*], which would impede the application of some of the production processes that require it as a necessary input and, as such, the maintenance of the economic system. It must be remembered that although [[alpha].sup.*] might be equal to the minimum of all the [[alpha].sub.j], this value represents the maximum growth rate possible for the entire economic system, so that it is really a condition of the maximum possible or the feasible maximum for balanced growth. In accordance with von Neumann's terminology, [[alpha].sup.*] would be the "coefficient of expansion of the whole economy" per unit of time.

In any case, this condition does not really determine what the total volume of output for each period will be. Conditions [13] to [16] simply indicate that, in balanced growth, the economic system itself must have the same [[alpha].sup.*] for all the goods and all the processes, and which, of necessity, must be the lowest of all the existing [[alpha].sub.j] to be able to guarantee future production. But there can be [[alpha].sub.j] > [[alpha].sup.*] in the economic system. In such a case, the system generates an excess of production of goods that will not all be used in the production processes of the following period. This excessive production of goods is labelled free goods as their price is equal to zero. This is why von Neumann introduces his Rule of free goods: given [11], and [p.sub.j] being the price of the good j, j = 1,K,n,

[17] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[18] And for the good j which in [17] obtains > we apply [p.sub.j]=0.

Indeed, for free goods it will be that [[alpha].sub.j] > [[alpha].sup.*] in other words, [for all]j such that

[19] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Furthermore, expression [14] has an additional implication.

If Q'= ([q'.sub.1],K,[q'.sub.m]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] it is obvious that Q'B' is a replica of QA.

If QB represented the total output obtained during that period, Q'B' represents the total amount of goods produced during that period and, in virtue of Supposition (v), they will be the input for the following period, so that they are the goods that will be needed to maintain the production during the following period. In other words, it represents only what will be used to guarantee future production. Furthermore, von Neumann also operates with Suppositions (iv) and (ix), such that the relationships among the intensities of the factors that remain constant Will have to be verified in each period, as well as the relationships among the production processes. This implies that for balanced growth to exist in the sense of von Neumann's definition,

[20] [a.sub.ij]/[a.sub.ij+1] = [b'.sub.ij]/[b'.sub.ij+1] y [q.sub.ij]/[q.sub.ij+1] = [q'.sub.ij]/[q'.sub.ij+1], i=1, K, m, j=1, K, n,

and this, in turn, implies that

[21] [there exists]h > 0 such that Q' = hQ.

As such,

[22] B' = [[alpha].sup.*] 1/h A,

which then implies that

[23] [b'.sub.ij] = [[alpha].sup.*]/h [a.sub.ij], i = 1,K,m, j = 1,K,n

As von Neumann himself suggests, there will be no changes in the production structure. The intensities of the factors remain constant. This is why B' is really a replica of A. It reflects the maximum distribution of goods possible but, at the same time, it has to be able to generate balanced growth in the following period.

If QB is the total of what is produced, given the different [[alpha].sub.j] that exist and which could be employed in subsequent periods, Q'B' represents what will really be employed, since the economic system only "requires" [[alpha].sup.*] QA. In accordance with [10] and [11], there will be an excess in production, or free goods in QB that will no be used for the goods that have [[alpha].sub.j] > [[alpha].sup.*].

Since [[alpha].sup.*] = min{[[alpha].sub.1],K,[[alpha].sub.n]}, a matrix B' is guaranteed, which is a replica of A We could go on using the same production processes indefinitely, guaranteeing a future output that is a replica of the one obtained in each of the previous periods. Since B' is a replica of A and the economy always grows at the same rate [[alpha].sup.*], in von Neumann's model technology was kept constant throughout all of the periods to ensure balanced growth. Once the production process to be carried out and the intensity with which it should be done had been decided, the coefficients of the different inputs [[alpha].sub.ij] remained fixed for each production process and the output that should be generated had to be the one that guaranteed that each and every [b'.sub.ij] was equally proportional to their respective [a.sub.ij]. The intensities of the relative factors remained constant, all the goods grew indefinitely at the same rate, but that rate then remained fixed in the first period, given the initial technology, and was determined by the technological capacity of the first period. Goods could be created in excess, but they would not be introduced as inputs for the following period, but would be free goods. Furthermore, there was a limit to growth which was fixed precisely by the good that was most difficult to reproduce. In von Neumann's model of balanced growth there seemed to be no place for any technological change or any technical progress and the maximum possible growth was restricted by the good with the lowest growth rate.

