Eigen-Values and Eigen-Vectors, Perron-Frobenius Theorem: Economic Applications
Eigen-Values and Eigen-Vectors, Perron-Frobenius Theorem: Economic Applications
Production prices, proportionate growth, maximum rate of profit and growth, and Italian economist Piero Sraffa’s standard commodity can be easily presented and analyzed by referring to the Perron-Frobenius theorem, which needs mathematical definitions concerning eigen-values and eigen-vectors.
Given a square matrix A composed of real numbers αij, consider the vector x such that Ax = αx. Obviously the vector x = 0 satisfies the equality Ax = αx for any real number α. The aim is to determine the non-zero vectors x and the real numbers α such that Ax = αx.
Consider a square matrix of order k. If there exists a real number α and a non-zero vector x such that Ax = αx, α is said to be an eigen-value of A and x is an eigenvector corresponding to α.
Determination of the Eigen-Values of A Any eigen-value α of A is such that there exists a non-zero vector x which satisfies Ax = αx = αIx where I is the identity matrix of the same order k as the square matrix A. Thus any eigen-value α of A satisfies (A – αI ) x = 0. Since A – αI is a non-zero matrix and since x is a non-zero vector, we must haveǀA – αI ǀ = 0, (recalling thatǀ A – αI ǀ is the determinant of the square matrix [ A – αI ]), i.e., α is a solution of the equation of the kth degreeǀ A – αI ǀ = 0, since the square matrix [ A – αI ] is of order k.
The equationǀ A – αI ǀ = 0 is called the characteristic equation of matrix A and has at most k real solutions. If the characteristic equation has no real solution, matrix A has neither real eigen-values nor eigen-vectors with real components.
Determination of the Eigen Vectors of A If the real number ᾱ is a solution of the characteristic equationǀ A – αI ǀ = 0 of the square matrix A, ᾱ is an eigen-value of A. Any non-zero eigen-vector x of A associated with ᾱ satisfies the matrix equation Ax = ᾱx = ᾱIx or (A – ᾱI ) x = 0. This equation represents in fact a system of k homogeneous linear equations with k unknowns, i.e., the components x 1, x 2, …, xk of the eigen-vector x. Sinceǀ A – ᾱI ǀ = 0, the rank of matrix A – ᾱI is at most equal to k – 1. One can then distinguish the following cases:
- Rank (A – ᾱ I) = k – 1: one of the components, say x 1, can be arbitrarily chosen and the k – 1 others are uniquely determined by the value chosen for x 1. The eigen-vector x is thus uniquely determined up to the multiplication by scalars.
- Rank (A – ᾱI ) ≤ k – 2: p components (p ≥ 2), say x 1, x 2, …, xp, can be arbitrarily chosen and the k – p others are determined by x 1, x 2, …, xp. One can then have p linearly independent eigen-vectors corresponding to the same eigen-value ᾱ.
Only one eigen-vector x (up to the multiplication by scalars) corresponds to the real eigen-value ᾱ when ᾱ is a simple root of the characteristic equation. But when the eigen-value ᾱ is a multiple root of order p of the characteristic equation, then:
- either one eigen-vector x (up to the multiplication by scalars) is associated with ᾱ; in this case ᾱ is said to be a semi-simple eigen-value; or several linearly independent eigen-vectors correspond to ᾱ.
One can also determine a vector y such that yA = ay. Call y the eigen-vector on the left of A associated with the eigen-value α. The vector x such that Ax = αx is then the eigen-vector on the right of A associated with α.
One can determine the eigen-values and the eigenvectors of the matrix:
The eigen-values α of matrix A are the solutions of a third degree equation ǀA – α I ǀ = 0 which can be written – α3 + (43/36)α2 –1/3 α + (35/1296) = –(α – 5/6) (α – 7/36) (α – 1/6) = 0. Thus A has three eigen-values 5/6, 7/36 and 1/6, which are the simple roots of the characteristic equation.
One can determine an eigen-vector on the right p and an eigen-vector on left q of matrix A associated with the highest eigen-value α.
with the eigen-value α =5/6 satisfics the matrix equation (A –5/6 1 )p=0, and its components p 1, p 2, and p 3 are solutions of a system of homogeneous equations which is an undetermined systen of rank 2. If one gives to p 1, the value 11, one has p 2=10 and p 3=8; i.e.p is an eign-vector on the right of A associated with α = 5/6 and any vector where μ is any scalar, is also an eign-victor on the right associted with α =5/6.
