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Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
275
Competition, Instability,
and Nonlinear Cycles
Proceedings of an International Conference
New School for Social Research
New York, USA, March 1985
Edited by Willi Semmler
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo
Editorial Board
H. Albach M. Beckmann (Managing Editor)
P.Dhrymes G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor)
H. P. Kunzi K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fUr Gesellschafts- und Wirtschaftswissenschaften
der UniversiUit Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Editor
Prof. Dr. Willi Semmler
Economics Department, Graduate Faculty, New School for Social Research
65 Fifth Avenue, New York, N. Y. 10003, USA
ISBN 978-3-540-16794-5 ISBN 978-3-642-51699-3 (eBook)
DOI 10.1007/978-3-642-51699-3
Library of Congress Cataloging-in-Publication Data. Competition, instability, and nonlinear cycles.
(Lecture notes in economics and mathematical systems; 275) 1. Business cycles-Mathematical
models-Congresses. 2. Competition-Mathematical models-Congresses. 3. Prices-Mathematical
models-Congresses. 4. Macroeconomics-Mathematical models-Congresses.1.Semmler, Willi.
II. New School for Social Research (New York, N.Y.) III. Series.
HB3711.C618 1986338.5'4286-20318
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© Springer-Verlag Berlin Heidelberg 1986
contents
Introduction
I. On Modeling the Cross-Dual Dynamics of Competition
P. Flaschel and
W. Semmler
R. Kuroki
R. Franke
L. Boggio
T. Fujimoto and
U. Krause
The Dynamic Equalization of Profit Rates
for Input-Output Models with Fixed capital
The Equalization of the Rate of Profit
Reconsidered
A Cross-Over Gravitation Process in Prices
and Inventories
stability of Production Prices in a Model
of General Interdependence
Ergodic Price Setting with Technical
Progress
II. Microfoundations of Macrodynamics and Limit Cycles
R.M. Goodwin
G. Dumenil and
D. Levy
W. Semmler
O.K. Foley
Swinging along the Autostrada
Stability and Instability in a Dynamic
Model of Capitalist Production
On a Microdynamics of a Nonlinear
Macrocycle Model
stabilization policy in a Nonlinear
Business Cycle Model
1-124
1
35
51
83
115
1.25-211
125
132
170
200
IV
III. Linear and Nonlinear Macrodynamics
J. Glombowski and
M. Krueger
M. Jarsulic
P. Flaschel and
R. Picard
E. Nell
Some Extensions of a Classical Growth
Cycle Model
Growth Cycles in a Classical-Keynesian
Model
Problems concerning the Dynamic Analysis
of a Keynesian Model with Perfect
Foresight
cyclical Growth: The Interdependent
Dynamics of Industry and Agriculture
IV. Econometrics of the Dynamics of Proportions and Nonlinear
Macrodynamics
M. Juillard
S.N. Neftci
The stability of the Reproduction Scheme:
Theoretical Discussion and Empirical
Evidence for the United States, 1948-1980
Testing Non-Linearity in Business Cycles
212-303
212
252
,269
289
304-340
304
324
Introduction
I. The Topic and the structure of the Proceedings
The papers in this book are the proceedings of a conference held at
the Economics Department of the Graduate Faculty of the New School for
Social Research in March 1985 in New York for which financial support
was provided by .the West German Consulate. The topic of the conference
was "Competition, Instability, and Nonlinear Cycles." A number of
mathematical economists from Italy, West Germany, France, Japan, and
the u.s. were invited as participants in this meeting.
The conference was preceded by two other conferences in which
several of the invited scholars had taken part. One, on "la
gravitation des prix," took place in Nanterre, France,in March 1984.
The other was held at the New School for Social Research in April 1984
on "Price of Production in Joint Production systems." Both conference
were concerned with classical prices of production systems and their
revival in the form developed by Sraffa and Pasinetti (1977).
In these conferences, considerable interest arose in more properly
modeling the dynamics of prices of production systems in a multi-
sectoral framework by utilizing modern mathematical tools of dynamical
systems. Such a discussion on the dynamics of the classical process of
competition and the stability of classical production prices was
originally initiated by several papers by Nikaido (1977, 1983, 1984)
and further pursued by several scholars (see Steedman, 1983; Boggio,
1980; Kuroki, 1983; Dumenil/Levy, 1983; Krause, 1983, 1984; Hosoda,
1985; Filippini, 1985).
The main idea of the conference at the New School for Social
Research was to explore the dynamics of production price systems and
in this way to further a firmer classically oriented microfoundation
for macrodynamic systems and nonlinear cycle theories.
The papers can therefore be divided into four parts. In section I
the dynamics of classical competitive processes and the stability
properties of production price systems with alternative assumptions on
the formation of prices and the determination of outputs is explored.
Section II elaborates on the microfoundations of macrodynamical
systems and nonlinear cycle theories. Section III presents some
classical/Keynesian macrodynamical systems, including some corrections
of sargent's Keynesian macroeconomic models. In section IV some
econometric methods are provided to test sectoral instabilities in
VI
input-output systems and to test nonlinearities and limit cycle
properties of macroeconomic models.
II. Notes on the Papers
Many papers in the first two section of the proceedinqs can be
understood as a response to Nikaido's papers on the dynamic
properties of the competitive process in classical economic theory.
The paper closest to Nikaido's approach is the one by Kuroki in the
first section of the book. Most of the papers initiated by Nikaido's
contributions and contained in the first section of the book are
concerned with the followinq cross-dual process for a qrowinq economic
system written in terms of a dynamical system of price and quantity
chanqes in a sequence of time:
[r (P,X)l - r(p,x»)
where Xl ,Pl are time rates of chanqe of outputs and prices, dl ,kl are
reaction coefficients, r(p,x)l the industry's profit rate, r(p,x) the
averaqe profit rate, D(p,X)l the averaqe or expected demand for
industry i and S(P,X)l the industry's supply.
Such a dynamical process, includinq both an excess demand function
and a chanqe in production levels due to differential profit rates,
can be found in classical economists such as A. Smith, D. Ricardo, K.
Marx, but also in L. Walras and A. Marshall (see Flaschel/Semmler in
this volume where a more elaborate introduction to classical dynamics
is also qiven). Nikaido's papers -- see Kuroki in this volume --
presented results on such a dynamics between output chanqes and
chanqes in differential profit rates, demonstratinq that in most cases
prices of production are not even locally stable where lonq- run
prices of production are formed throuqh the principle of capital flows
and an equalization of rates of return on capital. The dual of this
dynamic process of price formation represents the dynamic formation of
balanced qrowth proportions.
In classically oriented models such as those initiated by Nikaido
and elaborated in subsequent discussions, consumer behavior is kept as
simple as possible. Emphasis is instead qiven to the analysis of the
behavior of the firm respondinq to differential profit rates throuqh
capital flows. There exists, of course, a vast literature workinq
solely with excess demand functions,in which consumption behavior is
more specified (see Hahn 1982 for a summary). There are also studies
in qeneral equilibrium theory where a similar dynamic cross-dual
~I
process for a production economy is formalized. This appears in
particular in the work of Morishima (1960, 1964, 1976, 1977), Mas-
Colell (1974), which refer to Hartman's mathematical work on global
stability of differential equation system (Hartman, 1982) utilizing
distance functions as Liapunov functions. Another recent approach to
this problem can be found in the work of Sonnenschein (1981, 1982).
