Generalized Goodwin’s Theorems on General Coordinates

Abstract

This paper is concerned with providing foundations for the use of general coordinates in linear production systems, which Professor R. M. Goodwin originally proposed. The actually observed sectors of the Sraffa–Leontief system have bilateral relations to each other, which leads to complexities in analyzing prices or quantities. In the case that it is impossible to proceed with analysis directly in the original space, we may turn to an imaginary space, which is tractable, as long as the original space corresponds uniquely to the imaginary one adopted. This is a natural way for mathematics to proceed. Goodwin’s use of the method of general coordinates represents a “diffeomorphisrn” in this area of economic theory. The use of general coordinates in the Sraffa–Leontief system seems, at first glance, restrictive, since their use requires a diagonalizable input matrix. A mathematical meaning of diagonalizability (or the rank condition) is examined by means of linear perturbation. The eigenvalues of the linear perturbed system are seen to be distinct almost everywhere. Economic meaning is given to the diagonalizable input matrix as yielding a regular system. It is worth noting that this specification can be justified in essentially the same way as the notion of quasi-smoothness used by Mas-Colell in the neoclassical production set. As a by-product of this study, the perturbed version of the Sraffa–Leontief price system is analysed.

Reprinted from Structural Change and Economic Dynamics 2(1), Aruka, Y., Generalized Goodwin Theorems on General Coordinates, 69–91 (1991), http://www.sciencedirect.com/science/journal/0954349X. With kind permission from Elsevier, Oxford, UK.

Appendix

Proof of Proposition 3

Let μ _i be eigenvalues of a square matrix A of order n. Then we have a triangularization:

$$ {H^{ - 1}}AH = [{\mu_i}{{\ }_{diag}}]. $$

We also define such a diagonal matrix G that its diagonal elements are g _ii =τ i(i =1, …, n). Here we choose any τ fulfilling

$$ 0 \,<\, n\tau \,<\, {\min_{{\mu_i} \ne {\mu_j}}}\left| {{\mu_i} - {\mu_j}} \right|. $$

If μ ₁ = μ ₂ =  … = μ _n, we may choose τ arbitrarily. Furthermore, define F = HGH ⁻¹. Thus it follows that

$$ {H^{ - 1}}AH + \varepsilon G = {H^{ - 1}}[A + \varepsilon F]H = [{\mu_i} + \varepsilon {g_{ii}}{{\ }_{diag}}] $$

for a sufficiently small ε ≠ 0. The diagonal elements by definition are all distinct. Hence, any matrix can be made to have all distinct eigenvalues through fine tuning □

Proof of Theorem 1

Since the components of R(g, g′), a _i , b _i are determined by ε and f _kl . R(g, g′) can be expressed a polynomial of ε and f _kl . Arranging this polynomial with respect to ε for some 　 m ≥ n ² −1,

$$ {\upsilon_0}({\,f_{kl}}){\varepsilon^m} + {\upsilon_1}({\,f_{kl}}){\varepsilon^{m - 1}} + \cdots + {\upsilon_{m - 1}}({\,f_{kl}})\varepsilon + {\upsilon_m}({\,f_{kl}}) = 0. $$

Since ε is a sufficiently small non-zero number, R(g, g′) = 0 holds only if

$$ {\upsilon_0}({\,f_{kl}}) = 0,{\upsilon_1}({\,f_{kl}}) = 0,...,{\upsilon_{m - 1}}({\,f_{kl}}) = 0,{\upsilon_m}({\,f_{kl}}) = 0. $$

These are the condition that A + εF has multiple roots for a sufficiently small non-zero number ε. These are not identically fulfilled with any perturbation direction F, which, as a whole, constitutes a projective space of dimension n ² −1. Only a perturbation direction F specified by the above condition gives multiple roots. Thus such a perturbation direction determines an algebraic manifold V of a smaller dimension than n ² −1 in a whole projective space. Excepting these singularities, eigenvalues of A + εF are all distinct for any perturbation direction F. A graph of V is of measure zero. Hence we have all distinct eigenvalues almost everywhere.□

Proof of Corollary 1 to Theorem 1

${\upsilon_m}({\,f_{kl}})$ is really det|A|. If A has no repeated roots, det|A| does not vanish. Hence ${\upsilon_m}({\,f_{kl}})$ does not vanish. A + εF then will not have repeated roots for a sufficiently small ε(≠0). □

