1

Production of Commodities by Means of Capital*

Glen E. Canessa **

Abstract

The mechanics of the production of commodities by means of capital was analyzed extensively

by Marx in Capital. Later, their critics concentrated in pointing to the errors and

simplifications of his analysis, and the discussion ended at the now known as the

―Transformation Problem‖. Apparently this discussion has not been settled until today, neither

by those in favor of the theory of Marx nor by their opponents, even though varied solutions to

the problem has been proposed, many with the intention to defend, and many to demonstrate

the uselessness of the Marxist theory of value. In this article a definitive explanation of the

problem at the conceptual level is set out, which is exemplified with simple methods for the

calculation of values and prices

Contents

1. Introduction _____________________________________________________________________ 2

2. Basic Definitions _________________________________________________________________ 2

3. Production of Commodities by Means of Commodities ________________________________ 5

4. Production of Commodities by Means of Capital _____________________________________ 9

4.1 Exchange Between Sectors __________________________________________________ 13

4.2 Total Value Versus Total Price _______________________________________________ 13

4.3 Total Surplus-Value versus Total Profit _______________________________________ 14

4.4 Extension to a Three-Sector System __________________________________________ 15

4.5 Extension to Reproduction on an Extended Scale ______________________________ 17

5. Comparison to Other Transformation Methods _____________________________________ 19

5.1 Comparison to Bortkiewicz Method ___________________________________________ 20

5.2 Comparison to Carcanholo Method ___________________________________________ 21

5.3 Comparison to Winternitz Method ____________________________________________ 22

5.4 Comparison to Meek Method _________________________________________________ 23

6. Conclusions _____________________________________________________________________ 24

7. References ______________________________________________________________________ 25

* Since this article was originally written in Spanish, notation follows Spanish names. For example: m for

mercancía (commodity), d for dinero (money), etc. **

glen.canessa@novis.cl

2

1. Introduction

Labour value theory developed by Marx in Capital has been the subject of many criticisms and

defenses by renowned economists over more than a century (Böhm-Bawerk, 1896;) Bortkiewicz,

1907; Sweezy, 1942; Meek, 1956; (Morishima, Samuelson, etc.). However, at the end of this

long period its theoretical development is virtually null. The difficulties encountered to

implement the theory in its original state has led detractors and supporters of Marx to question

the validity, if not at least the usefulness of the labour theory of value (Baumol, Badhuri,

Laibman, Wolfstetter, Morishima, Samuelson, Steedman, Bronfenbrenner, Howard, etc.). A

clear example of the diversity of interpretations on the subject can be found in the various

solutions offered for the so-called problem of the transformation (Bortkiewicz, Winternitz,

Seton, Meek, Carcanholo).

This work is intended to demonstrate that most of the inconsistencies attributed to the Marxist

theory of value come from the inaccurate understanding of the basic categories of value and

price. A proper algebraic approach will make it possible to establish the valid principles of this

theory, first, unequivocally once and for all, and free of apparent internal inconsistencies. This

task is essential to clear the way for subsequent elaborations aiming to demonstrate the

usefulness of the labor theory of value as a basis for the scientific analysis of the capitalist

economic dynamics.

This work deliberately avoids a defense or attack to Marxist theory of value, or the scientific

personality of the own Marx. On the other hand, all the concepts used here are defined strictly,

in accordance with the basic and most unequivocal Marx approaches. As a result, is considered

unnecessary and undesirable both an initial dissertation as well as any previous comments on

the theories of value and surplus value. Neither is considered necessary to cite specific

locations of the works of Marx to support the presentation of the core of his economic theory,

which is widely known and in its most elaborate form is contained in the volumes I and III of

Capital.

Section 2 of this article defines the basic concepts of value and price of commodities, according

to Marx‘s definitions in the first chapter of Capital. On this basis, a method for the calculation

of values is devised for a system of simple (non capitalist) circulation of commodities, in section

3, and, in section 4, a method for the calculation of capitalist production prices, for a

corresponding capitalist system of production. As a result, two postulates concerning equality

between value and price of the total product, and between value and price of the surplus

product are verified. Section 5 compares the approach set out here with some other solutions

proposed in the literature for the problem of the transformation.

2. Basic Definitions

The value of a commodity is determined in the process of its production. Its magnitude is

measured by the amount of abstract work socially needed used up in this process. Assuming

that social work is homogeneous, the magnitude of value can be measured in simple units of

time, say hours, [h].

3

The price of a commodity is defined as the proportion in which it is exchangeable for a

particular commodity: the money-commodity. The magnitude of the price is therefore measured

in units of money-commodity per unit of the exchangeable commodity. Assuming that money-

commodity is gold, the price of wheat, for example, can be measured in ounces of gold per tons

of wheat ([oz gold/Ton wheat]).

Marx‘s law of value states that prices of commodities are also determined in the process of

production or, what is the same, that they can be derived from its values. This determination,

however, does not have a unique form, of absolute validity for all kinds of commodity

production modes.

