Eigenvalues and Eigenvectors in von Neumann and Related Growth Models : An Overview and Some Remarks

We take into consideration various relationships existing between eigenvalues and eigenvectors of suitable matrices or matrix pairs and the equilibrium solutions of the classical von Neumann growth model and of other related economic models.


Introduction
The von Neumann economic growth model (von Neumann (1945-46)) is perhaps one of the most investigated models in economic growth theory and in mathematical economics in general.Indeed, this model was (together with the models presented by Sraffa (1960)) one of the first attempts to introduce in multi-sectoral production schemes, the possibility of joint production.Moreover, contrary to the models of Sraffa with joint production, the von Neumann model is described by inequalities, which permits considerations of optimality and efficiency of the production processes.
The aim of the present paper is to focalize some links existing between the solutions of the classical von Neumann model, together with some related models, and the eigenvalues and eigenvectors of (real) square matrices or of (non necessarily square) matrix pencils.
The paper is organized as follows.
Section 2 recalls the basic results concerning the classical von Neumann model.
Section 3 is concerned with solution properties, in terms of eigenvalues and eigenvectors, of a Leontief-von Neumann model.
Section 4 recall the main properties of the Leontief-von Neumann model, proposed by Morishima in the study of his "turnpike theorem".
Section 5 is concerned with various "regular" von Neumann models, in the sense of J. Łos (1971).
Section 6 contains some conclusive remarks.
Throughout the paper the notation x y (x and y being two vectors of R n ) means x i y i , i = 1, ..., n; x ≥ y means x y, but x y; x > y means x i > y i , i = 1, ..., n.If y = [0] (zero vector), vector x is said to be, respectively, nonnegative, semipositive, positive.The notations x y, x ≤ y, x < y are evident.The same conventions are used to compare two matrices of the same order, say (m, n).We denote by [0] the zero matrix, so that the notations A [0] , A ≥ [0] , A > [0] mean that A is, respectively, a nonnegative matrix, a semipositive matrix, a positive matrix.If A is a matrix of m rows and n columns, A i , i = 1, ..., m, denotes its i-th row, whereas A j , j = 1, ..., n, denotes its j-the column.

Basic Results on the Classical von Neumann Growth Model
The literature on the classical von Neumann growth model is abundant.We quote only Bruckmann and Weber (1971), Gale (1960), Howe (1960), Karlin (1959), J. andM. Łos (1974), Morgenstern and Thompson (1976), Morishima (1964), Murata (1977), Nikaido (1968Nikaido ( , 1970)), Takayama (1985), Woods (1978).However, we point out that many mathematical treatments of the von Neumann model are incomplete and unsatisfactory (see, e. g., Giorgi andMeriggi (1987, 1988)).Some other treatments are correct but quite long and complicate.We follow the description of the model and the conventions adopted by Kemeny, Morgenstern and Thompson (1956).We consider a finite set of m processes that produces a finite set of n different goods.Each process operates at an intensity level x ⊤ = [x 1 , ..., x m ] , whereas p ⊤ = [ p 1 , ..., p n ] is a price vector.The model is characterized by a pair of two nonnegative matrices (A, B), both of order (m, n) : the rows of A represent the various activities, the columns of A describe the inputs; the rows of B represent the various activities and the columns of B describe the outputs.In other words, a i j is the quantity of good j technologically required per unit of process i.The output coefficient b i j simply represents the quantity of good j produced per unit of process i.

The basic assumptions on A and B
Then, following Kemeny, Morgenstern and Thompson, we precise better inequalitis (1), in the sense that we impose the following conditions: (2) Intuitively, (3) means that every good can be produced by some process, and (2) that every process uses some inputs.
We note that (2) is equivalent to The expansion rate or growth rate is denoted by α and the interest rate is denoted by β.Several authors call α the expansion factor and β the interest factor; then, in this case, (α − 1) is the expansion rate and (β − 1) is the interest rate.
