Table of Contents
A Historical Fit of a Model of the US Long Wave
Alexander V. Ryzhenkov
Institute for Economics and Industrial Engineering
Siberian Branch of Russia’s Academy of Sciences
17 Academician Lavrentiev Avenue, Novosibirsk 630090, Russia
Fax: +7 3832 30 25 80
ryzhenko@ieie.nsc.ru
Abstract
This paper refines a theoretical model of capitalist reproduction. A compact state-
space form of this model defines a hypothetical Law of capital accumulation. The state
variables are the unit wage, employment ratio, gross unit rent, man-made capital-
output ratio, natural capital-output ratio, indicated natural capital-output ratio, and
unit depreciation of the natural capital. An application of an extended Kalman filtering
to the US macroeconomic data 1958-1991 identifies unobservable components of this
Law. It is shown that long wave is a dominant non-equilibrium quasi-periodic behav-
ioural pattern of the US capital accumulation. Evaluating the historical fit through ap-
propriate summary statistics and long-range forecasting strengthen confidence in the
model.
Keywords: Capital accumulation, long wave, extended Kalman filtering, forecasting
Acknowledgements
This paper relates to a finished research project at the Economic Department of the Ga-
kushuin University. Professor Dr. Michiya Morita and the other Japanese colleagues
have facilitated that work. The University provided a financial support. A consultation
on VENSIM by Dr. Robert Eberlein has lightened a deeper learning of this advanced
simulation technique.
Introduction
Analysts have been pondering whether the current global slowdown is just a correction
of a brief period of overindulgence or a beginning of along-term economic recession. It
is reported that a former chief executive of Invensys, the UK-based engineering group,
said (The Financial Times, July 25, 2001: 15): “I had been projecting a mild downtum
in the US, which would have a limited impact on the rest of the world. In fact it has
tured into the most serious recession... in the US past 30 years.”
This paper asserts that the present US downturn is the beginning of the recession of the
long wave. Taking as a point of departure (Ryzhenkov 2000), next sections present an
upgraded system dynamics model that reflects these phenomena and enables futuristic
projections.
The model assumptions
A capitalist economy is restricted by natural resources. Produced capital is an embodi-
ments of knowledge and, similarly, natural capital is a stock of information. Fixed as-
sets, labour and natural assets are essentially complementary to each other and are also
substitutes to some degree depending on relative price changes. The other important
premises are such:
(1) two social classes (capitalists and workers); the state enforces property rights, yet
the cost of such an enforcement is not treated explicitly;
(2) three factors of production -- labour force, man-made capital, natural capital -- are
homogenous and non-specific;
(3) only one aggregated good is produced for consumption, investment and circulation,
its price is identically one;
(4) production (supply) equals effective demand;
(5) all wages consumed, the resource rent and a part of profits saved and invested;
(6) steady growth in the labour force that is necessarily not fully employed;
(7) a growth rate of the real wage rises in the neighbourhood of full employment;
(8) a change in capital intensity and technical progress are not separable due to a flow
of invention and innovation over time;
(9) a qualification of the labour force corresponds to technological requirements.
This model abstracts from over- production of commodities inherent in over-production
of capital during certain phases of industrial cycles. The assumption (5) corresponds to
the immediate aim of capitalist production.
The assumption (6) means that the labour force grows exponentially over time. This as-
sumption could be substituted by an assumption of an asymptotic growth or by another
hypothesis.
The model equations
The model is formulated in continuous time. Time derivatives are denoted by a dot,
while growth rates will be indicated by a hat. The model consists of the following equa-
tions:
P =K/s (1)
a =P/L (2)
u=wia (3)
=m, +m,(K/L) + m,y(¥) +m,F/L, (4)
where y (¥) = SIGN(¥ )ABS (¥)*j, m, 20,1 2m, =0, m,20, m, 20,0 <j
(K/L) =n tn,u ta,(v- v) +n,(Z/P), (5)
a, 20, n,20, n, 20, 1> v, >0,
v=LiN (6)
N =N,ett n=const 20,N, >0 (7)
W=-g trv +b(K/L) + GF/L, g 20,r>0 (8)
P=C +K +Y =wh+(1-WM+K +Y (9)
Fey-Z (10)
Z=eP,0<e <1 (11)
¥ =(0,(c -f) +0, f)y, y=V/P 20 (12)
x =i (13)
f=F/P (14)
c =X/P (15)
é= Bey/e-l), e e>0 (16)
K =kM =K{(1- w/a)P - Y)=K{(1- u)P - Y], (17)
where 0 <k St.
Eg. (1) postulates a technical relation between the capital stock (K) and net output (P).
The variables is called produced or man-made capital-output ratio. Eq. (2) relates la-
bour productivity (a), net output (P) and labour input or employment (L). Eq. (3) de-
scribes the shares of labour in net output (u).
Eq. (4) is an extended technical progress function. It includes: the rate of change of
produced capital intensity, K/L, the direct scale effect, my (¥) , and the rate of change of
natural capital intensity, F/L. ABS(x) is absolute value of x that is non-negative, xj is x
raised to the j-th power, SIGN(x) is a sign of x. The parameter j has been randomised
in a univariate sensitivity analysis (Ryzhenkov 2001).
Eq. (6) outlines the rate of employment (v) as a result of the buying and selling of la-
bour-power. Labour force grows exponentially in (7). In the Eq. (8), the rate of change
of the real wage (w) depends on the employment rate (v), as in the usual Phillips rela-
tion, and on the rates of change of capital intensity (K/L) and (F/L), additionally. The
capital intensity (K/L) is a proxy for qualification.
In the Eq. (9), a sum of net export, final private and public consumption is C =P[u + (1-
k)(1- u- y)]. The net formation of produced fixed capital is K = kM, where K is man-made
fixed assets. The gross accumulation of natural assets Y equals the gross resource rent
in monetary (or information value) terms. Eq. (9) and Eq. (17) show that profit (M = (1-
u-y)P) and incremental man-made capital (K) are not equal in monetary (or informa-
tion value) terms if the investment share k < 1. Considering the latter as a variable and
reflecting the workers saving is left for a future research.
In the Eq. (10), F is a net accumulation (loss) of the natural capital (F). Z is the net en-
vironmental damage in the Eq. (11), i.e., depletion and degradation of non-produced
natural assets (land, soil, landscape, eco-systems) due to economic uses above the re-
generation rate.
The rate of change of capital intensity (K/L) in the Eq. (5) is a function of the unit
wage (u), difference between real employment ratio and some base (‘natural’) magnitude
(v- vs depletion/degradation of natural capital in relation to net output (z/P). The rate
of growth of capital intensity depends on the environmental damage per unit of output
(an application of the principle ‘a pollution prevention pays’), in particular. A high wage
share and high employment ratio foster mechanisation (automation).
The indicated natural capital, x, may remain constant or change in the Eq. (13). The
Eq. (12) defines an investment policy that is aimed to develop the natural capital in ac-
cordance with the indicated natural capital (y is the investment ratio for the natural capi-
tal). A combination of proportional and derivative control over the investment in natu-
ral capital is used hereby.