In a simple (non capitalist) commodity production system, or simple circulation of commodities,

the price of any given commodity m, mp , is defined in a very simple form by its value, mv , and

by the value of the money-commodity, dv :

d

mm

v

vp .

Dimensionally, the value mv is expressed in work hours per some unit of the commodity m ,

[h/Um], and the value dv is expressed in work hours per some unit of the money-commodity,

[h/Ud]. For example: if the value of wheat is expressed in work hours per Ton, [h/Ton wheat],

and the value of gold (the money-commodity) is expressed in work hours per ounce, [h/oz gold],

then the price mp is expressed dimensionally in ounces of gold per Ton of wheat, [oz gold/Ton

wheat].

The price of a commodity, according to Marx, is the most developed expression of the relative

form of value, corresponding historically and logically to the emergence of money (metallic

money). In this situation, the commodity used as money has no price.

The money-commodity expresses its relative value in the developed form, which corresponds to

the whole set of exchange rates of this money-commodity to each of the remaining commodities.

This is equivalent to the set of all prices reversed.

The ordinary concept of price, which also expresses the rate of exchange between a commodity

and the money-commodity, but using the units of the currency sign, gives place to a definition

of a "price" for the money-commodity itself. To distinguish the two concepts we call monetary

price to the latter.

If the unit of measure of the money-commodity (arbitrary physical unit, e.g. unit of weight,

tipically) is associated to a currency sign (e.g. dollar), then the price of the commodity m, which

is expressed in units of money-commodity per units of commodity m, can also be expressed in

units of the monetary signs per unit of commodity m, converting the physical units of the

money-commodity to its equivalent in this sign. For example, if one ounce of gold is associated

to the dollar currency sign (1 oz Gold = $1) and the price of wheat is 3 ounces of gold per ton (3

[oz Gold/Ton wheat]), then the monetary price of wheat is $3 per ton (3 [$ /Ton wheat]).

4

The monetary unit, which initially expressed a certain value, in work hours, later becomes

independent and dissociates itself from that value (because of falsification of coins, for

example). As a result of this dissociation, the monetary prices of commodities, and also the

relationship between the currency sign and the unit of money-commodity, change. This last

relationship, varying in time, is commonly interpreted as the price of money (e.g. the price of

gold).

When the value of the money-commodity changes, the prices of all commodities change

accordingly, and their monetary prices change as well, unless the currency sign is already

dissociated from the physical quantity or value of money. On the other hand, if only the

relationship between the monetary symbol and the physical quantity of money changes, the

monetary prices of commodities change, even when their values and prices are the same. For

example, if price of wheat is originally 3 [oz Gold/Ton wheat] = 3 [$ /Ton wheat] and gold

decreases its value by half, the price will increase to 6 [oz Gold/Ton wheat] = 6 [$ /Ton wheat].

If only the monetary price of gold changes, from 1 [$ /oz Gold] to 2 [$ /oz Gold], the monetary

price of wheat changes from 3 [$ /Ton wheat] to 6 [$ /Ton wheat], even when its price is still 3

[oz Gold/Ton wheat].

The above example justifies the current interpretation of prices as purely conventional

quantities applicable equally to all commodities, including money. The monetary price of

commodities is indeed conventional in its absolute level, but not in its variations. On the

contrary, for the (metallic) prices, the absolute magnitudes and their variations are determined

by the values of the commodities, including among them the value of money-commodity.

From the definition of price it follows that for at least one price to exist it is necessary and

sufficient to have at least two different commodities, one of which is the money-commodity. In

general, as prices are relative magnitudes by definition, in a system of n sectors there are

1n prices and n values.

Defining relative price as the exchange rate between any two commodities, it follows that in a

system of n sectors there are 1nn relative prices, of which 1n are independent

quantities and the rest can be calculated from them.

The relative price of any commodity 1 with respect to any commodity 2, different from itself, is

equivalent to the ratio between their prices, or simply its price, when the commodity 2 is

money. When the commodity 1 is money, the relative price is the reverse of the price, as

defined here.

In a system of simple circulation of commodities, the relative price between any two

commodities equals the ratio between their prices and also the ratio between their values, so it

is independent, in general, of the value of the money-commodity.

5

3. Production of Commodities by Means of Commodities

The simplest system of production is that in which there are only two different commodities. In

order to have a price, one of the commodities must be the money-commodity. Thus, there is

only one price, that of the other commodity.

A production system of two commodities (two sectors) is described, in general, by their

technical relations of production, which can be expressed as

112111 ,, qtqq

222212 ,, qtqq ,

where ijq is the physical volume of commodity i used to produce commodity j , iq is total

volume of product for commodity i , and it is the amount of direct work necessary for the

production of commodity i . (Note that we deliberately avoid using ―+‖, ―–― and ―=‖ signs, to

affirm the idea that we are treating with physical magnitudes, dimensionally incomparable).

Similarly, the scale of the production, in line with technical laws, can be expressed in the

following functions of production

1111211111 ,, qEtEqEqE

2222222122 ,, qEtEqEqE ,

where 1E and 2E represent the scale of production in each sector.