, α 0, β 0 is an equilibrium solution for the von Neumann technology (A, B) if it satisfies the following system x ⊤ (Bp − βAp) = 0 (7) The following result is quite immediate.
A triplet (x, p, λ), with x ≥ [0] , p ≥ [0] , λ > 0, is an equilibrium solution for the von Neumann technology (A, B) if it satisfies the following system Bp λAp (10) The number λ is called by Łos (1971) equilibrium level and by Kemeny, Morgenstern and Thompson (1956) allowable level.The basic results on the von Neumann growth model described by ( 9)-( 10)-( 11) are contained in the following theorems.
(ii) There exists a number λ max > 0 solution of the problem ("technological expansion problem") (iii) There exists a number λ min > 0 solution of the problem ("economic expansion problem") (iv) λ max λ min > 0.
(vi) The set of equilibrium capital stock vectors x is a convex set; the set of equilibrium price vectors p is a convex set.
Curiously, the above results are scattered in several papers and books (see the works quoted at the beginning of the present section).There is not, as far as we are aware, a complete and self-contained proof of Theorem 1, which, however, contains classical results.For other questions concerning the classical von Neumann model, see Giorgi andMeriggi (1987, 1988).
Usually, in the current literature, the number λ max is denoted by α * and is called maximum growth rate, the number λ min is denoted by β * and is called minimum interest rate.In order to state the conditions which assure that α * = β * we need the following notions.The above notion is essentially due to Gale (1960) , contradicting the assumption that the model is technologically irreducible.If m = n and B = I, i. e. the joint production is excluded, the pair (A, I) is technologically irreducible (respectively technologically reducible) whenever A is irreducible (respectively reducible) in the usual sense of the theory of matrices (see, e. g., Debreu and Herstein (1953), Gantmacher (1959) and Section 3 of the present paper).
Definition 5.The pair (A, B) is said to be economically irreducible The previous definition is due to Robinson (1973) and obviously is a dual property with respect to Definition 4. We can also say that (A, B) is economically reducible when there exist two permutation matrices P and Q such that PAQ and PBQ have the following form: where each row of A 11 has at least one positive element.We can also say that (A, ) is economically irreducible.Moreover, any irreducible pair (A, B) satisfies also assumption (3), because if there exists a vector p ≥ and this contradicts the assumption of economic irreducibility.Note that the square simple production model (A, I) is economically reducible and the square model (I, B) is economically reducible (respectively economically irreducible) whenever B is reducible (respectively irreducible) in the usual sense of the theory of matrices.Robinson (1973) has observed that technological and economic irreducibility are independent properties.
2. There is only one equilibrium level λ = λ min = λ max = β * = α * , satisfying relations ( 9)-( 10)-( 11), if any one of the following conditions holds: Also for what regards Theorem 2, as far as we know, there is not a complete and self-contained proof.The result sub 1) has been proved by Kemeny, Morgenstern and Thompson (1956); see also Murata (1977).Condition a) is the famous condition due to von Neumann; it has been criticized as an economically unrealistic assumption, as it implies that every good is involved either as input or as output.Moreover, in his original paper (translated into English in von Neumann ), von Neumann did not assume (2) nor (3), so condition a) can give rise to absurd cases, such as A > [0] and . Moreover, the absence of ( 2) and ( 3) makes the von Neumann's proofs not always fully correct.See also Giorgi andMeriggi (1987, 1988)).For the proof of b) and c) see Gale (1960) and Robinson (1973).
We note that a more general condition assuring Jaksch (1977) proposed a condition on the pair (A, B), both necessary and sufficient to have α * = β * .It must be noted, however, that the condition of Jaksch is not, based, as Gale's and Robinson's conditions, on a qualitative structure of (A, B), but on "quantitative properties" of particular sub-matrices of A and B.
Theorems 1 and 2 assure the existence of a positive equilibrium level λ satisfying ( 9)-( 10)-( 11), and in particular λ = α * and λ = β * are two equilibrium levels; the same theorems, however, do not assure that it holds λ > 1.In this case β * is usually called minimum interest factor and α * is called maximum growth factor.
