ABSOLUTE PRICES

 

 

  Field of Economics (labor, commodity, price)

  Social Transformation

Distribution of Commodities

  Distribution of Economic Value

  Distribution as a System of Equations

The Problem of Transformation

  Physical Transformation

  Absolute Prices

  Coefficients of Relation

Field of Economics (Labor, Commodity, Price)

The social evolution -as for every other thing- occurs, is understood and described by means of the relationships -which are always reciprocal influences, or exchanges- among the composing parts of society and between society and the environment.
In economics, the physical transformation which occurs for the relationships with the surrounding environment is production of commodities, and the relationship between conscious action and the corresponding result becomes specifically a relationship between labor and commodity.
Social transformation regards the distribution of the outcome of the material transformations, and is the result of the negotiated exchange, determining the economic value, the price of the commodities.

The economic system, then, is constituted by a multiplicity of firms which produce and sell commodities, and that, for this reason, have in common three quantitative parameters which refer to three different conceptual levels, or to three different orders of value:

• labor, time of human existence employed to produce commodities, which is creative value,
• commodities, objects with specific characteristics, function or qualities, or material value (or use value),
• price, the economic value, or social value, the exchange relationship magnitude of the commodity with the other commodities.

Labor is the economic activity; commodities and services are the things of economics; and price is the magnitude of the quantified social relationships. These three types of value are different aspects of every economic system at any given moment of its existence.
The economic system, therefore, is composed of k firms, the A, B,..., K firms, which, during a given period of time, employ the quantities of labor L_a, L_b, ..., L_k, to produce the quantity of commodities and services A, B, ..., K, sold in exchange for the D_a, D_b, ..., D_k economic value.

In order to understand the meaning of these terms and the nature of the relationships among them, we must consider that physical transformation and social transformation are distinct but inseparable changes, reciprocally implicit but different. The two kind of transformation, therefore, must be considered first separately and then together, where production is the substantial aspect of economics, and distribution, which renders economic some physical transformations and not others, the essential one.

Social Transformation

Distribution of Commodities

Every firm furnishes society with the only kind of commodity it produces in exchange for commodities of different kinds from other firms. Knowing which and how many commodities each of them gives and receives during the considered period of time, the physical distribution can be represented as follows:

A (=) a_a A (+) b_a B (+)...(+) k_a K
B (=) a_b A (+) b_b B (+)...(+) k_b K
........................................................
K (=) a_k A (+) b_k B (+)...(+) k_k K
____________________________
GP (=) A (+) B (+)...(+) K

(NOTICE: the sign (=) stays for the sign of "identity", usually indicated with three horizontal dashes. The sign (+) means "functional sum")

Every commodity is indicated with a letter to differentiate it from the others, because diversity is the condition of the exchange, but their practical function as such does not matter (that's the reason why between commodity and service there is no essential difference) and is not further considered.
The example made by Sraffa in Production of Commodities by Means of Commodities (par. 2) can illustrate the above:

Piero Sraffa

An economic system is made up by a firm which produces 450 quarters of wheat, one which produces 21 tons of iron, and one 60 pigs. The wheat firm gives 210 quarters of wheat for 12 tons of iron and 18 pigs, the iron firm gives 15 tons of iron against 90 quarters of wheat and 12 pigs, and the pigs firm 30 pigs for 120 quarters of wheat and 3 tons of iron. Sraffa's representation is about the following:

240qr. wheat (+) 12t. iron (+) 18 pigs (=) 450qr. wheat
90qr. wheat (+) 6t. iron (+) 12 pigs (=) 21t. iron
120qr. wheat (+) 3t. iron (+) 30 pigs (=) 60 pigs
______________________________________________
450qr. wheat (+) 21t. iron (+) 60 pigs (=) GP

If as unit of measure of every commodity we fix its own totality, we have: W = 450qr. wheat, F = 21t. iron and P = 60 pigs, and the representation becomes this:

8/15 W (+) 4/7 F (+) 3/10 P (=) W
3/15 W (+) 2/7 F (+) 2/10 P (=) F
4/15 W (+) 1/7 F (+) 5/10 P (=) P
_________________________________
15/15 W (+) 7/7 F (+) 10/10 P (=) GP

It must be noticed that although the representation is formally the same, the classical framework requires that the objective value of the commodities should be calculated as the sum of live labor which produces them plus the crystallized labor transferred by the used means of production, so that distribution would not be represented by the produced commodities, of which we study the prices, but by those previously consumed in order to produce them. The commodities on the right side then would be physically others from those on the left, and even though we might suppose that the function was the same, there would be no reason for prices of being so. In order to force the sign of equal to maintain its own meaning, therefore, the classical economists assume the unrealistic hypothesis that the model was strictly static, retroactively static, one may say, hypothesizing that production would take place in the considered period of time, and distribution, instead, between the end of that period and the beginning of the next one, practically in no time.

