Controlling Employment, Profitability and Proved Non-Renewable Reserves
in a Theoretical Model of the U.S. Economy
Alexander V. RYZHENKOV
Institute for Economics & Industrial Engineering
Siberian Branch of Russian Academy of Sciences
17 Academician Lavrentiev Avenue
Novosibirsk, 630090, Russia
Fax: +007 383 330 25 80
E-mail: ryzhenko@ieie.nsc.ru
Abstract
This paper elaborates the original theoretical law of capital accumulation that (in its initial or transformed
form) gives birth to scenarios of macroeconomic evolution. The state variables are relative labour compensa-
tion, employment ratio, gross unit rent, produced capital-output ratio, proved non-renewable reserves-output
ratio, desired proved non-renewable reserves-output ratio, and depletion of non-renewable reserves per unit
of net output. This law was presented in the author’s earlier books and papers.
Worsening profitability and declining employment ratio over the long term characterise the inertia scenario
for the U.S. economy over 1991-2107. Excessive depletion of proved non-renewable reserves contributes to
these disadvantageous tendencies.
This paper demonstrates how a forward-looking investment policy based on proportional and derivative
control over proved non-renewable reserves brings about their extension, raises profitability and employment
over the long term in the first normative scenario in comparison with the inertia scenario.
Long waves with ups and downs of substantial amplitude in profitability, in employment ratio and in other
indicators represent the shortcomings of this normative scenario. These shortcomings necessitate elaboration
of closed-loop control over total profit and employment ratio in addition to closed-loop control over proved
non-renewable reserves.
The initial theoretical law is transformed into control law of capital accumulation. The present operational
application of this control law to the U.S. economy reveals and explains substantial lasting improvements in
profitability, employment ratio, labour productivity and real labour compensation in the second normative
scenario compared with the first normative scenario that are achieved together with extending proved non-
renewable reserves.
Keywords: Capital accumulation, proved reserves, long wave, closed-loop control, excess income levy, the
U.S. economy
Introduction
Strengthening the system dynamics paradigm necessitates building its solid core (Forrester, 1983). Adher-
ence to this core building motivated the earlier books and papers that introduced and refined a first hypotheti-
cal Law (HL1) of capital accumulation constrained by natural capital (Ryzhenkov 2000, 2001, 2002, 2003).
Its state variables are the relative labour compensation, employment ratio, gross unit rent, produced capital-
output ratio, natural capital-output ratio, indicated natural capital-output ratio, and expense of the natural
capital per unit of net output.
An application of HL1 in its probabilistic form together with extended Kalman filtering to the U.S. mac-
roeconomic data 1958-1991 identified unobservable components of this law. In this empirical application the
notion of natural capital was narrowed to proved mineral reserves since the available statistical sources did
not provide consistent data on other kinds of natural capital.' Despite these limitations, it was shown that
long wave was a dominant non-equilibrium quasi-periodic behavioural pattern of the U.S. capital accumula-
tion dependent on investment policy in proved non-renewable reserves.
The previous paper (Ryzhenkov 2005) has unveiled and explained the paradox that profit is the decisive
factor of long waves under capitalism and could be the key for smoothing them. That paper has demonstrated
that a more efficient social control over the oscillatory macroeconomic system requires a substantial reshap-
ing of primary income distribution that takes into explicit account this dual characteristic of profit. The pre-
sent paper elaborates this duality by introducing excess labour compensation levy and subsidy on pre-levy
primary profit.
The above mentioned previous paper has defined a second hypothetical Law (HL2) of capital abstracting
from explicit constraints of natural resources. The main state variables are the labour productivity, relative
labour compensation, employment ratio, and capital-output ratio; a comprehensive Phillips equation governs
a rate of growth of real labour compensation. An application of HL2 in its probabilistic form together with an
' The terms reserve(s), proved reserve(s), total resource(s) are in agreement with the UN Framework Clas-
sification for mineral reserves/resources. See ENERGY/WP.1/R.70 ECE-UN document [17 February 1997]:
United Nations International Framework Classification for Reserves/Resources — Solid Fuels and Mineral
Commodities: Geneva.
extended Kalman filtering to the U.S. macroeconomic data covering 1969-2002 and computer simulation
runs have shown that long wave resulted from the socio-economic relations was a viable pattern of capital
accumulation. HL2, unlike HL1, includes a hypothetical partial law for the labour supply as a non-linear
function of capital intensity. Although this partial law is not a necessary condition for long waves, it helps to
portray them more accurately.
The characteristic of the inertia scenario based on HL2 in its deterministic form is a strengthening of the
secular tendency of the general profit rate to fall. This is not accepted by the U.S. state and business leader-
ship pursuing a pro-growth stabilization policy at least since 2001.
This previous paper (Ryzhenkov 2005) has uncovered long-term advantages of the overt closed-loop control
over total profit depending on the difference between desired and current employment ratios. Based on ex-
amination of causal linkages, the supposed control law (CL) of primary distribution of income is derived as a
modification of the non-probabilistic form of HL2. The new equation, representing feed-forward control,
substitutes the comprehensive Phillips equation for real labour compensation of the initial HL2. A controlled
transition to a non-trivial stationary state defined explicitly alleviates the tendency of general profit rate to
fall, maintains deliberately high employment ratio and upholds total profit.
The comparison of the basal scenario (based on CL) to the inertia scenario (based on HL2) has demon-
strated that the supposed closed-loop control could bring about, each year, additionally | trillion 1996 dol-
lars of profit and 5 million of jobs in the U.S. (on the average for 2001-2057). This earlier paper has shown
robustness of the proposed CL as well.
The purpose of the current paper is a synthesis of the CL and HL] based on their deterministic forms.