The stock of environmental assets is not treated explicitly in this model. The natural
capital-output ratios -- real, f, and indicated, c, in Eq. (14) and Eq. (15) -- belong to the
state variables of the model.
We assume that the unit depletion (degradation) of the natural capital asymptotically
declines due to substitution and structural change as in (16) where for P> 0, € <0. The
higher the rate of economic growth, the faster the reduction of eco-intensity (or the
promotion of eco-efficiency in the narrow meaning). The Eq. (16) is, likely, a better ap-
proximation than e = const > 0. An approximation of a higher order can be easily imple-
mented in the future work.
Three profit rates are defined for this economy. The first is the average rate of retum
to man-made capital (1- u - y)/s. The second is a general one, it measures a ratio of the
economic surplus to the total value of produced and natural capital (1 - u - e)/(s + f).
The third is a biased profit rate (1- u)/s that is more easily calculated based on the statis-
tics with incomplete data on the natural resources.
The rate of net rent is the ratio of net unit rent to natural capital - output ratio, (y - e)/f.
The general rate of profit is a weighted average of the rate of retum to man-made capital
and the rate of net rent: (1 - u- e)/(s +f) =[s/(s + H](1- u - y)/s Hf/(s + Hlly - ef.
The model and law in a deterministic state space form
To get a compact model we need the following transformations.
§ =(k/P) =(K/L)- (P71)
=n, thu tn,(v- Ve) +n,(Z/P) - (m, +m, k/ L+m,y(v) +m,F/ L)
=n, t+nu +n,(v- vo) +n,¢ - (m, +m,(n, +n,u +n,(v- vo) + ne) +my (Vv) +m,(F -L))
Sine (1- m,)(n, taut a(v - vc) + n,€) > my (¥) - m, (F-K tn, tou +n,(v- Ve) +n,¢)
= -m,+(1- m,)(n, +nyut a,(v -v,) + ne) - m,y (¥) - m.( f-§ tn, tout a,(v =v) +n,e)
= -m,+(1- m,- m,)(n, +njut nv - Vo) +n,e) - my (¥) - m,( f-8);
¥ =(L/N)= K- (a, +2,0 +n,(v ~ vo) +n,e) -n
=, low-y 3
=k (a, tay ta,(v- Vo) +n,e) - n;
i=w-a
=-g trv +b(n, tnyutn,lv- ve)+n,e) + gF/L-(m, + mln, +n,0 +n,(v~ ve)tn,e) +
m,y (¥) +m,(F -L))
=-gtw-m, +(b- m, - m,)(n, +nyuta,(v- Vo) +n,¢) - my (¥) - m,( f-§) +q( f-8 +n,
+n,utnj(v- ve) +n,¢)
= -gtr-m, +(b+q-m, -m,)(n, +njutn,(v- vc) +n,¢) - my (¥) +(q- m,)( f -$);
f =F —p =Y* e -é-t
= (y-2)
£
- (m, tm (o,tnu +n,lv - vo) +n,e) +my (¥) +m, (F -£))-£
=(1- m,) Seal > my - m,(n, +nu +n,(v- v) +n,e) - my (¥) - (1- m,) (¥4n);
¢=X -P =i -K 49 =i eid 48;
§ =(0,(c -£) +f);
6= Ble /e-=(P/L+¥ +n)(q/e-1) =(k 12 4m, +(m, +m,-1)(n, +n,utn,(v-
Ve) +n,e) +m,y (¥) +m,( f -$))(e/e-1).
In a compact form, the model consists of the seven non-linear ordinary differential
equations (17) -- (23) that define the hypothetical Law of capital accumulation:
‘ Se (m,+(m,+m,- Lin, +n,u tny(v~ ve +n,e) +m,y (0) +m, f )s, (17)
= ~(a, tau tnyv- ve) +n,e)- ay, (18)
w=aCg trv-m, +(b+q-m, - m,n, tnyutn,(v- ve) +n,e) - my (¥) +(q- ms) ( f -§))u, (19)
v
f=(d- mS -m,- mo, +n0 tajv-v) +n,)-m¥(@) -(- mgm — (20)
c H(i Ay +8)c, (21)
¥ =(0,(0-f) +0 fy, (22)
e
§ +m, +(m, +m, -ia, taut n,(v- ve) +n,e) +myy lH) +m f-$))(e 2), (23)
wheres >0, 1 =v>0, 1 2u>0,f>0,c>0,1>y, 1 >e. The requirement for denomina-
tors to be positive is skipped. IfK >0,F >0 for every instant of time, the system (17) --
(23) defines a strongly sustainable development.
A nontrivial stationary state is defined as
Ea= (Sa, Var Ua, fay Cay Yara), (24)
where
Sa= So
va =(g +(1-b-g)(d- a),
ug = (d-n- n+ n,Va- Vo) - ean,)/n,,
fa = (1-ua- ea)/d - sak,
Ca=fa,
Ya=ea + dfa,
ea= ey
i=d;
the positive 8y is exogenous, ya is the gross unit rent, e is the unit depreciation of the
natural capital. This stationary state is not locally stable for reasonable parameters’ val-
ues.
The man-made capital stock, natural capital and net output increase thereby at the rate
K =F =X =p =¢d =——™! _ +n>0 thatis equal to the net rent rate and less
‘ie ee a Tony Sa
than (fork < 1) or equal to the average profit rate (for k = 1). The stationary average
profit rate equals (1 - uss y,ls, =d/k. The both quantities (d and d/k) are used as bench-
marks in a long-range forecasting below.
An Extended Kalman Filtering (EKF)
The Kalman filter is a particular powerful tool for estimating unobservable part of a
model (parameters and meta-parameters like variances) in one operation. Although the
Kalman filter itself does not estimate the unknown parameters of the model, it provides
a one-step-ahead prediction error with its covariance matrix. The prediction error de-
composition of the likelihood function utilises this information. Maximisation routines
can be used to determine the unknown parameters (Cuthbertson et al. 1992: 210-225).
The presentation of EKF below is an abridged version from (Peterson 1980) for a sta-
tionary white sequence with a time invariant covariance.
State equations:
x(n) =f [x(n-1)] + w(n).
Measurement equations:
z(n) = Hx(n) +v(n).
Index of data samples:
n=1, 2). Ne
Initial conditions:
x(0) =N[Xo, WT.
Equations errors (driving noise):
w(n) = N[0, QT’.
Measurement errors:
v(n) =N[0, RJ’.
Discrepancies v(n) and w(n) are modelled as zero mean uncorrelated random vectors
with time invariant covariance matrices
E{x(n,)v'(n,) } =0, ny #ng and ny =n
E{x(n,)w(n,) } =0, ny #ng and ny =n
E{v(n;)v (nj) } =R, m =n
E{v(n,)v'(n,) }= 0, ny ng
E{w(n,)w(n,)} =Q, ny=n2
E{w(n,)w(n,) } = 0, nm #n2.