Definition 6.The pair (A, B) (or the von Neumann model described by A and B) is called productive if there exists a vector From a strictly mathematical point of view we can say that (A, B) is productive if and only if (B − A) ⊤ belongs to the S-class, in the terminology of Fiedler and Pták (1966); see also Giorgi and Zuccotti (2014).We recall that a matrix A belongs to the S-class if there exists a vector x ≥ [0] for which Ax > [0] .It is easy to see that a matrix A is in S is and only if there exists a vector (Fiedler and Pták (1966)), i. e.A has all its principal minors positive.Equivalently, it can be proved that A ∈ P if and only if for every vector x [0] there exists an index i such that x i (Ax) i > 0. This last equivalence was discovered also by Gale and Nikaido (1965).Another sufficient condition for a square matrix A to be in S is that A is an N-matrix of the first category (Nikaido (1968)).Following Inada (1971), a square matrix A is termed N-matrix if all its principal minors are negative and it is said to be an N-matrix of the first category if, moreover, A has at least one positive element.Giorgi andMeriggi (1987, 1988) proved the following result.
Theorem 3. Let (A, B) satisfy assumptions (2) and (3) and let one of the conditions a). b), c) of Theorem 2 be satisfied. Then For other considerations on mathematical properties of the classical von Neumann model, see Giorgi andMeriggi (1987, 1988).

A Leontief-von Neumann Model
is an equilibrium solution of the Leontief-von Neumann model here considered, if the following relations hold true.
x ⊤ λx ⊤ A (12) p λAp (13) We continue to assume A [0] and that A satisfies (2); obviously B = I satisfies (3).There exist strict relationships between the equilibrium solutions of this model and eigenvalues and eigenvectors of the semipositive matrix A. It is well-known (see, e. g., Gantmacher (1959), Debreu and Herstein (1953)), that , given a semipositive square matrix A, there exists a real nonnegative maximum eigenvalue, i. e. the dominant or Frobenius eigenvalue, denoted λ * (A), such that λ * (A) | λ |, λ being any other root of the characteristic equation of A. The Frobenius eigenvalue is associated to a right-hand eigenvector p which is semipositive; the same statement holds for the left-hand eigenvector x ⊤ .
We recall that a square matrix A, of order n, is said to be decomposable or reducible (in the usual sense of Linear Algebra) if, after suitable permutations of its rows and of its corresponding columns, can be put in the form where A 11 and A 22 are square and at least one of matrices A 12 , A 21 is the zero matrix [0] .The sub-matrices A 11 and A 22 in ( 15) may be themselves reducible matrices, so that in this case we obtain a so-called "block-diagonal form" or also a more general decomposed form, due to Gantmacher (1959) and called Gantmacher normal form.Conversely, a square matrix A is called an indecomposable matrix or irreducible matrix, if it is not possible, by interchanging its rows and the corresponding columns, to reduce it to the form ( 15) with the specified properties on its sub-matrices.
have a unique solution if and only if λ = λ * (A).Moreover, it can be shown (see Gantmacher (1959), vol.II) that if A ≥ [0] has a dominant eigenvalue which is a simple root of its characteristic equation and problems ( 16) and ( 17) have a solution with λ = λ * (A), then A is indecomposable.However, when A [0] is decomposable the above results in general do not hold.It is possible, in this case, to obtain only "partial results".For example, if we suppose that A [0] has been reduced to the form with A 11 and A 22 square irreducible matrices and every column of A 12 a semipositive vector, then if the Frobenius eigenvalue of A 11 is greater than the Frobenius eigenvalue of A 22 , the left-hand Frobenius eigenvector of A (associated to λ * (A 11 )) has all positive components.The right-hand Frobenius eigenvector will be only semipositive: more precisely, its first k components will be positive, k being the order of matrix A 11 , and the other (n − k) components will be zero.By way of symmetry, if the Frobenius eigenvalue of A 22 is larger than the Frobenius eigenvalue of A 11 , then the associated right-hand Frobenius eigenvector of A has all positive components.On the other hand, the associated left-hand Frobenius eigenvector will be semipositive, in such a way that its last (n − k) components will be positive and its first k components will be zero.
