1

Contents Stabilization of Industrial Cycles by .................................................................................................................. 1

Profit Sharing Policies Localized near Stationary States ................................................................................... 1

Abstract .......................................................................................................................................................... 2

Introduction .................................................................................................................................................... 2

1. The model of industrial cycles Z-1 ............................................................................................................ 3

1.1.The ancestors of Z-1 ............................................................................................................................ 3

1.2. Model Z-1 of industrial cycles as capital accumulation cycles .......................................................... 5

1.3. Super-critical Andronov – Hopf bifurcations and self-sustained industrial cycles in Z-1 ............... 8

2. “Reverse engineering” in (de)stabilization policies ................................................................................... 9

2.1. A compact measure of intensity of workers' competition for jobs as control parameter in Z-1 ..... 9

2.2. Super-critical Andronov – Hopf bifurcation and self-sustained industrial cycles in Z-1 .......... 13

3. Extending Z-1 by mechanistic profit sharing into Z-2 ............................................................................. 15

3.1 General form of Z-2 with mechanistic profit sharing ........................................................................ 15

3.2. Profit sharing with stationary employment ratio lower in Z-2than in Z-1 .................................... 16

3.3. Profit sharing accompanied by opportunistic employment targeting in Z-2 .................................. 18

3.4. Profit sharing accompanied by targeted employment ratio in Z-2X................................................. 18

3.5. Policy optimization for improper structural setting in Z-2X ............................................................ 19

Conclusion ................................................................................................................................................... 21

Appendix 1 for Z-1 ...................................................................................................................................... 23

A.1.1. Z-1 with b as control parameter .................................................................................................... 23

A.1.2. Z-1 with '( )b bf v v as control parameter ............................................................................ 24

Appendix 2 for Z-2 .................................................................................................................................... 26

Appendix 3 for Z-2 .................................................................................................................................... 28

References .................................................................................................................................................... 29

Stabilization of Industrial Cycles by Profit Sharing Policies Localized near Stationary States

©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 Lavrentiev Avenue Novosibirsk 630090 Russia E-mail address: ryzhenko@ieie.nsc.ru

2

Abstract This paper illustrates how dangerous linear thinking and linear control could be if over-

stretched. It takes a three-dimensional Goodwinian model of industrial cycles as experi-

mental tool and demonstrates that effective stabilization of industrial cycles by standard prof-

it sharing policies is feasible mostly near stationary states. Yet stabilization fails in bringing

model economy to a higher target employment ratio distant from an initial stationary one. It

has been found out that if an initial displacement from a stationary state with high target em-

ployment ratio is not minuscule accumulation rate and other variables behave erratically and

leave a region of economic viability. The paper calls for organic profit sharing through pro-

portional and derivative control over growth rate of surplus value connected with target em-

ployment ratio and with growth rate of this ratio by appropriate feedback loops. Workers’

competition for jobs will be much weaker at the same stationary state (with target employ-

ment ratio X = 0.95) under organic profit sharing than under mechanistic one. Only truly dia-

lectic system dynamics approach is capable to find out badly needed robust non-linear con-

trol through designing interwoven feedback loops with appropriate gains.

Introduction

Different types of stabilization policies have been elaborated for growth cycles in the litera-

ture. Some of them choose policies focused on governmental taxes and expenditures, and/or

on interest rates and money supply, other select deeper policies involving workers’ competi-

tion for jobs and primary national income distribution, in particular, between wages and

profits. A prominent approach with a long history has the focus on profit sharing as a means

for growth cycles stabilization.

The industrial cycles are middle-term cycles with a typical duration between roughly 5

and 12 years. They are characterised not only by regular fluctuations of positive growth rate

of net output but by negative growth rates of net output in crises. This distinction makes

solving problem of stabilization of industrial cycles more difficult than that for growth cy-

cles.

Two- and four-dimensional Goodwinian models with standard profit sharing (SPS)

were developed by Lordon [1] and by Fanti and Manfredy [2], respectively. One of these

models’ main paradoxes resides in stabilization policy that governs economy to lower em-

ployment ratio in the long term than before the policy onset. Still there is no conscious tar-

geting of employment ratio in these papers. This is because of predominance of the dogma of

natural rate of unemployment taken uncritically from the mainstream economics. The divide

these papers establish between long term steady state growth and jobs creation deserved

careful consideration [3].

Two closely related “neoclassical” models of economic growth (1st with hidden, 2

nd,

more general, with intended) economies of scale are considered in [4]. The main variables

are relative wage and employment ratio, whereas a ratio of investment to profit is constant.

The spurious efficiency wage hypothesis [5, 6] supports equations for a growth rate of output

per worker. Workers’ competition for jobs (neglected in [1] and [2]) is stabilizing and their

fight for increased wages is destabilizing as revealed. In each model, a stationary state is lo-

cally asymptotically stable in a system of two ODEs. Deceptively, there is no possibility for

endogenous industrial cycle.

3

A 3rd

extended model, containing the greed feedback loops, reflects the destabilizing

cooperation and stabilizing competition of investors. In a system of three ODEs, rate of capi-

tal accumulation has become the new phase variable. Its targeted long-term decrease raises

profit rate together with reducing relative wage and capital-output ratio. Oscillations imitat-

ing industrial cycles are endogenous. Crisis is a manifestation of relative and absolute over-

accumulation of capital. Limit cycle with a period of about 7 years results from supercritical

Andronov – Hopf bifurcation.

The present paper takes the latter model as experimental tool and demonstrates that ef-

fective stabilization of industrial cycles by standard profit sharing policies is feasible mostly

near stationary states mostly because of strong positive non-linear dependence of a growth

rate of real wage on employment ratio in a Phillips equation. Consequently intensity of

workers’ competition for jobs is very sensitive to changes in the employment ratio that re-

stricts the region of successful application of linear control to vicinity of stationary states.

Yet stabilization fails in bringing model economy to higher target employment ratio

distant from an initial stationary one. It has been found out, in particular, that if an initial

displacement from a stationary state with high target employment ratio is not minuscule ac-

cumulation rate and other variables behave erratically and leave a region of economic viabil-

ity.

This restriction requires application of organic profit sharing instead of rather mechanis-

tic surrogates considered. Socially efficient stabilization policies necessitate linking class

distribution of national income with employment benchmarks through a system of appropri-

ate feedback loops with pertinent loop gains contrary to economic systems with mechanistic

profit sharing taken as a subject of research in the present paper. Intensive workers’ competi-

tion for jobs will be substantially weakened under stronger social cohesion and greater social

efficiency of capital accumulation than considered in the present paper.

1. The model of industrial cycles Z-1

1.1.The ancestors of Z-1

P-1 adds production factors substitution with a help of a “neoclassical” CES production

function [8] to Goodwin growth cycle model M-1 [7].

Table 1 lists variables of P-1 and subsequent models, considered in the present paper.

Table 1. Main variables in Z-1 as generalization of P-1 and P-2

Variable Expression

Net product q

Fixed production assets k

Capital-output ratio s = k/q

Employment l

Employment in efficiency units le

Output per worker a = q/l

Labour force tenn 0 , ≥ 0

Wage w

Total wage wl

Relative wage (unit value of labour power) u = w/a = wl/q

4

Profit M = q – wl = (1– u)q

Profit rate R = (1– u)/s

In [8, 9] a CES production function is applied for determining net product

/1

))(1()(),( eeκe lmkmclkFq , (1)

where, according to “neoclassical” interpretation, is distribution parameter, 0 < < 1, c is

efficiency parameter and is substitution parameter.

