Discussion Paper No.276
Marxian Growth Model with Production Delay
Toichiro Asada
Chuo University

Akio Matsumoto
Chuo University

Ferenc Szidarovszky
University of Pécs

March 2017

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Marxian Growth Model with Production Delay
Toichiro Asada,yAkio Matsumotozand Ferenc Szidarovszkyx

Abstract
In this study we demonstrate the possibility of persistent oscillationsof national income in ala Marxian growth model with consumption andinvestment sectors. Three main results are shown. First, the existence ofthe steady state is con…rmed. Second, the discrete-time dynamic model
augmented with pollution e¤ects generates simple as well as complex dynamics. Finally, the continuous-time dynamic model with production
delay gives rise to regular as well as erratic dynamics. These …ndings indicate that the Marxian growth model with nonlinearity and production
delay may explain various dynamic phenomena of economic variables.

Keywords: Marxian growth model, Pollution e¤ect, Production delay, Complex dynamics, Discrete-time dynamics, Continuous-time dynamics

The authors highly appreciate the …nancial supports from the MEXT-Supported Program
for the Strategic Research Foundation at Private Universities 2013-2017, the Japan Society
for the Promotion of Science (Grant-in-Aid for Scienti…c Research (C), 25380238, 26380316,
16K03556) and Chuo University (Joint Research Grant). The usual disclaimer applies.
y Professor, Department of Economics, Senior Researcher, International Center for further
Development of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, asada@tamacc.chuo-u.ac.jp
z Professor, Department of Economics, Senior Researcher, International Center for further
Development of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, akiom@tamacc.chuo-u.ac.jp
x Professor, Department of Applied Mathematics, University of Pécs, Ifjúság u. 6., H-7624,
Pécs, Hungary, szidarka@gmail.com

1

1

Introduction

Applying the recent development of nonlinear dynamic theory to a Marxian
two-sector growth model, we aim to reconsider the provocative result, "theinevitable breakdown of capitalism," that scienti…c socialism advocates. We …rst
demonstrate convergence to a steady state of the growth model and then thepossible birth of persistent oscillations that is inevitable phenomenon observedin an actual economy.
It has been well known that the two underpinnings of Marxian economics are
the theory of surplus value and the theory of historical materialism. The former
measures worker exploitation by capitalism that might be a source of the instability of the capitalist system. The latter describes the law of society that social
phenomena obey as natural phenomena obey the law of nature. Accordingly
social evolution to the socialist society from the capitalist society through theend of capitalism is historical inevitability. It is often mentioned that JapaneseMarxian economist, Nobuo Okishio, formulated the theory of surplus value andproved many Marxian theorems. Among others, Okishio (1955) con…rmed thatthe exploitation of surplus labor is the necessary condition for the existence
for positive pro…t, which was later called "Marxian Fundamental Theorem" byJapanese mathematical economist, Michio Morishima. Recently, mathematical formalization of the historical materialism has begun (see Yamashita andOhnishi (2002), Ohnishi (2014) and , for example). Using the basic structure of
the neoclassical optimal growth model, a two-sector growth model with labor
being the primary factor of production is constructed. It shows historical transition from the beginning of the capital accumulation caused by the Industrial
Revolution to the end of the capitalism at which the capital accumulation arrives at the saturation point and thus stops further development. In the existing
literature of Marxian economics, however, not much has yet been revealed withrespect to the circumstances under which capital accumulation exhibits persistence cyclic oscillations that are frequently observed in actual economic data.
In particular, many researchers have been involved to explain why an economy ‡uctuates since the pioneering works by Kalecki (1935), Samuelson (1939),
Kaldor (1940), Hicks (1950), Goodwin (1951), to name only a few. The main
purpose of this study is to demonstrate the possibility of persistent ‡uctuationsof national income as well as capital accumulation under Marxian circumstances
when both negative e¤ects on production caused by the capital accumulationand production delays are taken into account.
The rest of the paper is organized as follows. In the next section, we reconstruct a simpli…ed Marxian two-sector growth model and con…rm saturationof the capital accumulation. In Section 3, we examine dynamics in discretetime scales after introducing time-to-build technology and pollution e¤ect of
the capital. In Section 4 the same dynamic model is cast into a continuous-timeframework focusing on a destabilizing role of the production delay. Section 5o¤ers concluding remarks.

