The new business cycle three-dimensional model is formulated on the basis of new system approach known as the Method of Systems Potential. Cyclical dynamics with catastrophe jumps (alike to N.Kaldor (1940) model) and some stochastic properties is described. Properties of such cycles are similar to the properties of the typical business cycle.business cycle model, catastrophe theory, economic potential

Grigorii Pushnoi

English

Research Papers in Economics

  1
Business Cycle Model on the Basis of Method of Systems 
Potential. 
(Report for the Second Internet Conference on Evolution Economics and Econophysics; Organized 
by International Bogdanov Institute; 01.11.04 – 14.11.04). 
Grigorii S. Pushnoi. 
   2
A brief description of Method of Systems Potential (MSP) was given in the 
previous report [8]. Deduction of evolution equations for the normal evolution
1 of 
Complex Adaptive Systems (CAS) was considered. We assumed that the Economic 
System is some Complex Adaptive System
2 which evolve on the basis of universal 
laws of such systems. These laws were considered in paper [8].  
The goal of this short message is to demonstrate how abstract ideas and 
conclusions of MSP-approach can be successfully applied in the economic theory. 
Interpretation of basic system variables and evolution parameters of MSP as applied 
to the economic CAS would be given below. Solution of this problem leads to 
reformulation of evolutionary equations in terms of the economic variables and 
economic parameters. Such reformulation of CAS-structure and CAS-dynamics in 
economic terms gives some new explanation of the Economic System dynamics. The 
new business cycle model follows from such reformulation. We show that this model 
can be interpreted as some quantitative version of with well-known N.Kaldor’ [5] 
model. 
Let us choose the following state-variables of CAS: “potential”, Φ , 
“conditions of realization”, U , “realizable part of potential”,  R Φ . MSP evolutionary 
equations in these variables turn into the following equations: 
RR a Φ=⋅ Φ  ;            ( 1 )  
() R da d Φ+ ⋅Φ= + ⋅Φ  ,          ( 2 )  
R UU ν +Λ⋅ = ⋅Φ  ,           ( 3 )  
To apply these equations to the Economic System abstract values, Φ , U , and 
R Φ  should be connected with some macro-economical variables characterizing the 
properties of economy as united system. Such properties could be for example output 
Y , fixed capital,  K , human capital, and (or) some other characteristics of the 
Economic System.  
We start from considerations of simplicity and naturalness. Our deductions 
and reasoning were built in the following way. Activity of the Economic System is a 
process of production
3.  Economical index reflecting value of production activity is 
the gross output. As the value of production activity depends directly upon realizable 
part of economic potential we came to the following conclusions: 
Supposition 1: Issue of final product (gross output) could be considered as 
economical index of realizable part of the Economic Potential. 
R Y Φ= .            ( 4 )  
The definite value of costs can be used more or less effectively i.e. these costs 
can ensure more or gross output in economics. We suppose that there is the only one 
                                                           
1 The ‘normal evolution’ of Complex Adaptive System is such way of development when only two forces determinate 
the process: (1) activity of a System as only source of increments in ‘potential’ and ‘conditions of realization’ of this 
System and (2) entropy influence as destroying force [8]. 
2 Determination of CAS and description of its structure and function is given in paper [8]. 
3 We use this term in a very wide sense i.e. “production” here include any economically necessary activity of economic 
agents – trade, marketing and management in particularly.   3
highest possible gross output with certain given expense for each value of expenses. 
This highest possible (or ‘potential’) output characterizes maximal ability of given 
Economic System to produce the gross output
4. It follows that: 
Supposition 2: Potential output,  max Y , is an economical index of Potential of 
the Economic System: 
max Y Φ= .            ( 5 )  
Finally last supposition concerns the economical interpretation of term 
“conditions of realization”. 
Supposition 3:  Value of fixed capital,  K , could be considered as index of 
“conditions of realization” in the Economic System. 
This statement requires explanations. 
First, supplies of fixed capital in the country with the capitalistic economy are 
necessary conditions of productive application of available country’s resources. 
Thus, low developed countries that own supplies of resources could not use 
them because of absence of necessary supplies of capital. Involving of any resources 
to the production process requires some definite capital investments. In this sense the 
capital is condition of productive application of these resources, i.e. condition of 
realization of available economical potential. 
Second, the equation (3) describes the process of accumulation of some value. 
Formally this equation looks the same as equation of accumulation of fixed capital if 
identify parameter Λ  with depreciation rate, and parameter ν  with norm of gross 
investments: 
KKY ν +Λ⋅ = ⋅  .           ( 6 )  
Third, statistics testifies that capital productivity 
Y
K
 is stable enough value 
during long periods of time. For example in USA economy this ratio in fact did not 
changed from 1909 to 1929, and in average was equal approximately  0.3 ≈ . 
Constance of this ratio lies in the base of acceleration principle of contemporary 
economic theory. 
This allows write down the following approximate equality: 
KY
a
KY
≈=

