AGGREGATES AND ECONOMIC CATEGORIES

Labor and Leisure, Means of Production and Consumer Goods, Capital and Income

Economic Categories

Capital and Income

Economic Efficiency

Maximum Rate of Profit

Constant Capital

Income

Wage

Profit

Representation of Economic Relationships

The Perfect Competition Hypothesis

General Representation

AGGREGATES AND ECONOMIC CATEGORIES

While the distinction between means of production and consumer goods does not have any importance for the nature of the economic value, it is instead decisive for the magnitude of prices.
The difference between the quantity of time of social labor indicated by the price of commodities and the quantity of labor which produces them, more generally, is due to structural reasons (the magnitude of the categories and aggregates that the very existence of economics implies for the system as a unit), and to fortuitous reasons inherent to multiplicity and the natural diversity of people and places.

Labor and Leisure, Means of Production and Consumer Goods, Capital and Income

Regarding structural reasons, the quantification of relationships implies a separation into pairs of distinct categories at the three levels of value:

• On the level of the activity there is the distinction between labor and leisure.
• To this, on the material level, corresponds the distinction between the aggregate of means of production, MP -commodities used by labor- and that of the final goods of consumption, BC -used during leisure-.
• On the economic level, the distinction is between capital, C, and income, E. Capital is the economic value of the commodities destined to produce commodities, while income is the economic value expended mostly during free time for non economic purposes, for pleasure, for the satisfaction of personal needs and desires.

There is therefore a kinship between labor, means of production and capital on one side, and leisure, consumer goods and income on the other. Any of them refers to the others, but being on different levels, the distinction between the couples follows different criteria.
First of all we see that there is no precise and immediate coincidence between the magnitude of capital and the price of the means of production, nor between income and consumer goods; namely, it is not C (=) MP, nor E (=) BC, even though, certainly, C + E (=) MP (+) BC. Part of the income, indeed, both wages and profits, can be saved, and since every considered quantity of money corresponds to produced and sold commodities, the magnitude of saving corresponds to the magnitude of the new investments, MP_n. The net product therefore is composed by final consumer goods and means of production: E (=) PN (=) BC (+) MP_n.

Moreover, we see that the distinction between means of production and consumer goods is not intrinsic to the function of commodities, but that, instead, it depends on the way one perceive income.
Profit is the difference between the revenue obtained from the sale of the produced commodities minus the totality of expenditures for their production, comprehending the value of the means of production and the wages. Wage therefore is considered (and really is), at the same time, income by the workers and capital by the businessman, and the commodities which realizes them are consumer goods for the workers and means of production, just like food for animals and the fuel for machinery -as Sraffa says- for the entrepreneur. Marx calls constant capital the value of the actual means of production and variable capital the monetary amount of wages. Smith and Ricardo call gross income the sum of wages and profits, and net income the profit. (This consideration, alone, should show that the idea of the transfer of labor incorporated into the means of production can not be accepted; how could we decide, indeed, which commodities would transfer their 'embodied labor'?)

Finally, the function of commodities, the utility and the pleasure they give, cannot be employed as a criterion to draw the distinction between means of production and consumption goods. Spaghetti sold at the supermarket, for instance, is a final consumer good if he who buys it also eats or gives it away, but if it is a restaurant that buys and cooks and sells it, it must be considered means of production. On the other hand, a drill is intrinsically a means of production, a tool for producing other things, but if it is paid with income, used during leisure and the result is not sold, it must be considered as a final consumer good. And it is plainly not true that consumption of commodities bought with income always results in well-being or pleasure.

When we study the nature of prices, the economic value is emphasized, therefore we must consider specifically capital and income, which magnitude, however related with the corresponding terms on the other levels of value, is precisely reported in the accounting books of the firms. If, therefore, the distinction between capital and income is due primarily to the qualitative distinction between means of production and consumer goods, namely, to the physical function of the object, it is then the precise quantitative distinction between capital and income to fix in a formal way, in an accounting way indeed, which is arbitrary and conventional, the belonging of commodities to one, or the other, or both the aggregates.

Leaving aside the function of commodities, we also must leave aside the sensible degree of physical transformation that labor imposes upon the means of production. The end of a given productive process, from the economic point of view, is determined by the sale of the corresponding result, not by how or how much the object has been modified. Even if a commodity is only bought, transported and displayed for sale, it must be considered, therefore, as the other means of production and as such reported. In the representation of the activities of the system, it must be indicated by different letters before and after its physical transformation -although for the senses it maintains an unchanged material value- and listed with the other commodities of the gross product. The quantity of labor intervened between the two acts of exchange, as little as it might be in respect to the quantity of social labor indicated by the prices of the means of production, is the creative value of the re-sold commodities, which also constitutes the increase of the gross product value. For this reason, there is no essential difference between commercial and industrial capital.

