Endogenous Industrial Cycles in a Reshaped “Neoclassical” Model
©Alexander V. RYZHENKOV
Economic Faculty
Novosibirsk State University
1 Pirogov street Novosibirsk 630090 Russia
Institute of Economics and Industrial Engineering
Siberian Branch of Russian Academy of Sciences
17 Academician Lavrentiey Avenue Novosibirsk 630090 Russia
E-mail address: ryzhenko@ieie.nsc.ru
Abstract
Two closely related mathematically sophisticated “neoclassical” models of economic growth (first with
hidden, second, more general, with intended) economies of scale are considered. The main variables are
relative wage and employment ratio, whereas a ratio of investment to profit is constant. The efficiency
wage hypothesis supports equations for a growth rate of output per worker. Workers’ competition for jobs
is stabilizing and their fight for increased wages is destabilizing as revealed. In each model, a stationary
state is locally asymptotically stable in a system of two ODEs. There is no possibility for endogenous in-
dustrial cycle. Reality requires negation of this incorrect denial.
A third extended model reflects the destabilizing cooperation and stabilizing competition of investors.
In a system of three ODEs, rate of capital accumulation becomes the new main variable. Its targeted long-
term decrease raises profit rate together with reducing relative wage and capital-output ratio. Oscillations
imitating industrial cycles are endogenous. Crisis is a manifestation of relative and absolute over-
accumulation of capital. Limit cycle with a period of about 6.75 years results from supercritical Andronov —
Hopf bifurcation. The reality disagrees with the efficiency wage hypothesis applied in the analyzed models.
Keywords: rate of capital accumulation, relative wage, employment ratio, efficiency wage hypothe-
sis, industrial cycle, supercritical Andronov — Hopf bifurcation, limit cycle
To Sesquicentennial (2017) of K. Marx’
Capital: Critique of Political Economy, vol. 1.
1. Introduction on K. Marx’ industrial cycles and their denial in modern “neoclassical” models
The present paper considers the notion of industrial cycle as a dynamic process typical for industrial capi-
tal with a specific turnover in a macroeconomic context without disaggregating national economy in in-
dustry and other sectors. According to this understanding based on the first, second and third volumes of
K. Marx “Capital”, industrial capital moves in such diverse (still interconnected) fields as industry, agri-
culture, construction and other sectors.' The notion industrial capital is mostly restricted in this paper to
the notion of productive capital as the single source of surplus product and surplus value in capitalist
economy. Uncovering specific regularities of industrial cycle in industry and other sectors of capitalist
economy will require additional efforts far beyond restricted scope of this paper.
The term industrial cycle rooted in the K. Marx works became out of fashion. It was substituted by
such terms as trade cycle and business cycle. The present paper reintroduces the original term with intent
to stimulate the reader interest in the Marxist economic theory. “The course characteristic of modern in-
dustry, viz., a decennial cycle (interrupted by smaller oscillations), of periods of average activity, produc-
tion at high pressure, crisis and stagnation, depends on the constant formation, the greater or less absorp-
tion, and the re-formation of the industrial reserve army or surplus population... Effects, in their turn,
' An attempt to restrict the notion of industrial cycle to fluctuations in indicators of industry would be a
wrong denial of universal (or general) over-production and mistakenly viewing this over-production as basically
partial (in particular sectors, spheres or different branches of production).
1
become causes, and the varying accidents of the whole process, which always reproduces its own condi-
tions, take on the form of periodicity.” [1: 419-420].
K. Marx assumed in 1870, on the basis of the recurrence of crises in 1847, 1857 and 1867 that the
period of the industrial cycle was ten years. Later he stressed that the period of the cycle was likely to
shorten as the pace of technical progress increased (see for details [2]).
The K. Marx economic theory infers from the laws of capitalist production that crises are the most
dramatic manifestation of the underlying contradictions of the capitalist system, they manifest them-
selves through apparently contingent causes of each individual crisis [2].
It has been the shocking revelation for ordinary citizen by an analysis company that “economists, as
a consensus, called exactly none” of the last seven recessions, dating back to 1970. In other words [3], the
consensus view of economists has an accuracy rate of zero per cent: “...economists seem to be motivated
by a fear of bucking the professional consensus. Their models are not built to capture dramatic shifts and
their forecasts tend to move in tiny steps.”
One case is typical: in fall 2008 when the recession was already unfolding in the USA for more than
half a year even the journal of the Union for Radical Political Economics published a paper [4] endorsing
an implicit claim that trade cycles are the thing of the past in an influential paper of a renowned “neoclas-
sical” economist [5].
Anticipating (or reflecting) next global recession it would be wise to investigate the logic of both
papers hidden behind conformism (“a fear of bucking the professional consensus”). The system dynamics
methodology is a promising tool for such endeavour. J. Forrester [6: 362, 370] wrote: “There is lack of
courage in the field to open oneself to severe debate and criticism. However, we will never change in-
tended and widely supported detrimental policies without intense debate ... System dynamicists must go
behind the symptoms of trouble and identify the basic causes.”
The “neoclassical” school falls short of these requirements. Dialectic (historic) materialism is
deeper than subjectivism (alleged objectivism) of the “neoclassical” school.
F. van der Ploeg [5: 228] elaborated the famous Goodwin version [7] of the Marxist growth cycle.
He came to the conclusion: “When the assumption of a fixed capital-output ratio is replaced by the as-
sumption that firms maximize profits subject to a C.E.S. production function, whilst retaining the wage-
bargaining equation of R. Goodwin (1967), the perpetual class struggle cycles of Goodwin’s model are
replaced by either damped conflict cycles or monotonic convergence to balanced growth equilibrium”.
Still F. van der Ploeg [5: 229-230] recommends for a more realistic theory allowing natural and
warranted growth rate to differ by introducing an independent investment function taking into account
profitability and other factors.
The efforts in transforming the perpetual cycles of R. Goodwin (1967) into damped conflict cycles
by “neoclassical” tools have been supported by L. Aguiar-Conraria [4]. He undoubtedly tried a step in the
right direction to “endogenizing productivity growth”. This paper reveals the strength and the weakness
of [4] that contains grains of truth that need refinement and extension.
Unfortunately, L. Aguiar-Conraria [4] is very misleading taking it as a whole. Instead of following
F. van der Ploeg’s recommendation of introducing a realistic investment function, the assumption on con-
stancy of the accumulation rate has not been relaxed, the destabilizing effects of endogenous productivity
growth have been treated superficially, thoughtful remarks [5: 229] on decisive role of workers’ competi-
tion for jobs for asymptotically stable growth trajectories have not been elaborated. On other hand, impor-
tant shortcomings of F. van der Ploeg’s [5] have been amplified. Such papers do not advance Marxian
economics, listed among the keywords in [4], contrary to our expectations.
The reader well remembers that the original Phillips nominal wage-change equation in [9] is non-
linear where factors are the employment ratio, its rate of change and the rate of inflation. R.M. Goodwin
[7] simplified this equation in his original model to a linear one for a rate of change of real wage depend-
ent only on a single variable — the employment ratio.
? Starting with a rather problematic quasi-empirical relation between net output per worker and real wage
only (cf. equation (35) in section 2.2), a CES production function was derived in [8]. Hardly anybody reviews its
innumerable statistical applications in spite of specification errors.
2
F. van der Ploeg [5] applied a non-linear Phillips equation similar to the linear counterpart used by
R. Goodwin [7], besides that a new linear term was added which reflects the growth rate of output per
worker as a factor in real wage-change equation. This term implies ceteris paribus that the higher growth
rate of output per worker compels capitalists to promote a growth rate of real wage that is detrimental for
profit and profitability. L. Aguiar-Conraria [4] simply removed this hypothetic term without explanation.
This paper’s sections 3.2, 4.1 and 4.2 state a number of propositions and corollaries. Appendix A
contains formal proofs of three most complicated ones only. Throughout this text a variable’s time deriva-
tive is denoted by a dot, its growth rate — by a hat over the variable’s sign. An ordinary differential equa-
tion is shortened as ODE.
The rest of this paper is organized along the following main themes.
Section 2 reviews briefly properties of the Goodwin model abbreviated as M-1. F. van der Ploeg
dynamic model based on a “neoclassical” CES production function [5] denoted as P-1 is considered next.
The conservative oscillations in M-1 are stabilised in P-1 mostly due to introduced workers’ competition
for jobs. More technically, an intensive form of P-1 is built into a two-dimensional system of non-linear
ODEs, local asymptotical stability of its non-trivial stationary state is exposed. The stationary employment
ratio is lower than that in M-1 because of the surmised direct dependence of the growth rate of wage on
growth rate of output per worker.
Section 3 is devoted to L. Aguiar-Conraria model [4] denoted P-2 as extension of P-1. P-2 with addi-
tional scale effects provides check of P-1 structural stability. Proportional and derivative control over capital
accumulation, implicit in P-2, is revealed. As extension of P-1, P-2 reflects workers’ joint struggle for
higher wage that does not destroy the local asymptotical stability of a stationary state. Conceptual weakness
of P-2 resulting from the efficiency wage hypothesis and other “neoclassical” beliefs is uncovered.*
Section 4 offers a modernized model Z-1 that extends P-2 through an implicit Marx’s balancing
feedback loop involving endogenous rate of capital accumulation. Z-1 sheds light on industrial cycles as,
first of all, capital accumulation cycles. It is shown that capitalists’ investment cooperation weakens
(competition strengthens) stability of a stationary state in Z-1. Targeted reduction of the stationary rate of
accumulation increases stationary profit rate and output-capital ratio, yet it reduces stationary labour
value. Self-sustained industrial cycles are born in result of a super-critical Andronov — Hopf bifurcation.
