Bródy’s Stability and Disturbances

Abstract

Bródy’s conjecture regarding the instability of economies is submitted to an empirical test using input-output flow tables of varying size for the US economy, for the benchmark years 1997 and 2002, as well as for the period 1998–2011. The results obtained lend support to the view of increasing instability of the US economy over the period considered. Furthermore, our analysis shows that only a few vertically integrated industries are enough to shape the behaviour of the entire economy in the case of a disturbance. These results may, on the one hand, provide empirical evidence on the speed of convergence of Marxian iterative procedures ‘transforming’ labour values into production prices; on the other hand, they may usefully be contrasted with those derived in a parallel literature on aggregate fluctuations from microeconomic ‘idiosyncratic’ shocks.

Appendices

Appendix 1: On the Sraffian Multiplier

The concept of the Sraffian multiplier, for a closed economy of single production with circulating capital, homogeneous labour and two types of income (wages and profits), was introduced by Kurz (1985).¹⁰ This multiplier is an n × n matrix that depends on the (i) technical conditions of production, (ii) income distribution (and commodity prices), (iii) savings ratios out of wages and profits and (iv) consumption pattern s associated with the two types of income. Moreover, it includes, as special versions or limit cases, the usual Keynesian multiplier, the multipliers of the traditional input-output analysis and their Marxian versions.¹¹

Although in a quite different algebraic form, the Sraffian multiplier had been essentially introduced by Metcalfe and Steedman (1981) in a model with the following characteristics: open economy of single production with circulating capital, non-competitive imports , homogeneous labour and uniform rates of profits (and growth ), propensity to save and composition of consumption. Furthermore, Mariolis (2008b) (i) showed the mathematical equivalence between the Sraffian multiplier(s) derived from Kurz (1985) and Metcalfe and Steedman (1981) and (ii) extended the investigation of the latter to the case of pure joint production .

Assume that there are no non-competitive imports and that the price side of the system can be described by (see Sect. 2.2.3)

$$ \mathbf{p}\mathbf{B}=\mathbf{w}\widehat{\mathbf{l}}+\mathbf{p}\mathbf{A}\left[\mathbf{I}+\widehat{\overline{\mathbf{r}}}\right] $$

(6.1)

where w (w _j > 0) denotes the 1 × n vector of money wage rates, $\widehat{\mathbf{l}}\kern0.5em \left({l}_j>0\right)$ the n × n diagonal matrix of direct labour coefficients, $\widehat{\overline{\mathbf{r}}}$ (r _j > −1 and $\widehat{\overline{\mathbf{r}}}\ne \mathbf{0}$) the n × n diagonal matrix of the given (and constant) values of the sectoral profit rates and p is identified with e. Provided that [B − A] is non-singular, Eq. 6.1 can be rewritten as

$$ \mathbf{p}=\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}} $$

(6.2)

where ${\mathbf{V}}_{\mathbf{B}}\equiv \widehat{\mathbf{l}}{\left[\mathbf{B}-\mathbf{A}\right]}^{-1}$ denotes the matrix of additive labour values, and ${\tilde{\mathbf{H}}}_{\mathbf{B}}\equiv \mathbf{A}\widehat{\overline{\mathbf{r}}}{\left[\mathbf{B}-\mathbf{A}\right]}^{-1}$.

Also assume that the quantity side of the economy can be described by (consider Sect. 2.2.1.4)

$$ \mathbf{B}{\mathbf{x}}^{\mathrm{T}}=\mathbf{A}{\mathbf{x}}^{\mathrm{T}}+{\mathbf{y}}^{\mathrm{T}} $$

or

$$ {\mathbf{x}}^{\mathrm{T}}={\left[\mathbf{B}-\mathbf{A}\right]}^{-1}{\mathbf{y}}^{\mathrm{T}} $$

