This is the hardest part of Marx‘s analysis. I am going to have to simplify a great deal.

The point of departure is the analysis of the commodity. What is a commodity? A commodity can be seen from two points of view. 1) The commodity is a use-value, that is it is a physical object with useful properties. 2) The commodity is an exchange-value, that is it is a quantitative-relation in which a certain proportion of commodity A exchanges with another commodity B ( x commodity A =  y commodity B) . The question emerges: what determines the exchange-value of commodity A ? what determines that commodity A and B can be measured in a definite magnitude, and hence exchanged in definite proportions? in short how can commodity A and B be exchanged as equivalents?

I am going to simplify the steps he actually takes, and use argument I have built up for purely pedagogical and simplifying purposes.

A commodity has a dual-character: use-value and exchange-value. The exchange-value of A requires that it can be exchanged with another B in definite proportions. Hence, they must share and be reducible to some quality which can be quantified, or measured, in order for x commodity A to exchange equally for y commodity B.      An example, two different physical objects must be reducible to the property of being bodies in the abstract and hence having position, weight, dimensions, density etc. in order for them to be quantitatively compared and distinguished.

Commodity A and B are different use-values. They are qualitatively and quantitatively different physical objects. Therefore, the exchange-value of commodity A cannot be determined by the use-value of A or B. But then what quality do these commodities share in common, and how do we measure it so as to consider them equivalents?

The commodity A and B share the common property of “being products of abstract human labor”, human labor that is considered without regard to its mode of expenditure. Hence, commodities A and B are qualitatively reducible to the measure of such labor i.e. social labor-time, and the value of commodity A is determined by the quantity of social labor time that went into producing it .

(To clarify, not necessary to read: commodity A and B cannot be qualitatively compared on the basis of the concrete labor that went into the production of the commodities, because the objects still differ in physical properties and particularities, and the labor process, laborer and means, also differ in the same way. Hence, the commodity must be considered as the product of human labor without regard to its use-value and the concrete labor that went into it i.e. in the abstract. Such abstract labor is a property that is measured with social labor-time i.e. the average labor time that went into.  If I produce something in double the time it takes on average, that does not the value of the commodity twice as expensive, because it is reduced to and compared with average social labor. )

To make a jump. The value of the commodity A, determined by social and abstract labor, is not visible and must express itself in an objective form of representation. The form of representation of social labor is money. The price of a commodity A is the exchange-value of the commodity A as represented in money, or the quantitative-proportion in which x commodity A  exchanges for y amount of the “general commodity” money (x commodity A =  $ y).

Capitalism is a system of production and exchange of commodities , values which are represented in money. Yet, is this a comprehensive definition of capitalism?        Capitalism is the unity of the labor process and the process of valorization, in which the imperative for profit requires the exploitation of labor-power and the production of commodities, as well as the appropriation of surplus- labor in the act of exchange in the form of profit. What on earth does all that gabagook mean?!?

Capitalism presupposes not only a certain level of the accumulation of capital, but a division between capital and labor, between capitalist and a laborer. The formal division is that the laborer is dispossessed, and separated from the means of production (land, tools, grain etc.). Hence, the laborer is forced to sell their labor-power to those who stand on the other side possessing the means of production i.e. the capitalist. The capitalist uses their labor-power, that is their capacity to work, in exchange for a wage. Yet, the exchange of labor is only a formal contract that belies real relations of power, a systematic relation of the exploitation of labor in production.

How so, how is the seemingly voluntary contract between capital and labor nothing but a subjective representation of production ? How is capitalist mode of production really a system of exploitation in which the capitalist exploits labor-power?

Everything turns on this distinction between the use and the exchange-value of labor-power, between necessary labor and surplus labor, between necessary labor-time and surplus labor-time. Let us use a model of the working day to draw out these terms and distinctions (exchange value of labor power, necessary versus surplus labor, necessary versus surplus labor time), so as to be able to depict the capitalist mode of production as a system of exploitation of labor-power.

The worker needs to work let us say 5 hours to produce their wage. Let us say that the commodity is worth 5 hours. Then the working day will only produce enough value for the worker renumeration, and the portion that goes to reproducing the worker in the wage is called necessary labor. The working day only pays back the wage. “My god!” the capitalist exclaims, “I will never do that sort of business again, why I get nothing back !”. Now let us say that the worker needs to work say 3 hours and the product is worth 5 hours. Then 3/5 of the labor-time will be consecrated to the laborer in the form of wages, and 2/5 will go be pocketed by the capitalist. The capitalist proclaims “my god, by good fortune and wit I have stumbled on wealth!”.

The exchange-value of labor-power is the amount of social labor performed to reproduce the capacity to labor. Such a quanta of social labor is expressed in money terms in the wage. The working day is split up into two components, necessary and surplus labor, in which social labor is performed to reproduce labor-power, pay the wage, and another in which surplus labor is performed over and above necessary labor. Necessary labor is measured by necessary labor-time, and “surplus -labor is measured by “surplus” labor-time. In the examples above, the exchange-value of labor-power was respectively 5 and 3 hours of social labor, which is the necessary labor-time, the measure of necessary labor. 0 and 2 hours were the measure of the surplus labor, the surplus-labor time.