Linearization about old estimate (estimated state):
F(n) =2f
ox
k =&(n —I|n —1)
Filter Equations
Predicted state:
Rin —1) =f[X(n —In -1)].
Predicted measurement:
2(nfo 1) =H&(nhn -1).
Residuals (innovations):
8, (n|n —1) =z(n) —€(n]n -1).
Predicted state covariance:
2x (nf -1) =F(n)z, (n —IJn -1)F’(n) +Q.
“Nim, c] denotes a normal, white process with mean m and covariance matrix c.
Predicted measurement covariance:
2, (n\n -1) =H, (nin -1)H’ +R.
Normalised predicted measurement residuals:
8 (nfn -1) =, (n)n -1),/Z,(nfa 1)".
Updated state covariance:
, (nln) =12, aha =) +HR7HY.
Filter Gain:
Kin) =2, (an -1)H'2, “(n\n -1).
The gain depends on the estimate %(n —1,n —1) and thus cannot be pre-computed. The
full equation has to be solved in real time.
Updated state estimate:
(nin) =R(n{n -1) +K(n)6, (nn -1).
Define
a: vector containing all the unknown parameters in f, R, Q, V.
p(Zy : ): probability density of zy =(z(1),..., z(N)) forgiven a.
&(N : a) =In p(zy : a) : log likelihood function (Schweppe 1973: 434).
&(N) =value of « which maximises &(N : a) for particular zy. Thus &(N) is the
maximum likelihood estimate given the observations z(1),..., z(N).
Log likelihood is defined recursively:
&(n) =E(n -1) — San 18, (an
Initial conditions:
x, (nln 1).
1
1}.
)—z ln
2(0)0) =xp,
2, (0/0) =,
&(0) =0.
The matrices R , Q, W are positively (semi)-definite.
The basic one-step prediction problem is to calculate (n\n —1) , the estimate of x(n),
using 2(1),..., z(n-1). The Vensim output contains a file 1Step.err that equals
—6, (n\n -1) =A(n|n —1) —z(n). For the my stochastic model in the state space form, N
= 34, dim z(n) =7, dim x(n) =8, the indicated natural capital-output ratio, c, is not ob-
servable.
Confidence Tests from the Optimal Filter
The whiteness of 5, (n\n —1) is sometimes viewed as a basic property of an optimum
estimator. It means the correction 6, (nn —1) made using z(n) cannot be predicted.
A statistical test for whiteness of the prediction residuals using an estimator based on
the EKF belongs to the class of model behaviour tests. The EKF-based estimator does
involve a model simulating in computing a likelihood function. The VENSIM profes-
sional soft-ware has served for performing such an extended Kalman filtering (EKF) in
this paper below.
It is argued (Peterson 1980) that the whiteness property of residuals (innovations) pro-
vides an independent test of model validity since is not employed directly in maximis-
ing the log likelihood. The whiteness may be tested by computing correlation measures
of the residuals. Each residual (in the case of multiple-dimensional measurements)
should have a correlation coefficient of one with respect to itself, zero correlation with
respect to lagged value of itself, zero cross correlation with all other residual processes.
Examination of the two basic assumptions of the extended Kalman filtering (EKF)
Two crucial initial assumptions of the EKF are the conjectures that
= there are no structural errors in the model;
- the “disturbance terms” in dynamic and measurement equations are random vari-
ables.
The EKF is usually supported and lightened by auxiliary assumptions on a simple dis-
tribution law imposed on input and measurement “disturbance terms” as random vari-
ables, for example, in a form of the Gaussian multivariate normal distribution. The EKF
is maintained by another subordinate assumption that the discrepancy terms are not
only random variables, but even random variables of the special kind called white noise.
It has been already shown in the literature (Blatt 1983: 339) that “in all actual models,
specification errors are the price one must pay for having any sort of workable model at
all.” Truncation error, lag structure error, aggregation error, omitted variable error,
unknown variable error belongs to the mostly important specification errors that are
practically unavoidable (Ibid., 342).
The residuals from the mis-specified relationships are not due entirely, or largely, to
pure random influences. On the contrary, these residuals contain a highly systematic,
non-random component while data limitations make it impossible to eliminate these er-
rors.
Typically, a process of maximising the log likelihood function by a hill-climbing algo-
rithm with random multiple starts cannot be finished. This process should be normally
terminated by a researcher at a simulation run relying not only on logic but intuition as
well. Therefore to find a genuine optimal solution is hardly possible in practice. Lin-
earization of models’ equations, integration technique, rounding-off contributes to the
systematic errors additionally. So besides specification errors, identification errors in
the parameters values are unavoidable.
In macroeconomic applications, the input w(n) and measurement disturbances v(n) are
not white processes but rather structural processes; i.e., they are time-correlated dis-
crepancies. For example, the shorter quasi-cycles are the most important factor of auto-
correlation of these discrepancies in the model of long wave that abstracts from other
fluctuations in its deterministic part. Of course, we cannot exclude quasi-periodic pat-
terns longer than the long wave relying on the available data.
The model discrepancies, termed in the literature as disturbances, have uncertain val-
ues, their mean values can change over time. It has been pointed out (Barlas 1989: 60)
that ‘noise terms of system dynamic models are not necessarily independent and /or
normally distributed, behaviour patterns generated by typical system dynamic models
are highly auto-correlated, and they are in general non-stationary in the mean and some-
times non-stationary in variance. Such characteristics of system dynamic models violate
three fundamental assumptions of standard statistical tests: normality, independence and
stationarity. Therefore, standard hypothesis tests (such as t-test, F-test, x - test) are not
directly applicable to system dynamic behaviour validation... ”
The presence of non-white (time-correlated) disturbances, errors of specification and
identification makes passing the test for whiteness essentially impossible for a system
dynamic model of a macroeconomic system based on real data.
Pre- whitening of the noise and dealing with non-random input and measurement errors
through augmentation of a model is recommended for physical applications. Unfortu-
nately these procedures are either not warranted in macroeconomics by available data or
are to be continued ad infinitum in view of non-linear interactions of multiple quasi-
periodic processes rooted in circulation and turnover of the social and individual capital.
The EKF, based on abstraction and idealisation, is inconsistent to some degree with
macroeconomic data. Specification and identification errors, non-randomness and/or
non-whiteness of discrepancies exclude whiteness of residuals in practical macroeco-
nomic applications. Still parameters of the stochastic dynamic model, expected values
of the system states, residuals and their covariance matrices can be roughly estimated by
the EKF, Revealing a time-pattern in residuals could give additional insights into actual
regularities thereby. If the test for whiteness of residuals cannot be passed fundamen-
tally, a researcher is to apply other validity and consistency tests that are more appropri-
ate.
A Macroeconomic A pplication of the Extended Kalman Filtering
The compiled macroeconomic data in the Table 1 cover the period 1958-1999. The in-
formation on natural capital relates to a significant part of natural capital, namely the
stocks of proved mineral reserves for 1958-1991.