(6.3)

and

$$ {\mathbf{y}}^{\mathrm{T}}={\mathbf{f}}_w^{\mathrm{T}}+{\mathbf{f}}_p^{\mathrm{T}}-\mathbf{I}{\mathbf{m}}^{\mathrm{T}}+{\mathbf{d}}^{\mathrm{T}} $$

or, setting $\mathbf{I}{\mathbf{m}}^{\mathrm{T}}=\widehat{\mathbf{m}}\mathbf{B}{\mathbf{x}}^{\mathrm{T}}$,

$$ {\mathbf{y}}^{\mathrm{T}}={\mathbf{f}}_w^{\mathrm{T}}+{\mathbf{f}}_p^{\mathrm{T}}-\widehat{\mathbf{m}}\mathbf{B}{\mathbf{x}}^{\mathrm{T}}+{\mathbf{d}}^{\mathrm{T}} $$

(6.4)

where x ^T denotes the activity level vector, y ^T the vector of effective final demand, f ^T _w the vector of consumption demand out of wages, f ^T _p the vector of consumption demand out of profits, Im ^T the import demand vector, d ^T the autonomous demand vector (government expenditures , investments and exports ) and $\widehat{\mathbf{m}}$ the diagonal matrix of imports per unit of gross output of each commodity.

If f ^T denotes the uniform consumption pattern (associated with the two types of income), s _w denotes the saving ratio out of wages and s _p denotes the saving ratio out of profits, where $0\le {s}_w,{s}_p\le 1$ (and s _w and s _p are not both zero and not both unity), then the consumption demands out of wages and out of profits, in physical terms, amount to (see Eqs. 6.2 and 6.3, which imply that $\widehat{\mathbf{l}}{\mathbf{x}}^{\mathrm{T}}={\mathbf{V}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}$ and $\mathbf{A}\widehat{\overline{\mathbf{r}}}{\mathbf{x}}^{\mathrm{T}}={\tilde{\mathbf{H}}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}$)

$$ {\mathbf{f}}_w^{\mathrm{T}}=\left(1-{s}_w\right){\displaystyle \sum_{j=1}^n\left({w}_j{l}_j{x}_j\right)}{\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}=\left(1-{s}_w\right)\left(\mathbf{w}\widehat{\mathbf{l}}{\mathbf{x}}^{\mathrm{T}}\right){\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}} $$

or

$$ {\mathbf{f}}_w^{\mathrm{T}}=\left(1-{s}_w\right)\left(\mathbf{w}{\mathbf{V}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}\right){\left(\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)\right.}^{-1}{\mathbf{f}}^{\mathrm{T}} $$

(6.5)

and

$$ {\mathbf{f}}_p^{\mathrm{T}}=\left(1-{s}_p\right)\left(\mathbf{p}\mathbf{A}\widehat{\overline{\mathbf{r}}}{\mathbf{x}}^{\mathrm{T}}\right){\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}=\left(1-{s}_p\right)\left(\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}\right){\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}} $$

(6.6)

respectively.

Substituting Eqs. 6.5 and 6.6 into Eq. 6.4 leads to (take into account Eqs. 6.1 and 6.3 and that $\left(\mathbf{w}{\mathbf{V}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}\right){\mathbf{f}}^{\mathrm{T}}=\left({\mathbf{f}}^{\mathrm{T}}\mathbf{w}{\mathbf{V}}_{\mathbf{B}}\right){\mathbf{y}}^{\mathrm{T}}$, $\left(\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}{\mathbf{y}}^{\mathrm{T}}\right){\mathbf{f}}^{\mathrm{T}}=\left({\mathbf{f}}^{\mathrm{T}}\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right){\mathbf{y}}^{\mathrm{T}}$):

$$ {\mathbf{y}}^{\mathrm{T}}=\boldsymbol{\Lambda} {\mathbf{y}}^{\mathrm{T}}+{\mathbf{d}}^{\mathrm{T}} $$