The focus is again on pro-growth stabilization policy that brings about shifts in distribution of net value
output between the two main social classes under an explicit constraint of proved mineral reserves. The
restricted presentation of U.S. natural capital by proved mineral reserves up to the year 1991 only is due
to well-known limitations of the BEA statistics (Survey of Current Business, April 1994).* Although these
data are somehow aged in the mean time, their current usage is acceptable for achieving exclusively
methodological purposes of this paper.
> The BEA estimated value of resource reserves and changes in reserves for the period 1958-91 for major
mineral resources. The minerals valued include the fuels (petroleum, natural gas, coal, and uranium), the
metals (iron ore, copper, lead, zinc, gold, silver, and molybdenum), and other minerals (phosphate rock, sul-
fur, boron, diatomite, gypsum, and potash). Petroleum and gas account for the lion's share of mineral produc-
tion in the USA.
The present author agrees with other experts that informed planning and decisions concerning sustainability
and resource development require a long-term perspective and an integrated approach to land-use, resource,
and environmental management worldwide. This paper offers important generic building blocks for imple-
menting this integrated approach that can be realized by international co-operative efforts.
1. The main model assumptions
This paper considers the American economy as restricted to a certain degree by proved mineral reserves
while total (non-renewable) mineral resources are not explicit constraints. Import from the rest of the world
can substitute extracting non-renewable resources in the USA at least partially. It is assumed in this paper
additionally that renewable and environmental resources do not constrain production explicitly.
Produced capital (fixed assets) and proved mineral reserves are essentially complementary to each other
and are also substitutes to some degree depending on the factors distributive shares and employment ratio.
These two different kinds of use values are measured in the same monetary units as the net output (in con-
stant U.S. 1996 dollars).
The other important premises are such:
(1) two social classes (capitalists and labourers); the state enforces property rights, yet the cost of such an
enforcement is not treated explicitly;
(2) three factors of production — labour force, produced capital (fixed assets), proved mineral reserves —
are homogenous and non-specific;
(3) only one aggregated good is produced for consumption, investment and circulation, its price is identi-
cally one;
(4) production (supply) equals effective demand;
(5) all labour compensation consumed; the gross mineral rent and a part of profit saved and invested;
(6) steady growth in the labour force that is necessarily not fully employed;
(7) a growth rate of real labour compensation 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 in-
novation over time;
(9) a qualification of the labour force corresponds to technological requirements.
The 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 produc-
tion. It may be a key to explanation of the fact that the rate of profit on capital of order of 12 or 15 per cent
per annum is compatible with a rate of economic growth of two or three and half per cent per annum.
4
The assumption (6) means that the labour force grows exponentially over time. This assumption may be
substituted by an assumption of an asymptotic growth or by another hypothesis (Ryzhenkov 2005).
The focus of research is on pro-growth stabilization policy that brings about shifts in primary income
distribution between two main social classes.’ This research envisages in particular reshaping of pri-
mary distribution of income that may enhance sustainable development.
2. The detailed assumptions and 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 of the (implicitly open) U.S. economy consists of the following equations:
P=Kis; qd)
a=PIL; (2)
u=wila; (3)
=m, + m,KIL + m,y(0)+ m,F7 L, (4)
where y(*)= sign(?)|0
‘,m, 20,12 m, >0,m,>0,m, > 0,0</;
KIL=ntnu t+n(v-v)+n,Z/P, (5)
where n, 20,1,20,n,20, 1> v. > 0;
v= LIN; (6)
N=Ne", n=const = 0, N, >0; (7)
w= —g+rv+bK/L+ gF/L, g>0,r>0; (8)
P=C+K+Y=wlh+(1-)M+K +Y; (9)
F=Y-Z; (10)
> Different types of property income payable out of the value added created by production (profit be-
fore taxes, interest, resource rent) belong to primary income of owners of capital (capitalists). Primary
income of labour consists mainly of wages and salaries of employees, employers’ social contribution
and imputed labour income of self-employed. See for details (Commission of the European Communities
— Eurostat et al., 1993: 157-182).
Only net value added created by production is considered in this paper. The primary income of gov-
ernment is not treated separately, having been included in property income on this stage of research.
Z= eP, 0<e <1; (1)
v=[o(e-f)+ox/ly, y= YIP 20; (12)
X =i; (13)
S=FIP, f >0; (14)
c=X/P,c>0; (15)
é=Ple,/e-1), e> e>0; (16)
K =kM=K{(1 —wla)P- YJ=k{(1-—w)P- YI, (17)
where 0<k<1.
Equation (1) postulates a technical relation between the produced capital stock (K) and net output (P). The
(produced) capital-output ratio is denoted by s. Equation (2) relates labour productivity (a), net output (P)
and labour input or employment (Z). Equation (3) describes the labour compensation share in net output or
relative labour compensation (w).*
Equation (4) is an extended technical progress function. The rate of change of produced capital intensity,
KIL, the direct scale effect, m,y(>), and the rate of change of natural capital intensity, F/L, are factors de-
termining the rate of change of labour productivity.
Equation (6) defines the employment ratio (v) as a result of the buying and selling of labour-power. Labour
force grows exponentially in (7). In the equation (8), the rate of change of the labour compensation rate (w)
depends on the employment ratio (v), as in the usual Phillips relation, and on the rates of change of produced
4 The reader may be well informed that payroll taxes belong to other taxes on production and imports; they
do not include compulsory social security contributions paid by employers or any taxes paid by the employ-
ees themselves out of their wages or salaries (Commission of the European Communities — Eurostat et al.,
1993: 172). Among other subsidies on production there are subsidies on payroll or workforce, these may con-
sist of subsidies payable on the total wage or salary bill (Ibid.: 174). Net receipts from taxes on production
and imports are treated as primary incomes of governments (Ibid.: 157). Economists generally believe the
employer tax is ultimately paid by labourers in the form of relatively lower wages (Economic Report of the
President, 2007: 90).