3. TECHNOLOGICAL PROGRESS IN VON NEUMANN'S MODEL

In the previous section, we demonstrated that the suppositions that the relative intensities of the different production processes remain constant and that the intensities of the factors of each production process also remain constant over time, required that all of the goods produced should grow at the same rate in every period, so that the economic system could be replicated indefinitely and, thus, afford balanced growth. In this section, we shall demonstrate that the supposition of constant intensities of the factors involved does not imply that Technological Progress cannot exist. Under certain suppositions, we demonstrate that a balanced growth rate in von Neumann's multi-sector model is compatible with the existence of Technological Progress. We also demonstrate that the growth achieved under the supposition of Technological Progress has a minimum rate that is equal to the rate of balanced growth achieved without Technological Progress.

To do so, we consider that one period has already lapsed and that Conditions [13] to [16] for balanced growth have already been verified, so that a B' already exists, such that it verifies Expressions [19] and [20] regarding the constant nature of the relative intensities of the production processes over time.

A new period begins in which there is a technological innovation that allows us to obtain the matrix C of the coefficients [c.sub.ij], i = 1,K,m, j = 1,K,n.

Definition: Technological Progress is defined as the introduction of new production methods into the economic system that generate more amount of output (with the same volume of input) that the obtained with the old technology of the previous period, due to the increase on the productivity of some or all of the production factors. To be more specific, Technological Progress will exist when the output of each good surpasses the volume it would have had if its production method had operated with the old technology of the previous period. That is to say:

[24] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[25] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[delta].sub.j] being the technological innovation in the production processes of the good j that fulfil the Definition of Technological Progress, which, expressed as matrixes, gives us

[26] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Apparently, the definition of Technological Progress, represented by the existence of [[delta].sub.j] is a very open one, as it only imposes the restriction that the output of each good must be higher than its input with a coefficient that is greater than the one that would have existed in the case of no Technological Progress being applied ([alpha]). To demonstrate that with Technological Progress a balanced growth rate can continue to exist such that in each period the output matrix is a replica of the one obtained in the previous period, we have chosen the matrix B', obtained in the previous period and which fulfils the condition of balanced growth, as our input matrix. With this matrix, what is being implicitly considered is that, in the new period, and regardless of the Technological Progress that is applied, the intensities of the factors of each production process must always be constant, from which we deduce that [[delta].sub.j] really represents a Hicks Neutral Technological Progress.

For balanced growth to exist, according to J. Von Neumann and, therefore, for the entire system to be able to grow indefinitely, the diagonal matrix of the [[alpha].sub.i](1 + [[delta].sub.j]) must be such that we verify that

[27] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must be proportional to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j = 1,K ,n,

with [q".sub.i] being the intensity with which the production processes in the period of time that is beginning and [c'.sub.ij] the output coefficients obtained in this period will be repeated, but which fulfil Supposition (iv) and, therefore, guarantee balanced growth for the following period.

This can only be possible if all the components of such a diagonal matrix are equal. That is to say, that balanced growth exists with Technological Progress if [there exists][[delta].sup.*] > 0 such that

[28] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

fulfilling [q".sub.i] y [c'.sub.ij] Suppositions (iv) and (ix), and where

[15] [[alpha].sup.*] > 1,

[16] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and since for [25] [[delta].sub.j] > 0, Expression [28] is verified for

[29] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From these expressions we deduce that a maximum rate of balanced growth can continue to exist although a technological progress that changes the output/input from one period to another is included. The only condition is that all the goods be subject to technological advances that imply increases in their output. Just as in the case of the value of [[alpha].sup.*], the value of [[delta].sup.*] must be equal to the lowest of all the [[delta].sup.j] to be able to guarantee the level and the distribution of output required as input for the following period. In any case, the level of Technological Progress that guarantees balanced growth is the one that has the value [[delta].sup.*]. This condition, however, does not imply that the Technological Progress has to be equal for all the goods. Different improvements may exist for different goods, with different increases in the production for each one. This means that the economic system permits that for Expressions [25] and [29] there can exist

[30] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this case, there will be an excess of goods that will not be used in the production processes of the following periods. Once again, there will be free goods for the application of the following von Neuman Rule

[31] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[32] and for the good that, in [31] obtains > then [p.sub.j]=0 is applied.