(6.7)

where

$$ \boldsymbol{\Lambda} \equiv {\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}\left[\mathbf{p}-\left({s}_w\mathbf{w}{\mathbf{V}}_B+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right)\right]-\widehat{\mathbf{m}}\mathbf{B}{\left[\mathbf{B}-\mathbf{A}\right]}^{-1} $$

Provided that $\left[\mathbf{I}-\boldsymbol{\Lambda} \right]$ is non-singular, Eq. 6.7 can be uniquely solved for y ^T:

$$ {\mathbf{y}}^{\mathrm{T}}=\boldsymbol{\Pi} {\mathbf{d}}^{\mathrm{T}} $$

where $\boldsymbol{\Pi} \equiv {\left[\mathbf{I}-\boldsymbol{\Lambda} \right]}^{-1}$ is the Sraffian multiplier linking autonomous demand to net output . Consequently, the change on the money value of net output, Δ ⁱ _y (‘output multiplier’) induced by the increase of one unit of the autonomous demand for commodity i, is given by

$$ {\Delta}_y^i\equiv \mathbf{p}\boldsymbol{\Pi } {\mathbf{e}}_i^{\mathrm{T}} $$

If all the eigenvalues of Λ are less than 1 in modulus, then the dynamic multiplier process defined by (see Chipman 1950)

$$ {\mathbf{y}}_t^{\mathrm{T}}=\boldsymbol{\Lambda} {\mathbf{y}}_{t-1}^{\mathrm{T}}+\Delta {\mathbf{d}}^{\mathrm{T}},\kern0.5em t=1,2,\dots $$

(6.8)

is stable (also consider Sect. 2.2.2). In that case, the number $- \log {\lambda}_{\max}\left[\boldsymbol{\Lambda} \right]$ is called the asymptotic rate of convergence (see, e.g. Berman and Plemmons 1994, Chap. 7) and provides a (rather) simple measure for the convergence rate of y ^T _t to Π(Δd ^T).

In the hypothetical (or heuristic) case where $\widehat{\mathbf{m}}=\mathbf{0}$, Λ reduces to a rank-one matrix, i.e.

$$ {\boldsymbol{\Lambda}}_0\equiv {\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}\left[\mathbf{p}-\left({s}_w\mathbf{w}{\mathbf{V}}_B+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right)\right] $$

the non-zero eigenvalue of which equals

$$ 1-{\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}\left({s}_w\mathbf{w}{\mathbf{V}}_B+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right){\mathbf{f}}^{\mathrm{T}} $$

or, invoking Eq. 6.2,

$$ 1-\left[{s}_w+\left({s}_p-{s}_w\right){\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}{\mathbf{f}}^{\mathrm{T}}\right] $$

Thus, the Sraffian multiplier reduces to

$$ {\boldsymbol{\Pi}}_0\equiv {\left[\mathbf{I}-{\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}\left[\mathbf{p}-\left({s}_w\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right)\right]\right]}^{-1} $$

or, by applying the Sherman-Morrison formula ,

$$ {\boldsymbol{\Pi}}_0=\mathbf{I}+{\left(\left({s}_w\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right){\mathbf{f}}^{\mathrm{T}}\right)}^{-1}{\mathbf{f}}^{\mathrm{T}}\left[\mathbf{p}-\left({s}_w\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right)\right] $$

(6.9)

It then follows that (i) y ^T is not uniquely determined when

$$ \left({s}_w\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right){\mathbf{f}}^{\mathrm{T}}=0 $$