In a more compact form, production functions can be written in the form

112111 ,, QTQQ

222212 ,, QTQQ .

Q1

Q2

Q11

Q21

Q22

Q12

Sector 1

Sector 2

Q1n

Q2nT2

T1

Figure 1: Diagram for Two Sector Reproduction

6

In general it will be true that

nQQQQ 112111

nQQQQ 222212

where inQ is the net product of commodity i . In general, this net product is destined in one

part to accumulation and in other part to unproductive consumption. The unproductive

consumption reproduces the labour-power required by the production process. In conditions of

simple reproduction all the net product is consumed unproductively.

To convert from production functions to value equations, it is necessary to multiply each

volume of commodity by their unit value iv

1112111221111 vQQQTvQvQ n

2222212222112 vQQQTvQvQ n ,

from where we have that total net product, nn QQ 21 , , equals total direct labour, or

212211 TTvQvQ nn .

The two previous equations can be reduced to

11121221 vQQTvQ n

22212112 vQQTvQ n

from where the values 1v and 2v can be determined, as functions of the technical coefficients of

the production functions, resulting that

nnnn

n

QQQQQQ

TQTQv

21211212

1221

1

nnnn

n

QQQQQQ

TQTQv

21211212

2112

2

where 21 TTT .

From the equivalence between labor and value of the net product, the reverse process of

replacing direct work by its equivalent in terms of net product, can be done in the original

production functions. If i is the proportion of total labour assigned to the production of

commodity i , then

2211111 vQvQTT nn

2211222 vQvQTT nn ,

where 121 , for simple reproduction conditions.

Then, production functions can be expressed in terms of pure commodities, in the following way

7

121211111 , QQQQQ nn

222221212 , QQQQQ nn ,

or, hiding the division between volumes of commodities, in the simpler form

12111, QQQ

22212, QQQ .

In this form of the functions of production, usually used in the literature, direct work is

represented implicitly in terms of a set of commodities able to be consumed unproductively,

whose proportions correspond to the total net product.

Now, translating production functions to value equations, we get the general result

11221111 vQvQvQ

22222112 vQvQvQ

which are two equivalent equations, from where we can get

12

21

2

1

Q

Q

v

v

;

and, to calculate values 1v y 2v the following equation is required

TQvQv nn 2211 .

In Sraffa1 example: 11Q = 280 [qt wheat], 21Q = 12 [T iron], 1Q = 400 [qt wheat], 12Q = 120 [qt

wheat], 22Q = 8 [T iron], 2Q = 20 [T iron]. With these values it results

10

1

2

1 v

v [T iron / qt wheat] .

The expression 21 vv , if commodity 2 is money-commodity, is the price of commodity 1. If

commodity 1 is money-commodity, the reverse expression, 211212 QQvv , is the price of

commodity 2. It follows from the definition of money that both cases are mutually exclusive

and, therefore, there will always exist only one of these prices.

If the division of sectors is such that commodity 1 represents means of production and

commodity 2 represents means of subsistence, then the overall system is reduced to

nQQQQTQ 112111111,

nQQTQ 22212, .

In this case, only the commodity 2 is consumed unproductively. Then there are the following

alternatives: a) if 01 nQ , then reproduction is on an extended scale (increased reproduction),

or b) if the entire net product is consumed unproductively, reproduction is simple, and then the

1 Production of Commodities by Means of Commodities, Cambridge University Press, 1972.

8

entire net product has the form of commodity 2 and, therefore, 01 nQ . The latter not

meaning, as will be seen below, that sector 1 does not generate a net product.

Value equations, for simple reproduction, are

112111111 vQQTvQ

222112 vQTvQ n ,

and in this case the following result can be obtained

12

11

Q

Tv

nn Q

T

Q

TTv

22

212

.

The expression for 2v simply confirms that the value of net product is equivalent to the total

direct labour.

Net product for each sector is

1121 vQT

1122212212 vQvQTvQTTT nn .

The following discusses an example of this particular case, which is the most commonly studied

in the literature.

Let there be the following functions of production

TFQTTF 10001200,800

QTQTTF 2000800,200 ,

where iron, measured in tons, [TF]2, is the means of production and wheat, measured in

quarters, [QT], is the expression of direct labour.

The equations for values are

ftf vvv 10001200800

ftf vvv 2000800200

where fv is the value of iron, in work hours per ton of iron, [h/TF], and tv is the value of

wheat, in work hours per quarter of wheat, [h/QT].

Both equations can be reduced to an equivalent form

ft vv 2001200

2 Units notation comes from the original spanish version: TF stands for Toneladas de Fierro (tons of iron); QT

stands for Quintales de Trigo (quarters of wheat).

9

tf vv 1200200 ,

from where it follows that

TFQTv

v

t

f6 .

To calculate each value, that is, their magnitudes in work hours, we need to know the total

labour employed in the production process. If total labour employed was 4000 hours, then this

is the value of net product, 2000 quarters of wheat, and therefore the value of wheat, tv , is

QThQT

hvt 2

2000

4000

and, consequently, value of iron is

TFhvv tf 126 .

