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
same reason. From these facts we conclude that the changes and thus all cyclic trajectories
are perpendicular to the equilibrium vector. The scalar value z’z remains constant on the cyclic path.
The similarity transformation by the matrix Z serves to translate the equilibrium
point to a vector of unit magnitudes. We may therefore consider the elements of this matrix W, that exhibit the non-equilibrium response triggered by a non
equilibrium state or triggered by any perturbation of the equilibrium, both as absolute or relative deviations. The main force of Goodwin’s approach — working
with pure numbers without dimension, scale-free elasticities — is maintained and may be exploited when interpreting the numerical illustration presented below.
3. Numerical results
3
.
1
. Data and growth rate To illustrate such a wave matrix, the system was implemented with the data used
to compute equilibrium quantities and growth rate for the United States in Brody 1966. Appendix A lists the basic data: a seven-order 1958 flow coefficient matrix
and approximate product life-spans for each sector, used to estimate a capital coefficient, or stock, matrix.
While the data were very crude the early computations produced quite reasonable results. In particular, the equilibrium growth rates computed in 1964 at the
Harvard Economic Research Project were close to 4, comparing well with the average rate
6
observed over the next 20 years from 1959 to 1978. Input-output computation do have an error-correcting facet. Even if the flow coefficients and
thus also the eigenvectors are correct only in their first digit, the eigenvalues already possess a 2-digit precision. Nevertheless this peculiarity holds only for the
synthetic, average, aggregated or macroeconomic indices. Thus one may expect dependable results only for the computed cyclic frequencies, that is cycle-lengths,
not for all the results
7
. The error of the price and quantity elasticities, some of
7
This peculiar but very useful fact is granted by the theory of the Raleigh – Aitken quotient. See Bodewig 1952.
6
For the 1960s, average growth stood slightly above this mark but it did drop below 4 during the next decade.
which will be presented below, are strongly affected by the probably sizable errors of the original data.
3
.
2
. Elasticities The elements of the wave matrix are measure-invariant and therefore also all the
eigenvalues. In Table 1 only the third quadrant of the matrix is exhibited. This is the quantity response by branches indicated in the rows triggered by an increase
in the price of the respective branches set out columnwise. This quadrant see Table 1 as a matrix, is always positive semidefinite but not all
of its off-diagonal elements are negative. It is interesting to see that agricultural and service-sector price increases are associated with increases in the outputs of most
other sectors. The years 1959 – 78 actually witnessed a very rapid autonomous, self-boosting growth of services.
An increase of wages seems to uniformly lower output in all other sectors. Households, on the other hand, respond most sensitively to increases in agricultural
and service prices, less so to those of other sectors. Household is also the only sector that seemingly had an own quantityprice elasticity \ 1. Equipment and
construction prices do not seem to affect other branches.
Given the crudeness of the data and the simplified behavioral assumptions, we should not attribute too much significance to the actual size of these first estimates
of individual elasticities. The response of the household sector seems to be also particularly exaggerated. This abrupt reaction may stem from a conceptual fault in
the data: a very small stock multiplier, used for the household row
8
. This part of
Table 1 Quantity response to price changes, USA 1958
Price Quantity
M E
C F
S H
A −
88 33
3 −
1 Agriculture
− 1
53 25
Materials 20
4 54
− 102
Equipment −
107 63
5 2
16 −
1 22
67 5
15 −
109 −
1 2
22 Construction
Fuel and energy 23
2 −
1 −
2 17
68 −
106 21
2 −
1 Services
− 1
5 90
− 115
Households 224
− 151
− 10
− 4
− 59
− 403
224 29
13 Sum total
12 26
102
8
The neglect of human capital-in tabulations and otherwise-seems to be a constant source of economic misjudgement and miscalculation. A larger multiplier should have been used for the probably
much greater capital intensity required to produce labor. Yet this would have necessitated recomputing the original data. It was decided to leave the data intact to show that the approach yields acceptable
answers even if the data set is very rough.
the computation should yield much clearer results when a more realistic, or actual, stock matrix is used instead of our makeshift one. But, alas, I did not yet find detailed
and good stock data and my results for perturbation or sensitivity analysis are also very meager for the time being.
