Structural Change and Economic Dynamics 11 2000 157 – 166
A wave matrix
Andrew Brody
Institute of Economics, Budao¨rsi u´t
45
,
1112
Budapest, Hungary
Abstract
A wave matrix is presented consisting of price and quantity elasticities. Its eigenvalues determine the feasible cycles of a dynamic Leontief system. A numerical illustration and a
20-year trajectory are computed with the aid of a seven sector USA matrix for the year 1958. A short discussion of further work required closes the paper. © 2000 Elsevier Science B.V.
All rights reserved.
JEL classification
:
C67; E37 Keywords
:
Growth; Cycles; Elasticities; Forecasting www.elsevier.nllocatestrueco
1. Introduction
Richard Goodwin’s late research brought a fusion of the linear Leontief system with his own logarithmic growth theory
1
. In Goodwin 1989 he then showed the possible fluctuations of an input – output system around a non-focal, not attracting
equilibrium path. A similar non-linear system of differential equations is investi- gated, proposed in Brody 1989. The basic task of the paper is to find a suitable
linear approximation to an extended dynamic Leontief system, in which the realized rates of profit determine expansion and the relative disequilibrium of markets
governs price formation. The approximation involves two steps to handle the nonlinearity. The first is an exact reduction to stationary quantities, the second is
an approximation that does not seriously distort the motion while the system remains in the vicinity of the equilibrium path.
E-mail address
:
brodyecon.core.hu A. Brody.
1
Goodwin 1967. This is a variant of the ecological models of Volterra and Lotka. It may be also considered as a mathematical restatement of Marx’s cycle theory.
0954-349X00 - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 5 4 - 3 4 9 X 9 9 0 0 0 1 8 - 1
In the next section thus an input – output ‘wave matrix’ is constructed to model economic cycles. It is linearized around the equilibrium von Neumann Ray of the
system. In the third section some illustrative numerical results are presented and a 20-year forecast, based on USA data for 1958, is exhibited and discussed.
2. Construction of the wave matrix
The proposed system of cross regulation, where markets act on prices in the first equation, and profits determine growth in the second, has been set up by starting
from simple accounting identities
2
. It is a skew-symmetric dual system.
Price changes the logarithmic derivatives of the p prices are governed by excess
demand: p;
i
p
i
= − q
i
−
k
a
ik
q
k
−
k
b
ik
q;
k k
b
ik
q
k
1
while expansion of outputs the logarithmic derivatives of the q quantities depend on the rate of profits:
q;
k
q
k
= p
k
−
i
a
ik
p
i
−
i
b
ik
p;
i i
b
ik
p
i
1b
Here a
ik
are the flow and b
ik
the stock coefficients of the Input – Output system. In the first equation the market surplus is divided by total capital. In the second the
apparent profit is divided with capital intensity. It is well known
3
that such a system permits a positive equilibrium path, the so called von Neumann Ray. The economic interpretation of this solution is not
strictly symmetric. It implies constant prices but growing quantities. To circumvent this theoretical asymmetry in the definition we reduce the quantities to render their
equilibrium values constant. This can be done by continuously ‘discounting’ them with the uniform equilibrium growth rate. Thus instead of the possibly non-equi-
librium quantities q
i
, we take their transforms
x
i
=
exp − ltq
i
2 We may assume, without loss of generality, that at the initial moment t = 0. At
t = 0 the two quantities coincide, and the reduction is exact. Essentially, we split changes in q into two components, an equilibrium growth rate and a fluctuation
around the growth path. By focusing on x instead on q we explain the deviations from the equilibrium growth rate l.
2
The model may be considered also as an ensemble of stylized uniform behavioral equations or be deduced from extreme conditions of long run maximization.
3
The first proof has been furnished by Neumann 1937 and Neumann 1945. The dynamic theory has been applied to the I.O. system by Leontief et al. 1953, and computed for the US by Brody 1966.