Net product from sector 1 has a value of 2400 [h], equivalent to 200 tons of iron or 1200

quarters of wheat. Net product from sector 2 has a value of 1600 [h], equivalent to 800 quarters

of wheat.

4. Production of Commodities by Means of Capital

In the case of capitalist production, the determination of the prices of commodities by their

values is not as simple as before. The price of a commodity depends not only on its value and

the value of the money-commodity, but, in general, on the values of all commodities and

compositions of capitals that produce them.

A capitalist production system of two commodities or sectors is described by production

functions of a general type

Cvc 11,

Vvc 22 , ,

where sector 1 produces means of production, and sector 2 produces means of subsistence. ic is

constant capital and iv is variable capital employed in production for sector i . C and V are

the products of each sector, all these physical magnitudes of commodities.

Unlike production functions for simple circulation of commodities, in functions for capitalist

production the volumes of commodities appearing at both sides of the production function has

not the same value. Precisely, the functions of capitalist production are the expression of the

process of expansion of value of capital.

According to Marx‘s theory, the magnitude of the expansion is proportional to labour-power

used by each capital, in a proportion called rate of surplus-value.

10

Assuming that this rate is the same in both sectors3, the equations of value are

Cvpvvcv cvc 111

Vvpvvcv vvc 122

where cv is the unit value of constant capital, vv is the unit value of variable capital and p is

the rate of surplus-value.

From these last equations, and knowing the total labour applied in the production process, T

[h], the values and the rate of surplus-value can be obtained.

From the first equation

1

1 1

cC

pvvv vc

;

Replacing this expression in the second equation, we get

1

21

1

2

vvcC

c

Vp .

On the other hand

TVvv ,

following that4

V

Tvv and

2112

1

vcCvc

vTvc

.

For simple reproduction, 21 ccC , and the expressions can be reduced to

121

vv

Vp

V

Tvv

212

1

vvc

Tvvc

.

In the same way as for simple circulation of commodities, in case of simple capitalist

reproduction total net product has the physical form of means of subsistence.

3 The rate of surplus value is the same in all sectors if it is assumed a workforce of homogeneous quality and

with a total mobility. 4 This equation represents the "sector" which produces labour-power, i.e. the consumption of workers

themselves.

11

The physical magnitude of net product is pvvV 121 . Now we can distinguish in net

product a part called surplus product, whose magnitude is pvv 21 , and another part called

necessary product, which restores labour-power utilised, whose magnitude is 21 vv . The

value of surplus product, pvvvv 21 , equals total mass of surplus-value.

The problem to determine prices is that a portion of the product of sector 2 must be exchanged

by means of production, so that workers and capitalists of sector 1 can get their means of

subsistence and, in return, capitalists of sector 2 can get the means of production to replace

those consumed productively. This problem is not as simple as for simple circulation of

commodities, because this exchange is not only an exchange of commodities, but an exchange of

products of capitals which "claim" equal rates of profit.

Now, production functions will not translate into equations of value, but in equations of prices

of production. Lets call cvr the rate of exchange between means of production and means of

subsistence, and vcr its inverse. Since there are only two commodities, there will be only one

price of production; this price will be cvr or vcr , depending on which is the commodity chosen as

money.

Choosing commodity v as money, production functions can be expressed in units of money-

commodity, as

2111 ccrvcr cvcv

pvvvcrcv 12122

Converting them in equations, en units of commodity v, we have

2111 1 ccrgvcr cvcv

pvvgvcrcv 11 2122

where g is the average rate of profit5.

Aggregating both sectors, we get

V

Cr

p

vvccr

pvvg

cvcv

12121

21

where 21 ccC and then6 21 vvV .

On the other hand

11

12

vcr

vcrg

cv

cv

5 The rate of profit is the same in all sectors if it is assumed that social capital can move freely from one sector

to another. 6 Note that before we used pvvV 121 .

12

1

11

12

11

12

1

vcr

vcrp

C

V

gcc

gv

g

p

C

Vr

cv

cv

cv

From where it results a second degree equation for cvr , whose roots are

2

12

2

211211

2

14

Wc

pWvccpcWvcpcWvrcv

where VCW , is the physical composition of total capital (whose dimension is [means of

production units / means of subsistence units]), and the root of interest for cvr is that obtained

with ―+‖ sign in the formula.

In the example shown in the previous section, when production is simple circulation of

commodities, workers own and consume all net product and, therefore, no mention of surplus

product or value of labour-power can be made, neither of surplus-value.

To use the same data in the previous example, now in a schema of capitalist production, we will

assume that capitalists pay workers for only half the net product appropriated with the total

product, i.e. the rate of surplus-value is 100 % ( 1p ).

Accordingly, for the same example, the new capitalist production functions are expressed in the

following table

TFc 8001 QTv 6001 TFC 1000

TFc 2002 QTv 4002 QTpV 20001

Using this data, the equation for cvr is

0120012002002

cvcv rr

from where

TFQTrcv 873.6153

which means that a ton of iron is exchanged for 6.873 quarters of wheat.