This function is homogeneous, i.e., there are constant returns to scale by the standard

definition. This definition overlooks scale effects maintained by specific feedback loops as

[4] demonstrated. Function (1) has also a property of constant elasticity of substitution (CES)

between labour power (in efficiency units) and fixed production assets1 according to their al-

leged “marginal productivities” (el

F and kF ) under static conditions:

ln( / ) 10 1

ln 1

ed k l

d

for el

k

F

F .

Parameters c, mk and me help to harmonize units of measurement, each of the latter two

equals 1, both are skipped for brevity. Function (1) allows considering variable capital-

output ratio s, unlike M-1.

For 1 )0( this function is transformed into the Cobb – Douglas function, par-

ticularly, q = 1

eck l ; for 0 ( ) it becomes the Leontief technology, where

Min( , )eq ck cl , so capital-output ratio s = const = 1/c. The first case represents perfect fac-

tors’ substitutability in [8, 9], the second case their perfect complementarity as in M-1.

A simplified Phillips equation defines the growth rate of wage

ˆ ( )w f v , (2)

where ( ) 0,f v for 1v ( )f v .

For certainty a specification satisfying these requirements is applied in all the models

2

( )(1 )

rf v g

v

. (3)

A static problem of profitability maximization is considered for the Phillips equation.

Equating the “marginal rate of technical substitution” with the factor price ratio

//l k w R necessitates in P-1 the shaky hidden assumption of “perfect” competition.2

The latter is utter idealisation even for free competition and is untrue for state-monopoly capital-

ism.

Dynamics in P-1 are typically converging to stable node compared with self-sustained

oscillations around a neutral centre in M-1. Check of P-1 structural stability in P-2 with addi-

tional scale effects is carried out in [9].

The definition of employment in efficiency units le in Table 1, unlike P-1, takes into ac-

1 The papers [8, 9] recite uncritically the incorrect “neoclassical” notion of capital-labour substitution [5].

2 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 [4] demonstrated.

5

count direct scale effect

0   ( / )te le kl k , > 0. (4)

This newly defined  el is the factor of CES production function (1).

A static problem of profitability maximization is considered again for the same Phillips

equation. The scale effect, intended in [9], violates the distribution of net product between

labour and capital according to their “marginal productivities” l and k in P-2.

Production function (1) is to be easily expanded in true final terms of l and k instead of  el

and k, where  el is intermediate variable for l. This expansion reveals that (1) in these true terms

is production function (k, l) with variable elasticity of substitution (VES).

It becomes clear that (k, l) is not generally homogenous in terms of l 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–). 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.

The growth rate of output per worker is presented for P-2 retaining the efficiency wage

hypothesis as extension of a similar equation in P-1

ˆˆ

ˆ1

w ka

=

= 1 ˆ[ ( ) ]

1f v k

. (5)

The efficiency wage hypothesis is clearly relaxed in P-2 in relation to P-1 (with = 0).

Still dynamics in P-2, as in P-1, are typically converging to stable node or focus.

According to [9: 524], “[the] stabilizing effect of introducing some flexibility in the

production function is much stronger than the destabilizing effect of endogenous productivi-

ty 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 stationary capital

output ratio as ≈ 1 for plausible accumulation rate 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

scientific truth, those cycles “of the Frisch type” are not real – they are artificial and ill-defined

[10: 227–233].

1.2. Model Z-1 of industrial cycles as capital accumulation cycles

An intensive form of P-2 is a system of two ODEs that generalize equations of P-1

(1 )

( )( ) 1

z u uu f v

s u

, (6)

ˆ

1– –(1 ) 1

( ) ( )( )

1

z u uv

s uv

u

. (7)

6

The paper [4] has turned rate of accumulation z, or the share of net investment in surplus

product, in a new phase variable. The following soon ODE (8), first, takes into account, in

agreement with the views of K. Marx [11], that net change of the share of investment in sur-

plus product has an opposite sign in response to relative wage gains.

The negative feedback of the 3rd

order containing the rate of accumulation z, employ-

ment ratio v and labour value u, implicitly expressed by K. Marx [11: 634], is added to P-2.

Net change of the share of investment in surplus product has an opposite sign in response to

relative wage gains as surmised:

 ( )1

bu

z b pz Z zz zu

, (8)

where b ≥ 0, p > 0, 0 1 b Zz z .

This equation, second, reflects objective interest of capitalists in the long-term increase

of the rate of profit; restrictions p > 0 and 0bz z serve a long run increasing profit rate.

Third, the product z(Z – z) reflects logistical dependence of z on z that bounds trajectories in

the phase space while a magnitude of Z codetermines amplitude of fluctuations. Thus, Z-1

extends the equations of P-2 by (8).

The same static problem of profitability maximization from P-1 and P-2 is considered

for given Phillips equation (2) again in Z-1. Although in Z-1, as in P-2, “marginal produc-

tivity of capital” exceeds the profit rate, the rudiment “neoclassical” equivalence of “margin-

al productivity of labour” and wage remains.

The system (6)–(8) has stationary state

  ), ,( bb b bE u v z , (9)

where

/(1 )

1/(1 )   1  –bb

du

cz

,

1 ˆ( )b bv f a , bz is from (8).

Stationary rate of growth of output per worker, capital intensity and wage is defined as

ˆˆ ˆ( / ) ( ) / (1 )b b ba k l w . (10)

Stationary rate of growth of fixed production assets and net product is determined

 ˆ /   .ˆ ˆ ( ) (1 )      b b bk q a d (11)

Stationary capital-output ratio and profit rate are specified as

1/[ / (1 )] /b bs u c , (12)

(1 ) / /b b b bR u s d z . (13)

There is stationary employment ratio – stationary relative wage trade-off in Z-1: the

higher the higher is the first and the lower is the second. For specification (3) of (2) we

have 0bv

g

and 0bv

r

.

In the “neoclassical” conception, the stationary relative wage bu , being the higher, ce-

teris paribus, the higher is , aspires to supremum when→∞ (Leontief technology with fac-

tors complementarity): ) 1 / ( )( bs d zup u c ; stationary relative wage bu , being the lower,

7

the lower is , aspires to infimum when→0 (Cobb – Douglas production function with per-

fect factors substitutability): ( ) 1binf u .

Increase in stationary rate of economic growth d affects relative wage bu negatively;

1bu is true only if d > 0.

Ceteris paribus, the higher is rate of capital accumulation bz , the higher are stationary

relative wage bu and capital-output ratio bs and the lower is stationary profit rate bR .

Figure 1 and Table 2 reflect a condensed causal loop structure of Z-1 near bE (9).

Figure 1 – A condensed causal loop structure of Z-1 at bE ; a total number of

feedback loops – 8, among them: 1st order – 3 (1 – negative, 2 – positive),

2nd

order – 3 (2 – negative, 1 – positive), 3rd

order – 2 (2 – negative)

Table 2. The intensive feedback loops in Z-1 at stationary state bE

Quantity Order Positive feedback loop Negative feedback loop

3 1st R1 of length 1

uu R2 of length 1

zz

B2 of length 1

vv

3 2nd

R3 of length 3

uzzu

B1 of length 3

uvvu

Relative

wage u

Employment ratio v

+

-

B1

R1

B2

+

-

Accumulation

rate z +

-

-

+

-

R2

R3 B3

B5

B4

8

B3 of length 3

vzzv

2 3rd

B4 of length 5

uvvzzu

B5 of length 5

vuuzzv

Note. R2 and R3 are greed feedback loops in Z-1.

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, B5, R2 and R3). Neither F. van der Ploeg [8] nor L. Aguiar-Conraria [9] rec-

ognized these loops. The effects of production scale are strengthened in Z-1 with respect to P-2

and P-1.