Economic Categories

Capital and Income

The division of money between capital and income can be represented in this way:

D_a = C_a + E_a
D_b = C_b + E_b
.......................
D_k = C_k + E_k
____________
L_t = C + E

Prices differ from the corresponding quantities of labor in order to satisfy the conditions of equilibrium of distribution required by the different relationships between capital and income of the various firms, that is, the diversity of the various C_i / E_i among them, and with the C / E of the system.

Economic Efficiency

The magnitude of income is a fraction of the whole social labor which can be expressed as follows:

E = e L_t

e = E / L_t

where e expresses the economic efficiency of the system, that is, as for every engine, or transformer of energy, the relationship between the output furnished energy and the total imput energy used for that aim.
It is a pure number that, if there is capital even in only one firm, is less than one for every firm of the system.

Maximum Rate of Profit

Following Sraffa's indications, the relationship between capital and gross product can be expressed like this:

C (1+R) = L_t

R is the maximum rate of profit of the system, that is, the rate of profit that would be realized if wages were zero. Still, it is a parameter existing even, inversely, with a zero rate of profit, although its magnitude, owing to the variation of prices, would be different.

On the formal level, we can observe the following mathematical relations:

C + C R = L_t

E = C R

R = E / C

R = e / (1- e)
e = R / (1+R)

or

(1- e) (1+R) = 1
(1- e) = 1 / (1+R)

The magnitudes of efficiency and of the maximum rate of profit are not in a direct relationship to the physical volume of the net product. On the contrary, knowing that productivity is higher when the proportion of the employed capital is higher, the magnitude of these two monetary categories is inversely proportional to the quantity of the available final consumer goods.
If capital is a small fraction of gross product, then, although on the material level the available commodities are not so abundant, the properties of the system (e and R) are such that the rate of profit can be very high, even admitting high wages. By increasing the magnitude of capital, with an increased availability of material value, e and R decrease correspondingly.
Since in the relation D_t = L_t the material value does not appear, even if owing to the constant increase of productivity commodities will submerge us, their economic value would remain the quantity of time of labor which produced them.

Constant Capital

The distribution of the means of production among firms depends on the physical characteristics of the produced commodities and on the technology adopted, so that for the generic firm I we have:

C_i = a_i D_a + b_i D_b + ... + k_i D_k

where the a_i , b_i , ..., k_i , coefficients indicate the distribution of the commodities A, B, ..., K, directly expressed by their economic value, or price, D_a , D_b , ..., D_k.
For the whole system we have:

D_a = (a_a D_a + b_a D_b + ... + k_a D_k) + E_a
D_b = (a_b D_a + b_b D_b + ... + k_b D_k) + E_b
.....................................................................
D_k = (ak D_a + bk D_b + ... + kk D_k) + E_k
___________________________________
L_t = (a D_a + b D_b + ... + k D_k) + E

Since income must be positive, it is:

E = [(1- a) D_a + (1- b) D_b + ... + (1- k) D_k] > 0

If the capital of every firm is indicated as a fraction of the whole capital: C_i = c_i C = c_i (1- e) L_t, the representation becomes the following:

D_a = c_a (1- e) L_t + E_a
D_b = c_b (1- e) L_t + E_b
.....................................
D_k = c_k (1- e) L_t + E_k
___________________
L_t = (1- e) L_t + e L_t

where it is always c_a + c_b + ...+ c_k = 1, remembering that a theoretical variation of distribution generally implies a variation of prices and, therefore, of the magnitudes of C and C_i and then of c_i, e and R.

Income

Income, in turn, is composed of wages, W, proportional to the applied labor, and profit, $, proportional to the invested capital, therefore:

D_a = C_a + W_a + $_a
D_b = C_b + W_b + $_b
................................
D_k = C_k + W_k + $_k
_________________
L_t = C + W + $

(NOTICE: in what follows, the sign $ stays for the sign which on the printed paper would be the upper case "P greek", and § for the lower case "p greek")

With a given technology, the hypothetical variation of prices depends upon a variation of the relationship between wages and profits.

Wage

On the material level, wage is realized by a quantity of commodities of different kind that the system gives in exchange for the only kind of commodity produced by the individual for the system. Monetarily, it is the price of the creative value considered as a commodity, that is, the workforce price:

W_i = w_i l_i L_t

The wage of every firm can be indicated, as we did for capital, as a quantity of money, without specifying its material composition.