Yet in most of t~ese studies such a dynamic cross-dual process is
found to be stable only under specific assumptions on consumption
behavior (see also Debreu 1982 and Schafer/Sonnenschein 1982).
In the modeling of classical dynamics initiated by Nikaido'spapers
where, in particular, the firms' behavior is more specified than in
the aforementioned types of studies, two different types of price (and
output) formation are considered.
In the first type of analysis, a cross-dual dynamics of price and
output formation is considered with reference to the cross-dual
process. Here profit rate.differentials determine output changes due
to the flow of capital, and subsequent imbalances in supply and demand
(or inventory changes) determine price changes in a multi-sectoral
system. The stability properties of such dynamic processes are
discussed in Flaschel/Semmler, Boggio, Franke, Kuroki, Dumenil/ Levy,
and Goodwin.Different stabilizing forces are considered in the papers.
Whereas, for example,in the paper by Flaschel/Semmler the rate of
change of extra profits is utilized as an additional stabilizer for
the production price system, the papers by Franke and Dumenil/Levy
model price reactions in response to inventory imbalances and refer to
consumption behavior and/or firms' control of outputs as additional
stabilizing force.
A second type of dynamical process is also considered in the
sections I and II of this volume. Morishima (1977) calls it a dual
adjustment process, more popular in Keynesian dynamics.This dynamics
consists of two adjustment procedures which are, however, considered
to operate independently of each other. In such theories, prices are
determined by markups on full cost and prices are decreased
(increased) when computed prices (cost of production plus a normal
markup,) exceed (fall short of) actual prices. On the other hand, it
is then normally assumed that quantities respond to quantity
imbalances solely or that supply is adjusted to demand
instantaneously, so that imbalances in supply and demand do not affect
the formation of long-run prices of production (see Krause/Fujimoto).
Other versions of these theories have attempted to build into the
Viii
dynamic formation of markups also a response to excess demand in
markets (see the paper of Boggio in this regard). By referring to the
second type of adjustment process, the paper by Krause/Fujimoto
obtains global stability results even allowing a choice of technique
during the price adjustment process by utilizing a theorem on strong
ergodicity in nonlinear eigenvalue systems. Unlike other papers, they
also specify the speed of convergence. As Krause has demonstrated in
other papers, this approach to the dynamics of production prices can
also be extended to other types of markup pricing behavior (Krause,
1984). On the other hand, Boggio's paper, which works with given
techniques and obtains local stability under specified conditions, is
of great interest for a theory of a price-setting firm facing entry
and exit barriers. As Boggio also seems to indicate the two types of
adjustment processes --the cross-dual and the dual adjustment
processes-- are not necessarily exclusive. The dual process can be
considered to represent a fast dynamics whereas the cross-dual process
can be interpreted as representing a slow dynamics.
In section two of the proceedings, the two types of dynamical
processes are utilized in several papers as micro-foundations of
macro- models and limit cycle models. Goodwin's paper explores the
price and quantity dynamics of a sraffa/von Neumann production price
system from the point of view of a 2n dimensional potential function
such as V(q,p,)= pq-paq whose gradients can be used to obtain quite
elegantly results on the cross-dual dynamical process outlined above
or on markup price dynamics and quantity dynamics separately. Self-
sustained macroeconomic cycles between unemployment and income shares
are derived in a multi-sectoral and an aggregated framework based on a
nonlinear differential equation system of the Lotka/Volterra type.
The macromodel of Dumenil/Levy is built on a cross-over dynamics
between differential profit rates causing output changes and
deviations of actual from normal inventories creating price changes.
They also take into account quantity constraints and rationing as well
output control by firms if supply exceeds demand. In spite of the
local stability of the equilibrium for production prices and the
balanced growth path, the complexity of their model, which is mainly
tested by means of computer simulations, unfolds a rich macroeconomic
dynamics including business cycles and depressions. In Semmler a
nonlinear macromodel in the tradition of Goodwin (1982), Kalecki
(1971) and Kaldor (1940) is presented which is built on a classically
oriented cross-dual microdynamics generating stable limit cycles
around a qrowth trend for the macromodel. The macromodel is
demonstrated to be robust aqainst small perturbations and random
shocks oriqinatinq in the micromodel.
Foley's paper, on the other hand, explicitly takes into account
external financinq of the firm and other financial variables. Its
microdynamics results from price-settinq firms and endoqenously
qenerated constra~nts for financial funds. A macroeconomic model is
obtained throuqh aqqreqation of individual firms' behavior, and the
existence of a limit cycle with qrowth and explicit formalization of
the monetary flows is proved to exist for certain ranqes of the
parameters. As also shown, the paper allows one to draw interestinq
implications for the efficacy of macroeconomic stabilization policy.
Part III of the proceedinqs is devoted to pure macroeconomic
dynamics of a qrowinq system, oriqinatinq in the classical/ Marxian
and Keynesian traditions. In Glombowski/Krueqer, a cycle model with
qrowth is discussed which includes, in addition to oriqinal Goodwin-
like models, technical chanqe and a formalization of macroeconomic
imbalances between supply and demand indicated by a utilization of
capacity variable. Since the model is written in a discrete version,
it unfolds a rich macroeconomic dynamics for the three nonlinearly
interactinq variables. Jarsulic also enriches Goodwin's model of
fluctuatinq qrowth by introducinq a plausible investment function and
variable utilization of capacity. The existence of a limit cycle is
proved by use of the Poincare-Bendixson theorem, and more complex
cases are, as in Glombowski/Krueger, studied by means of computer
simulations. The work of Flaschel/Picard is devoted to an analysis of
Keynesian dynamical models starting with a discussion of sargent's
interpretation of Keynesian dynamics (sargent 1979). By showing some
defects in Sarqent's dynamic models on Keynesian macroeconomics as
well as in the economic assumptions underlyinq the models, some
reformulations of Keynesian dynamics based on a different price
dynamics than sargent's model are provided. The last paper in this
section, the paper by Nell, gives a more historical account of a well-
known inter-sectoral dynamics: the interdependent dynamics of industry
and aqriculture. It utilizes a Nikaido-like model of capital flows
where the capital flows follow differential profitability in order to
explain the interdependency of cyclical qrowth in industry and
aqriculture.
The section IV of the proceedinqs is concerned with the econometrics
of the stability of multi-sectoral systems and of nonlinearities in
x
macromodels. Juillard tests empirically the stability in proportions
by making use of comparable input-output tables of the U.S. economy
for the period of 1948 to 1980. Here empirical evidence is provided
for the trends as well as the stability of departmental proportions on
the U.S. post-war economy. Last but not least, Neftci's paper
elaborates possible ways to test for nonlinearities in aggregated
macro-data and thus paves the road for testing empirically nonlinear
cycle models.
III. Notes on the Mathematical Methods
The majority of papers inthis volume apply advanced methods of
differential equations and nonlinear dynamic systems to study the
microdynamics of competitive processes or the macroeconomic dynamics.