1.3. Super-critical Andronov – Hopf bifurcations and self-sustained industrial cycles in Z-1

Parameter b from (8) has been taken as bifurcation parameter at first. It has been proved that

bE (9) is locally asymptotically stable for b < b0 and that the Andronov – Hopf bifurcation

does take place in the system (6)–(8) at b = b0. According to simulations, a supercritical bi-

furcation occurs at b = bcritical > b0 [4].

Most essential Propositions 1–2 and preceding Lemmas 1 and 2 for Z-1 are posted in

Appendix 1. Similarly, Appendixes 2 and 3 contain Lemmas and Propositions for subsequent

models complementing those in the main sections.

A starting year in numerical experiments is denoted for certainty as 1958. For = 0.75

and b = bcritical = 57.3987 > b0 = 54.3987, there is transition to a limit cycle vicinity (up to

years 2200–2230) from the initial phase vector x(1958). The period of oscillations near bE is

about )(/2 01 ba ≈ 6.648 (years).

Net product reaches its local maximum on completion of the boom with the onset of the

crisis. Ending the fall of net product q expresses completion of crisis, whereas achieving the

pre-crisis peak completes recovery. Depression is defined as a phase starting at the end of the

crisis and ending before recovery, when capital-output ratio s is (locally) maximal.

Positive declining profit rate 1

u

Rs

( ˆ 0R ) is the indicator for relative excess of capi-

tal. The latter can be circular and/or cyclical.

A deeper Marx’s analysis in the third volume of “Capital” distinguishes two forms of

absolute excess of capital:

1) of type I, 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;

2) of type II, 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.

The relative over-accumulation begins in Z-1 on the boom phase and ends at the closing

stages of the depression phase. In the simulation run, with one quarter lag absolute over-

9

accumulation of type II starts. The drop of surplus value begins in the final stages of the

boom, continues on the phases of the crisis, depression and ends at the beginning of the re-

covery.

Relative over-accumulation of capital comes after the 2nd

quarter of the boom. One

quarter later a cyclical maximal surplus value Smax is achieved, employment ratio v also be-

comes maximal, and then immediately absolute over-accumulation of capital of type II starts.

At a late boom stage profit peaks at a cyclical 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. It is on the phases of recovery and boom the three con-

sidered forms of over-accumulation of capital are overcome, and capital accumulation finally

temporally accelerates [4].

Positive feedback loops dominate over the negative feedback loops in a worst-case sce-

nario when b > b0 + 7.875. Such domination leads to collapse without prudent stabilization

policies. In particular, socially efficient stabilization policies elaborated in [3] could be effec-

tively applied. These policies 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 stationary relative wage or

stationary accumulation rate.

The discredited efficiency wage hypothesis [5, 6] underlining P-1, P-2 and Z-1, is the par-

ticular 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 [11] especially under tough international competition.

2. “Reverse engineering” in (de)stabilization policies

2.1. A compact measure of intensity of workers' competition for jobs as control parameter in Z-1

Each serious researcher after heavy analytical work wants to check its results. Afterwards the

logic of negation manifests itself. Any proposition (thesis) can be reversed dialectically into

the opposite that enables synthesis of theses on next step. This process has not to be only an

intellectual exercise – it ought to reflect the essential properties of real object.

The Propositions on Andronov – Hopf bifurcation (AHB) in Z-1 suggest looking for an

appropriate control parameter that can be instrumental either in stabilization of closed orbits

or in destabilization of them. The most appropriate candidate for such endeavour is a syn-

thetic parameter that measures intensity of workers’ competition for jobs at a stationary state.

The scrutiny of the elementary parameters listing suggests stationary employment ratio

bv . Fortunately, this parameter enters the Routh – Hurwitz conditions of a stationary state’s

local stability always multiplied by derivative of function f(v) calculated at same bv . This

valuable property, given positive monotonous dependence of '( )bf v on bv , permits taking

their product '( )b bf v v as, first, the compact measure of intensity of workers’ competition

for jobs, and, second, as the synthetic bifurcation parameter that can be reduced to the multi-

pliers in further analysis. In other words, the Routh – Hurwitz conditions condense infor-

mation contained in the Jacoby matrix with its redundancy.

10

It is reasonable to conceive a fuller measure of intensity of workers' competition for

jobs than the proposed compact measure above – not only for a stationary state. In the au-

thor’s opinion, a comprehensive measure of intensity of workers' competition for jobs is v

v

> 0 for v

v

< 0. Its stationary magnitude equals 22J in (28).

The comprehensive measure of intensity of workers' competition for jobs for a station-

ary state coincides with the preliminary measure multiplied by coefficients depending on .

The lower is comprehensive measure of stationary intensity of workers' competition for jobs

22J for given compact measure , the higher is (and the lower is elasticity of the factors

substitution).

Stationary state bE (9) moving along changes in is denoted as E . Thereby only the

stationary employment ratio is affected in the modified model denoted as Z-1.

Lemma 3. The bifurcation magnitude under conditions of AHB for parameter b = b0 and

all other conditions taken unaltered from Z-1 is determined as the product congruent with E

(9) in Z-1

0 '( )b bb f v v . (14)

Lemma 4. Increases in , are accompanied by gains both in '( )bf v and bv for the same

bu and bz , that are components of stationary state E .

Proposition 3. Increases (even tiny) in over 0 are stabilizing: they reinstate local as-

ymptotic stability (LAS) of stationary state E with accruals in bv . Closed orbits turn into

transients to stable node or stationary focus. Decreases (even tiny) in compared to 0 are

destabilizing. Closed orbits turn into transients to a non-economic region in the phase space.

Proposition 4. There is no guarantee that through change of a magnitude of control pa-

rameter the economy can be feasibly moved to stationary state E with target employ-

ment ratio X. For a substantial difference between target employment ratio X and initial bv a

transient can leave economic region in the phase space. Thus transition to a distant attractor

will require a profoundly different (upgraded or, in other words, organic) stabilization policy

from [3] superior to the suggested rather mechanistic one.

Let b = b0 = 54.3987. Then AHB takes place in a simulation run in Z-1 at the following

critical parameters’ magnitudes (variable’s index i corresponds to its initial magnitude):

00.7466 0.8095 critical i , 00.037 0.04critical ig g g ,

0.8675   0.8709bv v , 0.8606   0.9 5'( ) ' 2( 9)bf v f v .

Proposition 5. The dynamics of system (6)–(8) linearized in the neighbourhood of its

hyperbolic stationary state E (9) are LAS provided that 0 2 0 . Then sta-

tionary state E is also LAS in the non-linear system (6)–(8). Stationary state E is not sta-

ble for 0 in the linearized system (6), (7) and (8).

Corollary. If stationary state E (9) is LAS, it saves this property if becomes higher

than its initial magnitude i. If stationary state E is not LAS, it gets this property if be-

11

comes sufficiently higher than its initial magnitude i. If stationary state E is LAS, it loses

this property if becomes sufficiently lower than its initial magnitude i.

Figure 2 reflects transformation of closed orbits generated in result of AHB in Z-1 ei-

ther in convergent fluctuations if is sufficiently high or, if is sufficiently low, – in steady

fluctuations that are partially outside economic region since accumulation rate z exceeds the

upper limit of one in Z-1.