Dual nature of capital as the driver and barrier of capitalist production is demonstrated analytically and
numerically through Vensim simulation runs.
Section 5 concludes.
2. How conservative oscillations in Goodwin’s model (M-1) were “neoclassically” stabilised
2.1. The M-1 extensive and intensive forms
This subsection shortly reviews famous Goodwin’s predator-prey model for closed economy [7]. Labour-
ers are advancing capitalists as they receive wage after a particular circuit of capital is finished. Having
abstracted from the public sector and foreign economic relations, M-1 consists of the following equations:
q = Kiso; (1)
a= ql; (2)
u=wia, (3)
> The efficiency wage hypothesis has its origins in the controversial logarithmic equation for net output per
worker linearly and positively depending only on real wage [8: 228]. Adding capital intensity is this equation has been
a standard way for deriving a typical variable elasticity of substitution (VES) production function in the “neoclassical”
literature [10]. We will see that a VES production function in [4] has been implicitly derived through direct additional
dependence of the growth rate of output per worker on the growth rate of fixed production assets instead of the growth
rate of capital intensity (equation (49) in section 3.2). Still the “marginal productivity” principle of income distribution
is broken in [4] without noticing by its author. See section 3.2.
O0<u<l;
a=a=const > 0, (4)
So = const > 1; (5)
v=l/n, (6)
0<v<1;
n =me™ (7)
B =const > 0, my > 0;
w =f), (8)
qg=Ct+k =wl+(1-2M+k; (9)
k =2M=2(1-wg, (10)
(a+B)so<z<l.
Equation (1) specifies a technical-economic relationship between fixed capital k and net product g.
Capital-output ratio is denoted by so. Equation (2) expresses output per worker a as a ratio of net
product g to employment /. Equation (3) describes relative wage as wage share in net product w. Equation
(4) assumes a constant exogenous growth rate of output per worker o that equals to a growth rate of capi-
tal intensity k//, whereas capital-output ratio remains constant according to equation (5).
Equation (6) defines employment ratio v as a result of the sale of the labour power. According to (7),
the growth rate of labour force n is equal to constant B. Equation (8) links the growth rate of real unit wage
w with employment ratio v.
The use of current profit reflects absence of information lags for labourers regarding the actual rela-
tive wage. In other words, capitalists and workers receive information on relative wage in real time.
Balance equation (9) shows the end use of net product g, where C is non-productive consumption,
k is net fixed capital formation defined in the equation (10). Investment delays as well as discrepancies
between orders and inventories are not taken into explicit account. In result, net fixed capital formation
equals net investment. The attribute ‘net’ will be omitted, as a rule, below for brevity.
Surplus product that equals total profit M@ can be not only invested, but also be used to cover per-
sonal expenses of the bourgeoisie and via implicit taxes for unspoken public consumption. Consequently,
rate of accumulation z, measured as share of investments in surplus product, or as ratio of investment to
profit, is such that (a + B)so <z < 1. The left boundary is set to avoid a non-positive stationary relative
wage (see equation (13)). Notice that z = 1 originally in M-1, P-1 and P-2. The given definition of the rate
of accumulation z differs from defining it as a growth rate of fixed capital in [16: xvi] and elsewhere.
The presence of (a + B)so as a lower boundary for rate of accumulation z is a drawback of M-1 and
subsequent models, since in reality relative wage remains positive even when (a + B)so > z. This means
they do not pass this particular extreme condition test. Models in [11-13] contain endogenous capital-
output ratio and endogenous rate of accumulation in the absence of the specified lower bound as a real
necessity. Long-term decline in this ratio mitigates the tendency of profit rate to fall in the USA, Italy
[11-13] and, by the same reason, still hypothetically in other industrialized countries.
According to [7], an intensive form of deterministic M-1 consists of two non-linear ODEs. Here is
this system in a generalized form for 0 < (a + B)so <z < 1 in relation to the original form (for z = 1) and
for non-linear Phillips equation as in P-1 and P-2:
a =) — alu P'(Y)>0; (11)
v= [z(1- u)/so— (a + B)]v. (12)
A positive stationary state of the system (11)—(12) is defined as
Eg= Ue Va), (13)
oP Vg = f(a), 50> 1.
where ug =1—
A stationary growth rate of output per worker and growth rate of wage equals a. A stationary
growth rate of fixed capital and net product is kg = dG =a+f. A stationary rate of surplus value is
mg'=(1-ug)/ug. A stationary profit rate is (1 — ug)/so = (4 +B)/z,a>0, B>0.
Figure | and Table | exhibit a well-known causal loop structure of M-1.
Relative
SS vtcen
so
Bl
Employment
~ ratio v
Figure 1 — A condensed causal structure of M-1: total number of feedback loops — 3,
among them: 1“ order — 2 alternating, 2™ order — 1 negative
Table 1. The main extensive negative feedback loop in M-1
Loop B1 of length 8
Relative wage u —— Profit rate
GR fixed capital
GR employment ratio
Net change of v
Employment ratio v
R wage
GR relative wage
Net change of u
Note. Only a negative first partial derivative is explicitly shown as an arrow. GR is for “Growth rate of’.
Improved profitability promotes increases in employment ratio (“prey”) that facilitate higher rela-
tive wage (“predator”) to the detriment of profit rate in negative feedback loop B1.
An equivalent for (15) reflects proportional control over the net change of employment ratio
p= {ie - a8), =
So Zz
= (z/so)(ug — u)v. (14)
Denote derivative of f(v) at vgas f'(v)g. Typically, the higher is f'(V)g the faster is adjustment
speed of u. Similarly, the higher is z/so the faster is adjustment speed of v. The period of conservative
fluctuations in M-1 is closely approximated as
Ty n tn) [@*PPOerete =2nh he = pW)are - (15)
So
I-ug
This period clearly depends sensitively on the value of f'(v)¢ that can be estimated with less cer-
tainty than other relevant parameters. In a simulation run in next section with plausible f'(v)g and other
reasonable magnitudes, Ty-; = 21.59 years.
It is relevant to recall A.B. Atkinson's paper that finished with a rather pessimistic conclusion on
the relevance of Goodwin's model [14: 151]: “with reasonable values of the parameters, the period is con-
siderably longer than the 4 or 5 years of the typical recent trade cycle. A cycle with this period is only
likely to be generated if all profits are saved, if the capital-output ratio is very low (less than 2), and real
wages respond very strongly to a fall in unemployment. In fact the model as it stands may be better suited
to explaining the 16-22 year "Kuznets" cycle than the post-war trade cycle.”
G.W. Low wrote that fluctuations in capital stock, averaging 19 years, reflect the management of
investment in fixed capital [15: 337]. Thanks to the G.W. Low critical research this paper will not turn to
the inadequate presentation of business cycle in P. Samuelson’s multiplier-accelerator model proposed in
the late 1930-s and modified subsequently.*
G.W. Low’s extended system dynamics model suggests that labour and inventory management can
produce short-term cycles independently of the long-term fluctuations observed in fixed capital. Still
G. W. Low has used noise input to keep business cycles going in that model for interaction of labour
force and inventories. Although his model contains objective elements it is not a required solution of the
problem of endogenous business cycles modelling.
The present paper steps back in relation to G.W. Low’s research abstracting from inventories be-
cause there are deeper reasons than disequilibrium on product market for business cycles: the contradic-
tion between social character of production and private capitalist ownership on the means of production
and contradiction between value and use-value of labour power. These two fundamental contradictions
are to be taken into account before their ramifications closer to the surface of production relations can be
modelled as subordinated processes (including fluctuations in inventories and orders).
G.W. Low [15] helps in filtering out unsuccessful attempt to solve the problem of business cycle
modelling proposed in [17]. The authors focused on effective demand neglected his warning on _para-
mount importance of conservation principle [17: 428]: “Goodwin employs the same definition, but one
should notice that its meaning slightly changed, as profits now include the increment of inventories in
case of excess supply and do not include sales of previously produced commodities in case of excess de-
mand. This definition of profits does not cause serious problems as long as it is reasonable to expect that
actual surpluses can be sold later. This is exactly what would happen in a non-explosive cyclical process
with periods of excess surpluses being followed by times of excess demand and vice versa. Note, how-
ever, that we have neglected possible feedbacks from levels of inventories to changes in production.”
The model in [17] is focusing on the role of effective demand in shortening the cycle against the
original Goodwin’s model (M-1 in the present paper). That model falls short of recognizing the decisive
role of feedback loops containing the rate of capital accumulation in industrial cycles although Marx al-
most explicitly revealed these loops as this paper implicitly demonstrates on its first page [17: 423].
2.2. “Neoclassical hijacking” of M-1 in P-1
Table 2 lists main variables of P-1.
4 G.W. Low [15] has shown that, when internal flows are conserved, a model of the multiplier-accelerator in-
teraction does not produce short-term fluctuations, but does generate plausible, in his view, long-term cycles. The
paper limits do not permit us discussing such cycles considered in [19, 23].
6
Table 2. Main variables in P-1
Variable Expression
Net product q
Fixed production assets k
Capital-output ratio s=kiqg
Employment 1
Employment (in efficiency units) L=1e™ ,a>0
Output per worker a=qil
Labour force a= me" .p>0
Wage w
Total wage wl
Relative wage (unit value of labour power) u = w/a=wilg
Profit M=q-wil =(l-u)q
Profit rate R=(1-u)/s
In [4, 5] a CES production function is applied for determining net product
a= Ft 1.) =quom,b* +0-womd.y (16)
where, according to “neoclassical” interpretation, s/ is distribution parameter, 0 < ys < 1, c is efficiency pa-
rameter and 6 is substitution parameter.