This transformation requires us to augment the matrix of flow coefficients with the stock coefficients, the latter multiplied by the equilibrium rate. Thus the
equilibrium ray is turned into a singularity of the governing equation. To do this in a transparent manner we change to matrix notation. We then need the A = {a
ik
}
flow and B = {b
ik
} stock matrices. From these the equilibrium price, p¯, and quantity, x¯, vectors are computed
4
as the left and right hand eigenvectors of the
matrix A + lB. The vectors x¯ and q¯, are then the same eigenvectors, connected, as they are, by a growing scalar multiplier.
p¯ = p¯A + lp¯B 3
and x¯ = Ax¯ + lBx¯
3b
Let us now denote the full von Neumann Ray by forming the z¯ vector as the concatenation of the equilibrium vectors p¯ and x¯. The first n elements are the prices,
the second n elements the quantities. z¯ = [p¯, x¯], and also generally z = [p, x]
4 The corresponding hyper-matrix of the right hand extended flow coefficients is
skew-symmetric. When postmultiplied by a possibly non-equilibrium vector z it expresses the positive ore negative absolute deviations. These are the positive or
negative excess demands and the sectoral surpluses or deficits of profits over or under their equilibrium amount. Let us denote this matrix by the letter K,
habitually used for skew-symmetric matrices.
K = A + lB − 1
1 − A − lB 5
Kz = 0 if and only if z = z¯, the von Neumann Ray, on account of the definitional
equations 3 and 4. In a linear model the left hand stock hyper-matrix acting on absolute changes
consists of the usual stock matrices to be multiplied by the differential of z.
D = B
B 6
It would thus contain the symmetrically arranged stock matrices on which the changes themselves act. But we have to handle the non-linearity. It is not directly
the amount of excess, but its relative rate that triggers the changes. We also seek the change as a logarithmic differential z
; z. We thus require rates of change. A profit
rate, that is computed by dividing excess profits by capital intensities, and a rate of price changes, that may be computed by dividing excess demand by the total
marketable, that is: existing amount of the respective commodities and services.
4
The value of the equilibrium growth and profit rate, l, is best calculated as the reciprocal of the
unique, dominant and positive eigenvalue of the strictly positive matrix QB. Here Q = 1 − A
− 1
is the well known Leontief inverse.
These proportionalities are furnished by the divisors on the left and the right side in the dual system of Eq. 1 and Eq. 1b. In matrix form they may be collected
into the main diagonal of the left hand hyper-matrix. The complete non-linear left hand hyper-matrix may be written out as
S = diag Dz.z − D
7 Here we use a capital S, habitually used for symmetric matrices. Thus, as can be
seen, the matrix S remains symmetric even after the inclusion of the main diagonal that carries the non-linear ‘disturbance’ of the linear system. The operation ‘.’
designates an element by element division of the respective vectors. The dynamic system thus reads simply as
Sz;=Kz 8
but it is nonlinear, because the main diagonal of the left hand matrix still depends explicitly on z according to Eq. 7.
The trick that effects the linearization of the matrix S is to neglect its dependence
on actual, never exactly equilibrium, prices and quantities by substituting, instead, their equilibrium values:
S = diag Dz¯.z¯ − D
7b We shall show later that the exact and the linearized solutions almost coincide
and their spectra the cyclic frequencies are the same as long as we remain close to equilibrium.
But one problem still remains. The matrix S both in its original and in its
linearized form is singular. It can not be inverted in the usual sense. Yet it has the same null space as the matrix K, that is: on the von Neumann ray Sz equals zero.
As we are interested only in the non-equilibrium motion of the system, we may safely use a so called pseudo-inversion. In a pseudo-inversion the diads pertaining
to zero eigenvalues are neglected and only the reciprocals of the here positive other eigenvalues are considered in the process of inversion.
The system 8 thus has a zero eigenvalue
5
, belonging to the equilibrium path. Besides this it possesses only purely imaginary eigenvalues, the cyclic frequencies,
generating orbits around the von Neumann Ray. The system matrix is thus similar to a skew-symmetric matrix that has only zero and imaginary eigenvalues. An
additional trick, a further similarity transformation, renders the system matrix measure-invariant. Let us designate by the letter Z capital Z a diagonal matrix
with its diagonal occupied by the von Neumann Ray:
Z = diag z¯
9
We will call the matrix W the wave matrix of the system, if it is of the form
5
More precisely: it has two zero eigenvalues, one for equilibrium prices and one for equilibrium quantities. In an actual computation both have to be normalized to fit the system of existing prices and
quantities. Or, the initial equilibrium may be transformed to unit magnitudes. Computation may then start with the initial deviations.
W = Z
− 1
S
− 1
K Z 10
The inversion of S is to be performed as a pseudo-inversion, as explained above. The wave matrix W is thus a similarity transform of the system matrix of the
differential Eq. 8 and it 1. has a zero trace the elements on its main diagonal add up to zero,
2. has all its row sums equal to zero adding the elements in a row amounts to postmultiplying the matrix with its equilibrium vector,
3. produces a zero vector if premultiplied by the squared elements of z for the