It is known that the likelihood computation is exact only in the case of linear systems
with Gaussian noise for driving and measurement errors (Peterson 1980). This paper
reports only about quasi-optimal estimates obtained so far for the presented non-linear
model.
The data from the Table 1 feed the model fora shorter period 1958-91. It is assumed
that there are high measurement errors and great uncertainty about the initial system
state. The famous book (Morgenstern 1963) has prompted this assumption, Of course,
neither Morgenstem’s inquiry nor our previous discussion suggests that errors in the
economic statistics are purely random. In fact, the both types of errors contain quasi-
cyclical components related to fluctuations that are not treated explicitly.
The dozens of marathon runs culminated in the final optimisation output (File 1). In
this macroeconomic application of EKF, we consider the economic subjects’ measure-
ment errors (variances associated with the data stream they face) as the variables’
weights in the pay-off definition file pay-off.vpd. The filter control file specifies a co-
variance matrix for a driving noise and initial covariance of the state vector (kal-
man.prm in the Appendix).
EKF realised in the VENSIM software has enabled to estimate the unobservable
components of the compact model (17) - (23) and of an additional equation for the level
of labour productivity, a = INTEG (a, a)), after this model has been transformed into
the canonical stochastic form. The number of measured levels is seven since the indi-
cated natural capital (c) is not observable directly.
The given estimates of the model parameters and meta-parameters are not as exact as
required by optimal filtering. Still it may be interpreted an advantage rather than a
drawback if we recall that the economic application of the optimal filtering assumes that
learning agents use the given information optimally. It is clearly a very strong idealisa-
tion for the world of bounded rationality. Without this idealisation the identification is
even more complicated. It is a matter of combined theoretical and empirical research to
find out how efficiently the information is really used by the economic agents. If it can
be proved than a quasi-optimal usage of available information is more likely than opti-
mal, then the common pragmatic usage of quasi-optimal estimates (in this paper, in
particular) is justified theoretically.
It is not excluded that economic subjects could learn the system states from their own
practice with a higher precision than the precision of the statistics that mirror these
states. So magnitudes of (co-)variances in a lower range are relevant for other comput-
ing experiments outside the current presentation. Smaller weights in the definition of the
EKF pay-off would be especially reasonable if the quality of the data could be im-
proved.
The compact model with the identified quasi-optimal parameters is run for along term
forecast. This forecast projects the internal tendencies of capital accumulations into the
XXI century. Our forecasting uses no real data after 1991.
Scweppe wrote (Scweppe 1973: 254-5): “Pattern recognition can be viewed as a spe-
cial case of hypothesis testing. In pattem recognition, an observation z is to be used to
decide what pattern caused it. Each possible pattern can be viewed as one hypothesis.
The main problem in pattern recognition is the development of models for the z corre-
sponding to each pattern (hypothesis).” Our main hypothesis is that the real economy
exhibits the long wave as a dominant quasi-cycle. A subordinate hypothesis is that the
residuals are not white due to other quasi-cycles and non-random factors.
A Weak Behaviour Reproduction Test
According to (Sterman 1984: 52), “a historical fit of a model is a weak test... it is
nonetheless an extremely important one.” My model focuses on long-run cyclical
growth and explicitly excludes the business cycle in its original deterministic form.
Driving and measurement noise is the only exogenous input in the final non-linear sto-
chastic model. The behaviour of this stochastic model and its ability to replicate histori-
cal data, to capture the long wave and its tuming points are mostly the result of interac-
tion of the endogenous variables.
I have simulated the model for 1958-1991 with the Time Step of one year. This simu-
lation enables a comparison of the model output to the observed behaviour of the econ-
omy over the all 34 years. A simulation with the Time Step of 0.588893 (File 1.out)
calculates estimated states only at 29 time points and is omitted here therefore.
As expected, the residuals are not white and auto-correlated. Figure 1 shows the auto-
correlation plot for the variable u. The x-axis on this plot represents the number of lags,
where a lag is one SAVEPER (here one year). The X-axis starts at 0 where, by defini-
tion, the auto-correlation is 1. On the Figure 1, the auto-correlation exceeds 0.65 for the
lag of one year. There is also a high correlation between residuals for a(t) and data of
the employment ratio with the lag of one year v(t-1) as seen on the Figure 5.
The residuals are not due entirely, or largely, to pure random influences. On the con-
trary, these residuals contain highly systematic, non-random components. The pre-
dicted residuals are plotted against the independent variable on Figures 2-4. The sub-
script A stands for the actual data, S - for simulated. The dynamics of the residuals, i-e.,
one-step-ahead prediction errors, are determined at least partially by the shorter quasi-
periodic growth cycle(s), as expected. Thus although our model does not ‘pass’ the test
for whiteness of residuals, this ‘failure’ supports empirically long wave as the behav-
ioural pattern.
Tables 2 and 3, Figures 6 - 19 report on other aspects of the error analysis for the
seven variables. Our calculations use the Theil inequality statistics (Theil 1966, Sterman
1984). The root-mean-square percent error (RMSPE) provides a normalised measure of
the magnitude of the error:
[Lieie _ F
a AJIAT
where S; and A; are simulated and actual quantities, respectively.
0.65
0.3
0.05
0.4
0 25 5 75 10
Lag
Figure 1. Auto-correlation for the residuals for u
The mean-square-error (MSE) and inequality statistics provide a measure of the total
error and how it breaks down into bias, unequal variation, and unequal covariation
components:
= (S, -A,)? = (S A)? +(S, —S,)? +2(1—r)S5Sq
where 5 and A are means of S and A, i.e.
5 =") S, and
1 “
A=) A, tively;
aD , Tespectively,
v(t)
Figure 2. vy (t —1) versus the VENSIM 1 step.err for a(t), 1961-1972. Clockwise
0,002
1 step.err fora (t)
S
o &
ODL 0.92 093 at 0.95 0.06
-0,0005
vt-1)
Figure 3. vq (t —1) versus the VENSIM 1 step.err for a(t), 1972-1978. Clockwise
0.002
ans we’
1 step.er fora (t)
0
oj il 0.91 Bs 0.93 0.94
v(t - 1)
Figure 4. va (t —1) versus the VENSIM 1 step.err for a(t), 1979-1987. Clockwise
0.003 y= 0-122tx= 6-654
2
= 0.002 R 0.4622 -
co °
o ot
& 0.001
F j eheet
T y Py
5 0.02 . ° 9.06
-0.001 ve,
i?
-0.002 *
Simulated a (t-1)
Figure 5. ag (t —1) versus 1 step.err for a(t), 1959-1991
Sg and Sj equal the standard deviations of S and A, ie.,
= ys —K)? , respectively.
The term (S —A)? measures the bias between simulated and actual series, The term
(S, -S Ne is the component of the MSE due to a difference in the variances of simu-
lated and actual series, and measures the degree of unequal variation between the two
series. Finally, the term 2(1—r)S,S, is the component of the error due to incomplete
covariation between the two series.