With Technological Progress, therefore, a rate of balanced growth can continue existing on the condition that the production of all the goods increases as a result of the technological innovation.

But, from Expressions [27] to [32], an additional conclusion can be drawn: i.e., in balanced growth, given that the same [[alpha].sup.*] and [[delta].sup.*] are applied for all the goods, and since by the application of Suppositions (v) and (ix), Q' and Q" are proportional,

[33] [there exist]C' such that C' = [[alpha].sup.*](1 + [[delta].sup.*]) 1/g B'

where B' and C' are proportional, g, being the factor of proportionality between Q" and Q', C' is the matrix mxn of the coefficient of output c' = ([c'.sub.ij]) required to be able to replicate the production in the following period and is obtained with the initial technology ([[alpha].sub.*] and the technological improvement [[delta].sup.*], and in which

[c'.sub.ij] = [[alpha].sup.*] (1+[[delta].sup.*]) 1/g [b'.sub.ij].

We also verify that the intensities of the relative factors of the inputs remain constants. That is to say,

[34] [[alpha].sub.ij]/[[alpha].sub.ij+1] = [b'.sub.ij]/[b'.sub.ij+1] = [c'.sub.ij]/[c'.sub.ij+1]

This implies that the economic system continues to grow in equilibrium, continually adopting a Hicks Neutral Technological Progress. This means that different Hicks Neutral Technological Progresses can exist over time and the yon Neumann condition of balanced growth can still be maintained. In accordance with Expression [25], and by the application of the Rule of free goods, [31] and [32], it is not necessary that the same Hicks Neutral Technological Progress should exist for each good and in all periods of time. The Rule of free goods allows an excessive production of goods. If, in the following period, some of the output, which would now be input, is not required, it would be converted into free goods. As such, any increase in output, with the same amount of input, is assumable. Technological Progress generates an output matrix, C, that does not necessarily have to be equal to the one required by the economic system for to be replicated (C'). Growth will continue to exist as long as the minimum amount of inputs that allow all of the production processes to be carried out in their fixed proportions continues to exist. Nevertheless, whatever the sort of Hicks Neutral Technological Progress that is assumed, from Expressions [28] and [34] we deduce that the one that the economic system automatically adopts is the lowest one in each period.

On the other hand, and given that [[delta].sub.j] > 0, we also verify that the maximum rate of balanced growth with Technological Progress is, obviously, always higher than the maximum rate of balanced growth without Technological Progress. The maximum rate of balanced growth with Technological Progress has a lower limit, which is precisely the maximum rate of balanced growth possible without Technological Progress, given that:

[35] [[delta].sub.j] > 0, j = 1,K,n, [??] [[alpha].sub.*](1 + [[delta].sub.*]) > [[alpha].sup.*]

Technological Progress allows us to overcome the restriction that the maximum rate of balanced growth must be determined, precisely, by the good with the lowest growth rate.

The route of balanced growth, with and without Technological Progress

In a situation of balanced growth when no Technological Progress exists, the output grows, initially, at the rate [[alpha].sup.*] which is fixed by the initial technology. As no technological change exists, the same situation is repeated in all the periods, such that the output in equilibrium in each period is [[alpha].sup.*] times the output of its previous period. As such, if we define [Y.sup.*.sub.t] as the level of output in equilibrium that is obtained by applying the balanced growth rate in the period t, t = 0,K,k and if we assume the particular case in which [Q.sub.t-1] = [Q.sub.t] = [Q.sub.t+1] = Q (since the relative intensities of the production processes remain constant), the following is verified:

[9] [Y.sub.0] = QAX,

[35] [Y.sup.*.sub.1] = QB'X = [[alpha].sup.*] QAX = [[alpha].sup.*] [Y.sub.0],

[36] [Y.sup.*.sub.2] = [[alpha].sup.*] QB'X = [([[alpha].sup.*]).sup.2] QAX = [([[alpha].sup.*]).sup.2] [Y.sub.0].

