Summary : This paper will show how Capitalism can prosper up to determined limit and its reasons for crises. We also show a mathematical proof of why capitalism system isn't stable, and for survival, it's either necessary to achieve new markets or keep a more indebted society. For both cases the system won't be stable and this consequently involves to its end.
In capitalism, the goal is profit:
“ Profit at capitalism society : What features a capitalist society, its basic principle, is the search for profit. It doesn't mean that other societies don't have activities which can be profitable. However, this aspect is overstated at capitalist societies and, therefore, it's going to determinate the other facets of the system. This searching for maximum profit is the goal of every single capitalist. Its consequences can be felt at relations among men, at society ideology and until at everyone behavior. The purpose of profit isn't to give ways for capitalism subsistence. This is only a secondary aspect. Profit is like an end in itself.” 
“The capitalism is an economic system where production and distribution means are private ownership and for profits ; decision about offer , demand , price , distribution and investments aren't made by Government: the profit is distributed for owners who invest on companies, and wages are paid for workers by companies .” 
By this mean, the end of profit – the end of companies to be able to profit - will also involve the end of capitalism.
Now let's demonstrate that capitalism isn't a stable system, i.e., can't be existing indefinitely across time and, therefore, will need to be ended.
Let's consider a group of capitalist companies (“pool”) which will be identified by “i” index. This index marks which companies of pool we are going to work. The “i” index may vary from 1 to total number of companies of our pool, which can be companies of a district, a city, a country or the whole world.
Later, we will show when there's no outside markets or when it's not possible to buy through debt accumulation, the total profit of pool will always be negative, what will broke the main base of capitalism – the profit – ,making it not viable. This happens because there will be always some companies of pool in deficit which close all doors, decreasing the total number of companies of pool. And now cycle will repeat with a smaller number of companies at pool.
Let's start lending some simple and economic concepts and formulas found at many books and Economy website:
“... At first we should considerer that the profit of a company is given by difference between revenue and expenditure. Expenditure is the expenses for a company to create revenue…”  Therefore:
Profit = Revenue – Expenditure (F01)
L i = R i - D i
At formula above “i” indicates the “i-th” company.
In a simple way, formula F01 above describes that the profit of a company, which is the base of capitalism system, is nothing else than the difference between what's raised and what's spend.
Revenue: “At private companies, generally the revenue corresponds to goods sale product or services  (named in Brazil as billing).”
Revenue = SUM[ Sales ] (F02)
R i = SUM (j)[ Sales i,j ]
The formula F02 above represents the concept that what's raised is originating from goods sales or company services.
(The index “j” of SUM (j) means that sum is done through index “j” covering all sales of th company of pool.)
The revenue (as sum of all sales) can also - for analysis effects and without losing generality – be decomposed by three quotas:
Revenue = Sales to Workers (commerce) + Sales to Companies (as Suppliers) + External Sales (F03)
R i = VT i + VF i + EXT i
“ Sales to workers ” (VT) are sales done to workers (generally at commerce) where products or services are bought with wage received from companies of pool.
“ Sales to companies ” (VF) are sales done to other companies of pool – like supplier – where these products/services would be charged from costing of own pool companies.
“ External Sales ” (EXT) would be all remaining sales whose money used for purchase is neither originating from wages paid by pool nor pool company costing. (for example through exportation or banking loans, etc.). i.e., would be all sales whose money used for purchase is not from companies of pool.
Expenditure: “For Accountancy , it is the expense necessary for revenue ”. 
Expenditure = Wages + Other_Costs (F04 )
D i = S i + C i
In this Expenditure formula (F04), we highlight “Wage expenditure” from other expenditures of company (duties, expenses with raw material, etc.)
From equation F01, F03 and F04 we would derive:
L i = (VT i + VF i + EXT i ) – ( S i + C i ) (F05)
[Formula of profit for th company of pool]
The equation F05 describes that Profit of “i” company is composed by “i” company sales subtracted by wage paid and other costs.
Pool of Companies
In our demonstration, we consider a set (“pool”) of capitalist companies based on profit and we are going to list them (index them) by numeric index "i". Thus for a pool of a single company, the index "i" is always 1; for two companies "i" may vary from 1 to 2, and so on, in order to "i" can index all companies of our pool which can comprehend all companies in capitalist world as a whole.
Within our pool of company we can still sum at variable “S” all wages paid by companies of our pool.