.            ( 7 )  
From the equation (6) follows that capital productivity could not change much 
indeed, and it is approximately equal: 
                                                           
4 Let’s indicate also that potential output depends on the technological base of economy. Non-optimal resources 
allocation and other factors that prevent to full capacity utilization must be taken into account also. We do not discuss 
here these nuances in determination of potential output. Index of capacity utilization (widespread in the modern 
statistics as the index of business activity) and realization ration of economic CAS are almost identical terms. Logistic 
growth of realization ratio in MSP-model of Economic System dynamics indicates that degree of utilization of technical 
and organizational basic innovations must be taken in account. Our definition of potential output means the full 
utilization and optimal coordination of all production factors (including the technological factor also).   4
Ya
K ν
+Λ
≈ .            ( 8 )  
Forth, the equation for the fixed capital accumulation is essential for any 
economical model. But we have only three evolution equations. As we identified Φ  
and  R Φ  with  max Y  and Y  respectively, we could follow one of two ways: to identify 
the equation of accumulation of fixed capital with equation for value U  or consider 
the equation of capital accumulation as an additional fourth evolutionary equation. 
But that should mean unjustified expanding of system of evolutionary equations. 
Finally we did not found any other economical interpretation of value U . This 
value, for example could not be identified with human resources of the country, L, 
because productivity of labor as distinct from capital productivity is permanently 
increasing value while we should obtain the stable productivity of labor by 
substitution KL →  in formula (6) if rate of growth of labor is constant. 
According to the listed-above suppositions evolutionary equations of MSP 
lead to the  following economic model of long-run dynamics of the Economic 
System. 
1. Equation of the growth of gross output with the constant rate of growth: 
Ya Y =⋅  ;            ( 9 )  
2. Equation for fixed capital accumulation: 
KKY ν +Λ⋅ = ⋅            ( 1 0 )  
3. Equation for adjustment of actual to potential output: 
[] [] Y Y d
dt
Y Y d
− ⋅ − =
−
max
max .        ( 1 1 )  
The last equation is only the other form of record of equation (2) taking into 
account the equation (9). Equations (9)-(11) describe the definite economical model. 
The basic properties of this system of equations were considered in paper [8]. These 
equations are completely equivalent to evolution equations of MSP. 
Evolution parameters of the Economic CAS have very simple interpretation: 
ν - Rate of gross investment; 
a- Rate of growth of output;  
Λ- Depreciation rate of fixed capital; 
d   - Adjustment coefficient in actual-potential output interaction.   5
 
Figure 1.   The business cycle of the Economic System  
according to MSP-approach. 
All the properties of dynamics of CAS evolutionary equations solutions 
transfers to the properties of dynamics of solutions of economic model (9)-(11). 
Particularly when  d > Λ  the cyclical dynamics of economic variables taking place. 
The index of efficiency,  R, and wealth-density z , of the Economic System 
satisfy to the following relations: 
max Y
Y
R = ;            ( 1 2 )  
max Y
K
z = .            ( 1 3 )  
Efficiency grows as logistic function
5: 
( ) ( ) 1 R adR R =+⋅ ⋅ −  .         (14) 
                                                           