This function is linear and homogeneous, i.e., there are constant returns to scale by the standard def-
inition. This definition overlooks scale effects maintained by specific feedback loops as next sections
demonstrate. Function (16) has also a property of constant elasticity of substitution (CES) between /abour
power (in efficiency units) and fixed production assets” according to their alleged “marginal productivities”
din(kile) Vey pega Fle
d\n0 1+6 Fy
Parameters c, m, and m, help to harmonize units of measurement, each of the latter two equals 1,
both are skipped for brevity. Function (16) allows considering variable capital-output ratio s, unlike M-1.
(F,, and F;,) under static conditions: 0 < ¢=
For ¢>1 (6-0) this function is transformed in the Cobb — Douglas function, g = ck¥/,""; for
e>0 (6%) it becomes the Leontief technology, where g = Min€kcl,), so capital-output ratio
s = const = I/c. The first case represents in [4, 5] perfect factors’ substitutability, the second case their
perfect complementarity.
It is not taken into account in (16) that capital accumulation in not separable from technical progress
embodied in fixed production assets, as N. Kaldor pointed with good reason [18]. The concept of “mar-
ginal productivities” defies the very essence of technology or of technical mode of production. Productivity
of labour is a synthetic characteristic of the dual unity of concrete and abstract labour [1]. Productivity of
capital is rigged notion that follows from denial of its essence as the specific production relation.
I have found out that production function (16) has the same CES in terms of k and / too: F(k, /.) =
din(k/1) 1 ®, i A . -
———— = —— for 0; = —. This is because of the static character of this notion that
d\n0, 1+6 ®,
requires treating e® for particular ¢ as constant.
A modified Phillips equation defines the growth rate of wage
w= f()+pa. (17)
A growth rate of wage is the sum of bargained w” and stimulating Ww? terms
(k, I) with €, =
> The papers [4, 5] recite uncritically the incorrect “neocl
I’ notion of capital-labour substitution [8].
wasn, (18)
where the first is determined by employment ratio v as in a simplified Phillips equation
w"= fv), (19)
where f'(v)>0, for v->1 f'(v) >, and the second — by growth rate of output per worker according to
the hypothesis on capitalists’ “ability to pay” [5: 224]
w= pa, (20)
where 0 <p < I/e=1+6.
Honestly, capitalists’ ability to pay necessitates raising real wage no more than ability to drink
whisky necessitates increased alcoholic consumption. F. van der Ploeg [5: 222, 224] did not sufficiently
clear explain how capitalists are compelled to increase wage along with increases in output per worker.
Specifically, a higher growth rate of output per worker hardly promotes workers’ bargaining power simi-
lar to the employment ratio, as the thoughtful reader of [1], undoubtedly, understands.
A static problem of profitability maximization is considered for the modified Phillips equation
R= (1-u)/s=(q-wl/k =(a—w)/(k/D) > max. (21)
In result the growth in this model is wage-lead as profit and hence net product follow total wage
with a lag of about one year duration in the simulation run below.
The offered optimal for capitalist class technological composition of capital — the relation between
fixed production assets and labour power in efficiency units — is
kil, =(0-wd-w wu", (22)
correspondingly, optimal capital-output ratio is determined
s(u) =[W/d-w)]? /e. (23)
Net change of fixed production assets (abstracting from investment delays) is accomplished by ad-
vancing a part of profit (surplus product) M@:
k=zM=2Z(l-u)q,0<z<1. (24)
The growth rates of fixed capital is
a 1-
j= z(l—u) 05)
s(u)
The growth rates of capital intensity and output per worker were determined as
kitza+t{—_), (26)
d\l-u
a=a+u/6. (27)
For0<p<1 (not 0<p<1+6as in [5]), we uncover proportional control over the net change of
relative wage:
5 .
14 =———_[f(v)-(l-p)alu, 28
a p25 (I-p)a] (28)
5
where adjustment coefficient 0 <————— < | (again for 0 < p< 1).
(=p)+8
I’ve revealed combined proportional and derivative control over the net change of employment ratio:
._|z(-uv) 1 a
= -=—-(a+ 29
stu) 8 1l-u ( ps 09)
or
(30)
8
Inside the square brackets in (30) there is the difference between the current profit rate and stationary
one. Rate of capital accumulation z takes the role of the adjustment coefficient. The product 7 | v de-
—u
termines a changeable strength of derivative control over v relying on —7 .
Equating the right parts (28) and (29) zero enables finding nontrivial stationary state
Ep = (ups Vp)s G1)
81048)
where wu, = 1 — (2) ul and vy, = f[d-p)ol.
cZ
The stationary rate of growth of fixed production assets and net product is determined independ-
ently of rate of capital accumulation z as k _ q = atBp. The stationary employment ratio declines com-
pared to M-1 if 1 >p>0.
Stationary rate of growth of output per worker, capital intensity and wage is defined as
G,=(k/]),=Wa= a. (32)
The stationary capital-output ratio and profit rate are specified as
5,=[w/d-u, Tle, (33)
R, = (-u,)/s,= (a+B)/z. (34)
Ceteris paribus, the higher is rate of capital accumulation z, the higher are stationary relative wage
u, and capital-output ratio s, and the lower is stationary profit rate R pi
Figure 2 and Table 3 display a revealed causal loop structure of P-1.
Relative
ow, E wage iu
a
Cop E1inployment
<p ratio v
Figure 2 — A condensed causal structure of P-1 at stationary state E,; total number of feedback loops — 3,
among them: 1“ order — 2 (1 — negative, 1 — alternating), 2™ order — 1 negative
First partial derivatives that are negative (—) and with alternating polarity (A) are explicitly shown.
The other derivatives are positive. Negative feedback loop B1 is inherited from M-1, new negative feed-
back loop B2 is due to competition for jobs among workers. Besides these, P-1 includes single 1" order
feedback loop with alternating polarity Al for relative wage u inherited from M-1.
The P-1 intensive form is built into two-dimensional system of non-linear ODEs, local asymptotical
stability (LAS) of its non-trivial stationary state is exposed in [5]. Our analysis founds out that proportional
control over wu and v, already present in M-1, is retained, whereas designed derivative control over v
strengthens proportional control over v additionally in P-1.
Table 3. Three intensive feedback loops in P-1 at E,
Loops descendant from M-1 New loop
Al of length 1
Relative wage u —A+ Net change of u
B1 of length 3 B2 of length 1
Relative wage u ——> Net change of v Employment ratio v—> Net change of v
Employment ratio v
Net change of u
The CES production function (16) seemingly has a constant return to scale (according to the stan-
dard “neoclassical” definition with respect to the arguments k and /.).
Still the standard “neoclassical” definition of economy of scale ignores feedback loops between
growth rate of output per worker and other variables. Taking these feedback loops into account results in
the deeper definition of economy of scale [19: 356-357].
Without going into details, direct economy of scale (direct increasing return) manifests itself in a
positive partial derivative of growth rate of output per worker @ with respect to employment ratio v or
growth rate of employment ratio 1: oa. >0 (type D or oa. > 0 (type ID). Roundabout economy of scale
ov ov
(roundabout increasing return) manifests itself in a positive partial derivative of growth rate of output per
worker with respect to employment ratio v or growth rate of employment ratio » intermediated by other
04a Ox, Ca x; 0a Ox, aa x, :
variable or variables (x; ): —-—~ > 0 or—...—+ > 0, —_— > 0or—...— > 0, i€{2,...,/}.
(a) ax, Ov ax, Ov Gx, ax,” OO Beau}
Economy of scale (increasing return) is reinforcing if a positive feedback loop connects the growth
rate of output per worker with employment ratio and (or) its growth rate. Economy of scale (increasing
return) is weakening if a negative feedback loop connects the growth rate of output per worker with em-
ployment ratio and (or) its growth rate.
For brevity and without loss of generality assume that p = 0. Then the growth rate of output per
worker (27) is presented as
a=a+0/3= a+ a
ad+w 1 :
1s3 Tg tot FOI (35)
This presentation enables uncovering feedback loops containing @ in P-1, particularly at the stationary
state. Equation (35) reveals that P-1 is similar to the models applying the efficiency wage hypothesis stating
that increases in wage promote output per worker. For example, a Goodwinian model with an efficiency wage
mechanism from [20] reflects roundabout increasing returns Caw. >0 and 6a ow ov ov >0. In the case
Ow Ov ow Ov Ov ov
of Cobb — Douglas production function the efficiency wage hypothesis merges with constancy of relative
wage since growth rates of wage and output per worker are virtually the same. The efficiency wage hypothesis
does not recognize many aspects of the capitalist reality investigated in [11—13, 19, 23]. There is hidden
presence of reinforcing roundabout economy of scale in P-1 containing CES production function FY, /.),
or CES production function (A, /), although degree of homogeneity in both equivalent cases of (16) is
constant and equals one according to the standard textbook definition.
Equating the “marginal rate of technical substitution” with the factor price ratio ®, /, = w/R requires
in P-1 shaky hidden assumption of “perfect” competition.° The latter is utter idealisation even for free compe-
° In the “neoclassical” conception, under “perfect” competition the “marginal rate of technical substitution”
is equal to the relative unit costs of the inputs, so the slope of the isoquant at the chosen point equals the slope of
the isocost curve. This equivalence is structurally fragile as section 3.2 demonstrates.
10
tition and is untrue for state-monopoly capitalism. I will return in section 3.2 to this assumption in [5].
Besides that the assumptions of constant returns to scale and widespread atomistic (“perfect”) com-
petition are not compatible; it is necessary from the very beginning to assume a dynamic — not a static —
substitution [22: 485-486, 488-492] and to overcome specification errors in CES production function
stressed in [22: 496-498].