A part of payroll taxes may be used for providing health services (for example, Medicare spending in the
USA is financed by a combination of payroll taxes, general revenue, and premiums paid by beneficiaries).
These and similar processes that do not belong to primary distribution of income are not treated explicitly in
this paper.
6
capital intensity (K/L) and of natural capital intensity (F/L), additionally. The produced capital intensity (K/L)
is a proxy for qualification of the labourers.
In the equation (9), final private and public consumption is C = Plu + (1— k)(1—u — y)]. The net formation
of produced fixed capital is K = kM, where K is produced fixed assets.
The gross accumulation of proved mineral reserves Y equals the gross mineral resource rent measured in
net output units.> The letter y stands for the gross mineral unit rent.
There is a potential threat of complete depletion of remaining mineral resource (a sum of proven reserves
and undiscovered resource, in Sterman’s paper terminology).° With attentiveness to this caveat, this paper
assumes that in next hundred years the remaining aggregated mineral resource in the USA (and in the world)
will not be depleted wholly and that it will not become uneconomical. The equation (12) requires, in other
words, non-zero remaining mineral resources that are of economic interest or potentially economic.
Profit is defined as a residual (M = (1 — u — y)P). Equations (9) and (17) show that profit and incremental
produced capital (K) are not equal in monetary terms if the investment share k < 1. Considering the latter as
a variable and reflecting the labourers saving is left for a future research.
In the equation (10), F' is a net accumulation (loss) of the proved mineral reserves (F). In this equation, Z
is the depletion of proved mineral reserves — the decline in the stock (F) associated with extraction in the cur-
rent period. This outflow is mitigated by inflow Y that represents additions to the proved mineral reserves due
to improvements in recovery techniques, as well as owing to investment in resource exploration and devel-
opment in the current period.
* Constant 1996 U.S. dollars are used to measure ratios of extraction from and addition to proved reserves
to net output for each year. These ratios are e and y respectively.
® According to that paper (Sterman, 2002: 514): “As exploration adds to the stock of proven reserves, the
stock of undiscovered resource falls. Ceteris paribus, the smaller the stock of resources remaining to be dis-
covered, the lower the productivity of exploration activity must be (on average), and the smaller the rate of
addition to proven reserves will be for any investment rate. In the limit, if the stock of undiscovered resource
fell to zero, the rate of additions to proven reserves would necessarily fall to zero...How large the resource
base is, what the costs of backstop technologies are, and whether a backstop technology can be developed
before depletion constrains extraction and reduces economic welfare are empirical questions, not matters of
faith. The very possibility that depletion might matter cannot be assumed away, to be made untestable with
models in which resources are assumed infinite, the price system always functions perfectly, delays are short
and technology provides backstops at low cost.”
7
The rate of change of the capital intensity (K/L) in the equation (5) is a function of the following factors:
relative labour compensation (u), the difference between real employment ratio and some base magnitude
(v—v.), depletion of proved reserves in relation to net output (Z/P). The rate of growth of produced capital
intensity depends positively on the relative depletion of proved mineral reserves (an application of the princi-
ple 'a pollution prevention pays’), in particular.’ A high labour compensation share and high employment ra-
tio foster mechanisation (automation) as well.
The desired proved mineral reserves (X) may remain constant, decrease or increase exponentially in the
equation (13). The equation (12) defines an investment policy that is aimed to develop the proved mineral
reserves in accordance with the desired proved mineral reserves. The proved mineral reserves-output ratios —
real (f) and desired (c) — belong to the state variables in the equations (14) and (15). A combination of pro-
portional and derivative control over this form of natural capital is attainable hereby if the first parameter is
positive (0, > 0), whereas the second parameter in this equation is negative (0, <0). It is likely that this a
priory expected combination has been absent in the reality on the macro scale (see section 4 below) possibly
due to bounded rationality of economic subjects and (or) for another reason.
lL assume that the relative depletion of aggregated proved mineral reserves (e) asymptotically declines to the
minimal level (e, = const > 0) due to substitution and structural change according to the equation (16) where
é <0 for P> 0. Obviously, the higher the rate of economic growth and the lower the minimal depletion of
proved mineral reserves per unit of net output attainable in the USA, the faster proved mineral reserves are
relative depleted. In practice, for particular natural resources (for example crude oil) the relation 4 >P is
possible, still it is assumed that for the aggregated proved mineral reserves the relation |é| < P is valid.
The equation (16) is a better approximation than e = const > 0. Another approximation could be easily im-
plemented. A tendency to greater substitution of mineral extraction by import is modelled in extreme condi-
tion tests for the U.S. economy (see section 4 below again).
Three profit rates are defined for this economy. The first is the average rate of return on produced capital
(1 —u—y)/s. The second is the general one; it measures a ratio of the economic surplus to the sum of pro-
duced capital and natural capital in the form of proved mineral reserves (1 — u — e)/(s +f). The third is a bi-
ased profit rate (1 — u)/s that is more easily calculated based on commonly available statistics.
7 Extraction of minerals brings about directly or indirectly environmental damage of different kinds that is
still not modeled explicitly. The relative depletion of proved mineral reserves (e) is the quantity of depleted
proved mineral reserves per unit of net output.
8
The rate of net mineral rent is (y — e)/f. The general rate of profit is a weighted average of the rate of re-
turn on produced capital and the rate of net mineral rent: (1 — u — e)/(s + f) = [ss + JA —u —y)/s +[f(s +
DIY- Wf.