The price composition of total capital, pW , defined as the adimensional ratio

V

CrW cv

p

Turns to be

873.6pW ,

which means that, in terms of means of subsistence, or in terms of prices, the total social

capital is composed of 6.873 times more constant capital than variable capital.

The average rate of profit is

13

%7.121

pcvWr

pg .

The following table sums up the results of calculation for prices and mass of profit, ig , for

each sector7:

4.54981 c 6001 v 6.7741 g 6873C

6.13742 c 4002 v 4.2252 g 2000V

4.1 EXCHANGE BETWEEN SECTORS

Now we can analyze the exchange between sectors, according to the resulting prices and values

of commodities, calculated in the previous section for this same example.

The table above shows that sector 1 provides sector 2 with its surplus of means of production,

200 TF, whose value is 2400 h, at a price of cvr ×200 = 6.873×200 = 1374.6 QT. In return, sector

2 delivers to sector 1 its surplus of means of subsistence, 1374.6 QT, at a price of 1374.6 QT =

2749.2 h.

Then, the exchange of non equivalent values is verified, that which is necessary between

sectors in order to both capitals to obtain the same rate of profit.

In sector 1 a capital was invested whose price was 6098.4 QT, and there was produced a

volume of iron whose price is 6873 QT. Of this total volume of iron, a part (800 TF) whose price

equals 6098.4 QT is used, in its original form of iron, to restore the means of production of

sector 1 itself. The rest, 200 TF, equivalent to 1374.6 QT (in price) is delivered to sector 2, with

which it restores their means of production and, in return, delivers to sector 1 a mass of means

of subsistence with the same price. The mass of means of subsistence received from sector 2

covers the needs of workers (600 QT) and capitalists (774.6 QT) in sector 1. On the other hand,

the remaining means of subsistence produced (625.4 QT) are used by sector 2 itself, in their

original physical form of means of subsistence, to satisfy the needs of its workers (400 QT) and

its capitalists (225.4 QT).

4.2 TOTAL VALUE VERSUS TOTAL PRICE

A comparison of total value against total price involves dimensionally different magnitudes.

Total value represents a definite mass of social work hours, and total price represents a

definite amount of money-commodity. Quantities of such a different kind can only be compared

through a conversion of units (cf. Dickinson).

Since total product is made up of different commodities, produced by capitals of different

composition, its total price cannot be equal or proportional to its total value. In the example

above, total product is 2000 QT and 1000 TF, whose values are equivalent to 4000 h and 12000

7 For convenience, all these quantities are expressed in units of commodity v. In fact, the price of commodity

c is rcv, and commodity v has no price.

14

h, respectively, and their prices are equivalent to 2000 QT8 and 6873 QT, respectively. Total

value is then 16000 h, equivalent in value to 8000 QT, but total price is 8873 QT.

Total price, PT, is an aggregate of total prices for each commodity, PTi. PTi is the product of

price of commodity i, pi, and its mass, mi. Dimensionally, total prices are expressed in units of

money-commodity, and therefore they can be added.

Note that one commodity, money-commodity, has no price. However, it is possible to add the

mass of money-commodity, md, directly to total, for its units are, precisely, that of total prices.

d

di

iid

di

i mmpmPTPT

.

Total value, VT, is the sum total of values of all commodities. Dimensionally, it corresponds to

a total amount of work hours. To compare total value with total price, the conversion to be

made arises naturally. Using the value of money-commodity, vd, we can convert total price in

units of value

d

i

iidd mmpvPTv ,

but this quantity is not necessarily equivalent to total value, except in the trivial case where

each price of production equals the (simple) price for each commodity.

In the example above, assuming wheat as money-commodity, total price is 8873 QT and total

value is 16000 h. Value of wheat, vd, is 2 [h/QT], so

vd PT =17746 ≠ 16000 [h] =VT.

However, the next statement can be postulated for any capitalist system of production:

First Postulate

Total price always equals total value, in the sense that both magnitudes

amount to the same combined mass of (physical) commodities.

In the example above, total value of 16000 h is equivalent to 8000 QT, but could equally

amount to 5000 QT + 500 TF, or 3500 QT + 750 TF, etc. However, given the values of both

commodities and their relative share in total product (2 [QT/TF]), total value of 16000 h

amounts uniquely to 2000 QT + 1000 TF. Similarly, given the price of 6.873 [QT/TF] and the

relative proportion of both commodities in total product, total price of 8873 QT also amounts

uniquely to 2000 QT + 1000 TF.

4.3 TOTAL SURPLUS-VALUE VERSUS TOTAL PROFIT

A comparison of total surplus-value against total profit involves dimensionally different

magnitudes. Total surplus-value represents a definite mass of social work hours, and total

8 This is not really a price, since wheat is, in this case, the money-commodity.

15

profit represents a definite amount of money-commodity. Quantities of such a different kind

can only be compared through a conversion of units (cf. Dickinson).