Table 4 reports on reinforcing roundabout economies of scale of types I and II in P-1 revealed neither
in [5] nor in [4]. These three extensive positive feedback loops hardly posit any threat to LAS of E, (31).
Table 4. Endogenous productivity growth and _ reinforcing roundabout economies of scale in P-1
No. Order and polarity Extensive feedback Toop
T mR a ee m
1+ a—>t 9k/1I— 9 pv
2 1+ A Sl oA . =e Re
: a— tt 9 1 Bu— ok 9 Sv Ww
3 mA . = Gomme m
2,4 Ii Si Su sk Sh pv
Figure 3 displays dynamics in P-1 converging to stable node compared with self-sustained oscilla-
tions around a neutral centre in M-1. These models’ calibrating is purely illustrative.
0,95 0,95
> >
iJ 2°
. 0,94 3
€ €
a 30,94 {~~
B 0,93 cf
& &
eT 0,93
0,61 0,63 0,65 0,67 0,69 0,71 0,7 0,71 0,72 0,73 0,74
Relative wage u Relative wage u
2
0,031 0,047
0,042
» 0,029 4
s
= &
3 3 0,037
% 0,027 3
2 <
5 5 0,032
a oe
oO
© 0,025 0,027
0,023 0,022
PO an? DP CP Pan? Po? PnP DP OP Pn? Po? _
SPB? LVAD” a A” O'S a”. O¥ AS” oy oY WS
VPS AS VP VSP F ¥ VPS F verge
——gqhat ——d ——gqhat ——d
4
Figure 3 — A centre in M-1 (panels 1, 3) and stable node in P-1 (panels 2, 4), 1969-2069: employment
ratio and relative wage (panels 1, 2); growth rate of net product and its stationary magnitude (panels 3, 4)
11
Equation (8) is specified as f(v) = —g+r/(1 — v)’. The initial state: uo = 0.702, vo = 0.937, z = 0.156;
common parameters: g = 0.04, a = 0.012, B = 0.015, r= 0.0002, d =a + B = 0.027, additionally in M-1:
stationary wg ~ 0.666 and vg ~ 0.938 and f"(v)g= 1.676, initial so = 1.93; additionally in P-1 8 = 0.33,
€ = 0.752, n= 0.3, p = 0.1, stationary s, = 1.512, u, ~ 0.738 > ug and Vy = 0.937 < vg.
2 Up
3. Check of P-1 structural stability in P-2 with additional scale effects
3. 1. Proportional and derivative control in P-2
As the present paper demonstrates, L. Aguiar-Conraria [4] has misperceived endogenous output per worker
already present in P-1. Still his paper has extended the number of feedback loops involving growth rate of
output per worker substantially that deserves readers’ appraisal. Without these additional feedback loops in
P-2 producing endogenous industrial cycles in subsequent Z-1 would be more difficult.
The former definition of employment (in efficiency units) /. is in Table 2. It is redefined against P-1
taking into account direct scale effect
L=le™ (k/ky)’, a> 0. (36)
This newly defined /, is the factor of CES production function (16).
A static problem of profitability maximization (21) is considered again for the same modified Phil-
lips equation. In result the growth in this model is also wage-lead as profit and hence net product follow
total wage with a lag measured by few months that is shorter than in P-1.
New equations for the growth rates of capital intensity and output per worker extend (26) and (27):
a 1 a An
k/l=a+—| —|+ yk, 37
(; “) u GA)
G-ati/S+yk, (38)
Intensive form of P-2 follows is a system of two ODEs that generalize (28) and (29)
z(1-u) ou
u={f(v)-C-p)la+y a (39)
- ) (=p) +8"
. z-u) 1
=[d- 40
v=[(l-y) s@) 8T- (40)
Workers’ cohesion in the fight for increased relative wage is the stronger the more is Ou /Ou above
zero. On the one hand, the intensity of their competition for jobs is the stronger the deeper is 0v/ dv be-
low zero. In equation G9), f) —© for v1 again.
Proportional control over the net change of relative wage as in P-1 is strengthened by derivative
control over it in (41). Combined proportional and derivative control over the net change of employment
ratio already present in P-1 is mostly retained in (42) although the adjustment coefficient is (1— y)z now
instead of z in P-1. These properties are seen = the corresponding equations:
apse (1—p)a] -(—p)y
“Ce a
: (-w) d, 1a
v=(l-y)z -—jv-———-+v. 42
pal s(u) 2 5 l-u (42)
Thus the scale effect serves as additional fix in P-2 and does not endanger local asymptotic stability
of a new stationary state that will be soon defined.
Instead of assumption z = 1, now this parameter belongs to semi-interval defined as
0<znr= de® /<z=const <1. (43)
Here the left boundary follows from requirement that a stationary relative wage is positive and sta-
tionary capital-output ratio is higher than 1 (see Proposition 2 below). The upper boundary is a bit too
= cps’ (41)
12
high as it permits at the extreme (z = 1) zero private capitalists’ consumption. As z = const < | enables
comparisons of the modified models with the original models which used z = const = | the upper bound-
ary will remain unchanged.
Equating the right parts (39) and (40) zero enables finding nontrivial stationary state
Eq = (uta; Va)- (44)
~ \Si(1+8)
where u,=1- (#5) and v= f[(—p)(a+yB)/1-y)].
For this stationary state, the rate of growth of output per worker, capital intensity and wage is de-
fined
Gy = (k/1)q=Wq=(0+YB/-9), (45)
as well as the stationary rate of growth of fixed production assets and net product is determined
Ky= Ga= 4,+ B = (a+B)/I-y) =. (46)
The stationary capital-output ratio and profit rate are specified as
Sa [W-u,)}*/e, 47)
(l-u,)/sg= diz. (48)
There is stationary employment ratio — stationary relative wage trade-off in P-2: the higher y, the
higher is the first and the lower is the second.
In the “neoclassical” conception, the stationary relative wage ua, being the higher, ceteris paribus,
the higher is 5, aspires to supremum when 6 (Leontief technology with factors complementarity):
sup(ua) = 1 — d/(cz); stationary relative wage uz, being the lower, the lower is 6, aspires to infimum
when 8-0 (Cobb — Douglas production function with perfect factors substitutabity): inf(w.) = 1 — p.
Increase in stationary rate of economic growth d affects the relative remuneration of wu, negatively;
Uq < 1 is true only if d > 0 [5: 225].
Figure 4 and Table 5 exhibit intensive causal structure of P-2.
Relative
SS———— SS)
nad ~ wage
Pea) 3 \
\ te) \
- Employment |
ratio v /
/
/
Figure 4 — A condensed causal structure of P-2 at stationary state £,; total number of
feedback loops — 3, among them: 1“ order — 2 (1 — positive, 1 — negative), 2 order — 1 negative
Table 5. Three intensive feedback loops in P-2 at the stationary state E,
Loops descendant from P-1 New loop R1 of length 1
B2 of length 1
Employment ratio ,——> Net change of v
B1 of length 3 Relative wage u
Relative wage u ——>Net change of v Net change of u
Employment ratio v
Net change of u
We see that this model inherits two negative feedback loops from P-1 and includes the new positive
feedback loop that is destabilizing and results from workers’ joint struggle for higher relative wage.
3.2. Conceptual weakness of P-2 rooted in “neoclassical” beliefs
The positive dependence of @ on v in (27) and (38) did not receive clear explanation in [4, 5]. For fill-
ing this gap this research digs deeper. For brevity and without loss of generality assume again that p = 0.
Then the growth rate of output per worker (38) is presented for P-2 retaining the efficiency wage hy-
pothesis as extension of (35) in P-1
08+ + yk
Ba, LOM WEOYE
1+6
| fas + f(y +dy6]. (49)
1+6
The efficiency wage hypothesis is clearly relaxed in (49) in relation to (35) in P-1.
The three feedback loops reflecting endogenous productivity growth and scale effects in P-1 (Table
4) are also present in P-2.
Table 6 reports on four additional extensive feedback loops positing no serious threat to LAS of E,
(44). L. Aguiar-Conraria [4] does not recognize them. Two latter reflect roundabout weakening and stabi-
lising economies of scale of types I and II in P-2. An analogue of degree of homogeneity for VES produc-
tion function (A, /) is generally not constant and exceeds 1.
Table 6. Endogenous productivity growth and scale effects in P-2
No. Order and po- | Extensive feedback loop involving growth rate of output per
larity worker
* Lt é— > > 11 pu—k
5 T,+ A ne -— Ff
a uuu: ug
6 2, §— i 1 9u—k S19 >) av
q Bo &@— >t >i 9 us 9k 4k) 1— 9 > pv 9H
Fixed capital formation is beneficial for growth, according to (16), (38) and (49). However, the lat-
ter two equations express the direct scale effect not quite correctly. Refinement requires that the growth
rate of output per worker is linked to the growth rate of employment ratio instead of the growth rate of
fixed production assets [19]. In our opinion, @ is in reality directly dependent on the growth rate of capi-
tal intensity besides its dependence on the growth rate of employment ratio.
The scale effects in (16), (38) and (49) require 0 < y < 1 as assumed in P-2 contrasting with P-1,
where y = 0 implicitly. On the other hand, p = 0 in P-2 unlike P-1.
The principal concern will be with the property of homogeneity of the first degree of CES produc-
tion function (16) in relation to /, and k. This property holds in relation to / and k only if y = 0 as in P-1.
14
The scale effect, intended by L. Aguiar-Conraria, violates the distribution of net product between labour
and capital according to their “marginal productivities” ©; u ®, in P-2 for y > 0.