Proof of Theorem 3

In matrix the differential equation system is of the form:

$$ \frac{{dp}}{{dr}} = Ap + (1 + r)A\frac{{dp}}{{dr}}. $$

(2.25)

This is reformulated into the following form:

$$\begin{array}{lll} \frac{{dp}}{{dr}} = \{ {[I - (1 + r)A]^{ - 1}}A\} p \\ \quad = \{ {[I - (1 + r)A]^{ - 1}}{({A^{ - 1}})^{ - 1}}\} p. \\ \quad = {\{ {A^{ - 1}}[I - (1 + r)A]\}^{ - 1}}p \cr \quad = {[{A^{ - 1}} - (1 + r)I]^{ - 1}}p. \\\end{array} $$

By the assumptions, $[I - (1 + r)A]$ is non-negatively invertible because of the Frobenius Theorem. Then it is expanded in the following series:

$$ {[I - (1 + r)A]^{{ - }1}} = I + (1 + r)A + {(1 + r)^2}{A^2} +... $$

(2.26)

Employing the expansion, it is verified that ${[I - (1 + r)A]^{ - 1}}A$ and A are commutable:

$$ {[I - (1 + r)A]^{ - 1}}A \cdot A = A \cdot {[I - (1 + r)A]^{ - 1}}A\ \hbox{for}\ r \le R. $$

(2.27)

Therefore, by virtue of Lemma matrix ${[I - (1 + r)A]^{ - 1}}A$ has the same eigenvectors with a commutable matrix A. We denote ${[I - (1 + r)A]^{ - 1}}A$ by B. Let matrix H be constructed by the eigenvectors u of matrix A. The general solution of this differential equation system can then be of the form¹⁶:

$$ p(r) = {e^{Br}}p(0). $$

If A is non-singular, it follows that

$$ {H^{ - {1}}}AH = [{\mu_i}{_{\ diag}}], $$

from which it is easily transformed:

$$ {A^{ - 1}} = {H^{ - 1}}{[{\mu_i}{_{\,diag}}]^{ - 1}}H = {H^{ - 1}}\left[ {\frac{1}{{{\mu_i}}}{_{diag}}} \right]H. $$

(2.28)

By making use of (2.15) and (2.28), the similarity transformation of ${[I - (1 + r)A]^{ - 1}}A$ is derived in the following manner:

$$ \begin{array}{lll} {[I - (1 + r)A]^{ - 1}}A = {[{A^{ - 1}} - (1 + r)I]^{ - 1}} \\ \qquad\qquad\qquad\qquad = {\left[ {H\left[ {\frac{1}{{{\mu_i}}}{_{diag}}} \right]{H^{ - 1}} - (1 + r)I} \right]^{ - 1}} \cr \qquad\qquad\qquad\qquad = H\left[ {\left[ {\frac{1}{{{\mu_i}}}{_{diag}}} \right] - (1 + r)I} \right]^{ - 1}{H^{ - 1}}. \\\end{array} $$

Accordingly,

$$ {H^{ - 1}}{[I - (1 + r)A]^{ - 1}}AH = {\left[ {\frac{1}{{{\mu_i}}} - (1 + r){_{\,diag}}} \right]^{ - 1}} = M. $$

(2.29)

The general solution then is represented as

$$ p(r) = {e^{Br}}p(0) = H{e^{Mr}}{H^{ - 1}}p(0) = \sum\limits_{i = 1}^M {{e^{{m_i}r}}} {c_i}{u_i}, $$

where $M = [{m_i}{{ }_{\,diag}}] = {\left[ {\frac{1}{{{\mu_i}}} - (1 + r){{ }_{diag}}} \right]^{ - 1}},$ and c _i , is a constant determined by the initial condition c = H ^–1 p(0).

Applying p =e ^mr u to (2.25) as a particular solution, it follows that

$$ mp = {[{A^{ - 1}} - (1 + r)I]^{ - 1}}p. $$

This is arranged as follows:

$$ Ap = \frac{m}{{1 + m(1 + r)}}p. $$

By replacing $\frac{m}{{1 + m(1 + r)}}$ with μ,

$$ Ap = \mu p. $$

Thus it has been proved that the eigenvalues m of the differential system correspond to the eigenvalues μ of the Sraffa–Leontief price system by the relationship of $\frac{\mu }{{1 - \mu (1 + r)}}$. □