In general,

[37] [Y.sup.*.sub.t] = [([[alpha].sup.*]).sup.t] QAX = [([[alpha].sup.*]).sup.k] [Y.sub.0].

As such, for each period:

[38] [Y.sup.*.sub.1]/[Y.sub.0] = [Y.sup.*.sub.2]/[Y.sup.*.sub.1] = [LAMBDA] = [Y.sup.*.sub.t]/[Y.sup.*.sub.t-1] = [[alpha].sup.*], with [Y.sup.*.sub.0] = [Y.sub.0],

while the rate of accumulated growth for any period would be:

[39] [Y.sup.*.sub.t]/[Y.sub.0] = [([[alpha].sup.*]).sup.t]

On the other hand, with Technological Progress, even if we assume the same initial situation, and considering that the technological change takes place in the second period, the level of output in equilibrium, as well as the growth rate of each period and the accumulated rate would be:

[9] [Y.sub.0] = QAX,

[40] [Y.sup.*.sub.1] = QB'X = [[alpha].sup.*] QAX = [[alpha].sup.*] [Y.sub.0],

[41] [Y.sup.*.sub.2] = [[alpha].sup.*] (1 + [[delta].sup.*.sub.2])QB'X = [([[alpha].sup.*]).sup.2] (1 + [[delta].sup.*.sub.2])QAX = [([[alpha].sup.*]).sup.2](1 + [[delta].sup.*.sub.2])[Y.sub.0],

[42] [Y.sup.*.sub.3] = [[alpha].sup.*] (1 + [[delta].sup.*.sub.2])[Y.sup.*.sub.2] = [([[alpha].sup.*]).sup.3][(1 + [[delta].sup.*.sub.2]).sup.2][Y.sub.0].

In general,

[43] [Y.sup.*.sub.t] = [([[alpha].sup.*]).sup.t][(1 + [[delta].sup.*.sub.2]).sup.t-1][Y.sub.0],

where [[delta].sup.*.sub.2] represents the technological innovation that takes place in the second period and which fulfils [29]. Furthermore:

[44] [Y.sup.*.sub.1]/[Y.sub.0] = [[alpha].sup.*] [not equal to] [Y.sup.*.sub.2]/[Y.sup.*.sub.1] = [[alpha].sup.*](1 + [[delta].sup.*.sub.2]) = [Y.sup.*.sub.3]/[Y.sup.*.sub.2] = [[alpha].sup.*](1 + [[delta].sup.*.sub.2]) = [LAMBDA] = [Y.sup.*.sub.k]/[Y.sup.*.sub.k-1] = [[alpha].sup.*](1 + [[delta].sup.*.sub.2]),

[45] [Y.sup.*.sub.t]/[Y.sub.0] = [([[alpha].sup.*]).sup.t][(1 + [[delta].sup.*.sub.2]).sup.t-1] > [([[alpha].sup.*]).sup.t], [for all][[delta].sup.*.sub.2] > 0, [for all]t = 1,K,k

Expressions [35] to [45] allow us to verify that when no Technological Progress exists, the growth rate of each period is constant, so that the output/input ratio does not vary. There is a route of uniform growth throughout all of the periods, which describes a geometric expansion of the system. With Technological Progress, on the other hand, the coefficient of expansion varies in the period in which the Technological Progress is generated, so that the output/input ratio varies between that period and the previous one. A geometric expansion of the entire system is obtained again but, from the moment at which the Technological Progress takes place at a higher level and with a higher growth rate. The accumulated growth is greater with Technological Progress and the difference is precisely the Technological Progress generated in the second period.

GRAPH 1

A possible route of maximum balanced growth with Technological Progress in the second period (P.T.2) and without Technological Progress (S.P.T.).