5 This formula follows from (9) and (11). Logistic law means that potential output of our model includes the full 
capacity of technological factor of growth. Just such form of dependence corresponds to process of diffusion of radical 
innovations in economy. Many authors [1; 4; 7] proposed to approximate the long-term efficiency growth by logistic 
function. MSP explains why just such law of growth acts.   6
Efficiency,  R, is the function of wealth-density of a system. This function 
satisfy to the following ordinary differential equation
6: 
() () ()( ) ( ) 10 z Ra d z R d z a d R R ν ⎡⎤ ′ ⋅ −+⋅⋅ +− Λ ⋅−+⋅ ⋅ −= ⎣⎦ .    (15) 
The graph of efficiency index has the shape of sharp lopsided ridge, and the 
top of it is the point of long-run equilibrium of a system. But this is the unstable 
point. That is why the system when reaches the ridge, jumps to the lower evolution 
branch and after that moves along the lower branch. Point,  1 z , is the second unstable 
point of a system. In this point the system jumps to the upper evolutionary branch and 
moves along the upper branch to the unstable point,  0 z
7. This cycle recurs. 
Cycles composed from phases of smooth changes and catastrophic jumps are 
described in H. Varian’ [9; 10] model of business cycles. But in his modernizing 
version of N. Kaldor’s [5] model cycle does not recur because the long-term 
equilibrium point corresponds to the stable state. Exogenous shock is the necessary 
condition for generation of the cyclical process in H. Varian’ model. Later George, D. 
[3] suggested to change the H. Varian’s model in such a way that the equilibrium 
point becomes unstable. Cycles with unstable equilibrium point formally could be 
completely identified with the cycles generating in a system described by equations 
(9)-(11). 
But in all other respects with the exception of the formal similarity our model 
strongly differs from these models. 
Hysteresis loop in the plane  ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
max max
;
Y
Y
Y
K  after the scaling transformation 
transforms into the classical N. Kaldor’ loop of the business cycle in the plane 
() Y K;
8.  
The business cycle in MSP-model has the same properties as the typical 
business cycle according to W. Mitchell’ [6] investigations. These properties were 
listed and discussed in article [8]. Besides that it is asymmetric cycle that is also the 
property of typical business cycle according to G. Gabish’ and H. Lorentz’ 
examination [2]. 
Note that duration of different phases of cycle in our model depends on 
fluctuations of evolutionary parameters and other disturbances of the Economic 
System (exogenous shocks). Duration of prosperity phase depends on perturbations 
                                                           
6 Details of derivation and solution of this equation are contained in paper [8]. 
7 Instability of these points is the direct consequence of the existence of mechanisms that stabilize the steady state of a 
system. Deviation of a system from point of evolution branch (which correspond to its steady state) activates the forces 
which tend to come back the system. In neighborhood of points,  1 z  and  0 z  this forces generate catastrophe jumps in 
efficiency of a system (revival and crisis phases of the cycle). 
8 Moreover MSP-model like to N.Kaldor’ in some other aspects. Three equilibrium points (investment = saving) of 
N.Kaldor’ model correspond to three possible values of efficiency for any  0 1 z z z < < . Regulating force in N.Kaldor’ 
model (excess investment over saving) is equivalent of MSP stabilizing mechanisms that come back the system into its 
steady state. Cycle in both models consists of two catastrophe jumps and two phases of gradual change. Thus MSP-
model can be interpreted as some quantitative version of N.Kaldor’ model.   7
of a system very strongly since the duration of this phase tend to infinity as 
perturbations of a system tend to zero. Thus this cycle unites the properties of 
determined and stochastic processes that make the dynamics of MSP-model very 
complicated and multiform. 
References. 
1.  Ayres, R., 1988, A Theory of Growth Rate Discontinuities, J. of Environmental 
Economics and Management 15, pp. 439-461. 
2.  Gabish, G. and Lorenz, H. W. 1989.  Business Cycle Theory. A Survey of 
Methods and Concepts. Springer-Verlag, Berlin. 
3.  George, D., 1981. Equilibrium and Catastrophes in Economics. Scottish Journal 
of Political Economy, 28, pp. 43-61. 
4.  Haustein, H. D., The Pathway of Dynamic Efficiency: Economic Trajectory of a 
Technical Revolution, In “The Long-Wave Debate” (ed. T. Vasko), pp. 198-215, 
Springer, 1987. 
5.  Kaldor, N., 1940, A Model of the Trade Cycle. Economic Journal, 50, pp. 78-92. 
6.  Mitchell, W. C., 1927,  Business Cycles. The Problems and Its Setting. The 
National Bureau of Economic Research: New York. 
7.  Mosekilde, E.; Rasmussen, S., and Zebrowski, M., Technoeconomic Succession 
and the Economic Long Wave, In “The Long-Wave Debate” (ed. T. Vasko), pp. 
257-273, Springer, 1987. 
8.  Pushnoi, G. S., 2004, Application of Method of System Potential for Analysis of 
Economic System Evolution. Report for the Second Internet Conference on 
Evolution Economics and Econophysics. 
9.  Varian, H.R., 1979.  Catastrophe Theory and the Business Cycle. Economic 
Inquiry, 17, pp.14-28. 
10.  Varian, H.R., 1981. Dynamical Systems with Application to Economics. In: 
Arrow, K.J./ Intriligator, M.D. (eds.): Handbook of Mathematical Economics, v.I, 
Amsterdam: North Holland; pp. 93-110. 

Business Cycle Model on the Basis of Method of Systems Potential (english version).

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Business Cycle Model on the Basis of Method of Systems Potential (english version).

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