Among the methods utilized are the gradient method, Liapunov
functions, the Poincare-Bendixson theorem for limit cycles, discrete
versions of nonlinear dynamical systems with fixed-point iterations,
bifurcation theory, and nonlinear eigenvalue theory (ergodicity
theorems). Since the dynamics of higher order systems is very often
difficult to elaborate qualitatively with conventional methods,
several studies in this volume also utilize computer simulations to
study the dynamic properties of their proposed models. The
mathematical methods used in these papers have recently been
elaborated in several books on dynamical systems. The interested
reader may be referred to more extensive presentations and treatments
concerning time-continuous as well as discrete versions of dynamical
systems in Hirsch/smale 1974, Hartman 1982, Ioos/Joseph 1980,
Guckenheimer/Holmes 1983, and Marsden/Mccracken 1976.
W.Semmler
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Introduction1
THE DYNAMIC EQUALIZATION OF PROFIT RATES FOR
INPUT-OUTPUT MODELS WITH FIXED CAPITAL
Peter Flaschel' and Willi Semmler·'
'Department of Economics
University of Bielefeld
Universitaetsstrasse, 4800 Bielefeld, FRG
"Department of Economics, Graduate Faculty
New School for Social Research
65 Fifth Avenue, New York, N.Y. 10003
Recently there have been numerous new interpretations of the
classical competitive process (Hollander, 1973; Garegnani, 1983;
Roncaglia, 1978; Shaikh, 1980; Semmler, 1984). There have also been
several attempts to model classical competitive dynamics. within the
latter group two trends are emerging. One line of thought explores the
dynamics of the competitive process on the basis of classical theory
by referring to supply and demand analysis (Nikaido, 1977, 1983;
Flaschel, 1983; Flaschel/Semmler, 1985a, 1985b; Dumenil/Levy, 1983,
1984; Franke, 1985; Boggio, 1984; Steedman, 1984). The other direction
utilizes the theory of markup pricing to elaborate on the stability
properties of classically oriented production price systems
(Nikaido/Kobayashi, 1978; Krause, 1983; Fujimoto/Krause, 1985; Boggio,
1985; catz/Laganier, 1984).
In this paper we want to address problems of classical dynamics
which are related to the supply and demand mechanism. The description
of the classical dynamics of competition (Steedman, 1984) as well as
the modeling of it are still of a somewhat elementary and
heterogeneous nature, especially with regard to the basic dynamics
involved. Currently there are diverse results presented, ranging from
the demonstration of instability of the classical competitive process
(Nikaido, 1977) to the demonstration that it is locally (or even)
globally stable. Neoclassical economists on the other hand have
expressed considerable doubts concerning the stability of their
lWe want to acknowledge, with thanks, helpful discussions and
comments on an earlier version of the paper by R. Franke, A. Shaikh,
G. Dumenil, D. Levy, D. Foley, B. Schefold, J. Eatwell, E. Nell and V.
Caspari. For computational assistance we want to thank R. Zambrano and
P. Cooney.
2
Walrasian short-run models (Hahn, 1970, 1982). Moreover, for long-run
models of efficient growth and accumulation it has been demonstrated
in this context that an asymptotically stable adjustment process to
the steady state can rarely be maintained, once the world of
homogeneous capital is abandoned (Hahn, 1966, 1970; Kurz, 1968;
Burmeister et al., 1973).
since the discussion on classical competitive dynamics is still in
an early state, a fully acceptable formalization of the process of
competition including growth cannot yet be expected. Most
formalizations of classical micro dynamics utilize only a circulating
capital model, with or withoutcapitalist consumption. The model
proposed in this paper which is related to other works of the authors
(Flaschel/Semmler, 1985b, 1986) will include circulating as well as
fixed capital, but for reasons of simplicity it works with the von
Neumann assumptions on savings behavior and laborers' consumption. The
inclusion of fixed capital makes our model comparable with earlier
work on the dual instability of closed Leontief systems (Jorgenson,
1960; Zaghini, 1971; Aoki, 1977) where usually two uncoupled dynamical
systems with fixed capital are discussed, one for output proportions
and the other for relative prices.
We will show in section III that our proposed formalization of
classical competition formulates a cross-dual dynamical process in a
more plausible and general way than originally proposed for input-
output systems by Nikaido (1977, 1983). In particular, it presents
results of a more than local nature based on the uniqueness of the
equilibrium. Our proposed cross-dual dynamics is formalized in section
III and reconsidered by means of computer simulations in section IV.
It relies on the interaction of capital movements resulting from flows
of investments from areas of low to high rates of return, and on the
other hand, on output changes and their effects on market prices. We
are able to show that supply changes initiated by differential profit
rates and related capital flows bring about market price changes and
thus changes again in differential profit rates. This basic dynamic
process will produce stable fluctuations around the steady state of
input-output models with single production and fixed capital.
Furthermore, it will be demonstrated in section III and IV that this
cross-dual dynamics will be globally asymptotically stable if the
direction of the rate of change of differential profit rates is also
taken into account when capital flows from areas of low to high rates
of return. Before presenting this dynamical model a brief review of
3
classical as well neoclassical literature on competition will be
given.
I. Classical Competitive Processes
I. 1. On Classical Theory
The verbal analysis of stability in classical economics is
necessarily ambiguous compared with modern treatments of stability
problems which utilize formal tools such as dynamical systems,
Liapunov functions or the global Newton method (Hahn, 1982: Smale,
1981). On the other hand any interpretation of the classics using
mathematical tools runs the risk of not accurately depicting the
classical analysis. Therefore, we wish to present some details of the
classical competitive process before discussing its possible
formalizations.
Of course there exist differences among the classics concerning
their theory of competition, their concept of long-run prices and the
role of supply and demand. Yet, they have in common the fact that
long-run prices (natural prices in Smith and Ricardo, prices of
production in Marx) are considered to be centers of gravitation which
are determined independently of short-run demand forces. In smith the
natural price is composed of the vertically integrated natural wage,
profit, and rent components. In Marx the prices of production are
determined by i) the structure of production, ii) the income
distribution and iii) the turnover time of capital. Long-run changes
in demand may change the long-run price because of an increasing
difficulty or facility to produce, possibly requiring new techniques
of production. However, for the analysis of the stability of the
market process, demand is given as the average demand (Marx) or
effectual demand (Smith) and the center of gravitation for market
prices is considered fixed. The verbally expressed view in the
classics that market prices fluctuate around or gravitate to long-run
natural prices indeed relies on the fundamental assumption that the
cross-dual dynamics between market-price dependent differential profit
rates and profit-rate dependent output changes is somewhat stable.
The fundamental classical notion of a crossover dynamics by which
deviations of profit rates from the average profit rate entail output
changes and the deviation of supply and demand from long-run average
demand will cause market prices to move may be expressed in the
following way:
4
<-- (r1 (p,x) - r), i = 1. •• n
<-- (01 - S1 (p,x», i = 1. •• n
where P1 and x1 are the time rate of price and quantity changes, D1 is
the average demand for the industry i, S1 (p,x) the industry i's
supply, r is the average profit rate for the entire economy and
r 1 (p,x) is the industry i's profit rate. Thereby the following dynamic
process is formulated: profit rates above (below) the average lead to
an increase (decrease) in supply, and an increase (decrease) in supply
above (below) the average demand in turn will entail a fall (rise) in
market prices. This dynamic process forces the prices and quantities
back within certain boudaries.
Smith referred explicitly to such a twofold process:
"If at any time it (supply) exceeds effectual demand, some of
the component parts of its price must be paid below their natural
rate •.• if it is wages or profit, the interest of labourers in
the one case, and of their employers in the other will prompt
them to withdraw a part of their labour or stock from the
employment ••• If on the contrary, the quantity brought to the
market should at any time fall short of the effectual demand,
some of the component parts of its price must rise above their
natural rate ••• if it is wages or profit, the interest of all
other labourers and dealers will soon prompt them to employ more
labour and stock in preparing and bringing it to the market."