Production function (16) is to be easily expanded in true final terms of / and x instead of /. and k, where
/, is intermediate variable for /. It becomes clear that the expanded VES production function ®(A, /) is not
generally homogenous in terms of / and k, therefore the Euler theorem for homogenous functions cannot be
applied, except the Cobb — Douglas special case with a degree of homogeneity expressed as 1 + y(1-1).
Strictly speaking, the profit maximization condition equates the value of the “marginal product of
labour” with the wage rate under “perfect” competition that is very strong idealisation even of free compe-
tition. Neither [5] nor [4] is explicit on this crucial “neoclassical” assumption, their authors have simply
avoided discussion of this.
Seemingly, the Phillips equation is maintained by assumption of bargaining strength of the workers’
movement [5: 222]. Thus, at least, a countervailing power of trade-unions against monopoly and firms’
monopsony on the labour market is recognizable (cf. [16: xvii]). This recognition is not followed by in-
vestigation of monopoly capitalism in depth. Therefore applying the profit maximization condition devel-
oped for “perfect”? competition creates unresolved logical inconsistencies in P-1 and P-2. Yet the problem
is even more severe — it cannot be removed by simple refinement as the following proposition establishes.
Proposition |. The logically contradictory principle of distribution of national income (net product)
is introduced without any theoretical justification in P-2: on the one hand, wage is determined there by
labour "marginal product" w = ®,, and /w = /®,, on the other hand, profit rate is lower than the "marginal
product" of capital R = M/k = (1-u)/s = Fy < ®, if 1 >y > 0.
Corollary |. Profit is lower than imputed profit: M = (l-u)g = q — Bil < Ok.
Corollary 2. Net product is lower than imputed total income: g < Oj + ®,k.
Corollary 3. The “factor price ratio” exceeds “marginal rate of technical substitution”: w/R > ©, /®, .
So the “marginal rate of technical substitution” is not equal to the “relative unit costs” of the inputs; in other
words, the slope of the isoquant at the chosen point is not any more equal to the slope of the isocost curve.
An easy algebraic proof of this Proposition and its Corollaries is omitted. Proofs are skipped in
similar cases below as well.
The “neoclassical” marginal productivity principle of income distribution is broken in [4] in silence.
In fact, the relative wage wu expresses the value of labour power; the profit of capitalists Mis a trans-
formed form of surplus value S created by the working class: S= (1 — u)L = M/a.
Proposition 2: (a) from the requirement wu, > 0 the first restriction follows 1 > z > a/c > 0; (b)
from the requirement s, > 1 the second (stronger) restriction follows 1 >z> inf(z) =de/ bl.
Proposition 3: (a) stationary capital-output ratio s,, being the lower, the higher is 5, ceteris paribus,
aspires to infimum when 6-00 (Leontief technology with factors complementarity): inf(s,) = 1; (b) sta-
tionary capital-output ratio s,, being the higher, the lower is 6, aspires to supremum when 6—0 (Cobb —
Douglas production function with perfect factors substitutabity): sup(sa) = p1z/d.
Proposition 4. (a) Stationary relative wage wu, and stationary capital-output ratio sq, being the
higher, the higher is rate of accumulation z, achieve maximum when z= 1; (b) stationary employment
ratio vy is independent of z; (c) stationary growth rates 4, =(k/l), =W,, k, = , do not depend on z
too; (d) stationary profit rate d/z, being the lower, the higher is rate of accumulation z, achieves mini-
mum when z= 1. Therefore real capitalists eschew the well-known “golden rule” of accumulation that
requires from them z= | against their material interests (see for details [11]).
Proposition 5. The stronger is the solidarity of workers in struggle for the relative wage, the higher
is 5, and therefore strengthening this solidarity is a means of enhancing stationary relative wage (value of
labour power) ua. This increase has no influence on a stationary rate of return (1 — u,)/sa, as stationary
capital-output ratio s, declines along with accrual in 6.
Proposition 6. For 0<y< 1: (a) LAS of hyperbolic £, in system of (41) and (42) takes place
according to Routh — Hurwitz criterion, if Trace(Jx) < 0 [4: 522]; in particular, (b) Ez is locally stable
15
node for 0 <8< 6), (c) Ey is locally stable focus for 5; < 6 < 52, here 8) is such that [Trace(Jy/2F = \Vu
6 is such that Trace(J,) = 0.
Corollary. LAS of hyperbolic £, is guaranteed as far as
et £0) v0
(l-p)1+8) du,”
Notice that d, v, and u, depend on y (cf. [4: 522]).
LAS of hyperbolic £,, in particular, is true for y= 0, z= 1 and 6 < oo. Similarly, for y = 0 in P-1, LAS
of E, is guaranteed. Stationary state E, (44) becomes neutral centre instead of being hyperbolic similar to
M-1 if y = 0 as in P-1 and there is no workers’ competition for jobs when 6, s, >1, Trace(J,) — 0.
According to [4: 524], “[the] stabilizing effect of introducing some flexibility in the production
function is much stronger than the destabilizing effect of endogenous productivity growth. Only when the
production function is extremely close to a Leontief technology does the system generate perpetual (and
explosive) oscillations.”
Such oscillations with period of 24-45 years require unrealistically low s,,~ 1 for plausible z. If z= 1,
this model, similar to M-1 and P-1, can produce converging fluctuations with period of about 10 years. Thus
for keeping them in life exogenous shocks are necessary as in so-called real business cycles. Sticking to sci-
entific truth, those cycles “of the Frisch type” are not real — they are artificial and ill-defined [21: 227-233].
Y (50)
4. Model Z-1 of industrial cycles as capital accumulation cycles
The reader remembers that the apologetic “marginal productivity” principle underlies P-1. Strictly speak-
ing it must assume “perfect” competition in the factor market which is unrealistic especially under state-
monopoly capitalism. The attempt to bring P-1 closer to reality through modification P-2 in [4] has un-
dermined this principle. It follows from above Proposition | that the "neoclassical" beliefs have moved
P-1 and P-2 to dead end. "Is there no exit?" —"No, it is on the opposite side!"
The model that follows is to a large extent liberated from these erroneous beliefs, in particular, from
the “marginal productivity” principle that still remains as rudiment. The labour theory of commodity and
surplus value is applied as the foundation for Marxian interpretation and for reasonable, still uncompleted,
re-shaping of the preceding models.
4.1. The intensive form of Z-1 with endogenous rate of capital accumulation
The following K. Marx fragment tells how rate of accumulation is determined [1: 634]: “Ifthe quantity
of unpaid labour supplied by the working class, and accumulated by the capitalist class, increases so rap-
idly that its conversion into capital requires an extraordinary addition of paid labour, then wages rise, and,
all other circumstances remaining equal, the unpaid labour diminishes in proportion. But as soon as this
diminution touches the point at which the surplus labour that nourishes capital is no longer supplied in
normal quantity, a reaction sets in: a smaller part of revenue is capitalised, accumulation lags, and the
movement of rise in wages receives a check. The rise of wages therefore is confined within limits that not
only leave intact the foundations of the capitalistic system, but also secure its reproduction on a progres-
sive scale. The law of capitalistic accumulation, metamorphosed by economists into pretended law of Na-
ture, in reality merely states that the very nature of accumulation excludes every diminution in the degree
of exploitation of labour, and every rise in the price of labour, which could seriously imperil the continual
reproduction, on an ever-enlarging scale, of the capitalistic relation.”
The negative feedback of the 3" order containing the rate of accumulation z, employment ratio v
and labour value wu, implicitly expressed by K. Marx, is presented in Figure 5.
J. Robinson, as [17: 423] reminded us, commented in some detail on this part of Marx's theory. She
argued [16: 84-85] that Marx was mistaken in presenting his model as an explanation of the business cy-
cle: “This cycle Marx identifies with the decennial trade cycle. This identification is an error... There may
16
be in reality a cycle of the type which Marx analyses. But if so, it must be of a much longer period than
the decennial trade cycle...”
After demonstrating in preceding research [19, 23] based on K. Marx’s theory that a cycle with
much longer period of about three to five decades is quite possible indeed, the later investigation [13] does
not agree with J. Robinson’s downgrading of the decennial trade cycle identified by K. Marx. The present
paper reinforces support for the Marx industrial cycle.
7 Emp! ratio v|
OO
v
+ +
Growth rate of wage
F
. +
Growth rate of fixed capital k
+
Unit surplus value 1- u
Stationary unit
surplus value 1- uj,
=
——____» Divergence
Accumulation rate z ss
Figure 5 — The negative extensive 3° order feedback loop for rate of accumulation, employment ratio
and value of labour power implicitly expressed in K. Marx [1]
The papers [11-13] have turned rate of accumulation z in a new base (phase) variable. The following
soon equation (51) takes into account, first, in agreement with the views of K. Marx above, that net change
of the share of investment in surplus product has an opposite sign in response to relative wage gains:
2= -b-* 22-2) + P@-2), (51)
—u
where b> 0, p>0, 2, <2)S1<Z.
This equation, second, reflects objective interest of capitalists in the long-term increase of the rate
of profit; restrictions p > 0 and Z, < Zo serve a long run increasing profit rate. Third, the product z(Z — z)
reflects logistical dependence of 2 on z that bounds trajectories in the phase space while a magnitude of Z
codetermines amplitude of fluctuations.
Z-1 extends the equations of P-2 by (51). Z-1 includes the other equations of P-2 without z un-
changed. Z-1 also includes the other equations of P-2 with z — yet now this is phase (level) variable in-
stead of being constant in P-2 (as in M-1 and in P-1 above).