(6.9a)

and (ii) one eigenvalue of Π ₀ equals

$$ {\left[{s}_w+\left({s}_p-{s}_w\right){\left(\mathbf{p}{\mathbf{f}}^{\mathrm{T}}\right)}^{-1}\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}{\mathbf{f}}^{\mathrm{T}}\right]}^{-1} $$

while all the other eigenvalues equal 1. The former eigenvalue corresponds to a Kaldorian multiplier (see Kaldor 1955–1956) and could be conceived of as the system’s multiplier (associated with the case $\widehat{\mathbf{m}}=\mathbf{0}$). Furthermore, from Eqs. 6.2 and 6.9, it follows that when both wV _B and $\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}$ are semi-positive, (i) Π ₀ is semi-positive, (ii) its diagonal elements are greater than or equal to 1 and (iii) its elements are non-increasing functions of s _w and s _p (as in the single production case; see Kurz 1985, pp. 133 and 135–136). For s _w = s _p = s (and p = e), each column of Π ₀ sums to s ⁻¹. In the case of homogeneous labour and for s _w = 0 and s _p = 1, Π ₀ reduces to a Marxian multiplier defined by Trigg and Philp (2008).

Finally, it seems that only little can be said, a priori, for the case where $\widehat{\mathbf{m}}\ge \mathbf{0}$ (also consider Mariolis 2008b). For instance, the application of the previous analysis to the SUT of the Greek economy for the year 2010 (n = 63) gives the following results¹²:

• (i) The matrix [B − A]⁻¹ (and, therefore, V _B ) contains negative elements. Consequently, the system under consideration is not ‘all-productive’, and, therefore, it does not have the properties of a single-product system.

• (ii) The matrix $\widehat{\mathbf{r}}$ contains one negative element. The matrix ${\tilde{\mathbf{H}}}_{\mathbf{B}}$ contains negative elements, although some of its columns are positive. The vector $\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\left(=\mathbf{e}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right)$ contains one negative element, while all of its remaining elements are semi-positive and less than 1 (see Eq. 6.2). It then follows that there exist values of s _w , s _p , for which Π ₀ is not semi-positive (see Eq. 6.9).

• (iii) For every value of s _w , s _p , it holds $\left({s}_w\mathbf{w}{\mathbf{V}}_{\mathbf{B}}+{s}_p\mathbf{p}{\tilde{\mathbf{H}}}_{\mathbf{B}}\right){\mathbf{f}}^{\mathrm{T}}>0$ (see Eq. 6.9a). Hence, the dynamic multiplier process defined by Λ ₀ is stable, Π ₀ is uniquely determined, and the eigenvalue of Π ₀ that differs from 1 is approximately equal to ${\left[{s}_w+\left({s}_p-{s}_w\right)0.655\right]}^{-1}\left(>1\right)$. For instance, for s _w = 0 and s _p = 1, Π ₀ is semi-positive, its diagonal elements are in the range of 1–1.059, that eigenvalue is 1.527, and the arithmetic mean of the output multipliers, Δ ⁱ _y , is 1.707. By contrast, the largest eigenvalue of Λ is −18.555 and its dominant eigenvalue ratio is 0.911 $\left( rank\left[\boldsymbol{\Lambda} \right]=50\right)$. Matrix Π is not semi-positive, while its diagonal elements are all positive and in the range of 0.051–1.037. The dominant eigenvalue of Π is 1.219, the arithmetic mean of the moduli of its eigenvalues is 0.791, and that of Δ ⁱ _y is 1.032. It can therefore be stated that the foreign sector plays a key role in the Sraffian multiplier for the Greek economy.

Appendix 2: Price Effects of Currency Devaluation

In what follows, let us assume that there are no non-competitive imports and that the price side of the system can be described by

$$ \mathbf{p}=\mathbf{p}\mathbf{A}+\mathbf{s} $$

(6.10)

or

$$ \mathbf{p}=\mathbf{p}{\mathbf{A}}^{\mathrm{d}}+\varepsilon {\mathbf{p}}^{\mathrm{m}}{\mathbf{A}}^{\mathrm{m}}+\mathbf{s} $$

(6.10a)

where p (= e) denotes the stationary price vector of domestically produced commodities; A ^d, A ^m the irreducible and primitive matrices of domestic and imported input-output coefficients , respectively; A ≡ A ^d + A ^m, with λ _A1 < 1; ε the nominal exchange rate ; p ^m the given vector of foreign currency prices of the imported commodities; p = ε p ^m; and s (> 0) the vector of gross values added per unit activity level , which, in national accounts terms, equals the sum of consumption of fixed capital, s _C, net taxes on production, s _T, net operating surplus , s _S, and compensation of employees , s _E, i.e.