By dividing each of the components of the error by the total MSE, the “inequality pro-
portions” are derived:
ys =_(S=A)
a 2”
FSA)
U § (Sg —S,)
T =
;D SM)
ye = 20 1)SsSa
1 2°
za -&)
By the definition of the correlation coefficient
COV(S, A) :
r =——_*~ , where the covariance
SsSq
= sya, —
COV(S,A) =F US, S)(A, -A)]
1 _ a
=F > (SA) S K (Sterman 1984: 63).
Of course, U M 4u$ 4y° =1,s0U M ,U | and U© reflect the fraction of the MSE
due to bias, unequal variance, and unequal covariance, respectively.
To compare the means of the simulated and observed behaviour patters, we compute
the percent error in the means E1 as
This quantity tells how large is a discrepancy between the means.
Similarly, the percent error in the variations is calculated as
In addition to the sample estimates E1 and E2, the amplitudes of the real and simulated
quasi-cycles will be measured and compared (Table 3). A summary measure of overall
behaviour discrepancy is finally the discrepancy coefficient, a version of Henry Theil’s
‘inequality coefficient’:
Ids -S -A, +A)?
a -A) +,) boxe -s)
“Ss A = s
which is insensitive to additive constants being between zero (‘perfect predictions’) and
one (‘worst predictions’).
Table 2. Error Analysis of the Model for 1958-91
Variable | RMSPE | MSE M s G S Ss.
(%) (units?) U U U A S
) (2) (3) (4) (5) (6) (7) (8)
1.37 [3.31E-07| 0.010 0.304 0.686 0.005 0.005
7.13 | 7.91E-07| 0.000 0.533 0.467 0.003 0.002
3.95 |6.16E-05| 0.001 0.059 0.940 0.076 0.078
3.65 0.006 0.026 0.156 0.818 0.149 0.118
1.89 [0.000180] 0.013 0.175 0.812 0.010 0.016
1.19 [0.000123] 0.045 0.006 0.949 0.015 0.016
28.76 | 2.83E-05| 0.000 0.400 0.600 0.008 0.004
Totals of the columns (4)-(6) may not add exactly to 1 due to rounding.
(
Ke |< |e] a|rolase
Table 3. Indicators of Discrepancy for 1958-91
Vari- Min Max Range El E2 U
able [real | simu- [real ]simu- | real | simu-
lated lated lated
0.031 | 0.031 | 0.050 | 0.051 | 0.019 | 0.020 | 0.001 | 0.062 | 0.054
0.008 | 0.009 | 0.018 | 0.018 | 0.01 [0.01 | 2E-04 | 0.211 | 0.161
0.119 | 0.109 | 0.375 | 0.375 | 0.256 | 0.266 | 0.001 | 0.025 | 0.051
0.685 | 0.683 | 0.731 | 0.728 | 0.047 | 0.045 | 0.002 | 0.564 | 0.523
0.903 | 0.919 | 0.965 | 0.969 | 0.062 | 0.050 | 0.003 | 0.056 | 0.352
a
e
f
Ss 1.845 | 1.910 | 2.417 | 2.297 | 0.573 | 0.387 | 0.006 | 0.208 | 0.289
u
v
y
0.006 | 0.008 | 0.046 | 0.023 | 0.041 | 0.014 | 0.001 | 0.435 | 0.439
The RMSPE of s is 3.65%. Three variables, u, v, a have the RMSPE under 2%. Excep-
tionally, the RMSPE of e is 7.13%, the RMSPE of y is 28.76%, whereas the RMSPE for
fis only 3.95%.
While the small total errors in most variables show the model satisfactorily tracks the
major variables, the two larger errors of e and y might raise questions about the internal
consistency of the model or the structure controlling those variables. A preliminary con-
clusion is that the model underrates changes of y, e but represents well the dynamic of
the natural capital-output ratio f.
0.055
0.055
0.0425
0.0425
0.03
0.03
|
aan
==
WAY
V7
1958 1966 1974 1982 1990
Time (Y ear)
a: ekf10-related-ekf-51aa-debug-on —_—
a: C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 6. ag (thick) versus ag (thin), 1958-1991
q
-0.01
-0.02 5
-0.03
0.03
0.02
0.01
\
0
A\ | Mee 4
8 A962 ig 1970 \1974 1978 roa Foes 1990)
vara
Figure 7. 2S 8A | 1958-1991
aa
SSURHEAL N
oy
Va
1958 1966 1974 1982 1990
Time (Y ear)
s: ekf10-related-ekf-51aa-debug-on —_
s: C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real-NNP-28-5-1
Figure 8. s, (thick) versus sg (thin), 1958-1991
0.05 | ——_ag —s
0.025 +
i> ae
4 oar
0.03 —To6e T9704 1976 T3pa] 1988
us NM i
Figure 9. SA, 1958-1991
A
0.74
0.74
0.71
0.71
0.68
0.68
W U
1958 1966 1974 1982 1990
Time (Y ear)
: ekf10-related-ekf-51aa-debug-on —
: C:\USA \e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 10. uy (thick) versus us (thin), 1958-1991
ees
8\ 60 pr 64 6ENB 70
-0.02
® ig ig ig W ig i
V
Ay [ws
] 919 19
‘2 74 76 78 80 82 84 86%88A90
Figure 11, "SA | 1958-1991
Ua
0.98
0.98 Lt
LAY
0.94 D J ANG OO
0.94
V
0.9
0.9
1958 1966 1974 1982 1990
Time (Y ear)
v : ekf10-related-ekf-51aa-debug-on —_
v: C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 12. vy (thick) versus vg (thin), 1958-1991
“OMe \ATo6d 1970 1976 82 198
Pred 6 Wer 1988" |
Figure 13, YS =YA 1958-1991
VA
The error decomposition gives additional insights. Consider e. Plotting the two series
and their residuals (Figures 14 and 15) shows that they diverge point-by-point. It causes
unequal covariation. However the tracking of declining e is not very bad. The bias is
negligible, unequal variation (nearly 53%) accounts for approximately one half of the
total while unequal covariation (nearly 47%) accounts roughly for the other half. More
importantly, actual e fluctuates with the shorter quasi-cycle(s) that are not captured by
the model.
The largest RMS percent error, nearly 29%, shows up in the y. Still the bias is negligi-
ble. The model reflects the declining tendency of y rather well. Error decomposition
shows the majority of MSE to be due to the unequal covariance (60%) and unequal
variance (40%). Additions (Y) to the stocks of subsoil minerals (F) experience more
violent changes than the stock itself and depletion (Z). Proven reserves quantities some-
times change dramatically because previously uncertain reserves are found to be eco-
nomic (e.g., Alaskan oil in 1970). The experts conclude (Nordhaus and Kokkelenberg
1999: 86): “The basic problem is valuing non-reserve resources. BEA intends ulti-
mately to include unproved resources as a part of non-produced environmental assets.”