(Smith, 1974, p. 160)
The supply responds to gains or losses in profits relative to the
normal, and supply and demand discrepancies entail market price
changes. The latter adjustment follows from the following process
described by smith:
"It is only the average produce of the one species of
industries which can be suited in any respect to the effectual
demand; and as its actual produce is frequently much greater and
frequently much less than its average produce, the quantities of
commodities brought to the market will sometimes exceed a good
deal and sometimes fall short a good deal, of effectual demand.
Even though that demand therefore should continue always the
5
same, their market price will be liable to great fluctuations,
will sometimes fall a good deal below, and sometimes rise a good
deal above their natural price."
(smith, 1974, p. 161)
Such a cross-dual adjustment process for prices and quantities
initiated by differentials in profit rates and imbalances between
supply and demand is also formulated by Ricardo (Ricardo, 1951, ch.
IV). There is not much difference in smith's formulation of this
gravitational process and Ricardo's as the reader of Chapter IV in
Ricardo's "principles" will realize. Yet, Ricardo explicitly refers to
this twofold process by arguing that this process will be brought
about by competition among owners of capital. In his view the outcome
does not necessarily require actual capital flows but simply the
contraction or expansion of industries due to the flows of credit.
Although neither Smith nor Ricardo has given a proof that this cross-
dual adjustment process is stable, they argue that market prices
oscillate around their centers of gravitation2 •
Whereas smith and Ricardo seem to assume that the natural price is
fixed and given during the adjustment process (steedman, 1984), Marx
explicitly asks how the prices of production and the equalized profit
rate are brought about (1977, vol. III, p. 179). Marx also refers to
the aforementioned twofold process to give a plausible argument for
equalization of profitrates in the long-run and prices of production
are established: "The movement of capitals is primarily caused by the
level of market-prices which lift the profits above the general
average in one place and depress them below it in another" (Marx,
1977, vol. III, p. 208). And,
"Through this incessant outflow and influx, or briefly, through
its distribution among various spheres, which depends on how the
rate of profit falls here and rises there, it creates such a
ratio of supply and demand that the average profit in the various
spheres of production becomes the same ••• "
(Marx, 1977, vol. III, p. 195)
2As one can see in smith's writings on astronomy, he got the notion
of "gravitation" from Newtonian physics (see smith, 1980). We owe the
knowledge of this relation between Smith's notion of gravitation and
Newton's laws of the planetary movements to R. Hei1broner.
6
Through this twofold adjustment process, by which capital flows are
intitiated by profit rates above or below the average and since
"supply and demand determine the market price" (Marx, 1977, vol. III,
p. 208) there will be, as Marx assumes, a tendency toward the
equalization of profit rates and toward prices of production as
centers of gravitation for market prices. Marx, however, more than
smith and Ricardo, also stresses that market prices constantly
oscillate around production prices and supply and demand may fluctuate
around their equilibrium positions (Marx, 1977, vol III, p. 190).
Marx, however, was realistic enough not to claim that his
considerations on the equalization of the general rate of profit
through competition would necessarily imply the existence of the
latter in any real economy:
"This movement of capitals is primarily caused by the level of
market prices ... Yet, with respect to each sphere of production
- industry, agriculture, mining, etc. - the transfer of capital
from one sphere to another offers considerable difficulties,
particularly on account of the existing fixed capital.
Experience shows, moreover, that if a branch of industry, such
as say, the cotton industry yields unusually high profits at one
period, it makes very little profit, or even suffers losses, at
another, so that in a certain cycle of years the average profit
is much the same as in other branches."
(Marx, 1977, vol. III, p. 208)
smith and Ricardo as well as Marx never explored sufficiently the
stability properties of the crossover dynamics which they described.
Since the verbal formulation of the competitive dynamics in the
classics is ambiguous, the terms gravitation or oscillation can be
interpreted in various ways in mathematical terms by means of a
variety of stability concepts. Some possiblities of a stable behavior
of this process will be considered in the following.
I. 2. On Recent Studies of Classical Dynamics
Already at the turn of the century, the dynamics of equalizing
profit rates was discussed very controversially in the debate between
Bohm-Bawerk and Hilferding (see Sweezy, 1949). More recently, Nikaido
(Nikaido, 1977, 1983) presented formal studies of a different type to
disprove the Marxian hypothesis of the formation of a general rate of
7
profit and the stability of prices of production. By referring to a
two sector input-output model with simple reproduction, he considers
in particular the following dynamic system:
X = A (x + x) + [CO] (1)
(1' )
and either
Xi = 0 i (r 1 - r 2 ) (2)
or
Pi/Pi= 8i (Oi (r1 - r 2 ) - X1 ), i = 1,2 (2' )
where A denotes the input-output matrix (augmented by the
consumption good matrix for labor) and 01 (r1 - r 2 ) the investment
function for industry i which is dependent on sectoral profit rates
r 1 , r 2 (01 '>0'02 '<0). If there are no profit rate differentials, then
0i (0)=0. In (2) supply and demand are instantaneously equated at
prices which imply the profit rate differential r 1 -r2 • In (2'), due to
0<8<~, there is a lagged adjustment for prices because of an imbalance
between intended and realized investments 01 and x1 • Thus equation (2)
presupposes that the intended level of investment is equal to the
realized level, whereas (2') allows for lagged adjustment in relative
prices. This early version of Nikaido's refutation of the equalization
of profit rates shows strong results (Nikaido, 1977). In the case of a
higher organic composition of capital in the capital good producing
sector,the eigenvalues of the dynamic system (1') are both positive
and the system is totally unstable. For an organic composition in the
capital good sector which is lower than that of the consumption good
sector, we obtain eigenvalues ~1<0'~2>0. The equilibrium is thus a
saddle point. These strong negative results are to some extent
weakened in his more recent publication (Nikaido, 1983), where he
differentiates between the flows of money capital and real capital.
Moreover, capitalist consumption is endogenously determined in this
revised model. In the case of a higher organic composition of capital
in the capital goods sector, prices of production remain unstable. The
case of higher organic composition in the consumption goods sector
brings a potential for convergence of prices of production in the
simplest case where money capital moves alone (Nikaido, 1983, p. 339).
However, Nikaido's formalization of the dynamics of competition is
not convincing. As is evident, in the light of the characterization of
8
classical competition, his formalization does not satisfy the typical
requirement of a cross-dual dynamics as formulated in 1.1. In both of
Nikaido's versions of the classical and Marxian theory of competition,
only one side is formulated correctly; i.e., prices respond to
quantity imbalances due to (2) and (2'), but the change of the
relative levels of output Xl' X2 is not really made dependent on
differentials in profitability in investments. The instability of
prices of ptoduction in his model comes from the fact that the
quantity dynamics is unstable by itself3.
Another paper written in the same vein as Nikaido yields similar
results (Franke, 1984). Franke removes some inconsistencies in
Nikaido's approach and discusses the stability in a n-sector model.
However, due to a modeling of classical and Marxian dynamics similar
to Nikaido's, Franke also gets negative results for his n-sector
model. A general proof of instability of prices of production, is not
provided in his formalization; the instability conclusion is drawn
from some computer simulations. Boggio has presented another study
(Boggio, 1984) utilizing excess demand functions and making quantity
changes dependent on differential profit rates which determine by
means of a further equation system the differential growth rates.