The same static problem of profitability maximization (21) is considered for the given modified
Phillips equation (17) again in Z-1. Although in Z-1, as in P-2, “marginal productivity of capital” exceeds
the profit rate, the rudiment “neoclassical” equivalence of “marginal productivity of labour” and wage
remains in agreement with Proposition | in section 3.2.
Now the intensive form of Z-1 including the renewed investment function consists of the following
three non-linear ODEs: equations (39), (40) and original (51) for rate of capital accumulation z. Accord-
ingly, the interpretation has changed for equations (39) and (40).
The system (39), (40) and (51) has stationary state
E4= Ub» Vo» 26) (52)
where up and v, are the same as uw, and vz in (31) for z = zp, 0< 26> Zeoa)< 1, restrictions on z, also remain
the same as in Proposition 2. Equations (32)-(34) define the stationary auxiliary magnitudes for Z-1 as well.
Figure 6 and Table 7 reflect a condensed causal loop structure of Z-1 near stationary state E;, (52).
Relative
wage u
Bl
Figure 6 — A condensed causal loop structure of Z-1 at £»; total number of
feedback loops — 8, among them: 1* order — 3 (1 — negative, 2 — positive),
2"™ order — 3 (2 — negative, 1 — positive), 3 order — 2 (2 — negative)
Table 7. The intensive feedback loops in Z-1 at stationary state E),
uantity rder ‘ositive feedback loo legative feedback loop
i Ord Positive feedback loop Negative feedback |
a) 1® R1 of length 1 B2 of length 1
uu y—
R2 of length 1
Z>z
3 Pia R3 of length 3 B1 of length 3
u——z 9 z— u—v 9 vu
B3 of length 3
y— zz >
2 am B4 of length 5
uU—Z 929 VP Vu
BS5 of length 5
y—z 3 z— Su
There are three feedback loops inherited from M-1, P-1 and P-2 (B1, B2 and R1) as well as five new
ones (B3, B4, BS, R2 and R3). The three feedback loops from P-1 (Table 4) and the four feedback loops from
P-2 (Table 6) are characteristics of the vicinity of stationary state £) in Z-1. Table 8 reports on three additional
extensive feedback loops that together potentially threaten LAS of £;. The second of them (No. 10) reflects
reinforcing roundabout economy of scale of types I and II, the third (No. 11) — weakening and stabilising
roundabout economy of scale of types I and II. Neither F. van der Ploeg [5] nor L. Aguiar-Conraria [4] recog-
nized these loops. The effects of scale are clearly strengthened in Z-1 with respect to P-2 and P-1.
Table 8. Endogenous productivity growth and additional scale effects in Z-1 at stationary state E,
No. Order and polarity Extensive feedback loop
9 I+ a—>yti pi 2 92 92 9k
10 2,4 (— ot S11 92 92 92 Sk Sh) vw
1 2,- at S12 92 2 9k Bk/1— 9) > Ww
Capitalists’ investment cooperation is the stronger the more 0z/0z exceeds zero, and the intense
competition in this field involves 02 /6z < 0. The first is destabilizing, the second — stabilizing. Notice
that Propositions 1—6 and their Corollaries remain untouched.
Proposition 7. The dynamics of system (39), (40) and (51) linearized in the neighbourhood of its
hyperbolic stationary state E, (52) are LAS provided that 0 < b < bo < b) < «. Then stationary state Ep is
also LAS in the non-linear system (39), (40) and (51). Stationary state E;, is not stable for b = bp in the
linearized system (39), (40) and (51).
Corollary. (a) If the stationary state E;, (52) is LAS, it saves this property if b becomes lower than
its initial magnitude 5;. If the stationary state E; is not, it gets this property if b becomes sufficiently lower
than its initial magnitude 5;. If the stationary state E;, is LAS, it loses this property if b becomes suffi-
ciently higher than its initial magnitude b;. (b) The stationary state E,, is LAS for b = 0 and p > 0. (c) The
stationary state E, is LAS for a special case of Z-1 with b = 0, p = 0 as in M-1, P-1 and P-2.
In our particular simulation run stationary state E; is not stable in linearized Z-1: ap ~ 0.0028 > 0, ay
= 0.8932 > 0, az ~ 0.0032 > 0, aia2— do ~ 0.0000; correspondingly, b; =—37.7085 < b3 =—37.6209 <bo=
54.3987 < by = 54.4863. Still stationary state E,, is stable in nonlinear Z-1 up to Beriticat = bo + 3 = 57.3987.
4.2. Super-critical Andronov - Hopf bifurcation and self-sustained industrial cycles
Proposition 8. The Andronov — Hopf bifurcation does take place in the system (39), (40) and (51) in
a local vicinity of E; (52) at b = bo defined by equation (75).
It has been proved that E;, (52) is locally asymptotically stable for b < bo and that the Andronov —
Hopf bifurcation does take place in the system (39), (40) and (51) at b = bo. According to simulations, a
supercritical bifurcation occurs. The period of oscillations near E; is about 27/,/a,(by) ~ 6.648 (years).
For y = 0.75 and b = Deriticat = 57.3987 > bo = 54.3987, there is a transition to a limit cycle vicinity
(up to years 2200-2230) from the initial phase vector x for 1958.
Consider the conditions in which experimental limit cycle stands idealization of industrial cycle. In
(8) function f(v) = -gtr/(1 — v)? is used. The roughly plausible values prompted by [4, 5] have served in
simulation runs: a = 0.005, B = 0, y= 0.75, 5= 1, €= 0.5, n= 0.3, p=0,p =0.2,c=l, g = 0.04,
r=0.001, d= 0.02, sqa=sy = 1.342, ua=up = 0.776, Va = Vp = 0.871, 2=Zy = 0.12, Z = 1.5, uo = 0.83, 50=
1.764, vo = 0.9, 29 = 0.267, (1 —uo)/so = 0.0964.
For beritical = bo +3 = 57.3987 a supercritical Andronov — Hopf bifurcation happens giving birth to a
limit cycle that depends on the initial vector xo. In addition, for those same parameters and initial condi-
tions, limit cycles in the economic subspace (z < 1) arise even for Deriticat = bo +7.875 = 62.274. Starting
19
year in numerical experiments is denoted for certainty as 1958. To about 2200 and later on movement has
become regularly established near the limit cycle that cannot be reproduced with absolute precision.
Net product reaches its local maximum on the completion of the boom with the onset of the crisis.
Ending the fall of net product g expresses completion of crisis, whereas achieving pre-crisis peak com-
pletes recovery. Depression is defined as phase starting at the end of the crisis and ending before recov-
ery, when capital-output ratio s is (locally) maximal. Cycle phases are reflected in Table 9.
Table 9. Phases of cycle of capital accumulation in the vicinity of the limit cycle, 2221—2227.75
Phase of industrial cycle Phase period Quantity of quarters Beginning | End
Crisis — the 1 phase of the cycle 2221.25-2222 4 1 4
Depression 2222.25-2222.75 3 5 7
Recovery 2223-2223.25 2 8 9
Boom 2223.5-2227.75 18 10 27
2221.25-2227.75
Complete cycle 27 quarters or 6.75 years
The growth rate of net product ranges from —1 to 6 (%/y.), rate of accumulation — from 0.054 to 0.471
(the stationary 0.12 inside this interval), employment ratio — from 0.857 to 0.891 (the stationary 0.871 inside
this interval), relative wage — from 0.819 to 0.825 (above the stationary 0.776), capital-output ratio — from
1.661 to 1.712 (above the stationary 1.341), profit rate — from 0.102 to 0.109 (below the stationary 0.167),
surplus value — from 0.141 to 0.151, labour value of net investment — from 0.008 to 0.071.
During the allied industrial cycles with a period of 6.75 y. in preparing crisis the chronicle sequence is
the following (Table 10): (max 2220) declining rate of accumulation — (max 2220.25) falling rate of profit,
(max 2220.25) falling surplus value, (max 2220.25) declining employment ratio — (max 2220.75) falling
profit — (max 2221) fall of net product. The exit from the crisis involves: (min 2222) increasing net output
— (min 2222.25) increasing profit and rate of accumulation — (min 2222.5—2222.75) the increasing profit
— (min 2222.75—2223) increasing surplus-value — (min 2223—2223.25) increasing employment ratio.
The cyclical dynamics in this model are not strictly wage-lead as in P-1 and P-2: profit falls from
the local high about half a year before total wage, still the latter resumes growth about half a year before
profit. Net product follows total wage with time interval of about one quarter out of the bottom, whereas
production declines few weeks earlier than total wage. In contrast to the above Marx’ views [1], reduction
of wage w does not occur, however, in agreement with his views, employed and unemployed workers' con-
sumption per capita wv declines on the crisis phase and reaches its maximum in the late phase of boom.
Table 10. Phases of cycle of capital accumulation in the vicinity of the limit cycle, 2221—2227.75
Boom (end) Crisis Depression
Year 2221 2221.25 | 2221.75 | 2222 | 2222.25 | 2222.5 | 2222.75
Quarter 1 3 4 5 6 7
qd max | min |
v min
Vv
Ss max
u max
Profit rate R min min
Ss min
M min |
w min
steady growth
wl max | min | max |
wv max | min | max |
4 min
kla min
20
Table 10 (continued)
Recovery Boom
Year 2223 | 2223.25 | 2223.5 | 2224.5 | 2226.25 | 2226.75 | 2227 | 2227.25 | 2227.5 | 2227.75 (end)
Quarter 8 9 10 14 21 23 24 25 26 27
q max | max 2
~ 0 0 max min
v min min max
s min
u min
Profit rate R max
Ss min max max
M max | max 2
2220.5
w max
wl max 2
wv max 2
4 max
kla max
Note. A critical magnitude of parameter b for the Andronov — Hopf bifurcation at E;, (58) is Beritical= 57.3987.