But there is not need for these hypotheses. During the considered period of time there is a flow of activity of production and exchange, and the distribution and the price of the commodities of that period (not of others) are studied. The commodities given to the system by the firms, represented on the left of the functional equal signs '(=)', are the same obtained in exchange from the system in the same period of time, placed on the right side, and GP (=) A (+) B (+) ... (+) K is given by both the functional sum '(+)' of the first column, on the left of the equal signs, and by that of the last row.

The proper mathematical sign of equality fixes the identity relationship on the quantitative level between two different aspects of the same quality, or of two different things considered on the same level of value, and a mathematical relationship of sum or subtraction cannot be placed between different things as such, but only among their common qualities, or values, like weight, function, or price (if we sum a diamond with a stove according their weight, the stove is of greater value; if we sum them according to their economic value, then the diamond is heavier). The sum in column of the various coefficients of distribution, therefore has a mathematical meaning because any of them refers to only one kind of commodity, and:

a_a + a_b + ...+ a_k = 1
b_a + b_b + ...+ b_k = 1
...................................
k_a + k_b + ...+ k_k = 1

The signs of sum (+) in the rows among the various commodities obtained by the firms, instead, have a functional sense. In the representation we can only see that every commodity is exchanged with a certain proportion, small or large, of the gross product, but we cannot express, with only one number, what exactly this proportion is.

Distribution of the Economic Value

Commodities, though, have an economic value which is implicit in that distribution, namely, there is an identity relationship between material value and economic value which, if we express the latter as a sum of money, can be represented as follows:

A (=) D_a
B (=) D_b
..............
K (=) D_k
_________
GP (=) D_t

This representation indicates, for every firm and the system, the indissoluble identity between two different conceptual orders: commodity and price, or material value and economic value, a quantity of objects with a given quality or function and, in order to visualize the economic value, a sum of money.
Substituting the prices for the corresponding commodities, the signs of sum and equal acquire mathematical sense even in the rows:

D_a = a_a D_a + b_a D_b + ... + k_a D_k
D_b = a_b D_a + b_b D_b + ... + k_b D_k
........................................................
D_k = a_k D_a + b_k D_b + ... + k_k D_k
____________________________
D_t = D_a + D_b + ... + D_k

Every distribution of material value requires a series of prices such to satisfy the conditions of equilibrium of the exchanges, that is, the equality of sale and purchasing for firms and for the system. If we suppose a variation of distribution, the commodities' prices vary in such a way that the revenue obtained by every firm continues to be equal to the price of the commodities received in exchange.

The relationship D_t = D_a + D_b + ... + D_k, is given by both the first column, representing the revenue obtained from the sale, and the last row, representing the corresponding purchase. Since they are the same commodities, acquired at the same price at which they are sold, this relation holds true for every period of time and every considerable kind of distribution.

Since the magnitude of value of the economic system is obtained by the arithmetical sum of prices of the exchanged commodities, the relationship between the economic value of every commodity and that of the system can inversely be expressed like this:

D_a = P_a D_t
D_b = P_b D_t
....................
D_k = P_k D_t
__________
D_t = D_t

and the representation of distribution becomes the following:

P_a D_t = a_a P_a D_t + b_a P_b D_t + ... + k_a P_k D_t
P_b D_t = a_b P_a D_t + b_b P_b D_t + ... + k_b P_k D_t
................................................................................
P_k D_t = ak P_a D_t + bk P_b D_t + ... + kk P_k D_t
________________________________________
D_t = P_a D_t + P_b D_t + ... + P_k D_t

 

P_a = a_a P_a + b_a P_b + ... + k_a P_k
P_b = a_b P_a + b_b P_b + ... + k_b P_k
........................................................
P_k = ak P_a + bk P_b + ... + kk P_k
_____________________________
1 = P_a + P_b + ... + P_k

In this last representation, where the sums of money do not appear, we see that however the various P_i magnitudes could change, their sum is equal to the unit, to one. This means that the price always expresses, in precise quantitative terms, the relationship between the magnitude of every firm and that of the system. A variation of distribution is a variation of the quantitative relationship between many variable magnitudes and a single constant one: that of the gross product.

Distribution as a System of Equations

Mathematically, the representation of distribution of the economic value can be considered as a system of equations, where the coefficients of distribution are the known terms, and the rates of exchange, since we are studying their nature, the unknown magnitudes.