S = SUM (i)( S i ) (F06)
[Sum of wage paid]
And we can also sum at variable “C” all expenditures and costs (“unless wage”) that companies of pool spend:
C = SUM (i)( C i ) (F07)
[Sum of costing except wages]
Likewise we can sum quotations of F03 (VT i , VF i and EXT i ):
VT = SUM (i)( VT i ) (F08)
[Sum of sales bought with wages]
VF = SUM (i)( VF i ) (F09)
[Sum of sales as supplier for other companies of pool]
EXT = SUM (i)( EXT i ) (F10)
[Sum of sales with money from out of “pool”]
We know that the quotation paid for all workers of pool for, eventually, comparing products of “i” company is a fraction “f i ” ( f i <=1) of sum of all wages paid (variable “S” of formula F06); this, mathematically, we can write:
VT i = f i * S (F11)
[Total of Sales of i- th company paid with wages, f i <=1]
As the sum of all purchases done with salary of employees of pool (VT) can't be higher than total of wages (S), we write:
VT <= S (F12)
[The sum of sales compared to wage of pool is lower or equal to total of wage of pool]
From F08 and F11 we derive:
SUM (i)( f i * S) <= S (F13)
We conclude that:
SUM (i)( f i ) <= 1 (F14)
The total profit of our companies of pool can be also accumulated at variable “L” (as being the sum of profit of each company of pool):
L = SUM (i)( L i ) (F15)
[Total profit of pool of companies is the sum of Profit of each company of pool]
We also know that the sum of company sales as suppliers (VF, of formula F09) can't be higher than total of expenditures ( C, of formula F07 ) of costing of companies of pool, because the purchases done by pool companies are gotten as its costs:
VF < C (F16)
[The total of sales done as suppliers should be lower than sum of expenditures (unless wage) ]
Now, let's decompose sales from th company as Supplier from other companies of our pool (VF i, from formula F03), where purchases are paid as job costing not as wages. So we have:
VF i = g i,1 * C+ g i,2 * C + g i,3 * C .... + g i,n * C (F17),
VFi = SUM (j)( gi,j ) * C
[where g i,j is the portion from full job costing of j th company pool to pay i th company which is their supplier. (g i,j <=1) ]
From F09 and F16 we can derive:
SUM (i)( VF i ) <= C (F18)
[The sum of all sales done as supplier of pool can't overcome the total of pool company job costing expenses]
From F17 and F18 we can compute:
SUM (i)(SUM (j) ( g i,j * C ) ) < C (F19)
What allow us to conclude:
SUM (i)( SUM( (j)( g i,j ) ) <1 F(20)
We can substitute at F15 the formula F05 and we have:
L = SUM (i)( (VT i + VF i + EXT i ) – ( S i + C i ) ) F21
Substituting VTi at formula F11 and VFi of formula F17 we have:
L = SUM (i)( f i * S + SUM (j)( g i,j * C ) + EXT i – (S i + C i ) ) (F22)
We can rearrange quotas:
L = SUM (i)( f i * S - S i ) + SUM (i)(Sum (j)(g i,j * C) - C i ) + SUM(i)( EXT i ) (F23)
and then we decompose this Profit into three quotas:
P1 = SUM (i)( f i * S - S i ) (F24)
P2 = SUM (i)(Sum(j)(g i,j * C) - C i ) (F25 )
P3 = SUM (i)( EXT i ) (F26)
L = P1 + P2 +P3 (F27)
P1<=0 , because P1 = SUM (i)( f i * S ) – SUM (i)( Si ) = S * ( SUM(i)( fi ) - 1 )
As (of F14) : SUM (i)( fi ) <=1
it follows that
But P2 can't be positive because:
P2 = C * ( SUM (i)( SUM(j)( g i,j )) – 1 )
But from F20 we have that SUM (i)( SUM (j)(g i,j ) ) <1 which indicates that
So, we concluded that the positive quota of profit from our pool of company can only be provided from quota P3 (see F26 )
This implicates that profit should come from sales for people and companies which aren't at pool of companies, i.e., money coming from different pool company sources and this implicates in the need of a increasing accomplishment of new consumer markets (which don't below to pool of company) or through bank loan (in other words, from increasing indebtedness).
We can also conclude that, without indebtedness, the capitalist companies as a whole can have some profit if there is a non-capitalist economy which can sip resources and pay its profits. And this was particular case which makes capitalist arises. It can thrive at globalization like a flash in the pan because there were few capitalist companies, and market was global. But, like a flash in the pan, fire can't take so much time and are resting only ashes from a profitable time.
But both continuous expansion of new markets and increasing indebtedness aren't economically stable because market is limited and one day loans should be paid . For this reason, capitalism can't be remain indefinitely, and, as we can find below, it's already agonized, maybe with signals that its lifecycle is ending.
That's no other reason that industrialized countries proposed what it's named “ Washington Consensus ” :
“it's a set of measures – which it's composed by ten basic rules –prepared in November 1989 by economists from financial institutions at Washington D.C., like IMF, World Bank and US Treasury, based on text from economist John Williamson, from International Institute for Economy, and which became the official policy at International Monetary Fund in 1990, when it came to be “prescribed” to promote “macroeconomic adjustment" of developing countries which is passing by difficulties.”