3. The first hypothetic law in a compact form (HL1)
In a compact form, the model consists of the seven non-linear ordinary differential equations (18)—(24) that
define the hypothetical Law of capital accumulation (AL1):
- {m,+ (m+ m,—1)[n, + nyu + n(v— ve) + ne] + my) + m,f 3s, (18)
5
l-u-y
v={k —[n, + nut nv.) +n.e]—n}v, (19)
a= {-gtrv—m,+(b+q-m, —m,)[n, +nu+n,(v—v) + ne]
—m,y(0)+ (q=m,) (fF -8)}u, (20)
f= (em)? — m, m,n, + nu + nv —v.) + nel-m, yO)
—(-m 8+) ¥f 21)
ee (22)
H=[o(e-f)+ox/ly, (23)
y
+m, +(m,+m,— I)[n, + nwt n,(v— ve) + ne]
+m,y(o) +m(f-§)} (ee), (24)
wheres >0,12v>0,12u>0, f>0,c>0,1>y20,1>e2 0.8 The requirement for denominators to be
positive is skipped. IfK >0,F >0 for an every instant of time, the system (18)-(24) defines a strongly sus-
tainable development.
There is a continuum of the non-trivial stationary states in this model with the same positive relative labour
compensation, employment ratio and relative depletion of proved reserves. A particular stationary state is
defined as
8 For details see (Ryzhenkov 2003).
E,=(SasVa> Yas fas Car Yara)» (25)
where
v, =[g+U-b-gyd-n)yr,
u, = [d-n-n, nv, — ve) — e, n,n,
f, = -u,- e, Wd- s,/k,
Ca = Sus
Va= eq +d fa,
e, =e
a P
i=d.
The produced capital stock, proved mineral reserves and net output increase thereby at the rate
K,=F,=%,=3, =d
that is equal to the net mineral rent rate and less than (for 4 < 1) or equal to the average profit rate (for k = 1).
The stationary average profit rate equals (1 — u, — y,)/s, = d/k. The both quantities (d and d/k) are used as
benchmarks below. The stationary produced-capital output ratio s, > 0 in the equation (25) is chosen from
an infinite number of possible positive magnitudes, such that stationary proved reserves-output ratio remains
also positive ( f,, > 0).
An extended Kalman filtering
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 un-
known parameters of the model, it provides a one-step-ahead prediction error with its covariance matrix. The
prediction error decomposition of the likelihood function utilises this information. The Vensim professional
soft-ware has served for performing such an extended Kalman filtering (EKF) in previous papers (Ryz-
henkov 2001, 2002, 2005) that report about quasi-optimal estimates obtained.
The data from the official U.S. sources fed the model (1)-(17) for the period 1958-1991. The information
on natural capital relates to a significant part of natural capital, namely the proved mineral reserves.
EKF realised in the VENSIM software has enabled to estimate the unobservable components of HL1. The
magnitudes of the identified parameters and components of the state vector for the year 1991 follow:
b = 0.621, d = 0.0384, e, = 0.0054, g = 0.0532, i = 0.0374, j = 0.211, k = 0.267, m, = 0.0145, m, =
0.100, m; = 0.011,m; = 0.089, n = 0.020, n, =-0.242,n, = 0.353,n; = 0.5,n; = 0.011,0, =—0.030,
10
0, = 9.934, q = -0.008, r = 0.061, v, = 0.925; a) = 0.0512, s) ~ 2.052, cy = 13.98, f,~ 0.1091, e, =
0.0087, y, = 0.0083, v, ~ 0.9475, u,~ 0.6948.
The stationary state E,, defined by the equation (25) with almost full employment in a literal sense (v, =
0.991) is not stable, unlike a full employment stationary state in the well-known Solow neoclassical model.
The simulation run based on the HL1 produces an inertia scenario. The magnitudes of the parameters and
components of the state vector are taken in this run as presented above with only two exceptions; the new
exogenous growth rate of labour force is the observed average for 1990-2004: n = 0.011 (Board of Trustees
2006), the new exogenous growth rate of desired proved mineral reserves is set equal to the new stationary
growth rate of net output (i = d ~ 0.0295) that is lower than before.”
This scenario projects the internal tendencies of capital accumulations into the XXI century. It is explora-
tory rather than normative.
4. The inertia scenario based on HL1: a falling rate of capital accumulation
Capitalists accumulate profit and expand output. New more advanced machinery is a main source of in-
creased labour productivity. The economic growth is punctuated by recurrent slowdowns. A quasi—period of
fluctuations is about 37 years. The rate of produced capital accumulation is repeatedly slowed down by the
relative shortage of labour. Accelerated introduction of the living labour-saving machinery, shedding of la-
bour enables to overcome these temporally hindrances.
A long term decline in e and f reflects several factors, including the economy’s increased reliance on for-
eign resources and the increased efficiency in the use of fuels and other minerals. Still there is a visible secu-
lar tendency of the average profit rate (Figures 1 and 3) and employment ratio to fall (Figure 4) accompanied
by a similar tendency of growth rate of labour productivity (Figure 2). The net mineral rent rate is negative
over the long term (Figure 1). The rate of net mineral rent falls from zero to —0.079, that is substantially
lower then the positive benchmark (d). This and all other Figures in this paper relate exclusively to the U.S.
economy.
° Under the familiar (intermediate) assumptions, the labor force is projected to increase in the U.S. by about
0.9 percent per year, on average, 2005 through 2015. Thereafter, the labor force is projected to increase much
more slowly, averaging 0.5 percent over the 2015 to 2020 period, and 0.3 percent over the remainder of the
75-year projection period up to 2080 (Board of Trustees 2006). A subsequent work could easily apply these
estimations in modified scenarios.