Figure 2 – Dynamics of accumulation rate z: cycles with steady period and amplitude in re-

sult of AHB for g = 0.04, = 0.8095 and b = bcritical = 57.4 in Z-1;

these cycles are stabilized if g = 0.05 and = 1.0313 or go out economic region (z > 1)

if g = 0.03 and = 0.6071 in Z-1

Figure 3 – Stabilization of industrial cycle illustrated by employment ratio v striving to

z

2

1.5

1

0.5

0

1958 1966 1974 1982 1990 1998 2006 2014 2022 2030

Time (Year)

z : Z-1 3-dim AHB b 574 g-

z : Z-1 3-dim AHB b 574 g+

z : Z-1 3-dim AHB b 574

v v eq

1

1

0.9

0.9

0.8

0.8

1958 1990 2022 2054 2086 2118 2150 2182 2214

Time (Year)

v : Z-1 3-dim AHB b 574 g+

v eq : Z-1 3-dim AHB b 574 g+

12

stationary magnitude 0.8804bv for = 1.0313 in Z-1,

when b = bcritical = 57.4 as AHB in Z-1 requires

Figures 2 and 3 illustrate stabilization of industrial cycle at the same stationary accumu-

lation rate zb = 0.12 and at higher stationary employment ratio 0.8804 than initial one 0.8709

for g = 0.05 > g0 = 0.04 and = 1.0313 > 0 = 0.8095 in Z-1. Obviously, this stationary

state E is stable focus in this case.

Yet there can be wrong “angle of attack” in Z-1if the goal is hasty stabilization of in-

dustrial cycles at an elevated target employment ratio, say, 0.95bv X starting from x0

= x(1958) as above. It is easy to calculate the necessary g = 0.38 >> g0 = 0.04 and = 15.2

>> 0 = 0.8095; b0 = 949.89 for AHB at E (far outside economically possible).

Figure 4 is apparent evidence that this ambitious stabilization task is successfully solved:

the target employment ratio is achieved with high accuracy not later than in 1961 after very

moderate overshoot in the preceding few years. Still this success is illusory and even fatal be-

cause, firstly, the intensity of workers' competition for jobs becomes immensely tougher (re-

flected by suggested comprehensive measure 0v

v

at 0.95bv X ) and, secondly, be-

cause accumulation rate z goes through the upper limit of one at the very beginning of this ill-

defined stabilization policy.

Figure 5 illustrates this over-shoot of z in Z-1, impossible in reality, compared to its

practically feasible dynamics on a transient to closed orbits in result of AHB in Z-1. Because

of such reckless attempt of the surge in the accumulation rate, the economy will nose-dive like

an aircraft moving with “wrong angle of attack” against headwind.

v

1

0.95

0.9

0.85

0.8

1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978

Time (Year)

v : Z-1 3-dim AHB b X 1

v : Z-1 3-dim AHB b 574

13

Figure 4 – Dynamics of employment ratio v through industrial cycles born in Z-1 in result of

AHB when b = bcritical = 57.4 versus deceptive stabilization of industrial cycles illustrated by

employment ratio v striving to stationary magnitude 0.95bv X for = 15.2 in Z-1

Figure 5 – Dynamics of accumulation rate z through industrial cycles born in Z-1 in result of

AHB when b = bcritical = 57.4 versus deceptive stabilization of industrial cycles illustrated by

accumulation rate z for = 15.2 in Z-1

2.2. Super-critical Andronov – Hopf bifurcation and self-sustained industrial cycles in Z-1

Proposition 6. The Andronov – Hopf bifurcation (AHB) does take place in the system (6)–

(8) in a local vicinity of E (9) at 0 .

It has been proved that E (9) is locally asymptotically stable for 0 and that AHB

does take place in the system (6)–(8) at 0 . In a particular simulation run, stationary

state E is not stable in linearized Z-1: a0 ≈ 0.0028 > 0, a1 ≈ 0.8931 > 0, a2 ≈ 0.0031 > 0,

a1a2 – a0 ≈ 0.0000; correspondingly, 1 = 0.0093 ≈ 3 = 0.0093 < 2 = 0.8081 < 0 = 0.8095.

Still stationary state E is stable in nonlinear Z-1 up to AHB taking place at

0 –  0.0629   0.7466critical . There is a transition to limit cycle vicinity (up to years

2200–2230) from the initial phase vector x for 1958.

According to simulations, a supercritical bifurcation occurs. The period of oscillations

near E is about 1 02 / ( )a y ≈ 6.648 (years) – in fact about 7.25 years at the limit.

Consider the conditions in which experimental limit cycle stands idealization of indus-

trial cycle. The roughly plausible values prompted by [8, 9] have served in simulation

z

2

1.5

1

0.5

0

1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978

Time (Year)

z : Z-1 3-dim AHB b X 1

z : Z-1 3-dim AHB b 574

14

runs:= 0.005, = 0, = 0.75, = 1,= 0.5, = 0.3, p = 0.2, c =1, gcritical = 0.037 < gi =

g0 = 0.04, r = 0.001, d = 0.02, 1  .342a bs s ,  0.776a bu u ,  0.871a bv v (for g0), z =

zb = 0.12, Z = 1.5, u0 = 0.83, s0 = 1.764, v0 = 0.9, z0 = 0.267, R0 = (1 – u0)/s0 = 0.0964.

As said, for 0 –  0.0629   0.7466critical a supercritical AHB happens giving

birth to a limit cycle that depends on initial vector x0. In addition, for the same parameters

and initial conditions, limit cycles in the economic subspace (z ≤ 1) arise even for lower

0 –  0.1038  0.7057critical and for gcritical = 0.035, 0.8651bv , 0.8157'( )bf v –

all these lower than the above magnitudes.

To about 2100 and later on movement has become regularly established near the limit

cycle that cannot be reproduced with absolute precision. The period of settled cycles (closed

orbits) is about 7.25 years (Figure 6).

Figure 6 – Growth rates of profit rate, surplus value, profit, employment ratio,

fixed capital and net output in simulated industrial cycle with a period of about 7.25 years

for gcritical = 0.037 and b = 54.4 in Z-1

Interestingly, for the whole period u and s are above their stationary magnitudes, the

correspondingly profit rate is below its stationary magnitude, whereas v, z, q , k , a encircle

their stationary magnitudes with mean values very close to stationary ones. The mean value

of z = 0.205 substantially exceeds zgoal = 0.12. These peculiarities illustrate the distinct prop-

erties of closed orbits that distinguish them from dynamics closer to the LAS stationary state

like node or focus.

Under the conditions of AHB in Z-1 with control parameter’s magnitude b = 54.3987,

we have:

11 0.0521J , 22 –1.8101J , 33 1.75449J , 1 –0.0031bTrace J ,

Leads in cycle

0.080.080.080.080.080.08

-0.08-0.08-0.08-0.08-0.08-0.08

6

6

6

6

6

5

55

5

5

4

4

44

4

3

33

3

3

2

22

22

2

1 1

1

11

1

2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230

Time (Year)

Profit rate hat : Z-1 3-dim AHB b 544 g AHB+ 1 1 1 1Surplus value hat : Z-1 3-dim AHB b 544 g AHB+ 2 2 2Profit hat : Z-1 3-dim AHB b 544 g AHB+ 3 3 3 3vhat : Z-1 3-dim AHB b 544 g AHB+ 4 4 4 4Khat plan : Z-1 3-dim AHB b 544 g AHB+ 5 5 5 5Phat plan : Z-1 3-dim AHB b 544 g AHB+ 6 6 6 6

15

whereas real part of both complex-conjugate roots is zero. Thus for 0 very strong and de-

stabilizing investment co-operation of capitalists together with rather weak destabilizing co-

operation of workers in struggle for relative wage almost outweigh very stabilizing workers’

competition for jobs. Thus the latter fix becomes unreliable under that setting. SPS as the

well-known additional fix suggests itself.

3. Extending Z-1 by mechanistic profit sharing into Z-2

3.1 General form of Z-2 with mechanistic profit sharing

Denote Z-1 extended by standard profit sharing (SPS) as Z-2. Specifications (variants) of

Z-2 will be introduced below.

