Table 1   Economy State According to Cycle Phases

 
128
 




N
i
i
i
i
p
x
y
Y
1
)
(
.   
To create income Y for sectors or scopes of activity the amount of necessary resources 
(R) (natural resources, physical capital, human resources, etc., that is, material resources) will 
totally be:  



N
i
i
R
R
1
 
Financial resources are also distributed between sectors and activities, so that their total 
volume can be defined: 



N
i
i
f
F
1
, where f
i
 is finance used on the given period of time by i – 
sector. Money supply in the country can be designated as M = α F. If the price of i – resource is 
s
i
, then the necessary volume of financing for this sector is f
i 
= R
i
 s
i
, 






N
i
i
i
N
i
i
s
R
f
M
1
1


. 
The consumed resource is processed into the product which provides income (in our terms, it is 
created income), the correlation being as follows: 
i
i
i
r
f
y

. Then: 








N
i
i
i
i
i
i
i
i
N
i
i
i
p
x
r
s
R
p
x
r
f
Y
1
1
)
(
)
(
. 
The sector share (b
i
) or activity in economy showing the value of the sector and its dom-
inating role, can be defined as a share of created income (y
i
) in the country’s total gain (Y), as 
the quantity of employed in the sector (L
i
) in the general number of economically active popula-
tion (L) or as added value in its total volume in the economy. The share of each sector: 
1
,
)
(
1
1









N
i
i
N
i
i
i
i
i
i
i
i
i
i
i
i
i
b
p
x
r
s
R
p
x
r
s
R
Y
y
b
 
We have three formulas for economic growth, decline and recession representing these 
three functioning modes: 
recession
y
y
decline
y
y
growth
y
y
t
t
t
t
t
t












0
1
0
1
0
1
1
1
1
 
Let's write down national income for the initial and future period of time: 










N
i
it
it
it
it
it
t
p
x
r
s
R
y
1
1
1
1
1
1
1
)
(
 

 
129




N
i
it
it
it
it
it
t
p
x
r
s
R
y
1
)
(
 













N
i
it
it
it
it
it
N
i
it
it
it
it
it
t
t
p
x
r
s
R
p
x
r
s
R
y
y
1
1
1
!
1
1
1
1
)
(
)
(
, 
and 
 
it
it
N
i
it
it
N
i
p
x
r
f





1
1
 
growth
p
x
r
s
R
p
x
r
s
R
y
y
N
i
it
it
it
it
it
N
i
it
it
it
it
it
t
t















1
)
(
)
(
1
1
1
!
1
1
1
1
 
depression
p
x
r
s
R
p
x
r
s
R
y
y
N
i
it
it
it
it
it
N
i
it
it
it
it
it
t
t















1
)
(
)
(
1
1
1
!
1
1
1
1
 
From the analysis of the presented correlations it follows, that the economy tendency to 
extension is expressed in the increase of resources, their price, profitability, reduction of losses 
and probability of their occurrence. The depression arises, when used resources decrease on the 
economy sectors, financial flow is depleted, profitability goes down, and losses probability in-
creases sharply. Let's present economic growth on the basis of correlations introduced above: 
1. Losses should decrease. Initially income is more than losses and income should in-
crease in the system: 

























N
i
it
it
N
i
it
it
N
i
it
it
N
i
it
it
N
i
N
i
it
it
it
it
N
i
it
it
it
it
p
x
r
f
p
x
p
x
r
f
r
f
r
f
r
f
1
1
1
1
1
1
1
1
1
1
1
1
1
,
0
)
(
 
If financing of economic system is equal to losses, it is necessary for growth that profit-
ability surpasses probability of these losses, that is, at f
it
 = x
it
 it is necessary, that r
it
> p
it
.   
2. In the initial state of the system income is more than losses, however, increase of 
losses is observed in the following time interval. Then growth is possible if income gain com-
pensates this increase. 

 
130






























N
i
it
it
it
it
N
i
it
it
it
it
N
i
N
i
it
it
it
it
it
N
i
it
N
i
it
it
N
i
it
it
N
i
it
it
p
x
p
x
r
f
r
f
r
f
r
f
p
x
p
x
p
x
r
f
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
)
(
)
(
 
3. This is a situation when in the initial position losses exceed income, and they rise in 
such a way, that they surpass the losses of the previous period. Thus income increase of the sys-
tem is such that it cannot cover the negative losses, or the same situation provides a negative dis-
parity on income increase. The relation of two negative values will give positive value, as it is 
required according to the criterion confirming economic growth. However, such situation is pos-
sible only theoretically, because losses cannot exceed national income in the initial condition. 
Infringement of the introduced correlations will correspond to depression (recession). 
Now, let us suppose that economy consists of two sectors, in which losses are equal to 
zero in the starting point and profitability is not the function of finances r ≠ χ (f). We will accept, 
that profitability on the time interval [t, t+1] on each sector does not vary. These are convention-
al assumptions, but the time interval can quite be such that this parameter does not change so 
quickly, especially as it also reflects with what efficiency the productive activity is carried out 
and how the work is organized, etc. It is necessary for simplification of calculations
4
. Then eco-
nomic growth rate (g) can be written down as follows: 