In Production of Commodities by Means of Commodities (par. 3), of the general representation of the gross product distribution Sraffa says:

One commodity is taken as standard of value and its price made equal to unity. This leaves k-1 unknowns. Since in the aggregate of the equations the same quantities occur on both sides, any one of the equations can be inferred from the sum of the others. This leaves k-1 independent linear equations which uniquely determine the k-1 prices.

And in his example of wheat iron and pigs, indeed, Sraffa only expresses the prices relatively to the price of the wheat taken as a unity.
But it is not correct to say that any one of the equations can be inferred from the sum of the others. One must say, instead, that every equation representing a firm is obtained by the difference between the equation representing the system, or gross product, minus the sum of the equations representing the other k-1 firms.
This means that the system actually already has k independent linear equations, where the k^th equation is that which fixes the constancy of the gross product's value for every possible distribution, thus allowing for the determination of the k unknowns all together. In other words: varying distribution, the equations indicating the firms all normally vary too, while the equation representing the system is the only equation which remains constant. This, therefore, is the unchanging reference by means of which we can say how prices vary in relation to it.

In the previous example, the economic magnitude of the firms -which Sraffa does not express- is: P_w = 15/32, P_f = 7/32, P_p = 10/32:

P_w = 8/15 15/32 + 4/7 7/32 + 3/10 10/32
P_f = 3/15 15/32 + 2/7 7/32 + 2/10 10/32
P_p = 4/15 15/32 + 1/7 7/32 + 5/10 10/32
___________________________________
1 = P_w + P_f + P_p

15/32 = 8/32 + 4/32 + 3/32
7/32 = 3/32 + 2/32 + 2/32
10/32 = 4/32 + 1/32 + 5/32
__________________________
32/32 = 15/32 + 7/32 + 10/32

The same value obtained through the sale of a commodity could be expended to buy different combinations of commodities, and even if price is determined by a specific distribution, there are infinite possible distributions which equilibrium is satisfied by the same prices. One of these theoretical distribution is that where every firm, which sells its own commodity for the D_i price, acquires a physical fraction D_i / D_t of every commodity of the gross product. Using the term formulated by Sraffa, we can say that by fixing D_t = cost. as the k^th equation, one assumes the very gross product as a standard commodity. The increase or decrease of the price of the commodity, therefore, indicates exactly and immediately an increase or decrease of the fraction of the gross product with which it exchanges, and if some prices increase, others will decrease correspondingly, so that the global economic value, as well as the material value, does not change.

The Problem of Transformation

David Ricardo

By fixing as k^th equation the constancy of whatever unknown magnitude, we determine the proportion by which commodities exchange among themselves, the relative prices, but the magnitude of the gross product is missed, because it must vary in order to maintain the fixed constant. By fixing as k^th equation, as indicated by Ricardo, the constancy of price of the wheat (or, as for Bortkiewicz, the price of gold), the price of wheat remains constant by definition even if exchanging for different quantities of commodities, for different proportions of gross product, while the price of the gross product, measured in terms of quantities of wheat, changes even if composed of the same quantity of commodities.
This also occurs, in general, by fixing the constancy of value of whatever aggregate of commodities, and particularly, as the classical economists do, the value of the net product (supposed to be equal to the quantity of live labor applied in the considered period of time).
But in this way, varying distribution, the gross product value, even with unchanged techniques of production, becomes unpredictably inferior or superior to the quantity of labor objectively "incorporated" in it (and considered coinciding with the price calculated in a socialistic distribution). This is unacceptable even for the classical economists, because their framework requires the constancy of the gross product value, since its magnitude, being determined into and by production, is however a datum independent on any subsequent distribution.
Herein lies the formal origin of the problem of transformation from objective values to market prices, which means that the relationship between applied quantities of labor and the prices is missed or, in other words, that prices cannot be expressed in absolute terms.

The awareness of the constancy of the gross product value, therefore, forces the classical economists to notice the existence of a contradiction inherent in the premises of their theory, but it seems a subconscious awareness, since it does not permit to identify and dissolve the origin of the contraddiction and positively define the meaning of value.

Although Sraffa's standard commodity as a unit of measure of value is inspired by Ricardo's idea of the constancy of the wheat price, it is actually composed by the aggregate of commodities of the net product, by fixing in this way the constancy of the relationship implicit in the classical framework. This is confirmed by the reduction to dated quantities of labor (even though it must be noticed that such a reduction consists in attributing a non-existing temporal dimension to the mathematical method for the solution of the systems of equations called the method of substitution, applied to a system which only represent quantities of labor and the commodities of the considered period of time).
For Sraffa, therefore, as well as for Marx, labor remains the essence of value, but the neo-ricardians, and even Sraffa himself, do not recognize it. They focus their attention on the fact that the rates of exchange, or prices of equilibrium, are calculated within the representation of distribution without the need to know the quantity of labor necessary to produce them -giving indeed the title of Sraffa's book: Production of Commodities by Means of Commodities-, and this, together with the persistently insoluble problem of transformation, has induced them to believe and sustain that labor cannot be considered as the essence of economic value.