22

Marxian Growth Theory

Recapitulating the basic elements of the Marxian growth model examined by
Takahashi (2011), we consider an economy with two sectors in which capitaland consumption goods are produced in the capital goods sector and in the
consumption goods sector, respectively. It is assumed that both sectors arecharacterized by Cobb-Douglas production functions. Denoting production ofcapital goods (i.e., investment) by Y1 and production of consumption goods by
Y2 ;(1)
Y1 = A1 K1 L11 ; 0 < < 1and
Y2 = A2 K2 L12

; 0<

<1

(2)

where Ai denotes the production technology level, Ki and Li are capital andlabor in sector i (i = 1; 2) satisfying
Ai > 0; K1 + K2 = K > 0 and L1 + L2 = L > 0:
K and L are the total amount of capital goods and total population. L is
assumed to be constant only for the sake of analytical simplicity whereas thecapital goods is changing according to the capital accumulation equation,_K(t)= Y1 (t)

K(t)

(3)

where the dot over a variable means time derivative and > 0 is the depreciation
rate of capital. Equilibrium in the consumption goods market is described by
Y2 (t) = wL
where w is the wage rate. It is further assumed that production factors areconstant proportions of the existing total amounts,
K1 = K , K2 = (1)K with 0 <

<1

(4)

L1 = "L, L2 = (1")L with 0 < " < 1:

(5)

and
The next task is to construct a dynamic equation of the model by substituting (4), (5) into (3),1_
K(t)= A1 [ K(t)] ("L)K(t):

Dividing both sides of the last equation by L and introducing the new variable,
k = K=L; present an accumulation equation of the percapita capital,_k(t)= A1 [ k(t)] "1k(t)that is simpli…ed as_k(t)= ak(t)k(t) with a = A1
3

"1

> 0:

(6)

This is the fundamental equation of the Marxian growth model. The steadystate of the percapita capital is
a

k =

1
1

:

Under the well-behaved Cobb-Douglas production function, the stability of kis con…rmed, that is, k(t) approaches k monotonically as t ! 1; regardless of
the initial value of k(0) > 0 and k(0) 6= k . We summarize the results obtained:
Theorem 1 The steady state values are obtained as
K = k L; Y1 = a1 L and Y2 = a2 L
where
a1 = A1"1(k ) > 0 and a2 = A2 (1

) (1

")1

(k ) > 0and each variable monotonically converges to its corresponding equilibrium value
as t ! 1:
Since Y2 can be rewritten as
Y2 = a02 (1

")1

"

(7)

with

a

a02 = A2 (1

A1

)

1

L;

di¤erentiating equation (7) with respect to ", equating it to zero and solving
the resultant equation yields the optimal labor allocation ratio that maximizes
consumption Y2 ," = :Notice the resemblance between the Marxian and neoclassical (i.e., Solow-Swan)
growth models. The optimal labor allocation that maximizes consumption is determined in the Marxian model where as the optimal saving rate that maximizes
consumption is determined in the neoclassical model.

3

Pollution E¤ect and Production Delay

We have seen that all trajectories of k(t) monotonically approach the steadystate in the continuous-time framework. In a real economy, we often observe…rst that production process may require gestation (i.e., time-to-build technology) and/or information lag and second that production process could be described by strong nonlinear production functions. Considering these points, we
modify the model (6) by augmenting su¢ cient nonlinearities and a time delay.
Concerning nonlinearities, we consider a "pollution e¤ect" caused by increasing
concentration of capital according to a multiplicative term, (m k) with > 0.
4

The production technology is modi…ed such that per capita output is produced
according to the function,y = ak (m k)(8)
For a smaller k, the …rst factor is small and this term does not make any essential
contribution to per capita production. On the other hand, for a larger k close
to m, productivity is largely reduced due to a negative (i.e., pollutant) e¤ect on
per capita production. For the sake of analytical simplicity, we assume = 1,
= 1 and m = 1 with which the production function becomesy = ak(1k):(9)