The 2nd quadrant, not presented here, the response of prices on oversupply, proved to be negative semidefinite, as expected. It has a dominant negative diagonal but the
other elements were not all positive. The 1st and 4th quadrants priceprice and quantityquantity elasticities are of minor importance, but they do exist and will have
to be inspected more closely when better data warrant the exercise.
3
.
3
. Frequencies The purely imaginary eigenvalues of the wave matrix are, in increasing order
rounded to two decimals v =
0.1 0.23
0.34 0.90
1.2 8.9
The feasible cycle-lengths, computed as T = 2pv rounded to years are thus: T =
65 27
19 7
5 1
Thus the Kondratiev wave, the demographic generational swing, an equipment or building cycle, two shorter inventory cycles and a yearly seasonality are already
in the repertory of this simple model. If we search the actual GDP series from 1959 to 1978 we may find power-peaks only for the 5, 7 and 19 year cyclic components.
Neither the short cycle nor the longer ones are discernible in the relatively short time series. But other evidence suggests that they do exist and may be found in the longer
series
9
.
3
.
4
. Forecasting The next step is to compare actual annual GDP growth rates with those computed
on the basis of this model. Both the logarithmic and linear model were computed with second and third order Runge – Kutta formulas to 4 digit precision. This level
is more than adequate for illustrative purposes because the resolving power of most graphs does not exceed a 3 digit precision.
In Fig. 1, the results of the linear approximation are very close to the exact logarithmic trajectory. They are, for practical purposes, hardly discernible from the
latter. Considering that economists normally accept a 2-digit accuracy in estimating growth rates, it is permissible to consider the two systems as virtually equivalent. The
same is also true for the details of the computation sectoral prices and quantities, not presented here.
9
Quarterly data for the same period do show short fluctuations and also a relative maximum of ‘power’ for 5.25, 7 and 18.75 years But the demographic swing and the Kondratiev-like longer wave
leaves no imprint. Of course, such waves can be fitted, but their power is not ‘peaking’ when compared to the neighboring years.
Fig. 1. Computed and actual growth rates. The GDP growth data were copied from the NIPA homepage.
This comes as no surprise. In spite of their jagged outline, the trajectories do not deviate more than a few percentage points above and below their average. There-
fore the ‘bias’ of the linear approximation did not reach a
1 2
magnitude even in the worst cases. Moreover, because of the periodic character of the trajectory, the
deviations do not accumulate; thus the linear approximation is and stays close to the logarithmic path.
Nevertheless the forecasting power of the model is less than impressive. Though the computed path for the 20 years exhibits a very high correlation with the actual
one R
2
= 0.99, computed and actual yearly growth rates correlate much less well
and show an increasingly deviating pattern. Fluctuations in the overall level of economic performance grew strongly in the 1970s. There were two heavy recessions,
with troughs registered in November 1970 and March 1975, the latter attributed to the world-wide oil crisis. They are not reflected by the computation. For the second
decade the whole exercise ceases to be meaningful, even if producing the closest fit, by sheer chance, for the latter years.
With minor modification of the initial prices it is not difficult to secure a better forecast for, say, the first 10 years. Allowances for agricultural subsidies and energy
improves the fit, because they boost the presence of the longer cycles, hardly discernible in the computed paths. Though the possibility of cyclic or anticyclic
intervention is an important topic in itself, it was not attempted to improve the fit by simulating government policy intervention. Working with the data and develop-
ing a ‘feeling’ for their behavior I believe that better forecasts might be gained by adding more detail more sectors mean a greater richness of cycles and improving
the data for the stock matrix the present one is really a very rough, makeshift ‘approximation’.
No account was taken of monetary and fiscal intervention of the government, taxation and subsidies that might change significantly the conditions of supply and
demand. Intervention, besides changing often quite abruptly, is mostly displaying a Sisyphean character. It apparently can not dispense with cycles. In the best case it
may roll them away, perhaps even attenuate some of them, presumably the minor ones, only to find them coming back, sometimes with increased amplitudes. The
intensification of fluctuations in the 1970s is well known, much discussed in the pertinent literature and shown in the actual data of the graph.
4. Conclusion