Another helpful rate, as will become clear later, is what is called the organic composition of capital which is written as the portion of value which goes to constant capital, over the the portion which goes to variable capital (c/V).

Now we can actually better define the term “capital”. Capital is a sum of money invested in C and V which is used to produce S. It can be represented by a circuit :

M-C-M’

where the ‘ means greater than M (change of M >0).

which is further analyzed as :

M- C ….P….. C’ …M ‘

The chain means: 1) a sum of money M is used to 2) purchase productive capital C, which is analyzed as c and V, 3) and put in motion a process of production (…P…), 4) the result of which is the production commercial capital C’ which is worth more than the original components. 5) The commercial capital C’ is then sold in exchange for a quantum of money M’ which is greater than M , an excess called profit  realized in the act of exchange.

We are almost there! We have almost all the well-defined tools to attack the TRPF.

Yet, first we must make more complex our model of capitalism as a system of the exploitation of labor-power, of the production of surplus-value, and of the appropriation in the act of exchange of surplus-value in the form of profit.

It is true that taken as a whole,  total value= total price, and rate of surplus-value= rate of profit . Yet it would be a logical fallacy to conclude it applies to each and every case of the average. We have talked about the total production of capital. Now we have to talk about the distribution or competition of differing capitals.

Each capital can be distinguished on our model according to the rate of surplus-value, the rate of profit, and the rate of the organic composition of capital. Let us take the rate of S/V, the c/V and S/c+V as variant.Then there is a tendency towards the equalization of profit-rate and surplus-value across differing capitals and organic compositions. If a rate of profit or surplus-value is higher in one sector, capital flows in and the profit rate and surplus-value is depreciated and normalized. Let us illustrate this process:

1. The total capital is c+V+S, and c+V is the cost-price of the commodities, and S is the average-rate of profit. Each capital has in the process of equalization, the tendency to get as much S as is invested in the total sum of their capital. As a tendency they each get an equal piece of surplus-value , relative to the total quantity of capital. Hence, the values in the process of competition develop prices of production (cost price + average profit). Prices of production are centers of gravity for the differing composition and values of commodities that tend to equalize rates.

2. Therefore each differing particular capital c+V+S, has a tendency for its c+V to go toward cost price, and S to go towards average profit. There is a two-way mechanism in the prices change with respect to the organic composition of capital, and the amount of surplus-value appropriated is transfered between sectors , both of which to equalize rates so that each capitalist tends to get an equal piece of surplus-value and profit on the sum of capital they invested.

We have come back to the issue of the TRPF, and we have found that it is located in the problem of the distribution or competition of capitals. In short, what the capitalist does is to compete with other capitals, is to increasingly invest in c, so that c/V increases, what I called the “capital bias”. This at first means they can produce commodities whose value is “bellow” the average price, hence they can sell at the average price but it requires less-labor time, less variable capital per commodity. There is a transfer of higher surplus-value, and hence profit, but the tendency is for capital to flow in, there is the over-investment in C, and a depreciation of the production prices. The depreciation means lower rates of profit.

Let me explain in mathematical terms. If you actually look at the equation of profit:

ROP= S/C+V , then divide each term on the right by V, you have ROP= S/V/ C/V+V/V, you might notice S/V is rate of surplus-value, and C/V is organic composition, and V/V is 1. Now look at the equation as such: ROP= RSV/ OC+1 , so that if you increase RSV and OC , as a function of time , you will find that in the long-run ROP increases at a decreasing rate, hence eventually decreases.

The mechanism is much more complex than the simple, calculus above. For one, there are more or less contingent counter-tendencies which act against the TRPF. Such counter-tendencies includes 1) increase in extension of work-day 2) increase in intensity of work-process 3) cheapening of the means of production or constant capital 4) increased access and integration of capital networks and circulation 5) wars and 6) monetary factors (interest, etc.). Hence, the mechanism operates in a real conjunction of differing kinds of tendencies, not all of which are inscribed at the “economic” level. Yet, such contingencies  affect the rhythm of an invariant and subterranean process. The TRPF is a long-term or net-resultant process of capitalist production, of the capitalist mode of production.

What does this mean for a crises- theory, how can we use such a theory of production and the TRPF to explain crises, and the near constant repetition of crisis on a decennial basis. I will give you a very cycle to explain the effects of the mechanism, yet it will be found it explains well the conjectural and seemingly cyclic nature of the capitalist mode of production and its “business-cycles” .

1. Long term profit decline

2 .Long-term price-decline

3. Less invested in V relative to C

4. Less goods are sold relative to V, and less profit is made

5. Firms cannot pay debts, share-holders, forced to borrow, sell-low or file bankruptcy , banks forced to collect debts on other struggling-capitalists, there is a

trigger in which other firms are forced to borrow, sell-low or file bankruptcy.

7. Firms reduce the work-force or share relative to V , unemployment increases

8. Firms destroy capital and other assets , there is a decrease in productivity

9. There is a capital shortage , prices are cheapened , productive investments increases

10. Prices and profit increase and the steps are repeated over and over without end.

In the least eloquent words of Marx ever, “and the whole crap starts over again!”

I have an extensive bibliography of material on Marxist political economy. Please contact me with your contact info and I will get it to you.