Physical Transformation

The identity relationship between labor and the correspondingly produced commodities can be represented as follows:

L_a (=) A
L_b (=) B
..............
L_k (=) K
_________
L_t (=) GP

Labor determines specificity and the duration of the physical transformation. When labor terminates, even the considered physical transformation ceases, and the product has assumed form and the characteristics that labor has imposed on it. Labor and commodity are the two opposite and complementary poles of the transformation; they are different and distinguished entities, but also identical and inseparable, constituting the temporal and spatial aspects of the transformation itself. It is not matter, therefore, of embodiment of labor into the commodities, but rather of a relationship of complete coincidence, of identity indeed, between two different conceptual orders, or values, which are and remain well distinguished: labor, measured as a quantity of human existence, which is the essential value of the physical transformation, or creative value, and the commodity with its own physical or functional characteristics, the substantial value, or material value, of the same transformation. The commodity I, therefore is the material value of the labor L_i, and L_i is the creative value of the commodity I. One may also say that L_i and I are the two complementary magnitudes which characterize the physical transformation. The same holds true for the system as a whole.

Expressing the creative value of every firm as a fraction of total creative value: L_i = l_i L_t, we obtain:

l_a L_t (=) A
l_b L_t (=) B
..................
l_k L_t (=) K
_________
L_t (=) GP

Absolute Prices

Physical and social transformation are distinct. The identity relationship between labor and commodity can be seen without knowing anything about social transformations. On the other hand, the identity relationship between commodity and price, can be seen without knowing anything of the employed quantities of labor. Labor, indeed, depends only on what has to be produced and how it is produced, while prices form according to a logic for which labor is only one among many factors determining their magnitude. The possibility of conceiving different distributions of the same commodities -even though only one of them will take place- shows the constant character of the creative value and the variability of the economic value.
For the entire system, instead, no matter how distribution varies, there is a complete reciprocal implication between physical transformation and social transformation which is just what defines the extension of the field of economics. Commodity is the visible conjunction between them, the intermediate term which totally belong to both transformations, which, for this reason, must be considered inseparably united, so that:

D_t (=) PL (=) L_t

And the mathematic relation:

D_t = L_t

establishes the absolute value of money, of the currency used as a means of exchange in the considered system, always and immediately indicating a quantity of commanded social labor by he who spends it, i.e., time of quantified human existence; the quantity of time that anyone gives to the others, to the society as an economic system.
By imposing this relation as the k^th equation of the system, the k prices and the value of the currency which expresses them, are immediately determined in absolute terms.

If two different economic systems produce exactly the same quantity of commodities, but one of them with a double quantity of labor in every firm, the relative prices are identical, but the absolute value of the commodities produced by the quick workers will be half, so that the same quantity of social labor permits to buy a double quantity of commodities.

Coefficients of Relation

Distribution is the transformation resulting from the relation between every economic subject and the whole system, between every physical transformation and all of them, between the time of labor which produces a commodity and that of the system in which it is produced.
The quantitative relationship between the price of every commodity, or firm, P_i L_t, and the corresponding creative value, l_i L_t, can be indicated by a coefficient of relation p_i (meaning that the price of the commodity I is p_i times the corresponding quantity of creative value), and expressed as follows:

P_a L_t = p_a l_a L_t
P_b L_t = p_b l_b L_t
.............................
P_k L_t = pk l_k L_t
_______________
L_t = L_t

The price of every commodity is a fraction of the creative and social total value, and a multiple of its own creative value, and p_i is the coefficient which expresses the quantitative relationship between the two of them:

p_i = P_i / l_i

Since social value and creative value of a commodity normally do not coincide ( P_i is different from l_i, or P_i L_t is different from l_i L_t), and since for the system they are at once always equal to L_t, the magnitude of the coefficients of relation oscillates around one, and the economic value around its own creative value. If the coefficient of relation of a commodity is one, the quantity of social labor expressed by its price, although remaining conceptually different, coincides with the quantity of time of labor which produces it.
In a system where every firm obtains from the sale of its commodity a quantity of social labor equal to the quantity of labor employed, all the coefficients of relation are equal to one. When, instead, the coefficient of relation some commodities is greater than one, then other coefficients of relation must be less than one.