The exchange value of a commodity is a function of both creative value and the means of production employed, where the creative value, in distribution, is not considered as such, but as a factor determining the magnitude of the incomes, which being realized by commodities and indicated as sums of money, are expressed like the means of production in terms of social value.
Since labor is partially applied for the production of means of production and partially for consumer goods, even the economic value is divided proportionally between them.
The quantity of social labor indicated by the price of a commodity produced in a firm without means of production, but in a system where means of production exist, coincides only with wages, it is inferior to the quantity of labor which produces it, and the superior price of the firms where there is capital permits the reintegration of the means of production -the maintenance of capital over time- and to distribute a wage equal of those of the firms without means of production.
The flow of economic activity is continuous, even if discrete, and, like all the other aggregates, the monetary amount of the represented wages is that which is distributed during the considered period of time, therefore there is no need to formulate the hypothesis -as Marx does- that they are paid in advance, nor -as for Sraffa- postponed, even admitting that they actually might be, each time, advanced or postponed.

Profit

Profit -however realized by commodities- is proportional to the magnitude of the employed capital. In this vein we suppose that a wage, or salary -distinct from profit- is paid to the owner of capital for the time of labor that he expends in its role of entrepreneur. The activity of selling must be considered labor, while that of buying is labor if the acquired commodities are bought with capital, and not labor if bought with income.
Since for the businessman wages are money expended for production, the profit of the generic firm I must be indicated, in agreement with Marx, as follows:

∏_i = (C_i + W_i) r_i

The distribution of the revenue obtained by the firm from the sale of the produced commodity, therefore, must be indicated like this:

D_i = C_i + W_i +  ∏_i
D_i = C_i + W_i + (C_i + W_i) r_i
D_i = (C_i + W_i)(1+r_i)

Since W = w L_t and  ∏ = π L_t, for the system it is:

E = W +  ∏
e L_t = w L_t + π L_t
e = w + π

and from the relation indicating profit,  ∏ = (C + W) r, we obtain the relationship between the rate of profit referred to the gross product, π, and that which refers to capital, r. 

Since indeed  ∏ = π L_t, C = (1- e) L_t, and W = w L_t, it is:

π = (1- e + w) r
and since:
e = w + π ,
π = (1- w - π + w) r

r = π / (1- π)

π = r / (1+r)

(1- π) = 1 / (1+r)

Representation of Economic Relationships

The Perfect Competition Hypothesis

For the perfect competition hypothesis, prices are such as to permit every firm to reintegrate capital, to pay equal wages for equal quantity of labor, and to realize equal rates of profit on capital inclusive of wages and the cost of the means of production. In this case the rates of wage and profit of every firm coincide with those of the system as a unit, or average wages and profit, and the representation of distribution is the following:

D_a = (a_a D_a + b_a D_b + ... + k_a Dk + w l_a L_t)(1+r)
D_b = (a_b D_a + b_b D_b + ... + k_b Dk + w l_b L_t)(1+r)
.....................................................................................
D_k = (a_k D_a + b_k D_b + ... + k_k D_k + w l_k L_t)(1+r)
___________________________________________
L_t = (a D_a +b D_b + ... + k D_k + w L_t)(1+r)

or:

D_a = [c_a (1- e)L_t + w l_a L_t)](1+r)
D_b = [c_b (1- e)L_t + w l_b L_t)](1+r)
.........................................................
D_k = [c_k (1- e)L_t + w l_k L_t)](1+r)
_____________________________
L_t = [(1- e)L_t + w L_t)](1+r)

remembering that instead of D_i, one can also indicate P_i L_t, or p_i l_i L_t.

One could make a distinction, among the commodities not indicated as means of production, between means of subsistence and superfluous goods, or luxury goods, and represent them as aggregates. If in such a representation one fixes the condition that monetary wages are equal to the price of fundamental goods and profits to that of the superfluous ones, the system has k+2 independent equations which allow for the simultaneous determination of the k+2 unknown terms: the k absolute prices, and the rates of wage and profit.
A hypothesis of this kind might certainly have practical relevance, but a clear distinction between fundamental and superfluous commodities cannot be drawn and, even if that was possible, wages could be expended in superfluous goods and profit in means of subsistence (there could also be very low profits among small tradesmen, for instance, and high 'salaries': professionals, the important entrepreneur himself, etc.).
Renouncing this distinction, the system has k+2 unknown terms and k+1 independent equations, allowing us to see how prices vary depending on the different quantitative relationships between wages and profits. By fixing as the k+2^th equation the magnitude of the rate of wage, or of the rate of profit, between zero and R (or the relation r / R), we find consequently all the unknown quantities.
The variation of a price implies the variation of all the other prices and of the parameters of the system. This shows the reciprocal influence between the system as a unit and the multiplicity of its firms; the way in which, on the quantitative level, the properties of the system, which depends on the properties of its parts, act in turn upon the properties of the parts themselves.