This brief exposition will be continued in next subsection.
4.3. Dual nature of capital as the driver and barrier of capitalist production
Excess of capital arises from the same causes as those which call forth unemployment — complementary
phenomena, footing at the opposite poles.
ao a 1- n , ae . F
Positive declining profit rate R= — (R < 0) is the indicator for relative excess of capital:
Ss
(53)
In P-1, P-2 and Z-1, wand S have always the same sign: sign(a ) = sign(s ); therefore profit rate R
always declines when relative wage u and synchronically capital-output ratio s rise in these models. So
minimum of R happens synchronically with local maximums of u and s, and vice versa for cyclical dynam-
ics in Z-1.
A deeper analysis distinguishes two forms of absolute excess of capital.
1) If the fall in the rate of profit is not compensated through the mass of profit, when the increased
capital produced just as much, or even less, profit than it did before its increase:
M=S+a= g-—* <0 (54)
—Uu
therefore
S<-a, (55)
or
q< . (56)
l-u
2) Similarly, if the fall in the profit share (unit surplus value) is not compensated through the mass
of surplus labour, when the increased capital produced just as much, or even less, surplus-value than it
did before its increase:
S§<0=/-——<0 (57)
21
or
fs". (58)
It develops for constant labour force ( = const) in two identical requirements S$<0
Absolute over-accumulation of type I is sufficient for relative over-accumulation of capital. Indeed,
M= k+R<0 therefore for >0 R<-k <0> R <0.
Let the relative over-accumulation occurs: for k >0, R>0O and R<0.
Relative over-accumulation of capital is not necessarily accompanied by absolute capital over-
accumulation of type I, or the former is not a sufficient condition for the latter: for R < 0, M> Oif
k+R> 0 therefore for k >—R>0.
Absolute over-accumulation of capital of type I is sufficient not only for relative over-accumulation
but for absolute over-accumulation of capital of type II as well if a>0.
Relative over-accumulation does not imply absolute over-accumulation of type II. However, the
former leads the latter, both last approximately the same duration of time in Z-1 (Table 10).
The relative over-accumulation begins on boom phase and ends at the closing stages of the depres-
sion phase. In our simulation run, with one quarter lag absolute over-accumulation of type II starts. The
drop of surplus value begins in the final stages of a boom, continues on the phases of the crisis, depres-
sion and ends at the beginning of the recovery.
Absolute over-accumulation of type II leads absolute over-accrual of type I; absolute over-
accumulation of type I lasts much less than absolute over-accumulation of type II and relative over-
accumulation. Absolute over-accumulation of type I covers last quarter of boom and phase of the crisis;
the previous local Mmax is overcome at the initial stage of the boom, and one quarter before the end its
new high is reached (Figure 7, Table 10).
170 31
30
160 | 99
7
a | 282
3150 =
5 27g
140 26
25
430 pe _ 4
2219 2220 2221 2222 2223 2224 2225 2226 2227 2228
——q ——M
Figure 7 — Highs of profit M lead highs of net product g by one quarter in Z-1, 2219-2228
After reaching wmin simultaneously with Vmax, Smin aNd Smax, on the boom phase, profit rate is maxi-
mal and begins to decline. The absolute decline in profit ends in the first quarter of depression whereas
decline of profit rate stops at the end of depression; surplus value sinks to the bottom in the late phase of
depression and in the early phase of recovery. Employment rate v reaches the floor during recovery, and
the ceiling three quarters prior to the completion of boom along with minimal relative wage u; the maxi-
mal relative wage is reached at the end of depression. Notice that (Umins Umax) correspond to (Vnax» Yin)
in Z-1 unlike M-1.
22:
Per capita consumption of workers vw is highest at the end of boom and it is lowest in the early
stage of crisis. The previous maximum is achieved in the early stage of recovery, after that growth con-
tinues until the end of next boom. Consequently, the improvement of living standard of the working class
is, if using the figurative expression of Marx, a stormy petrel for the new crisis.
From cycle to cycle per capita consumption of workers wv grows mostly due to the long-term
growth of output per worker a. Wage w is rising throughout the cycle (there is lack of realism in Z-1 in
this aspect!).
Growth rate of employment ratio? outpaces employment ratio v by one phase, or a quarter-cycle.
Maximal ?is reached one quarter ahead of highs of v, R, g and k, two quarters — ahead of maximal 4.
The more workers produce surplus value, the faster their wages increase. This model relationship
corresponds, in our view, to the Marxist theoretical status of use-value of the labour power as the unique
source of surplus value. We see that use-value plays outstanding role in political economy! Elevated em-
ployment / pushes record profit M down in boom, and resurgent profit M "drags" employment /, suffered
from lack of demand, up in recovery. Phases of S and w almost matches (Figure 8, Table 10) with some
leadership of the first.
0,045
0,035 ZZ
3
= 0,025
3
0,015
0,005
0,14 0,142 0,144 0,146 0,148 0,15 0,152
Surplus value
Figure 8 — Surplus value S as a factor of growth rate of wage w in Z-1, counter-clockwise, 2219-2227
Relative over-accumulation of capital comes after the on quarter of boom. One quarter later a cycli-
cal maximal surplus value Smax is achieved, employment ratio v also becomes maximal, and then immedi-
ately the absolute over-accumulation of capital of type II starts. At a late boom stage profit peaks at a cy-
clical maximum Mmax and immediately absolute over-accumulation of capital of type I manifests itself.
Very soon after that (through 1—2 quarters) the economy enters crisis (Table 10). It is on the phases of
recovery and boom the three considered forms of over-accumulation of capital are overcome, and capital
accumulation finally temporally accelerates.
Figures 11 and 13 illustrate alleged socio-economic harmony in P-2 for the same segment of time, char-
acterized by jerky movements of main cyclical indicators in Z-1 (Figures 9, 10 and 12). Z-1 does not guaran-
tee capitalist reproduction on constant or increasing scale.
For b = by +3 trajectory on the phase plane converge to a limit cycle, which lasts about 6.75 y., for b
<bo +3 they converge towards stable focus or node; for b belonging to by +3 < b < bo + 7.875 = 62.27 and
growing amplitude of fluctuations on transition to closed orbits increases and reaches economically per-
missible maximum, when b = bo + 7.875 = 62.27 (Zmax= 1), for b > 62.27 fluctuations with greater ampli-
tudes go out the economic region (Za.> 1).
23
2219 2220 2221 2222 2223 2224 2225 2226 2227 2228
aN
—s—vhat —s-GRq —*—GRk
0,06
Figure 9 — Growth rates of employment ratio, net product and fixed production assets in Z-1, 2219-2228
2219 2221 2223 2225 2227
AGN
Figure 10 — Over-production, idle fixed capital and unemployment in crisis:
growth rates of employment ratio, net product and capital-output ratio in Z-1, 2219-2228
0,02
0,018 Serereeesereerers
0,016
0,014
0,012
——vhat 0
0,01
—sghat 0
0,008
——khat 0
0,006
0,004
0,002
O r T T T
2219 2221 2223 2225 2227,
-0,002
Figure 11 — Growth rates of employment ratio, net product and fixed production assets in P-2, 2219-2228
24
2219 2220 2221 2222 2223 2224 2225 2226 2227 2228
0,06 |— A Am
0,04 a \ Va \
oo mee 1s
4 iv
——Profit rate hat —s—Surplus value hat ©—«—Profit hat
Figure 12 — Occurrence of relative and absolute over-accumulation of capital of types I and II in Z-1:
growth rates of profit rate, surplus value and profit, 2219-2228
0,18
0,16
0,14
0,12
0,1 —+— Profit rate 0
0,08 —s— what 0
0,06 —+—Mhat 0
0,04
0,02
0
2219 2221 2223 2225 2227
Figure 13 — Socio-economic harmony in P-2: profit rate and growth rates of profit and wage, 2219-2228
We can describe the worst-case scenario where positive feedback loops (Figures 4 and 6) dominate
over the negative feedback loops (b > bp + 7.875). Such domination leads to collapse without prudent sta-
bilization policies not modelled in this paper. In particular, the stabilization policy elaborated in [13, 24]
could be effectively applied. This policy could raise a long term employment ratio to a target higher than
stationary ones in M-1, P-1, P-2 and Z-1 without lowering a long-term relative wage.
The discredited efficiency wage hypothesis, underlining P-1, P-2 and Z-1, is the particular Achilles
heel of these models and should be overcome in the subsequent research. It neglects a forcible reduction
of wages as the means attempted for cheapening commodities and for increasing profitability [1].
5. Conclusion
This paper rejects J. Robinson’s [16] pessimistic conclusion on the relevance of K. Marx's industrial cycle
theory. Similar to [17], the present paper disagrees with A. Atkinson's [14] denial of Goodwin's [7] rele-
vance for business cycle theory. Employing modifications deeper than those suggested by A. Atkinson
and later in [17] our modified model Z-1 passes the test earlier suggested by G. Low [15] on the length of
trade cycle period. It is demonstrated that at a critical positive magnitude of parameter 5 a super-critical
Andronov — Hopf bifurcation happens and a closed orbit (limit cycle) is generated in the phase space with
a sufficiently shorter period than a period of conservative closed orbits in M-1.
25
As this paper demonstrates further, P-1 and P-2 exclude the existence of big cycles in capitalist
economy, terminologically related to the names of S. Kuznets and N.D. Kondratiev in the literature, not to
mention their neglecting self-sustained industrial cycles. Short-term fluctuations in inventory stocks and
orders are also not considered.
In these models, intensification of competition of workers for jobs strengthens stability of capitalist
reproduction while increasing profitability, on the one hand, and reducing relative wage, on the other.