[GRAPHIC OMITTED]

GRAPH 2

A possible route of maximum balanced growth without Technological Progress, with Technological Progress in the second period (P.T.2), and with Technological Progress in several periods (P.T.V)

[GRAPHIC OMITTED]

The graphic representation of the routes of growth in both cases allows us to see the differences between balanced growth with and without Technological Progress.

If a technological change takes place, then from moment [t.sub.2] onwards there is new technology that permits the system to expand with the coefficient [[alpha].sup.*] (1 + [[delta].sup.*.sub.2]). A jump is seen, which generates values of [Y.sup.*] in each period that are higher than what they would be in the case of there not having been any technological change. As a result, the value of the ratio [Y.sup.*.sub.t]/ [Y.sup.*.sub.t-1] is higher from period 2 on, maintaining its new level. When we consider that there could be different technological changes in different periods of time, the results change slightly. With different technological innovations in each period, the property of uniform geometric expansion of the economic system is lost, since the output/input ratio for each period varies as a function of the technological advances that occur in each period. A route is established in which the expansion rate of the system can vary from period to period, as is reflected in Graph 2, in the P.T.V. line. In this case, [9], [40] and

[41] continue being fulfilled for [Y.sub.0], [Y.sup.*.sub.1] e [Y.sup.*.sub.2]. However:

[46] [Y.sup.*.sub.3] = [[alpha].sup.*](1 + [[delta].sup.*.sub.2])(1 + [[delta].sup.*.sub.3])[Y.sup.*.sub.2] = [([[alpha].sup.*]).sup.3][(1 + [[delta].sup.*.sub.2]).sup.2](1 + [[delta].sup.*.sub.3]) [Y.sub.0],

[47] [Y.sup.*.sub.k] = [([[alpha].sup.*]).sup.k][(1 + [[delta].sup.*.sub.2]).sup.k-1][(1 + [[delta].sup.*.sub.3]).sup.k-2] [LAMBDA] (1 + [[delta].sup.*.sub.k])[Y.sub.0],

and the growth rate for each period, as well as the accumulated rate, would be

[48] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[49] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

being,

[50] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if any [[delta].sup.*.sub.l] > 0 exists.

4. THE PROFITABILITY RULE

It must be remembered that [[alpha].sup.*](1 + [[delta].sup.*.sub.2]) is a maximum expansion coefficient of the economic system in every period that allows balanced growth. But the fact that it is a maximum value does means that the economic system may grow at that rate. In other words, Expressions [15], [16], [29], [31] and [32] impose a restriction. They reflect an entire group of viable growth coefficients in which [[alpha].sup.*](1 + [[delta].sup.*]) is the highest growth coefficient possible. To be able to determine the system's true coefficient of expansion, we require another selection criterion that would allow the choice of a growth rate from among all the feasible ones. The second approach introduced by von Neumann, expressed in similar terms to those of the Free goods Rule, is known as the Profitability Rule. For a period in which there is no Technological Progress, this rule would be represented by:

[51] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[52] and for the process i, which in [51] obtains < we apply [q.sub.i]=0, bearing in mind that:

[53] [p.sub.j] [greater than or equal to] 0,

[54] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This Rule indicates that, in equilibrium, only those production processes that have a profit that is equal to zero after the interest rate of the economic system [[beta].sup.*] has been paid, can be used. If the return were lower, the process would not be profitable and would therefore not be used, which would yield [q.sub.i]=0 for this process.

This implicitly indicates that each productive process can generate a certain profitability that doesn't have reason to coincide with the interest rate of the economic system.

[55] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Expressed in matrix, we let:

[56] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since different [[beta].sub.i] exist, to guarantee maximum growth, it is imperative that the profitability rate of selected production processes be the highest of all the available production processes, which implies that:

[57] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we consider that a Technological Progress exists in Period 2, the rate of profitability of the chosen production processes will be:

[58] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since, in this case, the Profitability Rule will be expressed as:

[59] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[60] and [q.sub.j]=0 is applied for that process that obtains < in [59]

with [[gamma].sup.*] being the maximum rate of profitability of the production processes in the period in which Technological Progress occurs.