For simple circulation of commodities, and when only one commodity is consumable

unproductively, surplus product takes the physical form of this commodity at some point.

Considering surplus product in this homogeneous physical form, total surplus-value and total

profit must necessarily be equivalents. In the example above, total surplus-value is 1000 QT.

Total profit is 774.6 + 225.4 = 1000 QT.

In the general case, when surplus product is heterogeneous, applying the same reasoning as for

total product we can conclude that total surplus-value and total profit does not necessarily

match numerically, when converted to same units, using the value of money-commodity, vd.

However, the next statement can be postulated for any capitalist system of production:

Second Postulate

Total profit (price of surplus product) always equals surplus-value (value of

surplus product), in the sense that both magnitudes amount to the same

combined mass of (physical) commodities.

Similarly to the first postulate, this second one states that, on the one hand, given the values of

commodities and their relative share in total surplus product, total surplus-value amounts

uniquely to the combined physical volume of that surplus product; and that, on the other hand,

given the prices and the relative proportion of commodities in total surplus product, total profit

(price) also amounts uniquely to the same combined volume.

In the example above, total surplus product has the physical form of a homogeneous

commodity, so the explanation becomes trivial.

4.4 EXTENSION TO A THREE-SECTOR SYSTEM

Systems of three sectors follow the original division of Marx in Capital, where sector 1 produces

means of production (c), sector 2 produces means of subsistence of workers (v) and sector 3

produces means of subsistence for capitalists (p). As will be seen below, this division simplifies

substantially the equations to solve.

Let‘s consider the following system:

Cvc 11,

Vvc 22 ,

Pvc 33 , .

16

Co

V

c1

v1

v2

c2

Sector 1

Sector 2

Ce

Ve

Sector 3Po

v3

c3

C

Vo

Figure 2: Diagram for Three Sector Reproduction

For simple circulation of commodities, functions of production are

32111, cccvc

32122 , vvvvc

32133 , pppvc .

Equations for prices of production, choosing worker‘s means of subsistence (v) as money, are

Crgkgvcr cvcv 11 111

Vgkgvcrcv 11 222

Prgkgvcr pvcv 11 333 ,

where cvr and pvr are, respectively, the price of means of production and the price of

capitalist‘s means of subsistence, and ki is the cost-price of commodity i . Assuming that all

constant capital circulates in the considered period, cost-price equals the whole capital utilised

for production of commodity i .

The equation for cvr is

0112

2

2 vrcWvrWc cvcv

whose solution is

17

2

12

2

211

2

4

Wc

vWcWvcWvcrcv

,

where VCW .

On the other hand

11

Wr

Pr

V

Cr

V

Pr

K

Prg

cv

pv

cv

pvpv

,

where K is total cost-price, in this case the equivalent of total social capital utilised in the

production process.

Note that P’ = P/V is not the rate of surplus-value, but the ratio between physical volumes of

commodities p and v, whose dimensions are [up/uv]. The rate of surplus-value, p’, on the other

hand, is a dimensionless quantity that, in general, relates the values of those masses of

commodities:

3

3

2

2

1

1

vv

pv

vv

pv

vv

pv

Vv

Pvp

v

p

v

p

v

p

v

p .

From the equation for sector 3

K

Pr

r

g

r

P

k

pv

pvpv

11

3

from where we obtain

21

3

kk

k

P

Krpv

And, consequently,

21

3

kk

kg

.

It is also posible to obtain pvr from the expression

2

333

1

1

kP

k

g

VP

k

P

gkrpv

.

4.5 EXTENSION TO REPRODUCTION ON AN EXTENDED SCALE

Let‘s consider the same three sector system:

18

Cvc 11,

Vvc 22 ,

Pvc 33 , .

This time, the equations for prices of production, choosing worker‘s means of subsistence (v) as

money, are

ocvcv Crgvcr 111

ocv Vgvcr 122

opvcv Prgvcr 133 ,

where Co = C + Ce , Vo = V + Ve , y Po = p’ V + Pe .

Ce, Ve y Pe are surplus product (excedent) in each sector, positive o negative, compared to

simple reproduction.

From the above equations, it results

111332211

vcr

Pr

vcr

V

vcr

Crg

cv

opv

cv

o

cv

ocv

1

VCr

PrVCrg

cv

opvoocv

VCr

PrVCr

Wr

prg

cv

epveecv

cv

pv

1 .

The equation for cvr is

0112

2

2 vVrcVvCrcC ocvoocvo

Whose solution is

2

12

2

2121

2

4

cC

vVcCvCcVvCcVr

o

oooooo

cv

.

On the other hand it results

2

3

22

33

kP

kV

vcrP

vcrVr

o

o

cvo

cvopv

.

It follows from the above equations that the average rate of profit does not depend on sector 3.

This is because commodity 3 is not a factor of production in any sector and also there is no

proportionality with the production of any sector.