If A [0] is decomposable it may be possible to obtain other semipositive left-hand eigenvectors or semipositive righthand eigenvectors, besides the eigenvectors associated to the Frobenius eigenvalue λ * (A).Let A ≥ [0] be square of order n; let us denote by S + (A) = {λ ∈ R : λ > 0 is an eigenvalue of A to which it is possible to associate a semipositive eigenvector } the so-called semipositive spectrum of A.
In the above definition we do not specify if the semipositive associated eigenvector is a left-hand eigenvector or a righthand eigenvector.This will appear from the context of the applications of this definition.(ii) If x⊤ is a semipositive left-hand eigenvector of A, associated to the eigenvalue (1/λ) > 0, then there exists an equilibrium triplet (x, p, λ), with x = x.
In order to prove Theorem 4 we need a previous result, proved by Łos (1971) in a general framework of topological spaces and given also by Gale (1972), without proof.This result, which is a theorem of the alternative, is an easy consequence of the well-known Farkas-Minkowski lemma (see Giorgi and Meriggi (1987)).For the reader's convenience we give a direct proof. Lemma } .
The set P−R n + , being the algebraic sum of two polyhedral cones, is itself a polyhedral cone.It is easy to see that inequality (20) has no solution if and only if x⊤ B P − R n + .Vector x⊤ B can be separated from the polyhedral cone P − R n + .Hence there exists p ∈ R n such that x⊤ B p > 0 and y ⊤ p 0 for any y ∈ P − R n + .From this, being Relation ( 21) implies λA p B p, and at the same time from ( 22) we obtain p ≥ [0] .
Proof of Theorem 4.
(i) Let (x, p, λ) an equilibrrium solution of a Leontief-von Neumann model (A, I), i. e. the said triplet satisfies relations (12), ( 13) and ( 14).We shall prove the existence of a vector x ≥ [0] such that Denoting by A (k) the k-th power of the square matrix A, let us consider the following sequences: From inequality (12) it results that sequence ( 23) is non-increasing; similarly from ( 13) we deduce that sequence ( 24) is non-decreasing.Sequence ( 25) is both non-increasing and non-decreasing, so it is a constant sequence.As sequence ( 23) has all nonnegative elements, it is convergent.Let us set From ( 26) we get x⊤ = lim Taking the limit for k −→ ∞ in (25), taking ( 26) and ( 14) into account and recalling that sequence ( 26) is constant, we get ) is an eigenvalue of A, with a (left-hand) semipositive eigenvector x⊤ associated.From ( 13), ( 28) and ( 29) we deduce that the triplet ( x, p, λ) is an equilibrium solution of the model.
Let us absurdly suppose that there exist no vectors p ∈ R n + such that the triplet ( x, p, λ) is an equilibrium solution of the Leontief-von Neumann technology.Then, by Lemma 2, it will exist a vector By multiplying both sides of (30) by λA and by adding in both sides vector x⊤ , we obtain By repeating k times the said operation, we get But from this inequality it follows that x⊤ = [0] , which is in contradiction with the assumption that x⊤ is a semipositive (left-hand) eigenvector of A.
On the other hand, we can associate the (left-hand) eigenvector x 2 = [1; 0] to the other eigenvalue 2; however, in this case there exists no semipositive eigenvector p 2 associated to (A ⊤ , 2).Indeed, in general it holds S + (A) S + (A ⊤ ).As already remarked, if A is irreducible (i.e. indecomposable in the usual sense of Matrix Theory), the equality between S + (A) and S + (A ⊤ ) holds.In this case λ = λ min = λ max and the equilibrium vectors x and p are both positive (and unique up to multipication by a positive scalar).In the example of the present remark A is reducible and we have S + (A) = {2; 3} , whereas S + (A ⊤ ) = {3} .