Here, asymptotic stability is obtained only for particular types of
capitalist consumption and specific reaction coefficients of a
simplified dynamical two sector circulating capital model. In general
the dynamical process is proved to be unstable.
The studies by Oumenil/Levy (1983, 1984), which formalize a
crossover dynamics as it is suggested in 1.1., are capable of
demonstrating asymptotic stability of prices of production for cases
where substitution possibilities exist, the reaction of investments to
differentials in profit rates and the reaction of prices to quantity
imbalances are not "too large" and where capitalists consume and save.
The double adjustment mechanism, by which prices change due to
quantity imbalances and investments respond to differentials in profit
rates, works as follows in their basic model:
°i • t - 1 - Y i • t - 1 ]
Yi • t-l
(3)
3In our view a related situation is described by the so-called dual
(in)stability theorem of closed Leontief models with fixed
coefficients (see Jorgenson,1960; Zaghini, 1971; Aoki, 1977).
9
f(r j • t _ 1 ) (4)
i = 1,2, j = 1,2,
1\ 1\
where Pi.t-1' Yj .t-1' are the growth rates of prices and quantities.
The excess demand is determined by:
(5)
Pt - 1 d
with Dt - 1 , Yt - 1 demand and supply of period t-l, ~t-1 the sum of
profits, a the proportion of profits consumed, d a constant
consumption good vector for the capitalists which is independent of t.
Furthermore
yj. t = Yj. t -1 (1 + gj. t -1) = J't Yj. t -1 f(rj. t -1)
with gj .t-1 the growth rates of output, which are determined by the
differential profit rate dependent investments. The scalar J't is an
index which adjusts the capital fund used up thereby to the available
capital fund Kt generated by the past period's capital fund Kt - 1 and
the profit in period t-1. Thus:
L Yj t_1 f (rj t-1)Pi t-1 a i" i • j' • • J
(6)
(In a more recent version, the case of underutilization of the
available capital fund is also allowed for, and thus Say's Law is
dispensed with; see Dumenil/Levy, 1984, p. 20). The results of their
basic model are as follows. With reaction parameters for quantity and
price changes within a certain range, production prices formulated for
a two sector model are locally stable and generate balanced growth as
dual of the equalization of profit rates. But their basic model is
unstable for simple reproduction and for the von Neumann growth path
(Dumenil/Levy, 1983). As one would expect, the system becomes more
flexible if a degree of substitution in demand is allowed for or if
the quantity of overproduced outputs is taken into account by firms.
For the two extreme cases a=O and a=l, the system becomes stable after
allowing for this additional degree of flexibility as is shown by
means of computer simulations (Dumenil/Levy, 1984)
The formalization of the dynamics of competition of the classics is
more satisfactory in this model than in other models. Dumenil and Levy
capture the crossover dynamics of classical competition sufficiently.
Some remarks about the character of their model are necessary,
10
however. The complexity of the possible results of their model depends
on the formalization that they have chosen. First, the budget
constraint for capitalist consumption and the capital fund constraint
for growth (6) create complicated feedback effects on prices and
quantities. Thus the question might be raised whether or not these
constraints produce unnecessary complications not necessarily of great
importance in a classical model of competition (since sufficient money
capital or commercial credit might always be available for capital
flows as, for example, Ricardo assumed). Second, their results are
also strongly determined by the existence of capitalist consumption
(a>O). This holds at least for their early version (Dumenil/Levy
1983) •
In their early version the dynamical model becomes stable for a>O.
For a=O there is no convergence unless additional flexibility is built
in (see Dumenil/Levy 1983, 1984). As known in the literature,
consumption and sUbstitution in consumption can be introduced in a
dynamical model as stabilizer (Morishima, 1977, ch.2). Since the
quality of the behavior of their model changes with this additional
complexity introduced into their model, it is somewhat left open what
the true stabilizing forces are in their interpretation of the
classical competitive process.
II. Neoclassical Theory of the Competitive Process
In the history of economic analysis it was not only the classics
that conceived the aforementioned dual adjustment process. One can
also find in Walras and Marshall a twofold process which was supposed
to stabilize capitalist market economies. A closer look at the main
works of Walras and Marshall shows that the widespread belief that
Walras conceived of a price and Marshall a quantity adjustment process
in their stability analysis does not seem to be convincing. A
classical approach to competition can be found in Walras, for example,
when he considers the process of groping in a production economy. He
writes, for example:
" .•• entrepreneurs use tickets ('bons') to represent the
successive quantities of products which are first determined at
random and then increased or decreased according as there is an
excess selling price over cost of production and vice versa,
until selling price and cost are equal." (Walras, 1977, p. 242)
and
11
"In fact, under free competition, if the selling price of a
product exceeds the cost of productive services for certain firms
and a profit results, entrepreneurs will flow towards this branch
of production or expand their output, so that the quantity of the
product (on the market) will increase, its price will fall, and
the difference between price and cost will be reduced; and if (on
the contrary), the cost of productive services exceeds the
selling price for certain firms, so that a loss results,
entrepreneurs will leave this branch of production or curtail
their output, so that the quantity of the product (on the market)
will decrease, its price will rise and the difference between
price and cost will again be reduced." (Walras, 1977, p. 225)
If one interprets the notion "cost of production" in the classical
sense, i.e., profit is included in the cost of production, the
crossover dynamics the classics conceived is very well formulated in
this passage by Walras. Moreover, if one adds more explicitly, as
Walras does, that prices themselves are in general determined by the
IIlaw of supply and demandll when he describes the process of IIgropingll
in a pure exchange economy one sees the similarity to the classics
clearly. With regard to price changes Walras contends that:
IIGenerally, however, the total demand will not equal total
offer, of each and every commodity •••• What will happen on the
market then? If the demand for any commodity is greater than the
offer, the price of the numeraire will rise; if the offer is
greater than the demand, the price will fall. 1I
(Walras, 1977, p. 169)
Even in Marshall's partial equilibrium analysis, the abovementioned
twofold adjustment process is considered as central for the
stabilization of markets. with regard to quantity changes due to
differential profit rates, Marshall writes:
IIWhen therefore the amount produced (in a unit of time) is such
that the demand price is greater than the supply price, then
sellers receive more than is sufficient to make it worth their
while to bring the goods to market to that amount; and there is
at work an active force tending to increase the amount brought to
12
sale. On the other hand, when the amount produced is such that
demand price is less than the supply price, sellers receive less
than is sufficient to make it worth their while to bring goods to
market on that scale .•. and there is an active force at work
tending to diminish the amount brought forward for sale."
(Marshall, 1947, p. 345)
On the other hand, concerning price changes due to imbalances of
supply and demand, we can read: "Market values are governed by the
relation of demand to stocks actually in the markets" (Marshall, 1947,
p. 372; see also p. 345 and Marshall's notion of short-run
equilibrium). Although Marshall develops more carefully the influence
of long-run changes of demand on the long-run supply price when
nonconstant returns to scale prevail, his formulation of the dual
adjustment process is not radically different from the formulation
that we find in the classics and Walras. In neoclassical analysis of
adjustments to equilibrium Walras' process of groping and tatonnement
in an exchange economy mostly only has survived. Morishima's work
(Morishima,1977) seems to be an exception.
utilizing, however, excess demand functions and the tatonnementprocess for the demonstration of stability in an exchange economy,
captures only one side of classical -- as well as Walrasian --
dynamics. This may be one of the reasons why the one-sided short-run
Walrasian price tatonnement or the so called "Law of Supply and
Demand" does not produce satisfying stability results in general
(Hahn, 1982). We conjecture that important stabilizing forces derive
from side of production and the behavior of the firm which therefore
must be integrated to give the "law of demand" a solid foundation.