N
i
it
it
N
i
N
i
it
it
it
it
it
it
r
f
p
x
r
f
r
f
g
1
1
1
1
1
1
1
)
(
, then for two sectors (and the third one): 
S
t
t
t
t
t
t
t
t
t
t
t
t
t
t
f
f
f
f
f
f
f
f
f
f
f
f
f
z
z
z
z
z
p
x
z
p
x
r
r
r
r
r
r



























2
2
1
1
3
3
2
1
3
2
1
2
1
3
1
3
2
1
2
1
1
1
1
2
1
2
1
1
1
1
2
2
1
2
1
1
1
1


 
                                                 
4
 Certainly, other conditions are accepted as equal, there are no changes otherwise it generates a number of factors, 
for example, organizational or institutional which will, certainly, affect economic growth rate and even on its pres-
ence. 

 
131
where: f
S
 –created financial resource for the development of a new combination, a new sector or 
an activity or technology (technological possibilities), α1, α2 – shares of the borrowed resources 
from the first and second sectors accordingly. 
 Using the agreed notations (designating β =z/f2 – the total losses in the system per unit 
of finances in the second sector), we’ll have the following expressions for growth of economy 
(g> 0) consisting of two and three sectors, with the appearance of the third one in the form of a 
new combination: 
1
2
2
1
r
r
f
f



– condition of system’s economic growth with introduced assumptions for 
profitability change per sectors in the course of time and losses absence for sectors in the initial 
point. For the growth to be observed, the incremental ratio of finance (income) in two sectors, 
which the economy consists of, should be more than the relation of total losses excess in the 
economy per unit of finances in the second sector over profitability in this sector to profitability 
in the first sector of economy. If losses at the moment of time t+1 are recognized as zero, then 
the correlation is 
1
2
2
1
r
r
f
f


.  In other words, incremental ratio of finances (income) in economy 
sectors should be more than the inverse ratio of profitability in these sectors. Minus sign takes 
into account the fact, that, at growth of total gain of the system, change of finances (income) in-
crement in one of the sectors can be negative, that is, financing (income) of the sector decreases 
then.  
Let's suppose that there appears a third sector as a new combination in the economy. 
Then, assuming the situation that this new combination distracts a part of finances (national in-
come or resource) from the first and the second sectors, we must admit that it also creates a sepa-
rate resource for itself. And the last circumstance can essentially outweigh the effect of resource 
loan. The development of world economic system confirms this fact. Consumption of resources 
increases with technological progress, the factor of population growth being significant, but not 
the only one in this dynamics. If the development assumed destruction of old resource with the 
equivalent use of the resource, and the resource was not lost, the scheme of the development ex-
pressed in general dynamics of resources would look differently, and the problem of economy, 
including the one appeared for the account of technological progress and waste-free technolo-
gies, would not be voiced by the governmental economists, politicians, and in the reports of large 
research teams in the forefront. 
So, accomplishing some analytical calculations, it is possible to come to the following 
expressions and modes of the third sector appearance. At system growth, the volume of created 
resource and, respectively, finances for the third sector should be: 


)
2
3
)
2
1
(
2
(
2
)
1
3
)
1
1
(
1
(
1
3
1




r
r
f
r
r
f
z
r
f
S







 

 
132
If the resource is created entirely with the appearance of the third sector, then α1 = α 
2=0 and the value of needful resource will be: 


2
2
1
1
3
1
r
f
r
f
z
r
f
S



 
Provided that the first sector gives the resource entirely, and the second one does not 
give anything, the value of created resource for the system’s third sector, for the growth to be 
observed (otherwise, recession can be observed), will make – α1=1; α2=0:  


2
2
3
1
3
1
r
f
r
f
z
r
f
S



 
If both sectors give the resource for the third sector formation, it will still be necessary 
to create its own resource (α1=1; α2=1): 
t
t
t
t
t
t
S
S
f
f
r
p
x
f
f
r
z
f
f
f
r
z
f
2
1
3
3
2
1
3
3
2
1
3









 
The resource for the third sector should surpass total resource of the first and the second 
ones in the initial point of time t on the value of expected losses in this sector balanced on the 
value inverse to profitability of this sector. In practice, many reasons interfere, so that these cor-
relations are not carried out. However, analytically obtained correlations are useful guideline for 
further reasoning and modeling, and for diagnosis of crises and evolution of economic system. 
Using elementary optimization models, which allow seeing resources distribution be-
tween the elements of the system (sectors), it is possible to designate proposition qualitatively 
important for understanding the development (growth) of economic system on a numerical illus-
tration.  
Mathematical formulation of maximization model 
5
 of national income of economic 
system in the elementary form is as follows: 
     
Z
x
j
j
j
N





1
max,
  
At restrictions: 
x
C
j
j
N



1
,
 
x
j
 0
 where Z – national income, 
             
x
j
   – capital invested in j
th
  sector or activity, 
             

j
j
t
T
T
r t



1
1
( )
  – average expected income on j
th
  sector or activity, 
                                                 
5
 This model was already applied by me in my earlier works, when with the application of a numerical method of 
gradient projection the problem of investments “portfolio analysis” was being solved.
  