SPS (honestly, rather mechanistic) is reflected in the literature by additional terms in an

extended Phillips equation

1ˆ ( )

( )

uw f v

s u

. (15)

Hereby a growth rate of wage becomes the sum of bargained mw and stimulating

bw

terms

ˆ ˆ ˆm bw w w , (16)

where standard Phillips equation (2) governs bargained wage term whereas the stimulating

term is governed by the difference 1

( )

u

s u

in (15).

It is intended although not guaranteed that these two terms are positive or at least non-

negative. In particular, ˆ 0bw turns profit sharing into the opposite and strengthens labour

alienation in the production processes (see subsection 3.5).

For avoiding ˆbw < 0 an additional constant can be added to this term and the same con-

stant subtracted from bargained term mw for same (15). Still this modification is reliable only

near a stationary state and does not guarantee ˆ 0bw on a transient to this state.

The expansion of growth rate of output per worker (5) is also transformed

ˆˆ

ˆ1

w ka

=

=1 1 ˆ( )

1 ( )

uf v k

s u

=

= 1 1 (1 )

( )1 ( ) ( )

u z uf v

s u s u

. (17)

The new equation for relative wage can be easily derived from (15) and (17):

ˆu uu = ˆ ˆ( )w a u =

=(1 ) (1 )

( )( ) 1 ( ) 1

z u u u uf v

s u s u

. (18)

ODEs for v and z – (7) and (8) – remain the same.

16

Stationary growth rate of real wage is the same as stationary growth rate of output per

worker

1ˆ ˆ( )

( )

bb b b

b

uw f v d a

s u

. (19)

Consequently, new stationary employment ratio is determined

1 1ˆ( , )b bb b

d dv f a f d

z z

. (20)

Stationary relative wage bu , accumulation rate zb and other stationary magnitudes re-

main the same as in (9)–(13).

SPS creates four additional extensive feedback loops in Z-2 (Tables 3 and 4). The ef-

fects of these additional loops are explored below.

Table 3. Two new negative extensive feedback loops in Z-2 with SPS

Loop N1 of length 5 – negative Loop N2 of length 6 – negative

Profit rate

Growth rate of stimulating wage term

Growth rate of wage

Growth rate of relative wage

Net change of u

Relative wage u

Profit rate

Growth rate of stimulating wage term

Growth rate of wage

Growth rate of relative wage

Net change of u

Relative wage u

Capital-output ratio s

Note. Only a negative first partial derivative is explicitly shown as an arrow.

All other first partial derivatives are positive.

Table 4. Two new positive extensive feedback loops in Z-2 with SPS

Loop P1 of length 6 – positive Loop P2 of length 7 – positive

Profit rate

Growth rate of stimulating wage term

Growth rate of wage

Growth rate of output per worker

Growth rate of relative wage

Net change of u

Relative wage u

Profit rate

Growth rate of stimulating wage term

Growth rate of wage

Growth rate of output per worker

Growth rate of relative wage

Net change of u

Relative wage u

Capital-output ratio s

The intensive form of Z-2 is comprised of (18), (7) and (8). Stationary bu and bz are

the same as in (9), whereas stationary bv is determined by (20) and can differ from bv in (9).

3.2. Profit sharing with stationary employment ratio lower in Z-2than in Z-1

Consider special case of (15) with = 0. Then stimulating term ˆ 0bw in prescribed agree-

ment with SPS.

Lemma 5. For in (15) andthe following properties of the basic func-

tional forms are satisfied (substances A, B, C).

17

A) New stationary employment ratio (21) as a special form of (20) is lower than the former

(9) if

( )bv =1 ˆ( )b

b

df a

z

< bv = 1 ˆ( )bf a . (21)

B) Derivative of the Phillips function (2) becomes lower at the new stationary employment

ratio than at the former: 0 [ ( )] ( )v b v bf v f v for ( )b bv v .

C) The compact measure of intensity of workers’ competition for jobs ( )v b bf v v also

falls: ( )

0v b bf v v

with detrimental consequences for stability of the stationary state as

workers competition for jobs weakens.

Proposition 7. For in (15) and

A) New stationary employment ratio (21) as a special form of (20) is lower than the

former (9)for.

B) Derivative of specific Phillips function (3) becomes lower at the new stationary em-

ployment ratio than at the former for.

C) The compact measure of intensity of workers’ competition for jobs ( )v b bf v v al-

so lower than for.

Proposition 8. Introduction of mechanistic profit sharing with > 0 and = 0 destabi-

lizes the stationary state bE that was LAS in Z-1.

There is a realised AHB at some appropriate b for given and = 0 in Z-2Figure 7).

A period of a closed orbit is about 9.25 years.

Figure 7 – Fluctuations of accumulation rate z in relation to its stationary magnitude in result

of AHB at E for bcritical = 40.24 > b0 = 27.82 when = 0.15 and = 0 in Z-2

For = 0.15 the new stationary employment ratio is lower in Z-2than for = 0 in Z-

1: 0.831 < 0.871. The other stationary magnitudes of the phase variables remain the same.

See Appendix 2 for formal proofs.

z z0

0.6

0.6

0.3

0.3

0

0

2238 2240 2242 2244 2246 2248 2250 2252 2254 2256 2258

Time (Year)

z : Z-1 3-dim AHB b PS eta omega zero b0 4024

z eq : Z-1 3-dim AHB b PS eta omega zero b0 4024

18

3.3. Profit sharing accompanied by opportunistic employment targeting in Z-2

Let stationary employment ratio 1 ˆ( )b bv f a does not depend on as in Z-1, i.e.,

b

d

z (22)

that is relevant for (15)–(20).

Here a conformist stabilization policy strives to achieve a fixed stationary employment

ratio that determines target bX v instead of being determined by that (a motto is: “Avoid

strong intervention in the natural course of market processes”). Then the stationary growth

rate of stimulating term in (16) equals zero and can be negative on a transient to the station-

ary state. This imperfection could be cured by adding a constant to the stimulating term and

subtracting the same constant from the bargained term in (16) for unaltered (15).

Proposition 9. For same bv , independent of and , new 0b for AHB is higher than the

former in Z-1 – in other words, SPS with unaltered bv stabilizes cycles in Z-2. See Appen-

dix 3 for a formal proof.

For b = bcritical = 57.4 from AHB in Z-1, = 0.15 (respectively, = 0.025) yields stable

focus bE in Z-2. Still convergence (mostly with declining labour share u, diminishing

employment ratio v and decreasing accumulation rate z) to the unaltered stationary state is

rather slow and lasts centuries (illustrated by Figure 8).

Figure 8 – Stabilization of industrial cycle illustrated by accumulation rate z and

its stationary magnitude zb for b = bcritical = 57.4 from AHB in Z-1, = 0.15 in Z-2

Closed orbits are re-established in result of supercritical AHB at bE (9) for b0 = 56.81 <

bcritical = 75 in Z-2. SPS reinforces non-linear effects in Z-2in relation to Z-1.

3.4. Profit sharing accompanied by targeted employment ratio in Z-2X

z z eq

0.3

0.3

0.15

0.15

0

0

1958 1966 1974 1982 1990 1998 2006 2014 2022 2030

Time (Year)

z : Z-1 3-dim AHB b PS mod b0 574 b cor

z eq : Z-1 3-dim AHB b PS mod b0 574 b cor

19

Let the policy makers direct the national economy to a target employment ratio that is higher

that the initial stationary one in Z-1: bX v with beneficial consequences for stability as

workers’ competition for jobs is stronger, i.e., derivative of ( ) ( )v v bf X f v and the compact

measure of intensity of workers’ competition for jobs is higher:

( ) ( )X v b v b bf X X f v v .

Then stationary state (9) becomes

), ,( b bXE u X z , (23)

where bu and bz are identical to the same in bE (9).