11
Worsening profitability obstructs the growth in productivity that inhibits profitability, in turn. 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 = 0.1103;
the growth rate of net output deviates from the benchmark (d ~ 0.0295) downward while the growth rate of
labour productivity drops below its benchmark (d —n ~ 0.0185).
0.2
0.15
ee
crammed
0.05 general profit rate
average profit rate
(y - ef
0 r ; T ; : :
- 2 20 2S BRH MK“ K MAH rR MDW
> o-S8 ss BSSSsesssessgassge
rr AN aAAA A qa ANINnan
-0.05
-0.1
Figure | The average profit rate (1 — uv —)/s, general profit rate (1 — u — e)/(s +f)
versus the benchmark (d/k), and net mineral rent rate (v — e)/f,
in the inertia scenario, 1991-2107
0.045
0.03
0.015
0
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure 2 The growth rates of net output (?) and labour productivity (4) versus benchmarks (d and dn,
respectively) in the inertia scenario, 1991-2107
The excessive depletion of the proved mineral reserves slows the economic growth (Figure 2). The secular
deceleration of the labour productivity and real labour compensation is relentless (Figures 2 and 3).
0.03
J
0.015
: 4
0
0.06 0.08 0.1 0.12 0.14 0.16
(1-u-y)ls
Figure 3 Average profit rate (1 — u — y)/s and growth rate of real labour compensation w,
counter-clockwise, in the inertia scenario, 1991-2107
0.95
—e—v
--n7veq
0.9
0.85
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure 4 Employment ratio (v) versus its stationary magnitude (v eq = v, )
in the inertia scenario, 1991-2107
Let us return to the equation (12) for the time derivative of the gross unit mineral rent. According to a
statistical estimation, the second parameter in this equation is negative as expected (0, < 0), whereas the first
parameter is paradoxically negative (0, < 0) too (Ryzhenkov 2001, 2002a).'° Therefore the expected combi-
nation of the proportional and derivative controls over proved mineral reserves has been probably absent in
the reality.
The observable excessive depletion of some proved mineral reserves supports empirically at least partially
this formal result.'' Positive feed-back loops foster the gap between the indicated and actual proved mineral
reserves that possibly destabilizes the U.S. economy. It is likely that voluminous imports of goods and ser-
vices, produced abroad with a high direct or total input of minerals (such as oil, gas, metal ores), offset this
destabilizing tendency only partially.’
' Perhaps a rationale for negative o, is a tendency to a greater substitution of extraction of non-renewable
resources inside the USA by import. A stronger U.S. reliance on foreign resources could be reflected by a
new hypothetic equation e,= 0.0054(128.14//c) or by another equation with a positive partial deriva-
Oe,
af! e)
and renewable natural resources for future generations may underline this tendency. See also footnotes 11-12
tive
>0, where e,is the element of the equation (16). A reservation of the national non-renewable
below.
'! In my opinion, the report (Energy Information Administration 2006) provides additional empirical sup-
port for the assertion on declining proved reserves of crude oil and of some kinds of natural gas over 1977—
2005 in the USA as a whole. Whether an excessive depletion of proved mineral reserves facilitates conserva-
tion of total non-renewable and renewable natural resources for future generations is not quite clear on this
stage of research.
PA stronger substitution of extraction of non-renewable natural resources inside the USA by import could
be modelled by lowering the magnitude of the constant e, from the equation (16) or by turning this constant
into declining variable. Extreme condition tests for e,= 0 or for e,= 0.0054(128.14f/c) when this variable
approaches zero asymptotically while the ratio f/c declines steadily nearly to zero in these both cases do not
alter results of this section qualitatively if the other conditions remain the same. Still decoupling the growth
rate of labour productivity from dynamics of proved mineral reserves per labourer (additionally setting
ms = 0 in the equation (4) in particular) due to a slowing growth and stagnation of production in the extrac-
tion industry in the USA (Z approaches zero) permits overcoming the tendency of growth rate of labour
14
In spite of these imports, the excessive depletion of proved mineral reserves counter-acts tendencies to a
stable stationary growth or to stably oscillating cyclical motion. The economic fluctuations can surpass viable
boundaries if multiple positive feed-back loops, involving proved reserves, continue to run unchecked.
To avoid this, a more efficient policy is necessary. The next section designs such a policy.
5. The normative scenario I based on modified HL1: extending the proved mineral reserves
The first normative scenario corresponds to a new investment policy in relation to proved mineral reserves.
Although the form of the equation (12) for the gross unit mineral rent remains the same, the sign of the pa-
rameter 0, is changed (now 0, = 0.0001 > 0 instead of 0,< 0). This numerical change is simultaneously
structural one. It addresses the critical shortcoming of the inertia scenario, namely the excessive depletion of
the proved mineral reserves.
It is reasonable to stress that the substantial numerical change in parameter 0, of the equation (12) is the
single alteration of the above compact model. Only 0.35 per cent of the net output is invested additionally in
the proved mineral reserves each year on the average in 1991-2107 in relation to the inertia scenario. Still
this partial redistribution of the NNP produces desirable additional positive effects over the whole period on
the average:
™@ the growth rates of labour productivity and net output rate are increased (Figures 2 and 6);
™@ the proved mineral reserves instead of been excessively depleted are extended (Figure | and Figure
5);
™@ the secular fall in average and general rates of profit is moderated (Figures 1 and 5);
™@ there are gains in the employment ratio and growth rate of real labour compensation (compare Fig-
ures 3 and 7, Figures 4 and 8).
In the normative scenario I, the economic growth is quasi-cyclical with a period of about 32-35 years. This
period is shorter than in the inertia scenario due to a higher average growth rate of labour productivity.