A Leontief-Morishima-von Neumann Model
For the reader's convenience in the present section we summarize the results of Morishima (1961Morishima ( , 1964) ) on a Leontief-von Neumann model in which an industry i can choose between m i different activities for producing good i.The notational convention of Morishima is opposite to the one of Kemeny, Morgenstern and Thompson followed in the previous sections.
The total set of activities can be described by an (n, m) matrix Â where , where m = ∑ n i=1 m i and n is the number of commodities.An activity, say the s i -th activity of industry i, is defined by an n-dimensional column vector stating the inputs of n commodities per unit output.As there is no joint production, the output matrix, of order (n, m) is written as: Next, let x s i be the output of good i produced by the s i -th activity of industry i and p i the price of good i.Let x ∈ R m be the m-dimensional column vector Morishima sets up his von Neumann-like model in terms of the usual inequalities If each industry selects a single activity from among those available to it, there are m 1 × m 2 × ...× m n possible sets of activities that could be adopted by the economy.They are arranged in a certain order and denoted by α, β, ..., µ, where µ = m 1 × m 2 × ...× m n .Let σ be the activity set (s 1 , s 2 , ..., s n ) in which industry i selects its s i -th activity (i = 1, ..., n) and define A σ any (n, n) matrix which represents a particular set of activities adopted (σ = α, β, ..., µ).Since A σ is nonnegative, it has a dominant eigenvalue λ * σ that is nonnegative.In particular, let A ε = [A e 1 , A e 2 , ..., A e n ] be an activity set such that λ * ε λ * σ (σ = α, β, ..., µ), and let the right-hand and left-hand eigenvectors of A ε , associated with λ * ε , be denoted, respectively by x * ε and p * ε (i.e. we have p * εi be the i-th component of x * ε and let y * ε be an m-dimensional column vector such that its s i -th component is x * εi when s i = e i and zero, when s i e i .Next Morishima makes the following two assumptions and proves the following theorem.and (L2)" is equivalent to: Before giving some comments on the (not proved) assertions of Łos, we remark that there exists a link between the maximum interest rate λ min = β * of a von Neumann model, where (L1)" holds, and the Frobenius root of the square semipositive matrix H.We denote by v(A) the value of the matrix game A, where A is a (m, n) pay off matrix.That is, if As B has semipositive columns, it follows The following assertion of Łos (1971) states that, under the regularity assumptions (K1) and (L2), it is possible to generalize to a von Neumann model, where (2) and (3) hold, the results of Theorem 4. (ii) If λ > 0 is an equilibrium level, then there exists a triplet (λ, x, p) which is an equilibrium solution of the von Neumann model and where x is a (left-hand) eigenvector of H, associated to its eigenvalue λ −1 .
(iii) Let x ≥ [0] be a (left-hand) eigenvector of H, associated to its eigenvalue λ −1 ∈ S + (H).Then, there exists a vector p ≥ [0] such that the triplet (λ, x, p) is an equilibrium solution of the von Neumann model.
Remark 2. A proof of Theorem 7, not short nor similar to the proof of Theorem 4, is given by Vahrenkamp (1980).
A proof of results similar to the ones of Theorem 7, is given by Kogelschatz (1981).This author gives a more compact proof, which relies on definitions and properties related to the concept of generalized inverse of a matrix of order (m, n).We recall the following basic facts (see, e. g., Rao and Mitra (1971)).
Let A be an (m, n) matrix.An (n, m) matrix A − is a generalized inverse of A or g-inverse of A if AA − A = A. Usually A admits infinite generalized inverses, unless A is square and non-singular: in this case A admits one generalized inverse which coincides with the usual inverse A −1 .
The matrix A can admit a right g-inverse A − r , i. e. a matrix A − r of order (n, m) such that where I is of order (m, m).The matrix A can admit a left g-inverse A − ℓ , i. e. a matrix A − ℓ (of order (n, m)) such that with I of order (n, n).