The stationary growth rate of wage is defined by (15). We have either for fixed

ˆ( ) bb

Xd

f X az

(24)

or for fixed

ˆ[ ( )]bbX

za f X

d . (25)

The analytics for (24) or (25) are substantially the same, for certainty (24) is chosen.

Proposition 10. Stationary state XE is LAS for b0 and even for bcritical from AHB in Z-1.

Proof.

Consider this fight back stabilization policy as a synthetic one. Use setting in Z-1with

bv X , at first, and, second, add standard profit sharing with bv independent of from

Z-2. Then proofs for two previous cases merge into required one here.

First. For former b0 for AHB in Z-1 take in (3)

X Xg g (26)

This accruals in parameter Xg are beneficial for the compact measure of intensity of

workers’ competition for jobs and consequently for LAS of XE (23) as for LAS of E (9)

in Z-1

Second. Consider X independent of . Addition of standard profit sharing reinforces

LAS of XE (22) as already proved for Z-2.

Proposition 11. XE (23) loses its stability and AHB takes place for (new) bcritical >>

(former) bcritical in Z-1 that is so high in Z-2X that it is not economically relevant any more.

Deliberate changes of magnitudes of the two tied control parameters have contradictory

macroeconomic consequences. Still for target X one-parameter policy optimization will be

added for finding their best constellations through already established functional relationship

(24).

Consider dynamic policy optimization for Z-2X over parameters and linked

byin a Vensim equivalent of Z-2X. This does not guarantee absence of violent move-

ments on a transient to a stationary state.

3.5. Policy optimization for improper structural setting in Z-2X

Since parameters and arelinked by (24) it will suffice to consider one-parameter dy-

namic policy optimization over parameters in a Vensim model. An optimization domain

is set with rather wide boundaries facilitating attaining of a high target employment ratio:

20

0

T

T

Maximise v X dt

(27)

subject to

X = 0.95,

[ ( ), ],restrictedx f x t

0 ≤ ≤ 2

for given (18), (7), (8) and initial 0 0 0 0( , , )x v u z .

Optimization over 1958–1976 yields = 0.0363, = 0.346 for b = 57.4. The new criti-

cal magnitude of bifurcation parameter b is so high that the task of stabilization seems suc-

cessfully solved (Figure 9): the employment ratio is raised from initial v0 = 0.9 to target X =

0.95 within a year, b0 = 951.05 for AHB at XE (far outside economically possible). Still this

solution is hardly practically feasible as accumulation rate z jumps in simulation from 0.267

in 1958 to 0.7703 in 1959 (Figure 10), that requires, in particular, a growth rate of net output

to be 0.15 in 1959 over 1958. In reality, such a huge jump is doomed to failure.

Figure 9 – False fast success in achieving target employment ration X = 0.95

through policy optimization for b = 57.4, = 0.0363 and = 0.346

under improper institutional setting in Z-2X

v v eq

1

1

0.9

0.9

0.8

0.8

1958 1974 1989 2005 2020

Time (Year)

v : Z-1 3-dim AHB b PS mod X 1

v eq : Z-1 3-dim AHB b PS mod X 1

21

Figure 10 – Over-shooting in accumulation rate z through policy optimization

for b = 57.4, X = 0.95, = 0.0363 and = 0.346

under improper institutional setting in Z-2X

Profit sharing turns entirely in its opposite since bw is permanently negative and thus

de-stimulating in contradiction to its proclaimed public purpose (on the average for 1958–

1976, ˆbw = –0.3424, ˆ

mw = 0.363, and w= 0.0206 in the simulation run). This problem can

be cured by adding a constant (for example, 0.35) to the stimulating term and subtracting the

same constant from the bargained term in (16) for unaltered (15). Yet this is only a palliative.

Taking into account latent socio-economic aspects reinforces the author’s preliminary

denial of practical success of the considered stabilisation policy. Unlike extremely fast con-

vergence of v to X, convergence of two other phase variables (u and z) to their stationary

magnitudes requires centuries.

The comprehensive measure of intensity of workers' competition for jobs is immensely

sharpened in Z-2X (reflected by 0v

v

at 0.95bv X ) in comparison to Z-1. What is

also alarming, instead of being weakened, labour alienation in the production would be rein-

forced by this policy.

Even for ˆ 0bw this energetic capitalists’ effort of punching above the economy’s

weight is conflict-ridden and doomed as earlier in the similar crack in Z-1because linear

thinking and consequently linear control is over-stretched. As an ancient proverb says, evil

appears as good in the minds of those whom gods lead to destruction.

Conclusion

z z0

1

1

0.5

0.5

0

0

1958 1974 1989 2005 2020

Time (Year)

cir : Z-1 3-dim AHB b PS mod X 1

cir 0 : Z-1 3-dim AHB b PS mod X 1

22

This paper accentuates the principle role of workers’ competition for jobs for capitalist re-

production on the increasing scale. This factor was neglected in [1] as well as in [2]. These

papers extended Goodwin model M-1 by standard profit sharing (SPS). Under such profit

sharing, essentially mechanistic, a long-term employment ratio declines, whereas the station-

ary relative wage is not affected.

The present paper takes three-dimensional Goodwinian model of industrial cycles Z-1

as experimental tool [4]. This model contains a Phillips equation for the growth rate of wage

inherited from [8, 9] that is strongly non-linear with respect to employment ratio.

Effects of workers’ competition for jobs on economic dynamics are studied in depth. It

is demonstrated that weakened workers’ competition for jobs is destabilizing in Z-1.

For checking robustness of SPS, reasonable extensions of basic model Z-1 [4] are in-

vestigated additionally. The general form of these extensions is denoted as Z-2.

SPS destabilizes industrial cycles in Z-2 with lower stationary employment ratio than

in Z-1. SPS with opportunistic targeting of employment ratio at the same level as in Z-1 sta-

bilizes industrial cycles still without great gains in social efficiency in Z-2, whereas SPS

fails when employment targeting strives to more ambitious goal in Z-2X.

Like stabilization failure in Z-1 without SPS for elevated targeted employment ratio

X, there is also almost the same wrong “angle of attack”, not resolved by ill-defined SPS, in

Z-2X. In striving to target X under improper structural setting can lead to nose-dive of the

model economy masked by a sky-rocketed accumulation rate z hitting the ceiling. Besides, as

it was for ambitious employment targeting in Z-1, intensity of workers' competition for

jobs is immensely strengthened again in Z-2X at 0.95bv X in comparison to Z-1.

The following hypothesis suggests itself: the higher is intensity of workers' competition

for jobs quantified by the compact and comprehensive measures suggested in this paper, the

stronger is disjointedness (and possibly even mutual hostility) among workers and/or be-

tween groups of workers. If this hypothesis is true, than besides the technic-economic obsta-

cles there are also powerful institutional and socio-economic barriers for hasty stabilization

of industrial cycles through a huge accumulation jump. A theoretical and empirical elabora-

tion (or refutation) of this hypothesis goes beyond the present paper.

This paper illustrates how dangerous linear thinking and linear control could be if over-

stretched. Only truly dialectic system dynamics approach is capable to find out badly needed

robust non-linear control through designing interwoven feedback loops with appropriate

gains.

Organic profit sharing implements proportional and derivative control over growth rate

of profit (and/or surplus value). This rate will depend on a gap between the indicated and

current employment ratios and on growth rate of this ratio as demonstrated in [3]. Organic

profit sharing will be applied in a radical modification of Z-1 outside the present paper. Quite

differently from Z-1 and Z-2X, workers’ competition for jobs will be much weaker at the

same stationary state (with target employment ratio X) under organic profit sharing than un-

der SPS.

Gathering storms in the world economy revive interest in stabilization policies for in-

dustrial cycles through organic profit sharing characterized by weakened workers’ competi-

tion for jobs, stronger social cohesion and greater social efficiency compared to Z-1 and Z-2.