Quasi-periodic ups and downs with substantial amplitude represent the shortcomings of this normative sce-
productivity to fall in 1991-2107 without over-extending the employment ratio (v) in other extreme condition
tests.
These modified inertia scenarios would represent a drastically declining share of the U.S. mineral produc-
tion in the world. As it is hardly realistically, the next sections apply e,= const > 0 only.
15
nario. The tendency to over-extending the employment ratio is especially unsafe as v tends to hit the top limit
of unity (Figure 8). These shortcomings require overt closed-loop control over capital accumulation.
0.18
a Se ee —y me — dik
0.09 general profit rate
average profit rate
0.06 = id
(y- ef
0.03 f
0
Sh > Dn aN Gh 0 oD O NSN A & OD 40 Sd A <a)
BS SC SIPS Oe Sta Sa Sad SRS SSSR SS SESS Pa
Figure 5 The net mineral rent rate (v — e)/f, average profit rate (1 — vu —y)/s and
general profit rate (1 — u — e)/(s +f) versus
their benchmarks in the normative scenario I, 1991-2107
0.045
cas fo r™~ JP
Phat
eae
ee ee ee ee ee a ee sii ahat
0.025 Wr iw” med =n
0.015
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure 6 Growth rates of net output and labour productivity versus
their stationary magnitudes in the normative scenario I, 1991-2107
0.026
0.021
what
0.016 +
0.011
0.1 0.125 0.15
(1-u-y)ls
Figure 7 Average profit rate, (1 — u — y)/s, and growth rate of real labour compensation, w,
in the normative scenario I, 1991-2107 (counter-clockwise)
0.96
va
0.92
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure 8 Employment ratio (v) versus its stationary magnitude (v eq = v, )
in the normative scenario I, 1991-2107
6. The normative scenario II: extending proved reserves, controlling employment and profitability
There has been substantial labour compensation moderation accompanied by accelerated growth of labour
productivity in the USA after 2001. In my opinion, this moderation has been caused (skipping reference to
other factors), first, by the economic crisis; second, it has been due to the state military-mobilising policy;
third, outsourcing, competition with China, India and other countries have played a role too.
It has been shown in my previous paper (Ryzhenkov 2005) that setting aggregate real labour compensation
at a level that is too high relative to labour productivity raises unemployment in a business-as-usual scenario.
I have identified open-loop and closed-loop policies affecting real labour compensation for improving labour
market performance and profitability against this business-as-usual scenario. The overt closed-loop control does
not cause over-extending of the controlled economy unlike the open-loop control.
That analysis is not complete since the previous papers have not explored changes in the wage-setting and other
relevant institutions implied by supposed closed-loop control over capital accumulation as a whole. In particular, it
is not yet clear, first, whether such a closed-loop control is to be achieved through coercive and/or voluntary coop-
eration; second, what arrangement of coincidence, coercion and co-adjustment is mostly suited to provide superior
social outcomes."
The section 7 of the present paper (below) gives a preliminary answer to these complicated questions. A socially
desirable transition to a target employment ratio, profit enhancement and an extending of proved reserves will be
realised through excess income levy.
It has been assumed in the previous research that the decision-makers (the state officials, owners of capital,
labourers) set timely (without delay) a desirable growth rate of total profit (/ = (1 — u)P) depending on a dif-
ference between an indicated (¥) and current (v) employment ratios:
“ a a ss
M =-——+P=0;(V-v),
l-u
where v < ) is typical for recessions and depressions, 0, > 0. This equation is to be modified now taking
into account gross unit mineral rent (y):
M=-—"t” 4 P=o0(-v), (26)
1l~u-y
14
where 0;> 0.
'3 What is a complex outcome of these so far hidden socio-economic interactions may even appear super-
fluously as a result from one-shot game of the state and social classes.
'4 The notation of (Ryzhenkov 2005) is changed for better fit with the present story.
18
A new partial dynamic law for the relative labour compensation is derived correspondingly from the equa-
tions (19) and (26):
i =(P-M)(1-u-y)-j= (@+9+n-M)(1-u-y)-p
k(l-u- > » *
[ED (KIL m+ PiLen Mi fu y)~3
Ss
[suns
s
Ss
The system of the ordinary differential equations (18), (19), (21)}-(24) and (27) represents the determinis-
tic form of the modified control law of capital accumulation. This system has a continuum of non-trivial sta-
-[M S204 y)-5. @7)
tionary states with the same positive relative labour compensation, employment ratio, relative depletion of
proved reserves; this continuum contains the following one, in particular:
En=(Sp.%b> Ub» Sb» Ch» Yb» eb )s (28)
where
Sh = Sq
fy = (1 up~ ep d= sp/k,
Ca = Sos
vy = ¥-d/oy,
_ d= = 13 (Vp — Ve) = epnis
n
m,
d=——! _ +n,
1-—m,-—m,
y= +4fy,
ey =e),
i=d.
At this stationary state, Ly =n. The stationary average profit rate is (1 — uy, —),)/s, = d/k. The target
employment ratio is the stationary employment ratio vy .
A new equation for a growth rate of real labour compensation follows from the equations (3) and (27):
wands (PY 2 gn (enoen— Ct Sage
u u u u
=a V4 (G4n—Myiceey z
u u
ge tg i( EY pyle Rg ey IF (29)
u u u u u
The following magnitudes of the two new parameters are chosen for simulations: 0; ~ 8.521, ¥ ~ 0.963.
Correspondingly, the new stationary employment ratio (v, ) equals 0.961 (compared with 0.991 in the previ-
ous two scenarios). Only 0.354 per cent of the net output is invested additionally in the extension of proved
reserves each year on the average in 1991—2107 in relation to the inertia scenario. This increment is about the
same as in the normative scenario I.