- Financial discipline
- Reduction of government expenditure
- Tax reform
- Market interests
- Market exchange
- Trade liberalization
- Foreign direct investment, with no restrictions
- Privatization of state corporations
- Deregulation (slackening of economic and labor laws)
- Right to intellectual property
Which was the precursor of current economic Neoliberalism :
“ from 1960's , it acquired a meaning of an economic doctrine which defends absolute freedom of the market and a restriction to state intervention about Economy, and this should happen only at necessary sectors and still in a minimum degree (Minarchy). It's in this second meaning that term is more used nowadays.  ” 
Both "Washington Consensus" and notorious “neoliberalism” are manners to open the market of countries which economy is protected so that strong companies from industrialized counties can get new markets to prosper and increase its profits, because as we could see, in addition to indebtedness, the only way of pool of company at its countries to survive is achieving new markets.
Currently, as China is getting into markets before taken with cheap and competitive products, it's foreseeable that big companies shorts its end more fastly, with unending crises, mainly at Europe and at USA according to news we see every day at newspapers.
No expansion and no indebtedness
We conclude that our pools of companies can't sum its positive profits if there's no increasing expansion of consumer market or continuous indebtedness.
Let's suppose that all markets have already been globalized and there are only capitalist companies in the world. Our pool now is the whole world.
The formula remains and, as we could see, without indebtedness, the full profit will be continuously negative, indicating deficit, but it doesn't mean that all companies of pool are in red!
But there are some of them are in red. In this case, these companies will close its doors and its consumer market will be taken from remaining companies.
This new market – left blank by companies which closed – would work as type of new consumer market to be explored, and once taken, the cycle will repeat and will be reduction of profits again, and new companies in red will have its door closed. It's like a constant and inexorable cannibalization of regime.
Before “dying” due to lack of profits, many companies prefer to be "cannibalized" by others; the process is known by business means by “fusion”, "incorporation" or "absorption". So, one of clear trace that markets is taken and capitalism is on its end would be the decreasing of origination fee for new companies and also a series of “incorporation” and “fusion” of companies, pointing a lack of market for all of them.
Robert Kurz would say that the problem of capitalism is the competition among companies and showed how his Kkurzian spiral works :
The logic which shows the “suicide autophagy spiral”, proposed by Kurz, is really simple and elegant:
• By aiming profit, the capitalism seeks for reducing all possible expenditures.
• The workforce is one of main expenditure items of an organization.
• To reduce expenditures, the workforce should be minimized: or through automation/mechanization of workforce or through modern management techniques which aims to reduce complement.
• Reduction of payroll sponsors an increase of unemployment.
• The increase of unemployment makes average income and consumer market lower.
• As company profit depends on power of Market purchase (consumers), with Market retraction, the competition among companies is even more stubborn.
• With increase of competition, the organizations are coerced to reduce expenditures and, among them, workforce expenditure.
This anthropophagic spiral, within its limit, would culminate on organization fully automatized where only one worker, the President of company, would press the button and the whole production would be performed.
Only one question would left: Who would consume it? In this hypothetic and weird situation, the only purchasers would be who is employed: President of VW would buy only one fridge, Owner of GE, which, would by a VW car.
In Robert Kurz' words:
“A global economy limited to the minor always be restricted is incapable of survive. If globalized competition decreases industrial production income more and more and destroys in an increasing proportion, the economy of regions follows logically, that worldwide capital minimizes its own action radius. In the long term, the capital can't persists on accumulation of so restricted base, spread it for the world likewise it's impossible to dance on beer cap .” 
But, as we have seen, the Kurzian spiral isn't the true root of the problem. It can, of course, aggravate the scene, speeding loose of profits up.
However, we should notice by our formula F23 that the lack of profits of company pools do neither depends on company administration nor its competitiveness, nor launching of good products, nor wage of employers, nor price of goods or services, nor expenditure of fees.
It doesn't depend on anything mentioned above! It's a structural fault of capitalist system which always makes companies to close the door by lack of profits and before closing doors will fire lot of people and brings on a lot of painful. The good administrative measures for efficiency and productivity can only postpone the 'death' of companies, but they can't block them forever.
The reader may think that in the end of last absorption, fusion and closing processes of companies will rest only one company, without competitors and so would survive and profit. But even in this limit-case, we can see it can't profit; so can't exist as capitalist company.
To illustrate it, let's use our F23 and then reduce it for only one company.
L = SUM (i)( f i * S - Si ) + SUM (i)( Sum (j)(gi,j * C) - Ci ) + SUM (i)( EXTi )
In the case of one company (i = 1), the whole population would work for this one company which sell all goods and services for this population. We can simply some portions of formula:
L = (f1 -1) * S - C1 + EXT1
P1 = (f1-1) * S <= 0
P2= -C1 < 0
P3 = EXT1
The only portion which can be positive is EXT1, but as there's no new markets, the profit can come only by indebtedness, which could be unsustainable.
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“economic profit is, at least for neoclassifcal theory , which dominate modern economy, the difference between full company revenue and all expenditures”
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 Washington Consensus
 Virtual Economy
 The End of National Economy