The research in this direction will be continued.

23

Appendix 1 for Z-1

Let –b bZ Z z . Jacoby matrix for stationary state bE in Z-1 is defined as

bJ . (28)

The standard characteristic equation of the third order is written as

3 2

2 1 0 0a a a , (29)

where the parameters are calculated based on the corresponding values of some Jacobi ma-

trix XJ

11 22 33 12 23 31 21 32 13 13 22 31 23 32 11 1 10 2 2 33( )Xa J J J J J J J J J J J J J J J J J J J ,

23 32 12 21 13 31 11 21 2 33 22 33[ ( ) ]a J J J J J J J J J J J ,

12 1 22 33( )– Xa Trace J J J J .

The Routh – Hurwitz necessary and sufficient conditions for LAS of XE in the line-

ar system: 0 0a , 2 0a and 1 2 0a a a . If XE is hyperbolic and LAS, it is LAS also in the

non-linear system.

A.1.1. Z-1 with b as control parameter

Lemma 1. Consider a characteristic equation based on Jacoby matrix (28). The respective

quadratic equation

1 2 0( ) ( ) 0( ) a a b ab b a , (30)

where 0 0consta ,

1( ) ba b e o , (31)

2( ) – ba cb h , (32)

1 0e

bo

, (33)

2 0c

bh

, (34)

always has two real roots:

bb

b

zu

s '( )

1

bb

uf v

1

b

b

ud

z

1 1

11 1

bb

b b

zv

s u

'( )1

1 1

b b

b

f v v

u

(1 )

1

b b

b b

dv v

z s

1

b bb b

b b

u zb z Z

u s

'( )

1 1

b b b b

b

bf v u z Z

u

1

b bb

b

b u zZ p

s

24

20

0,3

( ) 4 ( )

2

oc eh oc eh oh a ecb

oh

. (35)

Lemma 2. It is true that 1 3 0 3 0 2– min( , ) ma ( )x ,b b b b b b .

Corollary. The conjugate roots of the quadratic equation ( ) 0a b are 3 1 2( , )b b b and

3 0 1 2( , )b b b b . It follows from economic requirements that 0 2(0, )b b .

The Routh – Hurwitz necessary and sufficient conditions for LAS of bE in the line-

arized system are satisfied for 00 bb : 0 0a , 2( ) 0a b and 1 2 0( ) ( )a b a b a . As bE is

hyperbolic and LAS, it is LAS also in the non-linear system, q.e.d.

Proposition 1. The dynamics of system (6)–(8) linearized in the neighbourhood of its

hyperbolic stationary state bE (9) are LAS provided that 0 20 bbb . Then station-

ary state bE is also LAS in the non-linear system (6)–(8). Stationary state bE is not stable

for b 0b in the linearized system (6)–(8).

Proposition 2. The Andronov – Hopf bifurcation does take place in the system (6)–(8)

in a local vicinity of bE (9) at 0b b defined by equation (35).

See proofs of Lemmas 1 and 2 as well as Propositions 1 and 2 in [4].

A.1.2. Z-1 with '( )b bf v v as control parameter

The parameters of the characteristic equation based on Jacoby matrix (28) are defined as

0 '1–   0( ) ( ) ( )bb b b

b

za p u f v v

sa , (36)

where 0 < a ≈ 0,

11 ( )

( ) ( ) '( )1 1 1

1–1– –b b b

b b b b bb b b

bb

z z zpa p u u u f v v

s s u sZ z b

=

= e + o, (37)

where 0 0–1

e a

, o > 0;

21 1

(–( ) ' )1 1 1

b bb b b b

bb

b b

z za u p u f vZ z b v

s s u

= –c + h, (38)

where c > 0, h > 0.

Lemma 6. The quadratic equation based on the characteristic polynomial

1 2 0( ) ( ) ( ) ( ) 0a a a a , (39)

where 1 00 e

o and 2 0

c

h , always has two real roots:

2

0 2( ) 4

02

co eh a co eh a hoce c

ho h

, (40)

25

2

3 1 0( ) 4

2

co eh a co eh a h e

ho o

oce

. (41)

Lemma 7. It is true that 3 1 2 00 , where 1 ≈ 3 and 2 ≈ 0 .

The Routh – Hurwitz necessary and sufficient conditions for LAS of E in the line-

arized system are satisfied for 00 : 0( ) 0a , 2( ) 0a and 1 2 0( ) ( ) ( )a a a .

As E is hyperbolic and LAS, it is LAS also in the non-linear system, q.e.d.

A proof of Proposition 6. Parameter engaged in equation (14) serves as the bifurca-

tion (control) parameter. Consider the stationary state E of the system (6)–(8) as dependent

on :

) 0,(f xx . (42)

The determinant of the Jacoby matrix E (28) evaluated at the stationary state E (9)

differs from zero in our case for any possible stationary state (x) as 0( )a > 0. A stationary

state x is unique for given still changes of do affect E .

It is assumed the following properties are satisfied:

(a) the components of the function f(x,), corresponding to the system (6)–(8), are ana-

lytic (i.e. given by power series);

(b) the Jacoby matrix 0( )J has a pair of pure imaginary eigenvalues and no other

eigenvalues with zero real parts (in this case 1 2 0( ) 0a );

(c) the derivative 2,3[Re ( )]

= 1 2 1 2 0

21 2

( ' ') '0

2( )

a a a a a

a a

(it is the transversality

condition);

(d) the stationary state E is LAS (for 00 ).

Then, according to the Hopf theorem, there exists some periodic solution bifurcating

from 0( )x at 0 and the period of fluctuations is about 2 0

00

 ( )2

, 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 0( )x , still it does not clarify the stability

of orbits, which may arise on either side of 0 .

Applying information from the proof of Propositions 1 and 2 for Z-1 in Appendix 1, we

establish that conditions (a), (b), (d) of the Hopf theorem are satisfied at 0 . In particu-

lar, the characteristic polynomial for 0 is

02

0 03 2

2 0 1 0 2 1 0) ) ) [ ( )][ ( 0)]( ( ( a a a a a . (43)

.

It has the following roots:

1 2 0( ) 0a (44)

26

2,3 1 0( )i a (45)

It remains only to check that transversality condition (c) is also satisfied. Indeed, for

0 , 0( )a a , 1( )a e o and 2 –(  )a c h ,

the derivative

2,3[Re ( )]

=

=)(2

')''(2

21

02121

aa

aaaaa

= 2 1

21 2

( )

2( )

oa ha a

a a

≈ 2 1

21 2

( )

2( )

oa ha

a a

≈ 1

1

02 2

ha h

a

(46)

as a ≈ 0 and 2 0( )oa ≈ 0, q.e.d. A magnitude of this derivative equals –1.118 in our simu-

lation run (with very precise final approximation as –h/2). This huge absolute magnitude is a

manifestation of high sensitivity of dynamics to the chosen key control parameter .

The supercritical character of the Andronov – Hopf bifurcation has been established on-

ly experimentally in multiple simulation runs. An analytical proof of this property still re-

mains a challenge.

Appendix 2 for Z-2

Additions to former bJ (28) are in the 1st column only:

( )bJ = . (47)

Consider augmentation of the parameters (31)–(32) of the characteristic equation (30)

for some = const:

0 0( ) ( ) (1 ) '( ( )) ( ) 0bb b b b

b

za a J p f v v u

s , (48)

1 1( ) ( )1 1

b b

b b b b

u ud da a b p e p ob E ob

z u z u

, (49)

2 2( ) ( )1

  –1

–b b

b b b b

u ud dba c hba b

z u zh

uC

. (50)

Proof of Proposition 7.

A) New stationary employment ratio (21) as a special form of (20) is lower than the

former (9).