As pointed out by the experts, the reported BEA’s initial estimates of the natural capi-
tal should be considered preliminary and tentative at this time (Nordhaus and Kok-
kelenberg 1999: 168). In the absence of observable market prices for reserves, BEA es-
timated mineral reserve and flow values using five valuation methods. BEA’s results
show clearly the potential margin for error among different techniques, for they give
widely different estimates. In some cases, the net change in the value of reserves even
has a different sign under different valuation techniques (Table 4). Therefore the data on
y, e and f are suffering from particularly high measurement errors.
Table 4. A Fragment of the IEESA Asset Account, 1987, for Subsoil Assets, the Highs
and Lows of the Range Based on Alternative Valuation Methods
Opening Change Closing
stocks, F(t- | Total, net Depletion, | Capital Revaluation | stocks, F(t)
1) degradation, | formation, | and other
Z(t) Y(t) changes
(1) (2) = (3) (4) (5) = (6)- (6)
(3) 4) +5) (1)+(3)-(4)
270.06 1067 | 57.86-116.6 | -16.76-61.6 | 16.66 64.6 586 -119.6 299.46 950.3
2]
Notice opposite signs in column 2, 5.
Source: Survey of Current Business, April 1994: 41.
0.018
0.018 AN
0.0127
0.0127
N
NY
0.0075
0.0075 N
1958 1966 1974 1982 1990
Time (Y ear)
e: ekf10-related-ekf-51aa-debug-on =
e: C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 14. e, (thick) versus eg (thin), 1958-1991
Figure 15, °S—®4 | 1958-1991
ea
0.048
0.048
0.0265
0.0265
Mamet
0.005 BP MANS, |_|
0.005 N Raa
1958 1966 1974 1982 1990
Time (Y ear)
y : ekf10-related-ekf-51aa-debug-on —
y : C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 16. y, (thick) versus ys (thin), 1958-1991
Figure 17, 9S YA 1958-1991
Ya
Besides the large measurement errors, unequal variance is likely due to a visible change
in ecological policy after 1970-1973. The policy was drastically directed to raise eco-
efficiency. Thus our postulate of the dynamic law for e is more appropriate for the pe-
tiod after 1970, while e = const would be more realistic for 1958-1970. A simplifying
assumption on investment policy in the natural capital could be partially responsible for
the large difference in variation of these observed and simulated data too.
It has been shown via Monte Carlo experiments (Barlas 1989: 72) that U can be as
large as 36% even if a model has a perfect structure with moderate noise. In the Barlas
experiments with ‘synthetic real systems’, E1 does not go beyond 5%, whereas E2 be-
comes as large as 30% and U changes between 35 and 70%. Experiments show that,
even for models with no systematic errors, U can be as large as 70% (Ibid.: 77-78).
The analogous indicators in Table 3 are closer the lower values of these bounds: E1
is less than one percent in all cases, E2 exceeds 30% only for y (44%) and u (56%),
while U is higher than 36% also only fory (44%) and u (52%). Still the unequal vari-
ance term (U °) for u is only about 17.5% with the bias term (U™ ) less than 1.5% (Ta-
ble 2). Thus the discrepancy between ug and u,is mostly due to fluctuations with
higher frequencies and divergence on a point-by-point basis.
0.39
0.39 iN
~
»
0.245 Ny N
0.245
SN
~
0.1 ~
01 Ss
1958 1966 1974 1982 1990
Time (Y ear)
f: ekf10-related-ekf-51aa-debug-on —______—
f: C:\USA\e-f-y-58-91-s-u-v-a-48-99-for-real- NNP-28-5-1
Figure 18. f, (thick) versus fg (thin), 1958-1991
0.05 A . ts
1958 1962 1966 ipo 1974 1978 ac 6 1990
-0.05
Figure 19. ‘ahs, 1958-1991
A
The extraordinary measurement errors for y, e and f preclude a substantial improving
of the model fit, particularly in respect to these variables. Still the errors of the retro-
spective forecast are substantially lower than the relative standard deviations of the
measurement errors and higher than the relative standard deviations of the equation er-
rors (Table 5). The reader is not to overlook that the same data have been used to esti-
mate the parameters and to test the historical fit.
Table 5. The Accuracy of the Retrospective Forecast for 1958-1991
Variable —‘ Innovation The Forecast. JQ. /A (%) Ry / A (%)
mean 6, relative emor ° :
JMSE/A (%)
a -0.0001 1.3 1.3 2.6
c we we 0 te
e 2.46e-006 6.7 2.7 17.1
f -0.000605495 3.6 0.014 14.4
8 -0.0139 3.8 1.5 7.3
u 0.0015 1.9 0.01 71
v -0.0025 1.2 0.08 7.0
y 3.89e-005 39.6 14.8 43.6
Note. The unit of measurement of the innovation mean coincides with that of the respective variable. The
suboptimal magnitudes of dispersion Qj; and Rj, are taken from File 1.out of the Appendix.
The exploratory scenario 1991-2107: a falling rate of capital accumulation
The characteristic of this first scenario is its opportunity rather than normative orienta-
tion. An extrapolation results from the compact model, probabilistically identified by
EKF. EKF has created a VENSIM data file with estimated values of the model vari-
ables and constants. The 4"-order Runge - Kutta integration with a variable step size has
been selected. The parameters from this file are used in a simulation with the state vec-
tor given by EKF for the year 1991: ay =0.0512, So =2.052, Cy =13.98, fy =0.1091, e
=0.0087, Yo =0.0083, Vo =0.9475, Uy =0.6948. The reader sees that these magnitudes
differ from the observation posted in the Table 1 for the year 1991. The growth rate of
the indicated natural capital i ~0.0374 <d ~0.0384 while 0, <0 (these parameters are
defined in the equations (21), (22) and (24)).
There is no locally unstable stationary state with a limit cycle nearby. The long wave is
perceived as a quasi-periodic trend and stochastic attractor.
Capitalists accumulate profit and expand output. The economic growth is punctuated
by recurrent slowdowns. New more advanced machinery is a source of increased labour
productivity. As wages increase, profits relatively decrease. The spiral of accumulation
is almost periodically arrested by the relative shortage of labour. A quasi-period of
fluctuations is about 29-33 years. Accelerated introduction of the living labour-saving
machinery, shedding of labour enables to overcome these temporally hindrances. The
current downswing in the long wave manifests itself in the growing produced capital-
output ratio and unit wage, declining profitability and employment ratio.
There is a secular profit squeeze and deceleration of economic growth in spite of
the steady reduction of the eco-intensity and labour productivity growth. Worsening
profits slow the growth in productivity that inhibits profits, in tum. The both profit rates
(1 - u - e)/(s +f) and (1- u - y)/s tend to be lower and lower than the benchmark d/k ~
0.144 (Figure 20).