The matrices A − r and A − ℓ are generalized inverses of A, as it holds Theorem 8.A matrix A of order (m, n) admits at least one right g-inverse if and only if rank(A) = m.In this case it holds rank(A − r ) = m.A right g-inverse of A is given by Vol. 8, No. 1;2016 If the semipositive matrix B −1 A is indecomposable, by the Perron-Frobenius theorem it results that 1/β 1 is the dominant (Frobenius) eigenvalue of B −1 A. When the semipositive matrix B −1 A is decomposable and has a strictly positive lefthand eigenvector, 1/β 1 is the dominant eigenvalue of B −1 A (see Theorem 11).Similarly, when the square matrix AB −1 is semipositive, 1/α 1 is the dominant eigenvalue of AB −1 if AB −1 is indecomposable or if it is decomposable and has a strictly positive right-hand eigenvector.We can distinguish the following cases.Finally, the relation x ⊤ Ap > 0 can be obtained as in Howe (1960) or Nikaido (1968).

Final Remarks
Links between equilibrium solutions of the von Neumann model and (generalized) eigenvalues and eigenvector are analyzed also by Thompson and Weil (1970, 1971, 1972).See, for an account, the book of Morgenstern and Thompson (1976).The existence of equilibrium solutions of a von Neumann model (A, B), in terms of generalized eigenvalues and eigenvectors, has been considered also by Drandakis (1966).This author assumes A and B square, of order n, and such that (2) and (3) are satisfied.A scalar λ is a generalized eigenvalue and a nonzero vector y ∈ R n is a generalized right-hand eigenvector of (A, B) associated to λ if By = λAy.
Similarly, a nonzero vector x ∈ R n is a generalized left-hand eigenvector of (A, B), associated to λ, if x ⊤ B = λx ⊤ A.
(A, B) is an F-transformation if there exists a unique, simple, positive generalized eigenvalue λ of (A, B) with positive right-hand and left-hand eigenvectors y and x, respectively.Now, let α * = λ max the maximum growth rate for the von Neumann model (A, B), with A and B square and with (2) and (3) satisfied.Drandakis proves the following result, which is a sufficient condition for a pair (A, B) to be an F-transformation.
Theorem 14.If b ii > 0 for all i = 1, ..., n, if b i j < α * a i j for all i j for which a i j > 0 and if A is indecomposable, then: (a) The equilibrium vectors x * and p * (associated to α * ) are positive and unique, up to a scalar multiplication.They are, respectively, left-hand and right-hand generalized eigenvectors of (A, B).
) becomes x⊤ B λ x⊤ A and from (6) we get x⊤ B p λ x⊤ A p.

Definition 3 .
Given a vector x [0] , x ∈ R n , we call the support of x the set of indices corresponding to the nonzero components of x; formally: Definition 4. The pair (A, B) which characterizes the von Neumann model is technologically irreducible (or technologically indecomposable) if for each semipositive vector x ∈ R m such that supp(x ⊤ A) ⊂ supp(x ⊤ B) we have supp(x ⊤ A) = {1, ..., n} , i. e. x ⊤ A > [0] .Otherwise the model is technologically reducible (or technologically decomposable).
Gale (1960) andŁos (1971)  have considered a von Neumann-type model where the number of goods coincides with the number of production processes, i. e. there is no joint production and therefore m = n and B = I.Here we shall complete the treatment of the said authors of this model, model we may call a Leontief-von Neumann model.According to Definition 2 a triplet (x, p

Theorem 4 .
Let the pair (A, I) describe a Leontief-von Neumann model.Then: (i) If (x, p, λ) is an equilibrium solution of the model (A, I), then (1/λ) is an eigenvalue of A to which it is associated a left-hand eigenvector x⊤ ≥ [0] .The triplet ( x, p, λ) is an equilibrium solution of the model (A, I).

Theorem 7 .
Let (A, B) be the technology of a regular von Neumann model, with (2) and (3) satisfied.Then: (i) The scalar λ > 0 is a von Neumann equilibrium level if and only if λ −1 ∈ S + (H).