_ aaa ——— dk
general profit rate
average profit rate
Figure 9 The net mineral rent rate (v — e)/f, average profit rate (1 — u —y)/s and
general profit rate (1 — u — e)/(s +f) versus
their benchmarks in the normative scenario II, 1991-2107
The structural changes produced desired effects. The pro-growth stabilization policy eradicates long-term
economic fluctuations and extends proved reserves; moreover, it maintains relative labour compensation,
profitability and employment ratio in vicinity of their reasonably high stationary magnitudes even during the
most of the transition period (Figures 9-11).
20
0.045
0.04
0.035
I Phat
Serd
0 ———————
] a ahat
0.025 heidi
oo2 HN
i
0.015
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure 10 Growth rates of net output and labour productivity versus
their stationary magnitudes in the normative scenario II, 1991-2107
0.97 0.06
0.96
0.95
0.94 + + + + + + + + + -0.04
No SO Sh D® ND HD ND WN 0D 0 MD
Sr SPB PSP PSP RYN? NP KL
DP PPP Pa
Figure 11 Employment ratio v versus its stationary magnitude v, (scale on the left), growth rates of labour
productivity (a hat) and of real labour compensation (w hat) after excess labour compensation levy (scale on
the right) in the normative scenario II, 1991-2107
The transitional dynamics are contained within reasonable limits. In particular, the transitional motion in
the first ten years is shaped as converging fluctuations of the main variables that move to their stationary
magnitudes rapidly. The relative labour compensation decreases by —1.25 percentage points of net output in
1992 against 1991 for the almost complete compensating the initial step-wise increase in gross unit mineral
rent (0.32 percentage points of net output) and the subsequent step-wise increase in profit share due to pro-
21
growth stabilization policy (0.93 percentage points of net output).'> The real labour compensation is roughly
unmoving in 1991-1992 only and increases in all other years (faster on the average than in the other two sce-
narios).
Figures A.1—A.6 in Appendix add information on the all three above scenarios. These Figures support the
above contention on benefits of the strengthened social control over capitalist reproduction on the increasing
scale under the explicit (although not precisely known and fixed) constraint of proved reserves.
A progress in statistics and in the theory of ecological-economic reproduction will allow including remain-
ing non-renewable as well as renewable and environmental resources in similar models of sustainable devel-
opment in future research. Ideally, the upgraded modelling process has to take into explicit account:
the environmental consequences of mining, including the costs of environmental compliance and the ef-
fects of using best practices in the environmental management of mining;
effects of improved environmental management and restoration ecology associated with mining and min-
eral processing (Committee on Earth Resources, 1997).
As a result new information will be delivered to scientific community, policymakers and public about how
mining affects the environment and how environmental degradation can be minimized.
7. Enhancing sustainable development through excess income levy
This section discusses advantages of adopting a more pro-growth policy of primary distribution of income
through levy on excess income (labour compensation or profit).!° The notion of excess income levy is used
in this paper as a general notion for the reduction in pre-levy primary income.
The term excess labour compensation levy is for the reduction in pre-levy primary labour compensation.
The counter-part of excess labour compensation levy is subsidy (of the same quantity) on pre-levy primary
profit. In the opposite case, excess profit levy equals subsidy on labour compensation receivable. For sim-
plicity, it is the state that can levy surcharges on excessive income of labourers (or capitalists) and pay
equivalent subsidies to capitalists (or labourers). The levy year and base are sliding.
* Comparisons of the normative scenario I with the inertia scenario and the normative scenario II with the
normative scenario I yield these magnitudes (0.32 and 0.93, respectively).
‘© Chapter 3 “Pro-Growth Tax Policy” of the Economic Report of the President 2007 treats almost exclu-
sively taxation within the secondary distribution of income.
22
Let w,, is the pre-levy labour compensation used as the levy base. Its rate of change is determined accord-
ing to the bargaining equation (8). The after-levy labour compensation is denoted as before by w; its rate of
change (1) is determined by the equation (29) based on the above deterministic form of the modified control
law of capital accumulation.
The rate of excess labour compensation levy (as a fraction of unit) in non-probabilistic setting is
Xy =(W,, —W)-I [year] (30)
The overall excess labour compensation levy equals overall subsidy on pre-levy primary profit
Ly = XywL = Sp. (31)
The total profit is now
P-(WppL-Ty)-Y =P-wyL+Sp-Y
=P-wL-Y. (32)
The labour share in net output is higher than 50 per cent in the American economy, so the relative subsidy
on pre-levy primary profit exceeds relative excess labour compensation levy in this theoretical model.
Using the new stationary employment ratio from the equation (28) we get the stationary relative excess la-
bour compensation levy (as a fraction of unit)
X= [dn —w,]-I[year]= [-g+rv, +bK, iL, +qF, iL, —(d—n)]-1[year]
= [-g+r(¥ -—d/o0,)+(b+q-1)(d—n)]-1[year]
= [r(¥ —d/0;)-rv,]-1 [year] = r(v, —v,)-1 [year]. (33)
The stationary share of excess labour compensation levy in net output (i.e., unit excess income levy) is
Xp =X,,up. The stationary relative subsidy on pre-levy primary profit is Xj =X,,u, /(l1—upy — yp).
Depending on relation between the target employment ratio (vj )in the normative scenario II and the sta-
tionary employment ratio in the inertia scenario (v, ) there are, ceteris paribus, three cases:
1) if vy =v, the all three stationary levy (subsidy) ratios X,,, X;y and Xpare zero;
2) if vp > vq these ratios are positive;
3) if vy <vy these ratios are negative.