Similarly, an eigen-vector on the left q = (q1, q2, q3) of A associated wth the eigen-value α= 5/6 satisfies the matrix equation q (A – 5/6I ) =0 and its components q 1, q 2, and q 3 are solutions of a system of homogeneous equations which is an undetermined system of rank 2. If one assumes q 1 = 1, we have q 2= 3/5 and q 3 =3/4, so q = (1,3/5, 3/4) is an eigen-vector on the left of A associated wth α =5/6 and any vector q =(μ, 3/5 μ,3/4μ), where μ is any scalar, is also an eigen-vector on the left associated wth α= 5/6.
Different kinds of matrices can be distinguished by using permutation. A permutation of a square matrix A is the square matrix A obtained by the permutation of the rows of A combined with the same permutation of the columns.
Indecomposable Matrix A square matrix is said to be decomposable or reducible if there exists a permutation A of
Â is said to be a quasi-triangular matrix, i.e, containing only zeros at the intersection of the first rows and its last columns.
When a square matrix is not decomposable, it is said to be indecomposable or irreducible.
Matrices with Nonnegative Elements A is said to be nonnegative when αij ≥ 0 for any i and any j.
A is said to be semi-positive when αij ≥ 0 for any i and any j and A ≠0.
A is said to be positive when αij ≥ 0 for any i and any j.
Perron-Frobenius Theorem Any indecomposable semi-positive square matrix A possesses a positive eigen-value α A and a positive eigen-vector x corresponding to α (A ). The positive number a (A) is a simple root of the characteristic equationǀA – α I ǀ = 0 and is the only eigen-value of matrix A wth a positive eigen-vector. The absolute values of the k – 1 other eigen-values of matrix A are not greater than α (A ).
The positive eigen-value α(A ) (or equivalently α*) of the indecomposable non-negative square matrix with an associated eigen-vector x > 0 is called the dominant eigenvalue of matrix A.
Profit Rate and Production Prices with “Advanced Wage” Assumption Consider a square input-output system with quite standard assumptions except one: one shall assume that “wages are consisting of the necessary subsistence of the workers and thus entering the system on the same footing as the fuel for the engines or the feed for the cattle” (such is the assumption made by Sraffa in Production of Commodities by Means of Commodities [1960, p. 9]). In such a simplified case, which corresponds to classical ecomomists’ and the German political philosopher Karl Marx’s assumption of “advanced wage,” one can interpret the preceding matrix equation Ap = α p in the following way. When α < 1 (which appears in all productive systems generally considered), one can write α = 1/(1 + R ) with R > 0 and one gets: p = (1 + R )Ap. There appears, with uniform positive profit rate R, a vector positive system p ; each element of vector p, say pi, represents the “cost of production” of each commodity, since it takes into account not only materials used in the production of the commodity considered but also wages paid to workers. However, since we consider here an indecomposable production it is better to resort to the denomination of “production prices” and not “costs.” In matrix notation:
which means that p is the eigen-vector corresponding to the eigen-value 1/(1 + R ). In this way, one determines simultaneously the dominant eigen-value α of A (i.e., the rate of profit R ) and the corresponding eigen-vector p (i.e., the [ k – 1] relative prices) are necessarily positive from the Perron-Frobenius theorem.
Note that in the preceding example, one gets α =5/6 and consequently R = 20% with which means that one determines in such a case the uniform rate of profit and the whole structure of prices, or exchange rates. One must note that this allows determination of price structure, not exact level, since μ is any scalar. (Money quantitative theory may be used to find price level but this is quite another problem and there exist many other possibilities.)
Profit Rate, Production Prices with Wages Paid “Post Factum” Whereas classics assumed wages to be advanced by the capitalist, another assumption is that wages are paid as a share of the annual product, post factum according to Sraffa’s terminology. In such a case, quantities of commodities necessary for workers’ subsistence no longer appear in the technology matrix. The quantity of labor employed in each industry (with labor of uniform quality) must now be entered explicitly. In such a case, and getting back to the preceding example, the production price system must now be written (1 + r )Ap + wL = p, with w as a scalar for wages paid to workers and L a column vector representing labor needs for production of each commodity. In the particular limit case of w = 0, one comes back to the preceding case so represents the “maximum rate of profit” determined by the dominant eigenvalue of matrix A.