19

To state this result in a more general way, we can divide the production system in basic and

non basic sectors, as proposed by Sraffa. Basic sector comprises all commodities which impact,

directly or indirectly, the value of labour-power or, what is the same, the value of means of

subsistence of workers. Thus, in general, the average rate of profit depends only on productive

subsystem consisting of the basic sector.

Co

V

c1

v1

v2

c2

Sector 1

Sector 2

Ce

Ve

Sector 3Po

v3

c3

C

Vo

Basic Sector

Non Basic Sector

Figure 3: Diagram for Reproduction showing Basic and Non Basic Sectors

Restricting the system to the level of simple reproduction, production of the basic sector is

determined by production of non basic sector, in the sense that total production of surplus-

value must exactly match production of capitalist‘s means of subsistence.

5. Comparison to Other Transformation Methods

Usually authors start from a three sector reproduction system expressed in values, with no

reference to physical quantities involved. However, since physical units used to measure

volumes of each commodity are arbitrary, it is possible to choose units so that unit values are

all equal to 1 (one). Thus, numbers in the schemes represent also physical quantities of each

commodity, allowing us to apply our method of calculation of prices of production, as set out in

the previous sections.

In a three sector system, we will use uc, uv and up as the units for the commodities of sectors 1,

2 and 3, respectively. So, unit values of commodities are then, vc = 1 [h/uc], vv = 1 [h/uv] and vp

= 1 [h/up]. A direct consequence of this election is the numerical equality between P’ and p’.

20

PV

P

Vv

Pvp

v

p (adimensional).

To transform from values to prices some authors define proportionality factors between price

and value for each commodity, usually called x, y and z, for commodities of sectors 1, 2 and 3,

respectively. Thus, for example, x expresses the ratio between price of a unit of means of

production (c) and its value. The unit of means of production is uc and the value of this unit is

1 hour; the price of this unit can be expressed in terms of commodity 2, worker‘s means of

subsistence (v), or in terms of commodity 3 (p), capitalist‘s means of subsistence.

In the first case:

h

ucuvrx cv

1

where rcv is the price of commodity c in terms of commodity v.

In the second case:

h

ucuprx

cp

1

where rcp is the price of commodity c in terms of commodity p.

In the same way,

h

uvupr

h

uvucry

vpvc

11

h

upuvr

h

upucrz

pvpc

11 .

As commodity v is used preferably as money in our method, to compare our prices with results

from another methods of transformation we use the following relations

ucuvrr

r

y

xcv

vp

cp

upuvrr

r

y

zpv

vc

pc .

5.1 COMPARISON TO BORTKIEWICZ METHOD

Bortkiewicz used the following example

c1 = 225 [uc] v1 = 90 [uv] C = 375 [uc]

c2 = 100 [uc] v2 = 120 [uv] V = 300 [uv]

c3 = 50 [uc] v3 = 90 [uv] P = 200 [up]

21

From this data it follows that: p’ = 66.7% and W = 1.25.

Using our method, we obtain the following results:

rcv = 1.2 [uv/uc]

rpv = 0.9375 [uv/up]

g’ = 25% .

Expressing the amounts of the original scheme in units of money-commodity, commodity 2 or

commodity v, we get

c1 = 270 [uv] v1 = 90 [uv] g1 = 90 [uv] C = 450 [uv]

c2 = 120 [uv] v2 = 120 [uv] g2 = 60 [uv] V = 300 [uv]

c3 = 60 [uv] v3 = 90 [uv] g3 = 37.5 [uv] P = 187.5 [uv]

Bortkiewicz fixes factor z = 1, and using his method calculates factors x = 32/25 and y = 16/15.

His scheme of prices is

c1 = 288 [up] v1 = 96 [up] g1 = 96 [up] C = 480 [up]

c2 = 128 [up] v2 = 128 [up] g2 = 64 [up] V = 320 [up]

c3 = 64 [up] v3 = 96 [up] g3 = 40 [up] P = 200 [up]

Both solutions are equivalent, because

x/y = 6/5 = rcv

z/y = 15/16 = rpv

= 1 - = 1/4 = g’ .

In conclusion, prices calculated by Bortkiewicz are equivalent to prices calculated by us, with

the only difference being that Bortkiewicz‘s prices are expressed in units of commodity 3.

5.2 COMPARISON TO CARCANHOLO METHOD

Carcanholo used the following example

c1 = 300 [uc] v1 = 100 [uv] C = 500 [uc]

c2 = 80 [uc] v2 = 100 [uv] V = 280 [uv]

c3 = 120 [uc] v3 = 80 [uv] P = 280 [up]

from where: p’ = 100%.

Using our method of calculation we get the following results:

rcv = 1.363 [uv/uc]

rpv = 1.165 [uv/up]

g’ = 33.9% .