Strengthening workers’ struggle for higher relative wage, weakening their competition for jobs are desta-
bilizing factors for capitalist reproduction on the increasing scale.
We have also established that [4] contains the unintended significant result: the “neoclassical” rule
of net output functional distribution between labour and capital from P-1 [5] is not structurally stable (bet-
ter to say is conceptually fragile) in P-2 with the additional scale effects. The latter model does not even
attempt to justify social distribution of net product between labour and capital by the apologetic “marginal
productivity” rule.’
Despite some realism uncovered, P-1 and P-2 are empirically and logically controversial. In particu-
lar, these “neoclassical” models with a CES production function in terms of & and /. do not reproduce the
positive association of growth rates of net product and output per worker, the prevalent tendencies of rela-
tive labour compensation and rate of accumulation to fall, endogenous cycles and other regularities of
capitalist economy [11—13, 19, 23, 26].
Extended model Z-1 reflects periodic recurrence of relative and absolute over-accumulation of capi-
tal as well as general over-production, as immanent characteristics of industrial cycle. It is shown that
capitalists’ investment cooperation weakens (competition strengthens) stability of stationary state in Z-1.
Targeted reduction of the rate of accumulation increases profit rate and reduces value of labour power
contrary to the working class interests.
Contrasting with the textbook Solow (1956)-like “neoclassical” models the employment ratio is not
steadily maintained at a constant level in P-1, P-2 and especially in Z-1. Even the stationary rate of unem-
ployment is not “natural” — like the unemployment itself, it is the product of the capitalist mode of pro-
duction, not of Nature.
The unreliable efficiency wage hypothesis, inherited from P-1 and P-2, remains as structural element
within Z-1. Therefore Z-1 is to be liberated from this hypothesis and other remaining conceptual weakness
(including a CES production function in terms of k and /,, wage — “marginal labour productivity” rudiment
equivalence and the like) in subsequent research relying to a large extent on models developed in [11-13].
The flaws in the indigenous "neoclassical" models are mostly due to mixing notions of concrete la-
bour and abstract labour, as well as because of general negation of the K. Marx theory of commodity and
surplus value [1]. The “neoclassical” school and some its “radical” supporters stubbornly reject this theory.
The revealed contradictions between the vulgar concepts and reality as well as specification errors
identified in the main "neoclassical" equations violate one of the fundamental prerequisites for the intelli-
gent application of regression and other econometric methods [21, 26]. Thus, a necessary preliminary step
for subsequent serious statistical investigations has been accomplished.
Non-random, rooted in the material capitalist class interests, nature of the revealed "neoclassical"
contradictions and specification errors should be further considered in thorough econometric studies. Sci-
entifically advanced university courses on micro- and macroeconomics ought to be liberated from the
bourgeois apologetics hidden under superficial and pretentious objectivism.
Marx’ methodology is congruent with the methodology of vanguard system dynamics. By learning
the predecessor’s work seriously and without prejudice, the system dynamics field can flourish further
faster and better. This will be a factor of progressive change in the social production relations.
7 A similar unintended result has been masked in [25] as [26] demonstrates answering on subjective objec-
tions raised in [27]. The models presented in [26] refine and encompass the L. Boggio’s [25] “neoclassical” (very
contradictory) model with alleged increasing returns.
26
Appendix A
A proof of Proposition 6.
Jacoby matrix J, corresponds to stationary state E, (44). It is defined as
1=p(+8 , 38
Cet) 1 2 yoo fv) ——u,>0
Le (l=p)+6 84 (l-p) +5 (59)
2.7 ¢aj-— 2 l-p 1 448, co] - 1 fd, <9
a (I-p)+8 Iu, & * (l-p)+8 1-u, Ma
Notice that Jacoby matrix J, corresponding to stationary state E, (31) in P-1 is a special case of (59)
for y = 0. The reader can use it for checking propositions on LAS of E,.
A characteristic equation based on (59) is
A +bA+b=0, (60)
where
b, =-Trace(J,) =- A=PU+9 2, 1 PO), 1
(I-p)+8 ‘ss, (I-p)+6 1-u,
by = |Ja| = ut LOE T8 a 5 +d) d-1) > 0. (62)
Notation Fal is for determinant of Jacoby matrix Jz.
The characteristic equation (60) has the following two roots with a negative real part for typical pa-
rameters’ magnitudes
2
hor TraceJa) , Trace) ) J). (63)
2 2
Trace(J ,) >
They are real if initially (22) -4l2 0. Then the stationary state is stable node. Otherwise
it is stable focus with a period approximated by
Zz
Ty =2n! yel-( =e) (64)
A proof of Proposition 7. Let Z, = Z—z»,. Jacoby matrix for stationary state E, in Z-1 is defined as
1—p)(1+8 4 Su, Su,
C= py +0) pyd+9) — Ub f'%)—— Ul -pt thee
(l-p)+5 — s, (l=p)+8 z, (l-p)+6
Jim | Zee top iy N48 | __ PO) | _y oy WEP) % | 65)
Sy (I-p)+8 l-u,,° 8 (l-p)+6 1-u, Zz (1-p)+8 sy
-b yy (-p)+8) 7, Z, a _ ) 8u,24Z, b= p)you, Fog. -p
l-u, (l-p)+8 sy Ug (1—p) +8 (I-p)+8 5
The standard characteristic equation of the third order is written as
M +ad +ah+ay=0, (66)
27
where the parameters are calculated based on the corresponding values of some Jacobi matrix Jy
ay = —Vx|= (1 S22 S33 Sto S23 Sa ton S32 Sis —S3. I22 S31 —J03 S32. S412 S21 S33)
a =—-[Jo3 Jn + Jig Jar + Sus Jai Jit (S22. + 433) — Jon S33]
ay = —Trace(Jy) = (J+ Jo + J33)-
The parameters of the characteristic equation based on (65) are defined as
+6
ay(B= a= pF Om Gog ayes tO (67)
“b
1+8 1 l-p)(1+8 z
a()= [7 ia" a Canenerapats tod 1280
‘b
6
=. > DIM G5 Tyre Mo e-ab=e+ 0b, (68)
where o > 0;
_, 2» (=p)(+8) Lf). oe (=p) og
OT Cepyr8 Gapped eu, Ps, MO pe
=c-—hb, (69)
where h > 0.
Lemma |. The quadratic equation based on the above characteristic polynomial
a(b) =a, (b)a,(b)— a= 0, (70)
where
a,(b) =e + ob, (71)
a,(b) =c—hb, (72)
b =-2<0 (73)
oO
by == >0, (74)
always has two real roots:
(75)
Lemma 2. It is true that —oo < b; < min(bs, bo) < max(bs, bo) < by < 00.
Corollary. The conjugate roots of the quadratic equation a(b) = 0 are b3 € (b,,b2) and b3< by € (hb,
bp). It follows from economic requirements that bp € (0, b2).
The Routh — Hurwitz necessary and sufficient conditions for LAS of £, in the linearized system
are satisfied for 0 <b < bo: ao > 0, a,(b) > 0 and a,(b)a,(b) >a. As E, is hyperbolic and LAS, it is LAS
also in the non-linear system, q.e.d.
A proof of Proposition 8. Parameter b engaged in equation (51) serves as the bifurcation (control)
parameter. Consider the stationary state E), of the system (39), (40) and (51) as dependent on b:
x =0=flx, b). (76)
The determinant of the Jacoby matrix J; (65) evaluated at the stationary state E, (52) differs from
zero in our case for any possible stationary state (x) as ao = const > 0 (independently of b). There exists
a unique stationary state x, as changes of b do not affect E,.
It is assumed the following properties are satisfied:
28
(a) the components of the function f(x, 5), corresponding to the system (39), (40) and (51), are ana-
lytic (i.e. given by power series);
(b) the Jacoby matrix J,(bo) has a pair of pure imaginary eigenvalues and no other eigenvalues
with zero real parts (in this case. A} =—dy (bp) < 0);
d(Reay3(0)) _ —(a,'a +a)dy')+ dy!
db 2(a, +a”)
(c) the derivative > 0 (it is the transversality condition);
(d) the stationary state E, is LAS (for 0 <b < bo).
Then, according to the Hopf theorem, there exists some periodic solution bifurcating from x,(bo) at
b = boand the period of fluctuations is about 27/Bo (Bo = A2(bo)/i). If a closed orbit is an attractor, it is
called a limit cycle. The Hopf theorem establishes only the existence of closed orbits in a neighbourhood
of x; at bp, still it does not clarify the stability of orbits, which may arise on either side of bo.
Applying information from the proof of Proposition 7, we establish that conditions (a), (b), (d) of
the Hopf theorem are satisfied at b = bo. In particular, the characteristic polynomial for b = bo is
3 2 2
A+ Ay (Doh + Gy (Bo)A + ag=K [A + Ay (bo)] + 4 (BoA + 4 (b0)]
2
=[A + Gy (bo)][A + 4 (bo)] = 0. (77)
It has the following roots:
dy = a (bo) < 05 (78)
dog = tia (dy) - (79)
It remains only to check that transversality condition (c) is also satisfied. Indeed, for b = bo, a,(b) =
e+obanda(b) =c-—hb,
Reha) _—(om5~ha +0 wz Mm hey (80)
db 2(a, +a") 2a 2
as 0d) (by) = 0, q.e.d. A magnitude of this derivative equals 0.018 in our simulation run.
The supercritical character of the Andronov — Hopf bifurcation has been established only experi-
mentally in multiple simulation runs. An analytical proof of this property still remains a Hercules chal-
lenge. The tools from [28] could be appropriate “ammunition”.
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