 

 
133
             
r
j
 – income in the period of time t on j
th
 sector or activity per a unit of input (resource 
or capital investments); 
             T  – the whole period of system’s development described by the models, 
              C – total resources (capital) invested, 
              N – number of sectors or activities in economic system. Mathematical formulation of 
minimization model of total risk is as follows: 
min,


Kx
x
R
T
 at restrictions: 
x
C
j
j
N



,
1
 
x
j
 0
, 
D
x
T


, where 
K
ij
 [ ]

2
 – co-variations matrix
6
 for N kinds of activity, sectors. D –
minimum expected average income, R – system’s total risk.
7
 
 Let there be a volume of a resource equal to 100 units and five sectors or activities. 
Each of them is characterized by the return and by qualitative tendency of its development (see 
Table 2). According to the first model, economic growth corresponds to modern situation in the 
world economic system in greater degree (“irrational” or absolute growth). If the events were 
closer to the second model and problems of ecology, wars, regulations, including the financial 
sphere, leveled the risks, obviously, there would be a model of “rational” or “reasoned” growth. 
The sixth sector or activity corresponds to the appearance of a new combination, technological 
possibilities. And it is interesting to look at the economy structure when this appearance occurs 
at the same initial resource and at resource expansion (creation of the resource for a new combi-
nation simultaneously with its appearance). It is interesting to note, that appearance of a new 
combination (innovation) at the first stage hardly earns essential profit. Usually, the return is less 
than the investments. In this connection financing of a new combination appearance, that is, fi-
nancial resource provision, is impossible without certain adjustment of financial institutions 
which are allocated in the sphere which, at the first interval of time, will obviously be unprofita-
ble. Sometimes, the subsequent prospects are also illusive and are not clear for financiers and 
creditors. Therefore, financial system and banks should risk and advance capital to a new combi-
nation appearance. On the one hand, under conditions of crisis it is difficult enough to fulfill. On 
                                                 
6
 
Co-variation is defined by formula
:                                     
 



ij
i
i
j
j
t
T
T
r t
r t
2
1
1





( ( )
)( ( )
)
 
 
7
 It is necessary to note, that the drawback of the presented models is that they do not consider the dynamics of risks 
change within the limits of the given structure of economy, additional capital investments, and change of return de-
pending on the change of a consumed resource. However, they can be used to show, a new sector appears and how 
the structure of the system aspiring to the greatest national income reacts to it. The fact that modern economy is this 
very system does not cause doubts as maintenance of economic growth at reasonable risks is a kind of ideological 
fetish  of all actions, decisions and politicies. As a rule, the logic of the second model operates extremely restricted-
ly, that is, the total risk for the system is not estimated due to the complexity. And more than that, it is not mini-
mized. More likely, we speak about reasonable income growth at acceptible expected risk. Crisis occurrence in the 
course of growth proves that risks were not considered, and macroeconomic management cannot preventively stand 
the crisis due to institutional reasons. 
 

 
134
the other hand, the firms having difficulties with production and technologies change their plans, 
refuse some projects and manufactures, running risks, sometimes according to the principle “to 
have everything or nothing”, select new directions of actions, “try” new products and technolo-
gies. But whether it is a common feature for the given economy and whether it is a large-scale 
phenomenon to characterize a crisis point. Probably, no. Not for each economic system and not 
for each crisis, let them repeat periodically for certain economic systems with more or less regu-
lar frequency. In the period of rise, when profitability of present productions and the ones, start-
ed after crisis or at the stage of its overcoming, begins to grow, there are possibilities of redistri-
bution of the resource between activities in the real sector, getting additional financial resource at 
the expense of its concentration. Financing of science and applied design improve. Thus, the 
possibility of new combinations appearance essentially increases, as in the crisis point the major-
ity of possibilities, especially those referring to the development of science and its applied re-
searches is curtailed. These expenses get to the area of economizing at once, and stereotypic pro-
duction, or those products, which do not require large expenses and risks for realization are de-
livered to the markets Thus, in case the crisis revives the new combinations, they are only the 
ones connected with realization of economizing problems, especially referring to energy-saving 
and resource-saving and corresponding technologies and techniques. 

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