General Representation

When we consider the formation of prices, we must remember that commodities interest and manifest themselves only for their particular physical qualities, or material value, while the creative value, for the buyer, is invisible, as invisible is the proportion between labor and the means of production used, and that distribution takes place by means of the negotiated exchange. In reality, therefore, rates of profit of different firms are actually different.
For the possibility of buying and selling whatever one wants and, therefore, of moving capital in the most profitable sectors, rates of profit too different, too far from the average, cannot remain so too long. The same holds true for wages, since the workers tend to move where remuneration is higher. This would be the free market, to which the perfect competition hypothesis refers.
But since these movements of capital and labor take place just because in every period of time there are effectively different profits and wages, the perfect competition hypothesis, essentially always corresponds to reality, but never really, even in theory.
The perfect competition hypothesis, as that of the variation of the relationship between profits and wages for a same considered period of time, permits to leave aside the fortuitous reasons of the formation of prices in order to formally define the various economic categories and their quantitative relationships, and to analyze the structural properties of the system. The rates of profit and wages of the system, whose magnitude depends on the magnitude of the aggregates, fixes the point around which the rates of profit and wage of the firms oscillate, and if some of them is higher, then the others must be lower.
Even if the structural aspect remains the general factor of the formation of prices, the quantity of social labor that they indicate is different from the quantity of labor applied, from the prices of equilibrium of an hypothetical socialistic distribution, and also from those required by the structural situation of the perfect competition hypothesis.

In order to accurately represent the economic transformations occurring in a given period of time, one must consider that in that period there is only one available technology, and only one distribution takes place. Creative value, material value, and also the prices of every firm, therefore, must be considered data. The same applies for the magnitude of capital and the rates of wage and profit, which, for the most diverse reasons, are different among them and from that of the system (some firms, indeed, flourish and others go bankrupt).
The actual wages and profits obtained by the firms, therefore, should be considered as the product of a particular coefficient of diversity for the rates of wage and profit of the system (or average rates) in this way:

D_a = [c_a (1- e)L_t + ^_a w l_a L_t)](1 + °_a r)
D_b = [c_b (1- e)L_t + ^_b w l_b L_t)](1 + °_b r)
.....................................................................
D_k = [c_k (1- e)L_t + ^_k w l_k L_t)](1 + °_k r)
___________________________________
L_t = D_a + D_b + ... + D_k

(NOTICE: ^ stays for "lambda" and ° for "ro")

The coefficients ^_i and °_i, also oscillating around one, have a qualitative origin which resolves and is precisely measured by the quantity of money obtained and used; namely, they are the effect on the quantitative level of the different quality of things and relationships.

For many particular reasons, a commodity which in normal conditions has a low price, in other conditions can be very expensive, and vice versa. An important case which allows for temporarily higher profits is that of the increase in productivity caused by the introduction of a new technology. There are also different and changing social and political situations which favor some economic sectors at the expenses of the others.
Moreover, the ability of people to produce and negotiate is different, as is the fertility and availability of natural resources of different geographical areas. If a valley farmer produces a quantity of grain double than that obtained in the mountains with the same quantity of labor, it is unlikely that the mountain man will be able to sell his grain at twice the price. Or if a slow worker produces a commodity in twice the time normally employed, it is improbable, though not impossible, that he will be able to find someone ready to pay a double price for his product.

At this regard, we must notice that it is not possible, as Sraffa suggests (par. 10), to assume any difference in quality (of labour) to have been previously reduced to equivalent differences in quantity so that each unit of labour receives the same wage. Operating in this way and taking as unit of measurement of labor the "simple" labor (like that of a factory worker, for instance), the gross product value would still be expressed, although for other reasons, by a quantity of time of labor greater than the quantity that actually produces it. Varying distribution, indeed, it is the economic value which changes, not the creative value.
The quantity of creative value is a datum, and also the way in which distribution occurs, even if not determined by objective criteria of equity, is a fact. The magnitude of those coefficients can be a recognition for the contribution to production, or for the effective quality and utility of the produced commodity, but regarding distribution it can depend on any other reason.
Since money immediately expresses the command upon a quantity of human existence, these coefficients represent the status of equilibrium in the power relationships among people and among classes.

In the unceasing relationships among people and with nature, every human expression can be submitted to the quantitative order. There is therefore a multiplicity of economic systems with different quality, quantity, function and utility of the produced and exchanged commodities, the motivations of the activity which produces them, the juridical laws which rule the economic relationships and the ideological or religious system which support it. The concrete aspect of the field of economic, therefore, vary according to places and during time, and in this relative sense one can speak of different economies, but the priority of quantity above quality is the permanent and common characteristic of all the economic systems, and the representation given can be adapted to any of them.