To investigate this claim, we shall go to the opposite extreme in the
following section, namely to a model where we focus on production and
its profitability and where demand is treated in a very preliminary
fashion. (A further evaluation of the neoclassical literature is given
in Flaschel/Semmler 1985b).
13
III. Capital Movements and Output Reactions in an Input-output system
with Fixed capital
In the following a dynamical system is presented based on a n x n
input-output system with fixed capital. Neither the case of multiple
techniques nor joint production will be considered here. (The latter
models are treated in Flaschel/Semmler, 1985b, 1986, where also
further details with regard to the following brief treatment of our
adjustment process towards prices of production and steady growth can
be found). The dynamical system presented here has a unique
equilibrium, and it allows us to demonstrate global stability results.
Fixed capital is treated not as a joint product but on the input side,
as proposed by Brody (1970), Pasinetti (1973) or in conventional
dynamical versions of input-output models (Woods, 1978, p. 182ff).
Consider the following equilibrium price and quantity equations:
AX + g* Bx
pA + r*pB
x
p
x ?; 0
p ?; 0
(1)
with A the nxn matrix of capital consumed and B the nxn matrix of
capital advanced (see Brody, 1970 for details). We assume for
simplicity that A is irreducible and productive and that all columns
Bj of B and rows Bi of Bare semipositive (?;O). These properties of
the stock matrix B are a consequence of the property that flow matrix
A is irreducible if at least a small part of the capital which is used
up by a process must also be tied up in this process (aij > 0 => bij >
0; see again Brody (1970) for details].
We denote by I the identity matrix, by a prime (') the transposition
of matrices and vectors and by <x> the diagonal matrix derived from
a given vector x.
It is well known (see Woods (1978) for example] that there exists a
unique equilibrium solution x*, p* (up to scalars a 1 , a 2 ), r*, g* of
(1) for activities x and prices p such that
x* >0, p* >0, r* =g*
(since the two matrices (I-A)-lB, B(I-A)-l are strictly positive and
have the same dominant eigenvalue A = r*-l, g*-l > 0].
We want to investigate the stability of this equilibrium x*, p* with
regard to a cross-dual adjustment process as discussed in section 1.
For this purpose, we stipulate the following price and supply
reactions for this model of fixed capital:
14
x • <X>-l X = <d1 >(I - A - rOB) 'p
p • <p>-l p =-<dz>(I - A - r*B)x (2)
where x and p denote the time derivative of the vectors x and p [which
are transformed into vectors for growth rate of activity levels and
prices by means of the inverses of the diagonal matrices <x>, <p>
belonging to the vectors x, pl. For the vectors of adjustment speed
coefficients d 1 , dZ it is assumed that they are elements of the
positive orthant R! of Rn (d1 , dZ > 0).
The first part of system (2) states that the growth rate of activity
j is of the same sign as the term
PJ - p'AJ - r*p'BJ ,
i.e., the extra profit of sector j per unit of output with regard to
the steady state rate r* as norm (XJ is therefore proportional to the
sum of extra profits of sector j]. The second part of (2) on the other
hand, asserts that market prices Pi react positively (negatively) if
the current supply Xi falls short of (exceeds) expected demand which
is here defined by AiX + r*Bix, i.e., current input requirements plus
the extension of capital advanced r*Bix which is needed for future
growth (of the norm rate r*).
As our following stability proofs will show, the dynamics (2) need
only be considered on certain compact subsets of R!D, which are
invariant with regard to the stipulated dynamics, i.e., no trajectory
which starts in such a set can leave it later on. There is
consequently no need for nonnegativity (positivity) constraints to
keep the dynamics (2) economically (mathematically) well-defined. The
proofs of the following propositions will be kept brief, since they
are similar to those of the joint product system considered in
Flaschel/Semmler (1985b, 1986).
proposition 1: The point of rest x*, p* of system (2) is stable in
the sense of Liapunov, i.e., for every neighborhood U of z* = (x*,
p*)' E R!D there is a neighborhood U1 of z* in U such that every
solution z(t) with z(O) in U1 is defined and in U for all t>o.
Proof: Making use of the new notation z = (x, p)' E R!D we can
represent the system of differential equations (2) in the compact form
(3)
where Q is given by
[
0
Q -
-(I-A-r* B)
15
Note that the matrix Q is skew-symmetric which implies z'Qz = 0 for
all z E R2n.
To show stability, we will construct a so-called Liapunov function
around the given equilibrium z*, i.e. here, a differentiable function
V on R!D with the strict local minimum z* which fulfils
V = grad V(z(t»Z(t) ~ 0 (4)
[see Hirsch/smale (1974, ch. 9) for details].
We propose as Liapunov function the function V
given by
R!D --> R which is
V(z) = q'[(z-<z*>ln z) - (z*-<z*>ln z*)] (5)
where q E R!D is defined by qk dk 1 and (In Z)i by ln Zi'
It can be easily shown that the equilibrium z* of (2) is a strict
local minimum of the function V and that grad V(z)' = q' (I-<z>-l<z*».
For the derivative V of V along the trajectories of (2) we thereby
get
V = q' (I-<Z>-l<Z*»Z
= q'(I-<Z>-l<Z*»<z><d>Qz
(z - z*) 'Qz
o
since <q><d> = I, z* 'Q = 0 and z'Qz=O by the skew-symmetry of Q.
q.e.d.
The above proof shows that the function V is constant along the
orbits of the dynamical system (2). These orbits therefore remain in
the 'level surfaces' of the graph of V which enclose the minimum
z*=(x*, P*)' of V. Note that this need not imply that the orbits
of (2) must be closed.
The Liapunov function proposed above is well known from Volterra's
analysis of the n-species case of interacting populations (see Rouche
et al., 1977, p. 263). In an appropriately extended form this function
also allows for the treatment of the case of multiple activities and
the von Neumann inequality approach to joint production and process
extinction [if free goods are excluded from consideration, see
Flaschel/Semmler (1985b, 1986)].
To obtain asymptotic stability as well, we will now consider the
following modification of the original dynamics (2). Assume that the
activity levels not only change with the prevalence of extra profits
(or losses) but that this change is also dependent on whether these
extra profits are themselves currently rising or falling. The growth
16
rates of activity levels (x) consequently are considered to be
influenced also by the distribution of signs within the vector
d •
s = --(I-A-r"B) 'p = (I-A-r"B) 'p
dt
(6)
This vector shows the direction of change of extra profits or
losses.
By means of (6) we suggest a revision of the dynamics (2) of the
following type (x,p € R~):
A (7)
P
where ~>O is an adjustment parameter. We expect the influence of the
time rate of change of extra profits to act as an extra stabilizing
force.
Inserting the second equation of (7) into (6) gives
s = (I-A-r"B) '(-<p><d2>(I-A-r"B)x)
-(I-A-r"B) '<p><d2 >(I-A-r"B)x
S(p)x
(8)
It can be easily shown that the matrix S(p) is negative
semidefinite, [X'S(p)x~O for all x€R n and that in particular
x· 'S(p)x"=O].