Proof of Corollary 2

In the assumptions of Theorem 3. the principal eigenvalue is only one real value. Choose r closely to the maximum rate of profit R defined by the principal eigenvalue $\mu (A) = {{1 + R}}.$ Then, in the principal eigensector, the corresponding component of M in (2.29) will be:

$$ {\left\{ {\frac{1}{{\mu (A)}} - (1 + r)} \right\}^{ - 1}} = \frac{{\mu (A)}}{{1 - \mu (A)(1 + r)}} \,> 0, $$

since $\frac{1}{{1 + r}} \,>\, \mu (A) \,>\, 0.$ The particular solution of the principal eigensector will definitely be divergent. In a productive system, all the eigenvalues are smaller than 1 in terms of absolute values:

$$ 1 > |{\mu_i}|. $$

Consequently, the general solution will be unstable as the rate of profit becomes the maximum rate of profit R. □

Proof of Corollary 3

In our price movement equation system the initial value is the value when r is set to 0:

$$ p(0) = {[I - A]^{ - 1}}{a^0}, $$

which can be expanded in the following manner, given the assumptions of Theorem 3:

$$ p(0) = {a^0} + A{a^0} + {A^2}{a^0} +... \,.$$

Taking account of our requirement Aa ⁰ = μa ⁰, this will furthermore be rearranged into

$$ p(0) = (1 + \mu + {\mu^2} +...){a^0} = \frac{1}{{1 - \mu }}{a^0}. $$

Then it follows that

$$ p(r) = \frac{1}{{1 - \mu }}H{e^{Mr}}{H^{ - 1}}{a^0}. $$

H is constructed by the columns of eigenvectors u of A. Substitute a ⁰ into u ₁. The initial value will be expressed as follows:

$$ c = {H^{ - 1}}{a^0} = [1,0,...,0]\prime, $$

because of H ⁻¹ H = I. Hence other coefficients other than the first component of c are 0.

Therefore

$$ p(r) = \sum\limits {{e^{{m_i}r}}} {c_i}{u_i} = {e^{{m_1}}}{u_1} = {e^{{m_1}}}{a^0}. $$

This completes the proof. □

Proof of Theorem 4

Let m be the ratio satisfying the condition

$$ m = \frac{{{a_i}p}}{{a_i^0}} = \frac{{{a_i}Ap}}{{{a_i}{a^0}}} = \frac{{{a_i}{A^2}p}}{{{a_i}A{a^0}}} =...{\hbox{ for }}r \in (0,R]. $$

This ratio implies “the balancing proportion” for the prices against the changes of the rate of profit. The activity vector s preserving this critical proportion has to satisfy the following condition:

$$ m = \frac{{s{A^n}p}}{{{a_i}{A^{n - 1}}{a^0}}}\ \hbox{for}\ r \in (0,R]. $$

(2.30)

Substituting (2.12) into (2.30), it follows that, taking account of (2.27),

$$ s\{ {[I - (1 + r)A]^{ - 1}}A - mI\} {A^{n - 1}}{a^0} = 0. $$

Let z be the vector $s\{ {[I - (1 + r)A]^{ - 1}}A - mI\}$ and L the matrix $[{a^0},A{a^0},{A^2}{a^0},...,{A^{n - 1}}{a^0}]$:

$$ zL = 0. $$

(2.31)

Hence if L is of dimension n, the null space of L is of dimension n−n =0. Thus (2.31) has only the solution z = 0. Conversely, if z = 0, the dimension of L is n. L of dimension n implies that

$$ {a^0},A{a^0},{A^2}{a^0},...,{A^{n - 1}}{a^0} $$

are linearly independent. z = 0 means that s is a left-hand eigenvector of matrix ${[I - (1 + r)A]^{ - 1}}A$:

$$ s{[I - (1 + r)A]^{ - 1}}A = ms, $$

i.e.

$$ s[{A^{ - 1}} - (1 + r)A] = ms. $$

This completes the desired results:

$sA = s \frac {m} {1 +\,m(1+r)}$ for a left-hand eigenvector s,

$Au = \frac {m} {1 +\,m(1+r)}$ for a right-hand eigenvector u.

The second result is just the same as Theorem 3(ii). Thus the necessary and sufficient condition of this theorem is equivalent to normal modes of the Sraffa–Leontief differential equation system (2.14).