$$ \mathbf{s}\equiv {\mathbf{s}}_{\mathrm{C}}+{\mathbf{s}}_{\mathrm{T}}+{\mathbf{s}}_{\mathrm{S}}+{\mathbf{s}}_{\mathrm{E}} $$

(6.11)

By solving Eqs. 6.10 and 6.10a for p, we obtain

$$ \mathbf{p}=\mathbf{s}{\left[\mathbf{I}-\mathbf{A}\right]}^{-1}=\left(\varepsilon \mathbf{m}+\mathbf{s}\right){\left[\mathbf{I}-{\mathbf{A}}^{\mathrm{d}}\right]}^{-1} $$

(6.12)

where m ≡ p ^m A ^m. The price effect s of currency devaluation may be represented by the following dynamic version of Eq. 6.10a:

$$ {\mathbf{p}}_{t+1}={\mathbf{p}}_t{\mathbf{A}}^{\mathrm{d}}+{\varepsilon}_1\mathbf{m}+{\mathbf{s}}_t,\kern0.5em t=0,1,\dots $$

(6.13)

where ${\varepsilon}_1\equiv \left(1+\overset{\frown }{\varepsilon}\right){\varepsilon}_0$, $\overset{\frown }{\varepsilon }$ denotes the devaluation rate and

$$ {\mathbf{p}}_0=\left({\varepsilon}_0\mathbf{m}+\mathbf{s}\right){\left[\mathbf{I}-{\mathbf{A}}^{\mathrm{d}}\right]}^{-1} $$

(see Eq. 6.12). Although there are alternative approaches for modelling the response of sectoral gross value added to currency devaluation , the choice between them should also take into account the input-output data availability. Thus, for the purposes of an indicative estimation, it may be postulated, for instance, that

$$ {\mathbf{s}}_t=\left({\mathbf{p}}_t{\mathbf{A}}^{\mathrm{d}}+{\varepsilon}_1\mathbf{m}+{\mathbf{p}}_t{\widehat{\mathbf{S}}}_{\mathrm{CT}}\right)\widehat{\boldsymbol{\upmu}}+{\mathbf{p}}_t{\widehat{\mathbf{S}}}_{\mathrm{CT}} $$

where (see Eq. 6.11) ${\widehat{\mathbf{S}}}_{\mathrm{C}\mathrm{T}}\equiv \left[{\widehat{\mathbf{s}}}_{\mathrm{C}}+{\widehat{\mathbf{s}}}_{\mathrm{T}}\right]{\widehat{\mathbf{p}}}_0^{-1}$, $\widehat{\boldsymbol{\upmu}}\equiv \left[{\widehat{\mathbf{s}}}_{\mathrm{S}}+{\widehat{\mathbf{s}}}_{\mathrm{E}}\right]{\widehat{\mathbf{c}}}_0^{-1}$ and

$$ {\mathbf{c}}_0\equiv {\mathbf{p}}_0{\mathbf{A}}^{\mathrm{d}}+{\varepsilon}_0\mathbf{m}+{\mathbf{s}}_{\mathrm{C}}+{\mathbf{s}}_{\mathrm{T}} $$

which imply that Eq. 6.13 becomes

$$ {\mathbf{p}}_{t+1}={\mathbf{p}}_t\boldsymbol{\Delta} +{\varepsilon}_1{\mathbf{m}}^{*} $$