5. BALANCED GROWTH

The model assumes that [a.sub.ij], [b.sub.ij] y [c.sub.ij] are already given while the values of [[alpha].sup.*] y [[beta].sup.*] are unknown, so that [[delta].sup.*] and [[gamma].sup.*] will also be unknown. As such, the problem expressed in [5], [6], [17], [18], [31], [32], [51]-[54], [59] and [60] must be resolved by taking [16] and [29] into account. The resolution of this system is quite similar to that of yon Neumann's, so that we shall briefly outline, below, the case in which Technological Progress appears. Let [Q'.sup.*] be a vector of any sort of production processes ([q'.sub.1.sub.*],K,[q'.sub.m.sup.*]) in Period 1 (in which there is a Hicks Neutral Technological Progress), each of which verifies Expressions [5] and [6], so that:

[5'] [q'.sub.i.sup.*] [greater than or equal to] 0,

[6'] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In an analogous way, let [P.sup.*] be a vector of prices ([p.sub.1.sup.*],K,[p.sub.m.sup.*]) for Period 1 (in which there is a Hicks Neutral Technological Progress) each of which verifies Expressions [53] and [54], so that:

[53'] [p.sub.j.sup.*] [greater than or equal to] 0,

[54'] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let:

[61] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, let Q' = ([q'.sub.1],K,[q'.sub.m]) and P = ([p.sub.1],K,[p.sub.n]), be the hypothetical solutions. In such a case, under the conditions described in the Rule of Free Goods and the Rule of Profitability we obtain that:

[62] [phi](Q',[P.sup.*]) achieves its minimum value for [P.sup.*] if [P.sup.*] = P,

[63] [phi]([Q'.sup.*], P) achieves its maximum value for [Q'.sup.*] if [Q'.sup.*] = Q', and, since [29], [31], [32], [58], [59] and [60] are already given, we obtain:

[64] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[65] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As such, we therefore very that

[6] [[alpha].sup.*](1 + [[delta].sup.*]) = [[gamma].sup.*] = [phi](Q', P'],

just as what happens in the von Neumann model without Technological Progress. The maximum Balanced Growth rate with Technological Progress and the minimum Profitability of the system are equal.

6. CONCLUSIONS

In 1937, J. von Neumann published an innovative article in which he presented a model of economic growth that was not only a multi-sector model, but also one that did not impose any restrictions on the functional forms that determined the output in every period. However, other very restrictive suppositions were introduced: i.e., permanence in the intensity with which the different production processes were used and, secondly, that each process had a constant combination of inputs. This implied that for a "coefficient of expansion of the economic system" to exist, all of the goods should appear in the same proportions within the economic system. What was required in every period, was that there be a replica of what was required in the previous period. Specifically, the economic system was ever-lasting and was always replicated with the same proportions and at the same rate. In our study, however, we demonstrate that the expansion rate of the economic system does not necessarily have to be the same for all periods of time. If Technological Progress appears, this rate can certainly change. If, however, we wish to guarantee balanced growth under the suppositions of von Neumann's multi-sector model, the intensities of the factors must remain constant in all periods. This implies that the only type of Technological Progress that would be compatible with the von Neumann model is a Hicks Neutral Technological Progress. Any other sort of Technological Progress implies changes in the intensities of the input factors. So that for it to be feasible in the von Neumann model, his definition of Balanced Growth would have to be altered and an alternative definition that would allow the sustainable growth of the economic system found.

End Notes

(1) "Zur Theory of the Gesellshaftspiele", in Mathemathische Annalen.

(1) Even the articles written by John F. Nash on the matter were published alter those of the authors mentioned here. John von Neumann and Oskar Morgenstern had already presented a similar solution to "Nash Equilibrium", although it was designed exclusively for zero sum games.

(1) In spite of the mathematical complexities of the von Neumann model, there have been several attempts to relate some or several of the suppositions implied in his model.

(1) Column 1 indicates the inputs of good 1 required by each of the different production processes, from 1 to m.

(1) Likewise, Row 1 reflects the amount of each good, from 1 to n, that Process 1 requires.