0.14 4
0.135 dk
0124 -2-(1-u-y)/s
-3— (1-w-e)/(s+f)
0.11 4
7s
2000 2050 2100
Figure 20. The average profit rate, (1 - u - y)/s, and general profit rate, (1 - u - e)/(s +f),
telative to the benchmark (d/k) in the exploratory scenario
The net rent rate is negative over the long term. A depletion of the natural capital
slows the economic growth. The rate of economic growth deviates from the benchmark
(d) downward. The rate of net rent falls from zero to about -0.045, thatis lower then
the positive benchmark (d ~0.0384). The quasi-cyclical deceleration of the labour
productivity and real wages is relentless.
An univariate sensitivity analysis for 1991-2034 has shown that the long term busi-
ness upturn will not probably happen until 2012 or even 2018. It will proceed thereafter
up to the beginning of the next long term downtum in 2035-2040 (Ryzhenkov 2001).
The above scenario does not satisfy the all necessary conditions for sustainable de-
velopment: the society accumulates the produced capital; however its activity brings
about the excessive depletion of the natural capital. To avoid this, a more efficient
policy is necessary.
A real development will differ from the offered description because of remaining
specification and identification errors. Still the model parameters can be adjusted by
EKF and the forecast can be updated each period, based on new information.
The normative scenario 1991-2107: extending the natural capital
The second scenario corresponds to a rather strong criterion of sustainable development.
In particularly, the gross rent is increased step-wise in the year 2003:
y =(0,(c —f) +0, f)y +STEP(0.0018, 2003). (22a)
At the end of the year 2002 or beginning of 2003, y2993 ~0.00656; at the end of the year
2003 or beginning of 2004, yoo9q ~0.00757. This modification does not exclude other
possible alterations for achieving sustainable development. Still it addresses the critical
shortcoming of the exploratory scenario, namely the depletion of the natural capital.
In the normative scenario, the economic growth is quasi-cyclical with a period of
about 31-33 years. The maximum employment is firstly achieved in the year 1999, it
declines thereafter until the year 2011, then growth again until the year 2028.
The increase in the unit gross rent is achieved by a reduction of the unit wage by about
the same quantity. Yet the labour productivity and real wage of an employee increase
faster than in the previous (exploratory) scenario. The natural capital-output ratio de-
clines, while the total capital-output ratio goes up (see for details Ryzhenkov 2001).
Figure 21. The rates of growth of net output (P ), growth rate of natural capital (F )
versus the benchmark (d) in the normative scenario
It is reasonable to stress that the substitution of the Eq. (22) by the Eq. (22a) is the sin-
gle alteration of the above compact model. Only 0.18 per cent of the NNP is invested
additionally in the natural capital each year since 2003. Still this partial redistribution
of the NNP produces desirable positive effects over the whole period on the average:
™@ the rate of the economic growth rate is increased (Figure 21);
@ the natural capital is extended;
™@ the average and general rates of profit are raised without any apparent tendency to
fall;
™@ there are gains in the employment rate.
Conclusion
The suggested conscious alteration of the societal relationship with the natural capital is
probably pivotal for converting the unsustainable evolution of the exploratory scenario
into the strongly sustainable development of the normative scenario.
A drop in the unit wage is the condition for the higher employment ratio and profit
rates, growing natural capital in the normative scenario. A complete win-win solution
with an increased unit wage would require a substantial reduction in the produced
capital-output ratio (s). Historically, lifting this ratio is a leverage against an exces-
sive growth of the real wage. New patterns of the production relations and new tech-
nologies are prerequisite for greater equity.
The present downturn in the big quasi-cycle of conjuncture is not only a regularly re-
current phase of the long wave that strengthens the tendency of the profit rates to fall.
Additional pains of this downturn are characteristic of childbirth of the natural capital-
ism. The system dynamics approach could be helpful for shortening and lessening dis-
order and distress of this evolution. There is a potential for a greater social advantage of
the positive feedback between sustainable development and unbiased profitability, re-
vealed in this paper.
The long wave is comprehended as the stochastic attractor and non-equilibrium quasi-
periodic trend. The above analysis strengthens confidence in the presented model.
Appendix (The real data and sources)
Table 1. The author's estimates for U.S. real macroeconomic data, 1958-1999
£ u s v a y e
1958 0.37537 0.71458 2.16474 0.93162 0.03149 0.02143 0.01768
1959 0.35589 0.70956 2.05753 0.94542 0.03312 0.025 0.01728
1960 0.34513 0.71619 2.02956 0.94438 0.03333 0.01489 0.01712
1961 0.34072 0.71012 2.03005 0.93321 0.03413 0.02047 0.01703
1962 0.32338 0.70172 1.96848 0.94452 0.03579 0.01964 0.01646
1963 0.31121 0.69675 1.93987 0.94357 0.03679 0.01788 0.01647
1964 0.29774 0.69298 1.89615 0.94833 0.03813 0.0202 0.01595
1965 0.28434 0.68468 1.86284 0.95494 0.03962 0.02035 0.01545
1966 0.26946 0.68737 1.84462 0.96215 0.04117 0.01802 0.0154
1967 0.26558 0.68933 1.88948 0.96158 0.04122 0.01777 0.01591
1968 0.25116 0.69404 1.888 0.96446 0.04231 0.01337 0.01574
1969 0.2379 0.70784 1.92997 0.96493 0.04238 0.00928 0.01581
1970 0.26829 0.71733 2.03042 0.95015 0.04184 0.04639 0.01663
1971 0.25499 0.70482 2.04807 0.94048 0.04288 0.01131 0.01589
1972 0.23468 0.70488 2.03457 0.94396 0.04371 0.00819 0.01517
1973 0.21493 0.70169 2.03485 0.95124 0.04485 0.00813 0.01407
1974 0.21282 0.71508 2.29921 0.94375 0.04351 0.0094 0.01367
1975 0.20938 0.7058 2.30697 0.91533 0.04352 0.0074 0.0132
1976 0.19149 0.70207 2.22419 0.92306 0.04461 0.00623 0.01232
1977 0.18417 0.70181 2.17677 0.92962 0.04509 0.01332 0.01185
1978 0.17141 0.70391 2.15084 0.93948 0.0456 0.00845 0.01154
1979 0.16713 0.70826 2.20955 0.94155 0.04575 0.01244 0.01136
1980 0.17061 0.7217 2.3441 0.92829 0.04513 0.0136 0.01158
1981 0.16563 0.71519 2.35172 0.92384 0.04554 0.00966 0.01126
1982 0.16738 0.73149 2.41718 0.90281 0.04462 0.00815 0.01132
1983 0.1583 0.71291 2.27873 0.90411 0.04595 0.00842 0.01053
1984 0.14766 0.70398 2.14105 0.92487 0.04743 0.01072 0.01037
1985 0.14381 0.70729 2.10606 0.928 0.04797 0.01065 0.00991
1986 0.13807 0.71158 2.10566 0.93004 0.04827 0.00776 0.0094
1987 0.13395 0.71733 2.08067 0.93814 0.04856 0.00921 0.00903
1988 0.12582 0.71845 2.04407 0.94496 0.04952 0.00607 0.00871
1989 0.12148 0.70831 2.01227 0.94733 0.05011 0.00786 0.00823
1990 0.12008 0.71042 1.99851 0.9439 0.05037 0.00879 0.00808
1991 0.11898 0.71355 1.99378 0.93162 0.05028 0.00572 0.00815
1992 «0.71058 1.95891 0.92497 0.05132
1993 «0.70712 1.94959 0.9309 0.05196
1994 «0.70476 1.94537 0.93914 0.0526
1995 «0.70051 1.94455 0.94402 0.05325
1996 «0.69304 1.92755 0.94602 0.05426
1997 .. 0.68867 1.89862 0.95064 0.05524
1998 «0.69822 1.88958 0.95494 0.05656
1999 ws oe «0.95782 0.05769
Units of measurement: u, v, e andy [dimensionless], c, f and s [years], a [ billions of
chained 1996 dollars per 1000 civil persons employed per year]. In calculating fand a,
constant 1996 dollars are used for the nominators and denominators; calculations of u
and s are done with the nominators and denominators valued in current prices. The em-
ployment ratio v is for the civil labour force.