Propostion 2: The system (7) is quasi-globally stable, i.e., each
sequence of points on a trajectory of (7) in R!n converges to a point
of rest of (7) [see Hahn (1982, p. 751) for the details of this
stability concept].
We note that the points of rest of the dynamical system (7) must all
be proportional to the given equilibrium z"=(x" ,p")', since p=O
implies x=x· and s=O, i.e., x=O implies p=p •. Therefore this
proposition entails that all trajectories of (7) in R!n approach the
ray in R!n which is determined by the equilibrium z·=(x" ,p.) '.
Proof: According to Hahn (1982, p. 751), a process is quasi-globally
stable if there exists a Liapunov function V for which we can show
that (i) V- 1 ((-00, c]), c>O is an invariant set with regard to the
trajectories of (7), (ii) V- 1 ((-00, c]) is compact, i.e., all sequences
along the trajectories in this set must contain convergent
subsequences, and (iii) all limit points (of the above convergent
subsequences) are points of rest.
From the definition (5) of our former Liapunov function V, it is
obvious that this function fulfils the second of the above three
17
conditions. We shall show in the following that this function (5) also
fulfills (i) and (iii), demonstrating that our original Liapunov
function is a Liapunov function in Hahn's senE;e,Le., it implies quasi-
global stability for the revised dynamics (7).
Ad (i): The new dynamics (7) can be written in compact form as
" z ... <d>Q (-y ) z (9)
where Q(~) is given by
[ ~S(p)
- (I-A-r* B)
[see (3) with regard to the other symbols].
utilizing the Liapunov function (5) we find for (9), cf. the proof
of proposition 1:
V - ql(I - <Z>-l<Z*»Z
... (z - z*) I Q (-y) z
... ZlQ(-y)Z
... ~XlS (p)x :s 0
since z* IQ(-y)=O, Q(-y)=Q(O)+ y [S~p) :], ZlQ(O)Z
negative semidefinite.
0, and since S(p) is
The above inequality implies that the sets v- 1 «-~, c]), c>o must
all be (positively) invariant, since V cannot increase along the
trajectories of (7).
Ad (iii): Following Hahn (1982, p. 750), we must show that V is
convergent along the trajectories Z of (7) and constant if and only if
such a trajectory describes a point of rest. The first of these
conditions has already been shown [since V is monotonically decreasing
along all solution curves Z of (7) in R!D and bounded from below]. To
show the second condition, let us assume that Voz is constant for an
entire orbit z(t)=(x(t), pet»~ I, ~o in R!D. The condition V = 0 then
implies xlS(p)x = 0 for this orbit which because of V- 1 «-~,c]) c R!D
implies (I-A-r*B)x=O. This gives x=Q1x* for some positive scalar Q1
[which can be shown to be independent of the time parameter t].
Inserting this result into (7) gives p=o. i.e.,
x - <d1>(I-A-r*B) Ip ... const.
Multiplying this equation from the left by w1 ... x* I<d1>-1 gives w12
... o. Since, however, a constant growth rate xj>O is incompatible with
our result that the compact sets v- 1 «-~,c]) are invariant, we get Xj
18
. .
= 0 for all j. Hence x - 0, i.e. z z 0, i.e., the assumed orbit z(t)
is a point of rest.
q.e.d.
Remarks:
1) The above result of quasi-global stability can also be expected
to hold in the case where the average rate of profit
p' (I-A)x
r(x,p)
p'Bx
is used instead of the natural rate r*, if a sufficiently large
parameter ~ is chosen [at least for an appropriately chosen region
around the equilibrium ray z*]. This assertion has been tested by
means of computer simulations [see section IV], and it may follow from
the fact that the discrepancy between the rates r(x,p) and r* may be
kept small relative to the stabilizing adjustment parameter ~ > o.
[This possibility of extending our result has been communicated to us
by R. Franke]. Due to limitations in space, this modification of
proposition 2 will not be examined in the present paper but is left
for future investigations.
2) Brody (1970, p. 90) proposes the following iterative procedure
for the so-called transformation problem [from values to prices of
production] :
P't (I - A)X
P't+l = pitA + ------
P't Bx
In the light of our adjustment process (2) this one-sided rule can
be extended in the following way:
pit (I - A)Xt
P't+l = pitA + ------- P't B
P'tBxt
P't A + r(xt , Pt) P't B
P't (I - A)Xt
------ BXt
P'tBXt
= AXt + r(xt , Pt )Bxt
In time continuous form these adjustment processes can be written as:
.
x = - [I - A - r(x,p)B]x
. p - - [I - A - r(x,p)B]'p
19
setting r = r* the following compact form can be utilized:
with
G
and C = (I - A - rOB).
.
z Gz
Neglecting the role of the adjustment coefficients d 1 , d Z our
proposed adjustment processes can be written as (see system (2))
· x <x>(I - A - r* B) 'p
· p -<p>(I - A - r* B)x
or in compact form:
· z = <z>Qz
with
Q = [ -: :' ]
Comparing the matrices G and Q, one realizes that the adjustment
processes derived from Brody -- and in a more generalized form
presented in Morishima (1977) as Keynesian adjustment processes -- are
dual adjustment processes only. Here solely the diagonal of G
determines the stability property of the adjustment processes. (The
stability property of these dual adjustment processes can further be
analyzed by referring to the so-called Metzler matrix, Hahn, 1982, p.
752). Our proposed adjustment processes, however, represent a cross-
dual dynamics. In its basic formulation (2) with ~ = 0 the off-
diagonal of Q determines the stability property of the system.
It has been suggested by Morishima that a comparison of these two
adjustment processes may be of importance in the further analysis of
the transformation problem. Note, however, that Marx's transformation
procedure derives from a purely formal redistribution of profits
between sectors [also in the above reiterated form], whereas our
proposed adjustment process takes the stylized facts of capitalist
competition into account, i.e., pays attention to profit rate
differentials when moving capital between sectors and utilizes a
simple version of the law of demand as the reference for price
modifications. An iteration procedure for our proposed dynamics is
developed in section IV.
3) The gradient of the average rate of profit r(x,p) = p'(I-A)x/p'Bx
reads for p and x respectively:
gradpr(x,p)
gradx r(x,p) =
20
(I-A)x plBx - Bx pi (I-A)x
(pIBx)2
(I-A) Ip p'BX - B'p pi (I-A)X
(pIBx)2
We denote by u the vector BX/p'Bx and by v the vector B'p/p'Bx.
Furthermore, let gi denote the current growth rate in the production
of commodity i and rj sector jls current rate of profit. By means of
these expressions the above gradient can be reformulated as
grad r(x,p) = { <v> [ r 1 -:r(x,p) ], <u> [ gl -:r(x,p) ] }
rn - r(x,p) gn - r(x,p)
i.e., it measures the discrepancies of sectoral growth and
profitability with regard to the average rate r(x,p) •
The dynamical system
A A (x,p) = - (- gradx r(x,p), gradp r(x,p))
therefore is of a similar type as the cross-dual dynamics which we
have formulated in (2). Furthermore, this kind of dynamics is well-
known in the study of saddle points in non-linear programming [see
Arrow et al. (1958)].