Finally, it remains to examine the assumptions sa ⁰ ≠ 0, and a ⁰ ≠ (1 + R)Aa ⁰. It is easily verified that if sa ⁰ ≠ 0, a ⁰ ≠ (1 + R)Aa ⁰, then ${a^0},A{a^0},{A^2}{a^0},...,{A^{n - 1}}{a^0}$ are linearly independent. Suppose, on the contrary, they are linearly dependent. In the case n = 2, appropriating the non-zero numbers 1 and −(1 + R), it follows that a ⁰−(1 + R)Aa ⁰=0. a ⁰ happens to be a left-hand eigenvector of A. Then, by orthogonality of the left- and right-hand eigenvectors in an eigenvector space, sa ⁰ = 0. Conversely, if sa ⁰ = 0, a ⁰ is an eigenvector of A. While z ≠ 0. In the case n = 2, then, in view of Jordan normal forms, it can be expressed:

$$ {s^1} = m{s^1} + {s^2}. $$

Here s ⁱ is a generalized eigenvector. Post-multiplying the left-hand vector a ⁰ on both sides, it follows that

$$ {s^1}{a^0} = m{s^1}{a^0} + {s^2}{a^0}. $$

This implies s ² a ⁰ is 0 if s ¹ a ⁰ is 0. By induction on a cyclic subspace, it is proved that if a ⁰, ${a^0},A{a^0},{A^2}{a^0},...,{A^{n - 1}}{a^0}$are linearly independent, sa ⁰ ≠ 0 (rsp. a ⁰ ≠ (1+R)Aa ⁰). □

Proof of Proposition 4

By one of the Frobenius theorems, the principal eigenvalues $\mu (A), \widetilde{\mu }(\tilde{A})$ fulfilling $Ap = \mu (A)p, \widetilde{A}\widetilde{p} = \widetilde{\mu }(\widetilde{A})\widetilde{p}$ are positive and simple. Suppose, on the contrary, that for $\mu (A) > \widetilde{\mu }(\widetilde{A})$,

$$ \frac{{{{\widetilde{p}}_k}}}{{{p_k}}} > \frac{{\mu (A)}}{{\widetilde{\mu }(\widetilde{A})}}ma{x_{j \in \Gamma }}\frac{{{{\widetilde{p}}_j}}}{{{p_j}}}. $$

Then it follows that

$$ \frac{{\mu (A)}}{{\widetilde{\mu }(\widetilde{A})}} > ma{x_{j \in \Gamma }}\frac{{{{\widetilde{p}}_j}/{p_j}}}{{{{\widetilde{p}}_k}/{p_k}}} > 1. $$

Hence $\mu (A) < \widetilde{\mu }(\widetilde{A})$. This is a contradiction. Similar to the above, we can prove:

$$ \frac{{{{\widetilde{p}}_k}}}{{{p_k}}} \ge \frac{{\mu (A)}}{{\widetilde{\mu }(\widetilde{A})}}mi{n_{j \in \Gamma }}\frac{{{{\widetilde{p}}_j}}}{{{p_j}}}\ \hbox{for}\ \mu (A) < \widetilde{\mu }(\widetilde{A}). $$

□

Proof of Theorem 5

• (i) If it is the case that

  $$ \mu (A) + \varepsilon a \ > \frac{1}{{1 + r}} > \mu (A), $$

  a price system will be made unreasonable for a given rate of profit r. In other words, the reasonableness of a price system after perturbation will depend on not only a matrix F but also on the magnitude of a given rate of profit. Thus Proposition 5 supports statement (i).¹⁷

• (ii) The perturbed matrix A + εF is assumed to be non-singular. Define H(ε) to be a transformation matrix being constructed by the eigenvectors u(ε) of A + εF. All the arguments in Theorem 3 will be valid, if A is replaced with A + εF as well as μ with μ + εa. Hence

  $$ \begin{array}{lll} p(r) = H(\varepsilon ){e^{M(\varepsilon )r}}H{(\varepsilon )^{ - 1}}p(0) \\ \qquad = \sum\limits_{i = 1}^M {{e^{{m_i}(\varepsilon )r}}} {c_i}(\varepsilon ){u_i}, \\\end{array} $$

  where

  $$ M(\varepsilon ) = [m{(\varepsilon )_i}{_{\,diag}}] = {\left[ {{{\left\{ {\frac{1}{{\mu {{(\varepsilon )}_i}}} - (1 + r)} \right\}}_{diag}}} \right]^{ - 1}}, $$

  c(ε) is a constant determined by the initial condition c(ε)= H(ε)⁻¹ p(0). □