(1) In the particular case of [q.sub.i] = 0, it indicates that process i will not be used, i = 1,K,m.

(1) This avoids the situation in which [q.sub.1] = [LAMBDA] = [q.sub.m] = 0.

(1) According to von Neumann's definition of growth, the output of each good must be higher than the input by a given rate [alpha]. This does not suppose that each production process must generate an output that is higher than the required input, since it would be different for each good). Instead of employing this definition, an alternative definition of growth could have been used, in which the increase in output of each good would also vary among the different processes. This implies that [there exists][[alpha].sup.ij], such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In any case, operating with the latter definition would not alter the results of obtaining the rate of the maximum balanced growth, so that we chose to keep von Neumann's definition.

(1) If, instead of operating with [[alpha].sub.j] we had considered [[alpha].sub.ij], the [[alpha].sub.ij] that guarantees balanced growth would have been equal to [[alpha].sup.*] = min{[[alpha].sub.ij]} = min{[q'.sub.i][b.sub.ij]/[q.sub.i][a.sub.ij]}, i = 1,K,m, j = 1,K,n.

(1) In the same terms as before, and analogous of the von Neumann model, we opted to work with [[delta].sub.j] rather than working with a different [[delta].sub.ij] for each process and for each good. The use of either coefficient, however, does not alter the results on the maximum rate of balanced growth.

(1) If, instead of having worked with [[delta].sub.j] we had considered [[delta].sub.ij], the [[delta].sub.ij] that guarantees balanced growth would be equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1) Without Technical Progress and with balanced growth in each period, we obtain the ratio

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With Technical Progress and balanced growth in each period, we obtain the ratio

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1) We must point out that, from an analytical point of view, it is possible to include any sort of Technological Progress. The condition of balanced growth described in the von Neumann model implies, of necessity, that Hicks Neutral Technological Progress is the only sort of Technological Progress that can be considered, since balanced growth implies permanence in the relative intensities of the coefficients used in each production process. Any other type of Technological Progress contradicts this condition, since the von Neumann model considers that if the relative intensities of the coefficients used in the production process are modified, a new production process would be required. As a result, von Neumann's definition of balanced growth would not make sense.

(1) To construct the routes of balanced growth for Graphs 1 and 2, we considered [alpha] = 1.2, [[delta].sub.2] = 1.1, and, in the case of the route P.T.V., we also considered [[delta].sub.t] equal to 1.1, 1.2, 1.1 and 1.1 in Periods 3, 4, 7 and 10, respectively.

(1) Or, in analogous terms, when there are different technological progresses in different periods of time, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1) To avoid the situation in which [p.sub.1] = [LAMBDA] = [p.sub.n] = 0

(1) Higher profits would not be possible, since a positive return would prompt competitors to use the same production process and, thus, equilibrium would not exist.

REFERENCES

Champernowne, D. G., "A Note on J. von Neumann's Article", Review of Economic Studies, Vol. 13, 1945-1946, 10-18.

Dorfman, R., Samuelson, P.A. y Solow, RM., Linear Programming and Economic Analysis, 1958, McGraw-Hill.

Kemeny, J.G., Morgenstern, O. and Thompson, "A Generalization of Von Neumansn's Model of an Expanding Economy", Econometrica, Vol. 24, 1956, 115-135.

Koopmans, T. C., "Economic growth at a maximal rate", Quarterly Journal of Economics, Vol. 78, 1964, 355-394.

Von Neumann, J., "Uber ein Okonomisches Gleichungs-system und eine Verallgemeinerung, des Brouwerschen Fixpunktsatzes", K. Menger (comp.), Ergeb. eines Mathemat. Kolloq., Vol. 8, 1937. [English Translation: "A Model of General Economic Equilibrium", Review of Economic Studies, Vol. 13, 1945-1946, 1-9].

Dr. Raquel Diaz Vazquez * earned her Ph.D. at University of Vigo in 2001. Currently she is an associate professor of economics at University of Vigo, Spain.

* The autor gratefully acknowledge helpful comments received to previous versions of this paper from Esperanza Sanmartin Carbon.