This paper assumes that the annual value of the labour force of a proprietor is equal to
the annual earnings of a hired employee. Calculating the relative labour income (u) re-
quires two steps:
1) estimating the ratio of the total labour force to employees M = (CLF + AFP)/(CLF +
AFP - SE), where CLF is the civil labour force, AFP is Armed Force Population, SE is a
number of self-employed in all industries; this account does not covers the part of the
defence related personnel outside the AFP because of incompleteness of the official
statistics available for the author;
2) estimating the labour share u = (W+S+NWC)*M/NNP, where W, S, NWC are accru-
als for wage and salary income and disbursements for other labour income, NNP is net
national product.
The natural capital is represented only by the value of _ the closing stocks of proved
mineral reserves (Survey of Current Business, April 1994: 58-60). Current rent method I
and current rent method II are net present value methods based on the Hotelling valua-
tion principle. The stock valuations, given by the BEA current rent method I, in bil-
lions of 1987 dollars have been converted into billions of 1996 dollars through multi-
plication by the NNP price index for 1987-1996. This index is calculated as (NNP
1987, billions of 1996 dollars)/(NNP 1987, billions of 1987 dollars) = 5460/4029 =
1,355,
The book (Nordhaus and Kokkelenberg 1999: 59-105) and discussion paper (Ryz-
henkov 2001) offer explanations of an upward bias in the mineral-resource values cal-
culated with the Hotelling valuation principle, especially if the BEA current rent method
II is applied. To avoid this bias as much as possible, this paper does not use the stock
valuations based on the latter. Still due to the usage of the Hotelling valuation technique
by the BEA, the current rent method I gives biased estimates too.
The new BEA estimates of non-residential fixed assets for 1998 and revised estimates
for 1958-1997 are used in calculating produced capital - output ratio (Survey of Current
Business, April 2000). The other data are compiled from the site
http:/www.economagic.com and issues of “Statistical Abstract of the USA” for 1999
and 2000 published by the U.S. Department of Commerce.
The VENSIM optimisation files
The following specifications of the noise and optimisation pay-off are in agreement
with the VENSIM format:
pay-off.vpd
a/meas a variance
e/meas e variance
f/meas f variance
s/meas s variance
u/meas u variance
v/meas v variance
y/meas y variance
kalman.prm
a/ a dr variance/a ISC
c/c dr variance/c ISC
e/e dr variance/ e ISC
f/f dr variance/f ISC
s/s dr variance/S ISC
wu dr variance/u ISC
viv dr variance/v ISC
yly dr variance/y ISC
All variances from the file kalman.prm and variances of the measurement noise (pay-
off.vpd) for the same seven state variables have been included in the list of parameters
to be estimated.
File 1.out
:COMSY S After 2123 simulations
:COMSY S Best payoff is 930.511
:COMSY S User terminated multiple search session
:OPTIMISER=Powell
:SENSITIVITY =Payoff Value
:MULTIPLE_START=Random
:RANDOM_NUMER=Linear
:OUTPUT_LEVEL=2
:TRACE=2
:MAX_ITERATIONS=10000
:PASS_LIMIT=2
:FRACTIONAL_TOLERANCE=0.0003
:TOLERANCE_MULTIPLIER=21
:ABSOLUTE_TOLERANCE=1
:SCALE ABSOLUTE=1
:VECTOR_POINTS=1.24418e-306
0 <=b =0.621019 <=1
= 13.2528
= 0.0054249
0 <=g =0.0531989 <=1.5
0 <=m1 =0.0149761 <=0.02
0 <=m3 =0.0105916 <=0.1
0 <=m5 =0.0888489 <=0.3
=0.0199143 <=0.022
= 0.353022 <=0.5
=0.5 <=0.5
=0.0106352 <=1
0 <=r=0.0609304
0.001 <=fISC =0.001 <=0.004
0.001 <= MEAS f VARIANCE =0.001 <=0.004
0.0025 <= MEAS u VARIANCE =0.0025 <=0.01
0.0025 <=uISC =0.0025 <=0.01
0.0043 <= MEAS v VARIANCE =0.0043 <=0.0174
0.00435 <=v ISC =0.00435 <=0.0174
0.0234 <= MEAS s VARIANCE =0.0234 <=0.0936
0.0234 <=s ISC =0.0234 <=0.0936
0.05 <=j =0.211049 <=1
0.1 <=m2=0.1 <=0.75
0.125 <= TIME STEP =0.588893 <=3
0.2 <=k =0.267234 <=0.5
0.75 <=ve =0.92536 <=0.99
1.25e-006 <=aISC =1.25e-006 <=5e-006
1.25e-006 <= MEAS a VARIANCE =1.25e-006 <=5e-006
1e-009 <= aDR VARIANCE =2.93202e-007 <=0.001
cDR VARIANCE =0
cISC =1
1e-009 <=e DR VARIANCE =1.22235e-007 <=1e-006
1e-009 <=f DR VARIANCE = 1e-009 <=0.001
1e-009 <=s DR VARIANCE =0.000929877 <=0.01
1e-009 <=u DR VARIANCE =5.51477e-009 <=0.001
1e-009 <=v DR VARIANCE =5.16068e-007 <=0.001
1e-009 <=y DR VARIANCE =3.9415e-006 <=2e-005
5e-006 <=e ISC =5e-006 <=2e-005
5e-006 <= MEAS e VARIANCE =5e-006 <= 0.0002
5e-006 <= MEAS y VARIANCE = 3.41319e-005 <= 0.0002
5e-006 <=y ISC =5e-006 <=2e-005
i =0.0373606
nl =-0.24223 <=0.02
ol =-0.0299728
02 =-9,93389
q =-0.0084833
Notice: The File 1 contains the best payoff so far, the reason the optimiser stopped, and
the values of the search parameters needed to achieve that payoff. It has been assumed
additionally that c ISC equals one, c dr variance equals zero.
References
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