This is considered to be the AK production function taking the in‡uence ofpollution into account as a …rst approximation. Concerning the time delay,
we reformulate the model in discrete time in which economic activities anddecisions are made at discrete time intervals. The aim of this section is to verify
whether the dynamic emerging in the fundamental equation in continuous time_
are robust with respect to these modi…cations. Replacing k(t)with kt+1 kt
reduces model (6) to
kt+1 = akt (1kt ) + (1

)kt :

(10)

Introducing the new variablex=aa + (1)k

reduces equation (10) to the logistic equation
xt+1 = xt (1xt ) withA positive …xed point is obtained for= a + (1

(11)

> 1;
1

x =

):

:

(12)

The dynamics of kt generated by the fundamental equation in discrete time isequivalent to the dynamics of xt controlled by the logistic equation (11). The
dynamic structure of the logistic equation has been extensively studies so far
and it is now well-known that it can generate a wide variety of dynamics ranging
from monotonic convergence to chaotic ‡uctuations, depending on the value ofthe parameter : For 0 <1; the …xed point is zero, which is eliminatedfor further investigation since it is economically unrealistic. We will perform
numerical simulations under di¤erent values of > 1:
In the …rst simulation illustrated in Figure 1, we start with an initial condition x0 = 0:8 and is taken to be 1:5: The percapita capital quickly and5monotonically converges to the positive …xed point.

Figure 1. Monotonic convergence to x
The second simulation illustrated in Figure 2 is performed taking the same
initial value of x0 and = 2:9. The time trajectory shows dumping oscillations
approaching to the positive …xed point.

Figure 2. Oscillatory convergence to x
We still take the same initial value of x but increase the value of to 3:5: The
resulting third simulation is illustrated in Figure 3 in which the time trajectoryshows persistent oscillations (i.e., a period-4 cycle). The …xed point is locallyunstable since the derivative of the logistic equation evaluated at the …xed point
is greater than unity in absolute value,
dxt+1
> 1:
dxt
6

However the nonlinearity of the logistic equation prevents unstable trajectories
from diverging. In consequence the time trajectory neither converges to the…xed point nor diverges in the large.

Figure 3. Periodic oscillations around x
The fourth and last simulation was done with the initial value x0 = 0:45 and= 4:1 As is seen in Figure 4, the time trajectory exhibits persistent andirregular (i.e., chaotic) oscillations.

Figure 4. Chaotic oscillations around x
We can summarize the simulations results as follows:
Theorem 2 The steady state of k is de…ned by
k =

1
1+

for

>1

1 For
> 4; the dynamic equation (10) generates time trajectories going to negative in…nity. Such dynamics is economically meaningless and thus is also eliminated from further
investigation.

7

and follows the dynamics described as(1) monotonic convergence to zero for 0 <1;(2) monotonic convergence to k for 1 <2;(3) oscillatory convergence to k for 2 <3;(4) periodic oscillations around k for being close to 3;(5) aperiodic oscillations around k for being close to 4:

4

Hopf Bifurcation of Capital Accumulation

In this section we focus on the case in which economic activities and decisions
run continuously and then investigate the e¤ects caused by the production delayin the continuous-time scales.2 To this end, we return to equation (6) witha di¤erent type of production function that possesses essentially the similar
nonlinear property,ky = ak ewith > 0. This function takes a mound-shaped pro…le that has zero value at
k = 0; increases for k < = ; decreases for k > = and converges to zero as k !
1: It has the maximum value at k = = ; implying that the negative pollutant
e¤ect caused by increasing concentration of capital is de…ned for any k > = .
With this modi…ed function including production delay and assumption = 1,3
the dynamic equation becomes_k(t)= ak(t

)e

k(t

)

k(t)

Multiplying both sides of the above equation byvariable x(t) = k(t); simplify the expression,x(t)_= ax(t)e x(t)(13)

and introducing the new
x(t):

(14)

The steady state is de…ned by x(t)_= 0 and solves the following equation,
x ae x= 0:

The trivial steady state is x = 0 and the non-trivial steady state denoted by x
is
hai
(15)
x = log

where, to ensure a positive value of the nontrivial steady state, it is assumed
throughout the paper thata> :This inequality assumption is natural since the production technology a contributes to increase the production level (i.e., a > 0) and the depreciation rateis usually not greater than unity (i.e.,
1).
2 This section depends on Matsumoto and Szidarovszky (2011, 2013).
3 See Matsumoto and Szidarovszky (2013) for the case with