(6.14)

where $\boldsymbol{\Delta} \equiv \left[{\mathbf{A}}^{\mathrm{d}}+{\widehat{\mathbf{S}}}_{\mathrm{CT}}\right]\left[\mathbf{I}+\widehat{\boldsymbol{\upmu}}\right]$ and ${\mathbf{m}}^{*}\equiv \mathbf{m}\left[\mathbf{I}+\widehat{\boldsymbol{\upmu}}\right]$. Then the solution of Eq. 6.14 is

$$ {\mathbf{p}}_t={\mathbf{p}}_0{\boldsymbol{\Delta}}^t+{\varepsilon}_1{\mathbf{m}}^{*}\left[{\boldsymbol{\Delta}}^{t-1}+{\boldsymbol{\Delta}}^{t-2}+\dots +\mathbf{I}\right] $$

and p _t tends to ${\varepsilon}_1{\mathbf{m}}^{*}{\left[\mathbf{I}-\boldsymbol{\Delta} \right]}^{-1}=\left(1+\widehat{\varepsilon}\right){\mathbf{p}}_0$, since ${\lambda}_{\boldsymbol{\Delta} 1}<1$, while the price movement is governed by

$$ {\mathbf{m}}^{*}\left[{\boldsymbol{\Delta}}^{t-1}+{\boldsymbol{\Delta}}^{t-2}+\dots +\mathbf{I}\right] $$

which could be conceived of as the series of dated quantities of imported inputs.

In the extreme case where the gross values added are ‘insensitive’ to devaluation, i.e. s _t = s, Δ should be replaced by A ^d, m ^* should be replaced by $\mathbf{m}+{\varepsilon}_1^{-1}\mathbf{s}$, and p _t tends to $\left({\varepsilon}_1\mathbf{m}+\mathbf{s}\right){\left[\mathbf{I}-{\mathbf{A}}^{\mathrm{d}}\right]}^{-1}<\left(1+\widehat{\varepsilon}\right){\mathbf{p}}_0$. Finally, in the ‘intermediate’ case where ${\mathbf{s}}_t={\mathbf{p}}_t\widehat{\mathbf{s}}{\widehat{\mathbf{p}}}_0^{-1}$, Δ should be replaced by ${\mathbf{A}}^{\mathrm{d}}+\widehat{\mathbf{s}}{\widehat{\mathbf{p}}}_0^{-1}$, m ^* should be replaced by m, and p _t tends to $\left(1+\widehat{\varepsilon}\right){\mathbf{p}}_0$.¹³

Empirical evidence from the SIOT of the Greek economy for the year 2005 shows that (Katsinos and Mariolis 2012):

• (i) The P-F eigenvalue of A ^d is 0.321 and the damping ratio is 1.290.

• (ii) The P-F eigenvalue of ${\mathbf{A}}^{\mathrm{d}}+\widehat{\mathbf{s}}{\widehat{\mathbf{p}}}_0^{-1}$ is 0.949 and the damping ratio is 1.045.

• (iii) The P-F eigenvalue of $\left[{\mathbf{A}}^{\mathrm{d}}+{\widehat{\mathbf{S}}}_{\mathrm{CT}}\right]\left[\mathbf{I}+\widehat{\boldsymbol{\upmu}}\right]$ is 0.893 and the damping ratio is 1.248.¹⁴

• (iv) For $\widehat{\varepsilon}=50\%$, the cost -inflation rate (as measured by the gross value of domestic production) at t = 1 is 9.3 % (is 5.3 %), and the arithmetic mean of commodity prices associated with matrix $\left[{\mathbf{A}}^{\mathrm{d}}+{\widehat{\mathbf{S}}}_{\mathrm{CT}}\right]\left[\mathbf{I}+\widehat{\boldsymbol{\upmu}}\right]$ (with matrix ${\mathbf{A}}^{\mathrm{d}}+\widehat{\mathbf{s}}{\widehat{\mathbf{p}}}_0^{-1}$) reaches approximately 95 % of its asymptotic value at t = 14 (at t = 30).

These figures seem to be consistent with the finding that (other things constant) even ‘large’ devaluations would not imply great inflationary ‘pressures’.¹⁵