These three cases are a particular manifestation of the employment ratio — relative labour compensation
trade-off. In the second case, labourers, having a higher stationary employment ratio than in the inertia sce-
nario, pay levy to the state that provides subsidies to capitalists. In the opposite (third) case when the target
employment ratio is lower than the stationary employment ratio in the inertia scenario, capitalists pay levy to
23
the state that provides subsidies to labourers. In the first case, when these both employment ratios are equal,
the stationary relative levy (subsidy) is zero. Still excess income levy is pertinent even in this case since em-
ployment ratio varies on the transient to the stationary state.
If the society could raise the former stationary growth rate of labour productivity (d — n) to a higher level
(d—n+ 8), then, assuming that the target employment ratio remains the same, this increment in labour pro-
ductivity brings about the absolute reduction in the relative stationary excess labour compensation levy by
(1-b—q)8, where 1 — b—q>0 and 8>0. Moreover, relative excessive labour compensation levy may be
negative except few initial years on a transient to the stationary state if there is a substantial increment in the
stationary growth rate of labour productivity that raises v, to a level higher than v,, as in the above third
case.
The following Figures 12-14 in this section relate to the normative scenario II as in the previous section.
They illustrate a rather fast convergence of pre-levy labour compensation to post-levy labour compensation
as well as narrowing difference between pre-subsidy and post-subsidy profitability (respectively, Figure 12
and Figure 13). The scatter of employment ratio and unit excess income levy (Figure 14) illustrates similar
rapid convergence to the target employment ratio.
0.042
boa —_
a
3 0.038 a :
s JL we
unas we
0.034 , , T T T T T
1991 1992 1993 1994 1995 1996 1997 1998
Figure 12 Pre-levy (wp) and post-levy labour compensation (w) in the normative scenario II, 1991-1998
24
0.16
0.15 f
£0.14
es — Profitability initial 1
2 —s- Profitability final 1
= 0.13
0.12
0.14 et
SS oP _ PP MP NY
PPP SF Prag Brg oh
Figure 13 Rates of return on produced capital (1 — u —y)/s before and after levy
in the normative scenario II, 1991-2018
r T —y r
0 of 445 0.95 0.955 6 2965 0.97
Figure 14 The scatter of unit excess income levy (xp) against employment ratio (v)
in the normative scenario II, 1991-2018
Measurement errors and external influences not taken into explicit account complicate practical realiza-
tion of the supposed overt closed-loop control over capital accumulation. These limitations necessitate fu-
ture concretization of above results for the real economy with fuller account of risk and uncertainty.
Conclusion
A sustainable development point of view has shaped the author’s research aimed at reinforcement of the
system dynamics paradigm. This paper offers, firstly, elementary building blocks for generic models of na-
tional and international sustainable development. Secondly, it suggests, for the U.S. economy specifically,
strengthening the conscious societal control over the proved mineral reserves that facilitates converting the
probable unsustainable evolution of the inertia scenario into the strongly sustainable development in the
normative scenario I. Thirdly, this paper describes the deterministic forms of the enhanced closed-loop con-
trol over total profit and employment that could bring the substantial additional social gain as the given com-
parisons of the normative scenario II with normative scenario I demonstrate. As shown, the supposed altera-
tion of the feedback structure and information links in the U.S. economy is a reasonable prerequisite for these
possible improvements.
This paper offers, in particular, the novel outline of excess income levy within attainable bounds that may
enhance long term sustainable development. It suggests the appropriate levy base and appropriate levy rates
for primary distribution of income in the modern U.S. economy encouraging efficient investment into pro-
duced capital as well as into exploration and development of mineral reserves. Thus, this paper demonstrates
with a help of system dynamics methodology rather realistic opportunities for moderating substantial dynamic inef-
ficiency in modern capitalism and exposes its potential structural changes for maintaining resilience of the social
organism. To what extent does the uncovered excessive depletion of proved mineral reserves in the USA fa-
cilitate conservation of total non-renewable and renewable natural resources for future generations is still the
unresolved question.
The presented theoretical model ought to be generalized. This presupposes improved statistical account of
mineral, renewable and environmental resources on the national and global scales as well as deeper theoreti-
cal understanding of ecological-economic reproduction. Different policies of substitution of minerals import
for their extraction deserve more attention, than given in this paper, in particular.
26
References
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27
28
Appendix
Comparison of the three scenarios of the U.S. economic
growth
0.15
0.12
0.09
0.06
es
Wawa
yl
—normative II
—normative |
—— inertia |
ae,
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure A.1 The average profit rate (1 — uv —y)/s in the three scenarios, 1991-2107
0.05
1
normative II
normative |
inertia
a
anal
991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure A.2 The real labour compensation (w) in the three scenarios, 1991-2107
on | Pi \U Ly Y normative II
3 69 VL \* /\ a —— normative |
—— inertia
Figure A.3 The relative labour compensation (w) in the three scenarios, 1991-2107
0.6
0.5
— normative II
«© 0.3 normative |
—— inertia
0) T T T T T T T T T T
Nay
s
P&P oS
Xo oP WO Gh PO SL Sd
¢ OY oO” @ oO Oo s +
eek PF PS
Figure A.4 The labour productivity (a) in the three scenarios, 1991-2107
al) A A A
kV PAC AS,
. \ hoy, rd 4 Fd \ normative II
0.92 normative |
ee
0.86 T T T T T T T T T T
1991 2002 2013 2024 2035 2046 2057 2068 2079 2090 2101
Figure A.5 The employment ratio (v) in the three scenarios, 1991-2107
0.013
0.0105 i
es | —normative ll
> 0.008 normative |
—— inertia
0.0055
0.003
Nb nD gk 1 © A © 2 DPD vd
FPP HF KH FH HF a
Figure A.6 The gross unit mineral rent (y) in the three scenarios, 1991-2107
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