Price-Movements and Sraffa’s “Standard Commodity” Consider the preceding system (1 + r )Ap + wL = p. Obviously, prices cannot stay unchanged when distribution varies. But, the study of the price-movements that accompany a change in distribution is complicated by the necessity of having to express the price of one commodity in terms of another that is arbitrarily chosen as standard. For instance, when one studies the variations of price pj when distribution changes, and one has already chosen (implicitly or explicitly) a peculiar commodity price pi for “numeraire” or “standard,” it is impossible to determine if the change in pj /pi arises from the variation of the commodity which is measured or from those of the measuring standard.
This problem of an “invariable standard” has already been considered by the English economist David Ricardo who imagined, in a rather abrupt way and without any justification, that “gold” could be such an invariable standard. The presentation of the modern solution, given by Sraffa, can be given using preceding developments concerning eigen-vectors and eigen-values.
Remember that a particular commodity (i) is produced by using a row vector of intermediate consumption ai and a quantity li of labor. Its price is given, wth preceding notations, by equation pi = (1 + r)aip + li. When distribution changes, pi will change because of modifications of the value of the different “layers” composing the means of production of the commodity considered; such heterogeneity implies that if we consider the ratio of the value of such a commodity to its means of production, this ratio cannot remain unchanged when distribution changes; this is obvious since
So, such ratio cannot, in general, be invariant to changes in distribution. But does there exist a particular commodity, or a “basket” of commodities, a “composite commodity” u = (u1, u2, … u) satisfying this condition? If such is the case, the condition should be written where up is the price, or value, of the “composite commodity” and uAp price, or value, of means of production uA is used in the production process of u. And one knows from preceding developments that in the case of semi-positive indecomposable matrix A there exists a unique positive vector q which satisfies this condition since qA = α q with α = 1/(1 + R).
In the preceding example, the “composite commodity” is vector q = (μ,3/5,μ,3/4μ) where μ is any scalar.
Such “balanced” commodity would have the same proportions in all its layers. As noted by Sraffa, “It is true that, as wages fall, such a commodity would be no less susceptible than any other to rise or fall in price relative to other individual commodities; but we should know for certain that any such fluctuation would originate exclusively in the peculiarities of production of the commodity which was being compared with it, and not in its own.” Such commodity is the standard capable of isolating the price-movements on any other product “so that they could be observed in a vacuum” (1960, p. 18).
Consider a square indecomposable semi-positive matrix, as for instance the preceding example. In such a case, where the dominant eigen-value is α = 5/6, one can get:
- uniform rate of profit and prices α of production structure p determined by right eigenvector : p = (1 + R ) Ap
- uniform growth rate and levels of α activities structure q determined by left eigen-vector : q = (1 + G ) qA.
Both uniform growth rate and uniform profit rate are determined by maximum matrix eigen-value. Of course, there is no final consumption, since constituents of matrix A take into account labor subsistence needs and all profits are re-invested. Note that, in such case of “proportional growth,” Marx’s analysis concerning rate of profit is validated: rate of profit R is equal to the ratio of total surplus-value to the value of total advanced capital.
Such model of “proportionate growth” implies some similarity with Hungarian-born mathematician John von Neumann’s model. However, the von Neumann model is much more complicated. Among the main differences:
- In the von Neumann model, there is a difference between free goods and economic goods, when in the preceding simplified model all goods considered are economic goods; free goods (such as industrial wastes, superabundant commodities) are explicitly taken into account in the von Neumann model.
- In the von Neumann model, all production processes (activities) are not necessarily used (there occurs a problem of “choice of techniques”) when here all processes are used (input-output matrix is square but rectangular in von Neumann).
- In the von Neumann approach there occurs a possibility of joint production: the “cattle breeding” process produces simultaneously meat and wool. Traditional input-output systems are single production processes: one process produces meat, another wool.
- In the von Neumann approach, the uniform growth rate and profit rate are no longer determined by referring to the Perron-Frobenius theorem.
SEE ALSO Inverse Matrix; Linear Systems; Marx, Karl; Matrix Algebra; Sraffa, Piero; Vectors
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