01

b

b b

ud

z u

0 0

2

1 10

(1 )b

b b

dv

z u

0 0

20

(1 )

bb

b

ubd Z

u

0 0

27

3/20

( ) 1

2

ˆ

b

b

bb

v d r

z dg a

z

=

=

3[1 ( )]1

2

b

b

vd

z r

. (51)

B) Derivative of the Phillips function (2) becomes lower at the new stationary employ-

ment ratio than at the former:

3

2( ( )) 0

[1 ( )]v b

b

rf v

v

. (52)

Therefore ( ( )) ( )v b v bf v f v for ( )b bv v with detrimental consequences for stability of

the stationary state as workers’ competition for jobs weakens.

C) The compact measure of intensity of workers’ competition for jobs ( )v b bf v v al-

so falls:

[ ( ( )) ( )]v b bf v v

=

3

3

[ ( ( ))] (1 ( ))1 2( )

2 [1 ( )]

v b bb

b b

f v vd rv

z r v

=

= 3

( )1 ( )

bb b b

d dv

v z z

1 2 ( )0

1 ( )

b

b b

vd

z v

. (53)

Proof of Proposition 8.

It is suffice to consider the necessary condition for LAS of bE – the coefficient

2( ) 0a of (30). Under the conditions of AHB in Z-1 at bE for b0 with = 0 it was almost

zero still positive. Thus it is sufficient to demonstrate that its derivative with respect to the

specific control parameter is negative. Indeed,

2 1 2 ( )( ) 1 10

1 1 1 ( )

bb

b b b

va du

z u v

(54)

if3

( ) 12 (1 )

b bb

v vu

;

2 1 2 ( )( ) 1 10

1 1 1 ( )

bb

b b b

va h du

z u v

(55)

if3

( ) 12 (1 )

bb

vu

.

In Z-2,only inequality (54) is relevant.

Introduction of mechanistic profit sharing with > 0 and = 0 moves stationary state

bE (9) that was LAS for b = b0 under conditions of AHB in Z-1; new stationary state E for

28

b = b0 is not LAS in Z-2. For restoration of LAS of E , b must decline to a new b such that

a0 > 0, a2 > 0 and a1a2–a0 > 0.

In spite of changes not only in ( )bv but possibly even in signs of the elements of the

first column of augmented Jacoby matrix ( )b bJ J , the main properties of Z-1 remain

intact in Z-2. This confirms structural stability of Z-1 and Z-2.

Appendix 3 for Z-2

Notice: elements of bJ (28) containing bv are 21J , 22J and 23J , elements containing

'( )bf v are 12J , 22J , 23J ; only 22J contains both as a product of them.

Lemma 8. The quadratic equation based on the above characteristic polynomial

1 2 0( ) ( ) ( ) 0a a a a , (56)

where

1( ) ba E o , (57)

2( ) – ba C h , (58)

1 0E

bo

, (59)

2 0C

bh

, (60)

always has two real roots:

20

0,3

( ) 4 ( )

2

oC Eh oC Eh oh a ECb

oh

. (61)

Proof to Proposition 9.

Equations (55)–(61) and Table 5 trace changes in the coefficients of the characteristic

equation considered as linear functions with constant terms getting increments thanks to

SPS.

Table 5. Coefficients of characteristic equations in Z-1 and Z-2

Coefficients AHB in Z-1 for b0= 54.4 LAS bE in Z-2 with same bv

for b0 = 54.4

0a 0 0a 0 0a

1( )a e + ob > 0,

e > 0, o > 0 01

b

b b

ude p ob

z uE ob

1( )a = 0 1 0

eb

o

11

b

b b

udp

z ue E eb

o o o o

2( )a c – hb > 0,

c > 0, h > 0 –  01

b

b b

udc Chb

z uhb

,

particularly, if

c – hb > 0 (sufficient condition)

29

2( ) 0a 2 0

cb

h

21

0

b

b b

ud

z uc C cb

h h h h

Recognise that coefficient of characteristic equation a0 is the same in Z-2 as in Z-1.

New constant terms of 1( )a and 2( )a result from augmentation of the former ones

1

b

b b

E eud

pz u

e e

, (62)

               1

 b

b b

C c cud

zc

u

. (63)

These equations lead to following relations

0 0 0( )( ) ( )a EC a e e c c a ec e c c e c e

0 ( )a ec e c c e c e . (64)

Clearly

4 ( ) 0oh e c c e e c (65)

and

0o c h e .3 (66)

Thus the quadratic characteristic equation is specified as

20–  ( – – ) –   0–( ( ) )ohb oc eh o c h e b e e c e e c a (67)

therefore the new bifurcation magnitude of control parameter b from (8) is

20

0

20

( ) 4 ( )(new)

2

( )

2

( ) 4 ( ) 4 ( ).

2

oC Eh oC Eh oh a ECb

oh

oc eh o c h e

oh

oc eh o c h e oh a ec oh e c c e e c

oh

(68)

The reader sees that the following relation is true as stated:

20

0 0

( ) 4 ( )(new) (former)

2

oc eh oc eh oh a ecb b

oh

. (69)

Similar to Z-1 and Z-2, Z-2is structurally stable. Still Z-2Xreveals that over-

stretched SPS contradicts structural stability – therefore organic profit sharing is badly

needed for the resilient and sustainable reproduction on the extended scale.

References

1. Lordon F. 1995. Cycles et chaos dans un modèle hétérodoxe de croissance endogène //

Revue économique 46(6): 1405–1432.

3 Inequality (66) is true for > 0 so long as '( )

1b bf v v p

that is satisfied not only for the given

parameters magnitudes but as well for their broad variations in Z-2.

30

2. Fanti L., Manfredi P. 1998. A Goodwin-type growth cycle model with profit-sharing

// Economic Notes 27: 371–402.

3. Ryzhenkov A.V. 2015. Socially efficient stabilization policies for growth cycles /

Advances in Economics and Business 3 (11): 502–527. URL:

http://www.hrpub.org/journals/article_info.php?aid=3066

http://www.hrpub.org/download/20151130/AEB5-11804853.pdf

4. Idem. 2016. Endogenous industrial cycles in a reshaped “neoclassical” model// The

34th International Conference of the System Dynamics Society. Delft, Netherlands. July 17 –

July 21, 2016 [Electronic resource]. - Delft, 2016. - Mode of access (27.06.16):

http://www.systemdynamics.org/conferences/2016/proceed/papers/P1318.pdf

http://www.systemdynamics.org/conferences/2016/proceed/supp/S1318.pdf

http://www.systemdynamics.org/conferences/2016/proceed/abstracts/1318.html

5. Arrow K. J., Chenery H. B., Minhas B. S., and Solow R. M. 1961. Capital-labor sub-

stitution and economic efficiency // Review of Economics and Statistics 43 (3): 225–250.

6. Manfredi P., Fanti L. 2000. Long-term effects of the efficiency wage hypothesis in

Goodwin-type economies / Metroeconomica, 51 (4): 454–481.

7. Goodwin R.M. 1972. A growth cycle, in: C.H. Feinstein (ed.). Socialism, Capitalism

and Economic Growth, Cambridge University Press, Cambridge, 54–58.

8. Ploeg F. Van der. 1985. Classical growth cycles / Metroeconomica 37 (2): 221–230.

9. Aguiar-Conraria L. 2008. A note on the stability properties of Goodwin's predator-

prey model / Review of Radical Political Economics 40 (4): 518–523.

10. Blatt J. 1983. Dynamic Economic Systems. Brighton, Wheatsheaf Books.

11. Marx K. Capital. A Critique of Political Economy. Volume I. Book One: The Pro-

cess of Production of Capital. First published: in German in 1867. URL:

https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-I.pdf