We conjecture therefore that the study of the gradient of the
average rate of profit may contribute in important ways to an analysis
of the dynamics of classical competition [cf. also the remark of von
Neumann (1945/46) concerning the function r(x,p)].
4) We have seen in the proof of proposition 2 that w1ln x must be
constant for a suitably chosen vector w1 • Similarly, w2 lnp will be
constant for the vector p. l<d2>-1. This implies a particular rule of
normalization of activity levels x and prices p [which is different,
but not inferior to the customary one: LXj = 1, LPi = 1].
IV. Computer Simulations of the Models
In the following section, iteration procedures will be developed and
computer simulations will be presented in order to demonstrate the
oscillatory character of system (2) and the asymptotic stability of
system (7). Moreover, in addition to the model presented in section
III a model with multiple techniques will be considered and simulation
results given which have not yet been proved mathematically.
21
IV.l. A Model with 'Circulating Capital'
For the purpose of simulations we will here refer to a Leontief
input-output system and a turnover time of fixed capital equal to one
(A=B). In the light of Brody's considerations of fixed capital (Brody,
1970, ch. 1.2), we then get A+r*B=(l+r*)A=R*A. Computer simulations
are performed for R=l+r~R* and R*=l+r* •
IV.l.l. Oscillation in the Price-Quantity Dynamics - Modell
In the first step, we will deal with the dynamic system of single
production without the additional stabilizing term. For the purpose of
a computer simulation the system (2) of section III has to be
rewritten as discrete model. By referring to time derivatives (x,p)
and setting B = A, the system can be written as:
.
(10)
p
Assuming <d1 >, <d2 > to be constant, we can derive the following
system of difference equations which provides us with an iterative
procedure to approach the question of stability of the equilibrium
(x* ,po ,R*) :
(11)
In addition to our formal proofs, we shall also consider the case
wherel R* is replaced by the average rate R. utilizing the iteration
(11), two types of simulation were performed. In the first version the
rate R changes in each step, since it is calculated as the average
rate of profit.
PtAxt
In the second version of the iteration (11), the ratio R was kept
constant (and equal to R* in general). For both of these versions as
well as for model 2 of asymptotic stability to be discussed in the
next section, the following input-output system, which includes real
wage goods in the elements of the matrix, was used:
[
.4
A-
.3
.6 ]
.5
22
The maximum eigenvalue of the input-output matrix A is A=.8722, i.e.
R*=1/A=1.1399. The associated right and left hand eigenvectors with
the normalization~x~ = 1, ~p~ = 1 are as follows: x; =.7826, x:
=.6225, p;=.5322, p:=.8466. The ratio of x; to x: is 1.257 and the
ratio of p; to P: is .62.
As mentioned, for our first version we assumed R(Xt'Pt) = 1 + r t to
change in each iterative step. The start vector for outputs was x~
=:9, x~ =.6 and their ratio is 1.8. The start vector for prices was P~
=.6 and P~ =.6, i.e., p~/p~=1. As can be observed, the start vectors
are not very close to the vectors associated with the maximum
eigenvalue. We used the following diagonal matrices as adjustment
coefficients:
[.1 0]
o .2
d2 = [.2 0]
o .3
The results of the iterations with 200 steps are graphically
demonstrated for R(xt 'Pt) in figures 1, 2, and 3.
,~.--------------------------------.
,..1
,.21
,.2
1.1. -,.....:1,--+--\---/--1---1'----+--+--\--+--\--+1
1.'"
0 ....
w ~ ~ ~ ~ ~ ~ ~ ~ =
T ...
oscillation of profit
rates
Figure 1
~2~-------------------------------rn
u,
'.0
"" .j-j...----f-----\-----+-----\-----+---I
Q4
Q2
~ ~ ~ ~ ~ = ~ ~ ~ =
T"'"
oscillation of relative
prices
Figure 2
The computer simulations for our model with R(xt,pt) revealed what
has already been conjectured in sections II and III, an oscillatory
behavior of prices and quantities. In contrast to our mathematical
demonstration, we now observe an oscillatory movement with slightly
increasing amplitude in the course of time. The fluctuations of
relative outputs and prices increase with the number of steps
performed. Moreover, it was also observable that whenever relative
23
Oscillation of relative outputs
Figure 3
outputs or prices were close to their respective eigenvectors, the
duals showed the greatest distance from their respective eigenvectors.
In addition, with the increasing amplitude of these ratios the average
profit rate -- and thus the average growth rate -- was also
fluctuating with an increasing amplitude. The procedure was also
repeated for different adjustment coefficients and, as would be
expected, the increase of the adjustment coefficients made the
fluctuation stronger.
In general, we obtain the results that the actual price vector is
mapped into quantity imbalances and the quantity imbalances are mapped
into price changes. This process continues without any convergence.
Instead a kind of gravitation (with increasing amplitude) was the
general outcome.
In a second version of our model, R was taken as constant and equal
to R* = 1.1399, the natural rate of profit. By using the same data as
in the first version and also applying the same start vector to the
iteration (11), we obtained quite similar results to that of the first
version of our model. Prices and quantities were continuously
oscillating (with a slightly increasing amplitude).
Although according to the reasoning given in section III of the
paper, we would have expected some kind of regular oscillation for the
vectors x and p, the actual iteration with the formula (11) showed a
slightly perturbed behavior in this respect (an increasing
fluctuation). These slightly different results should be due to the
24
difference in the stability properties in the formulation of the
process in continuous time (10) and in discrete time (11).
An additional run was performed with an arbitrary R = 1.18. This
assumes that capital movements are oriented towards a profit and
growth rate of 18% and not towards the steady profit and growth rate
of 13.99% or an average profit rate r(xt,pt). The results of the run
did not differ significantly from the version tested earlier, with the
provision that the amplitude of the oscillations was increasing
faster.
This allows us to conclude that the orientation of capital flows
toward any arbitrary average profit rate in the neighborhood of the
steady profit rate R* does not change qualitatively the stability
properties of the dynamical system as proposed in (10) and as
reformulated in discrete time by (11).
IV.l.2. Convergence of the Price-Quantity Dynamics - Model 2
The additional adjustment term in the system (7) assumes that firms
respond with their investment not only to the deviations of the profit
rates from the natural or average value, but they also take into
account the rate of change of profit rates when deciding on
investments. System (7) can be rewritten by referring to the time rate
of change as in system (7) as follows
.
x <d1><x>[(I-R*A) 'p + ~S(p)x]
. (12)
p
System (12) suggests therefore the following discrete form for our
iteration procedure.
Xt + 1 Xt + <d1 ><Xt > [ (I-R* A) 'Pt + ~S (p) xt ]
Pt+l Pt + <d2 ><Pt>(AR*- I)x
(13)
Taking into account the determination of S(p)Xt as proposed in
section 2, the first part of the iteration (13) can be written as
with
Xt + <d1><xt>(I-R*A)'Pt + <dl><Xt>~s
_<dl><Xt>~(I-R*A) '<d2 ><Pt>(I-AR*)Xt
The average rate R(Xt'Pt) will also be used instead of the natural
rate R* in the following.
In the new iterative procedure, we use the same data for the matrix
A as in section IV.l.l which means that the maximum eigenvalue and the
associated eigenvectors are the same as before. The adjustment
25
coefficients d 1 , d 2 remain the same as in the model before. In
addition, we use ~ = .5 as adjustment parameter for S(p)xt .
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