8

6= 1:

Set z(t) = x(t)x ; then z(t) satis…es)e z(t):and its characteristic equation with a possible solution z(t) = e

t

u is

z(t)
_ =

z(t)

x

e z(t

1

)

+ z(t

The linearization of the last equation at z(t) = 0 is
z(t)
_ =

z(t)

(x

+ + (x1) z(t

1) e

)

= 0:

(16)

It is con…rmed that the nontrivial state is locally asymptotically stable for = 0
since > 0 implies =x < 0: For > 0; it is clear that = 0 does not
solve the characteristic equation. At a stability switch, we suppose that = i!
with ! > 0 is a solution,
i! + + (x1) (cos !

i sin !) = 0:

Separating the real and imaginary parts, we obtain two equations,(x1) cos ! =

(17)

(x

1) sin ! = !

(18)

and
Adding the squares of these equations yields
!2 =

2

x (x

2):

! can be positive if and only if x > 2 or equivalently if a > e2 : Let the positiveroot of ! be denoted by ! 0 ,p
x (x2) > 0!0 =and solving the real part equation for
m =

1
cos 1
!0

yields

1
x

+ 2m

1

for m = 0; 1; 2; :::(19)

Solving the imaginary part equation yields the same value of the delay in adi¤erent form.
Di¤erentiating the characteristic equation with respect to presents
1

(x

d
d

1)e

(x

1) e

=0

which is solved for (d =d ) 1
d
d

1

=
=

1

(x
1)e
(x
1) e

(x
9

1
1) e

:

Using the characteristic equation,(x
substituting1)e=( + );= i! and taking the real part yield
"
#
1
d
1
Re
= Re
d
(
+ )
=i!

=i!

1

=

Re

=

1
!2 +

!2

i !

2 > 0:

So at the smallest critical value 0 stability is lost and cannot be regained later.
We then have the following results:
Theorem 3 For the capital accumulation equation, the following hold:(1) If < ae2 , then k = x = is locally asymptotically stable.
2(2) If a > e ; then k = x = is asymptotically stable for < 0 and unstable for >(3) For a > e2 ; k = x = loses stability and undergoes Hopf bifurcation at = 0 ;where

0 is de…ned as
1
cos 1
k
1
p
=
:
0
k ( k
2)

We now perform simulations to con…rm the theoretical results. We take
= 5; = 1 and = 0:2 for which the positive steady state is ensured. Further
the critical value of the length of the delay is

0 =

1
cos 1
k
1
p
' 5:145:k (k2)

In the …rst simulation, we take

=3<

10

0 and thus have convergence to the0.

steady state k as shown in Figure 5.

Figure 5. Oscillatory convergence
In the second simulation, we increase the value of to 6 > 0 . which violates
the stability condition. As a result, a limit cycle emerges via a Hopf bifurcationaccording to parts (2) and (3) of Theorem 2.

Figure 6. Emergence of limit cycle
In the third simulation, we further increase the value of to 30; with which thedelay equation (13) gives rise to erratic behavior of k(t):11Figure 7. Chaotic oscillations

5

Concluding Remarks

In Section 2 of this paper, we have reconstructed a Marxian two sector growthmodel that is similar to the neoclassical growth model and then shown that any
trajectory converges to the positive steady state. From the Marxian economic
point of view, this convergence implies the end of capitalism in a sense that theeconomy ceases to grow. In Sections 3 and 4, we have demonstrated the existenceof persistent cyclic ‡uctuations in the same model if we take into two essential
elements, nonlinearity caused by the strong capital accumulation and a delayin the production process. In particular, dynamics in the discrete-time framework exhibits wide variety of dynamics ranging from monotonic convergence
to complex dynamics involving chaos whereas dynamics in the continuous-timeframework is characterized by the stability switching curve: the steady state is
asymptotically stable if the length of the delay is below the curve, loses stability
just on the curve and bifurcates to a limit cycle if above the curve. With theseresults the paper highlights the fact that nonlinearity and delay are essential
ingredients for the emergence of persistent oscillations even in the simpli…ed
Marxian economic growth model.

12References

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13