22

Expressing the amounts of the original scheme in units of money-commodity, commodity 2, we

get the following system

c1 = 409 [uv] v1 = 100 [uv] g1 = 173 [uv] C = 682 [uv]

c2 = 109 [uv] v2 = 100 [uv] g2 = 71 [uv] V = 280 [uv]

c3 = 163 [uv] v3 = 80 [uv] g3 = 83 [uv] P = 326 [uv]

Carcanholo calculates directly all prices for his system, as the following

c1 = 336.6 [uv] v1 = 82.3 [uv] g1 = 142.2 [uv] C = 561.1 [uv]

c2 = 89.8 [uv] v2 = 82.3 [uv] g2 = 58.3 [uv] V = 230.4 [uv]

c3 = 134.7 [uv] v3 = 65.8 [uv] g3 = 68 [uv] P = 268.5 [uv]

from where it results g’ = 33.9%.

Both systems are proportionally equivalent, and the average rate of profit calculated with both

methods is the same. Again, the difference between the two systems lies in the units used for

each one. The units used by Carcanholo, as well as those used by Meek (see below), are a result

of his postulate of invariance, which is the same in Winternitz case (see below).

5.3 COMPARISON TO WINTERNITZ METHOD

Winternitz used a three sectors system, without restrictions on the scale of reproduction. The

system is described by the following relations, in terms of values:

I. c1 + v1 + s1 = a1

II. c2 + v2 + s2 = a2

III. c3 + v3 + s3 = a3

In terms of prices, the distribution is modified as follows:

I. c1x + v1y + s1 = a1x

II. c2x + v2y + s2 = a2y

III. c3x + v3y + s3 = a3z

where x, y and z are price/value relations for each sector.

As the rate of profit is the same in all three sectors, it follows that

yvxc

ya

yvxc

xap

22

2

11

11

,

from where

21

2121

2

21122112

2

4

ca

cvaavacavaca

y

xm

,

and the average rate of profit can be determined as

23

111

1

vmc

map .

To determine prices of production Winternitz adds an equation which equals the sum of prices

to the sum of values, i.e.

aaaazayaxa 321321 .

Winternitz does not use a numerical example; however, comparing their equations deduced for

the more general case with the equivalents in our method, the correspondence can be observed

immediately. The final results for what Winternitz call prices of production will differ

necessarily from ours, by definition, and also because of the additional equation mentioned

above.

5.4 COMPARISON TO MEEK METHOD

Meek used the following example

c1 = 3 [uc] v1 = 4 [uv] Co = 11 [uc]

c2 = 18 [uc] v2 = 15 [uv] Vo = 48 [uv]

c3 = 9 [uc] v3 = 6 [uv] Po = 21 [up]

with p’ = 100% .

Using our method of calculation we get the following results:

rcv = 0.933 [uv/uc]

rpv = 1.035 [uv/up]

g’ = 50,95% .

Expressing the amounts of the original scheme in units of money-commodity, commodity 2 or

commodity v, we get

c1 = 2.8 [uv] v1 = 4 [uv] g1 = 3.46 [uv] Co = 10.26 [uv]

c2 = 16.8 [uv] v2 = 15 [uv] g2 = 16.2 [uv] Vo = 48 [uv]

c3 = 8.4 [uv] v3 = 6 [uv] g3 = 7.34 [uv] Po = 21.74 [uv]

Meek calculates the factors x = 0.864 ; y = 0.9275 ; z = 0.9583 ; and the average rate of profit

r =50.8% ;

And his prices system results

c1 = 2.592 [uv] v1 = 3.710 [uv] g1 = 3.202 [uv] Co = 9.504 [uv]

c2 = 15.552 [uv] v2 = 13.911 [uv] g2 = 15.052 [uv] Vo = 44.515 [uv]

c3 = 7.776 [uv] v3 = 5.564 [uv] g3 = 6.784 [uv] Po = 20.124 [uv]

Both results are approximately equivalent:

24

cvry

x 93,0

pvry

z 033,1

gr %8,50 .

Despite the minor discrepancies in the numerical results we can say that both calculation

methods give equivalent results for this example. Therefore, our pricing system is proportional

and equivalent to Meek‘s one. However, it is difficult to see what are the units in which Meek‘s

quantities are expressed9.

Let us remember that Meek reused Winternitz‘s system, replacing its additional equation with

the following

yvyvyv

zayaxa

vvv

aaa

321

321

321

321

.

According to Meek, the equation above expresses in its best form Marx‘s idea by which prices of

production represent a redistribution of total surplus-value in the form of profit. The particular

interpretation of Meek translates to his invariance postulate in the proportion of value or total

price of product and the value or price of total variable capital.

In Meek‘s method is also evident an error common to all previous solutions. This error was

noted by H. D. Dickinson (1956) precisely with regard to the solution proposed by Meek.

Dickinson rightly observes that in the systems of Bortkiewicz, Sweezy, Winternitz and Meek

an additional equation is not required, because the unknown quantities x, y and z are not three

independent variables, but what need to be determined are only their proportions: x:y:z.

Additionally, points out Dickinson, values and prices are magnitudes dimensionally different,

therefore it is incoherent to compare sums of prices to sums of values. What can only be done is

to compare ratios between prices to ratios between values, as only Meek does.

6. Conclusions

1. Our method has been demonstrated useful to calculate values or prices based solely on

technical relations of production, and strictly according to definitions from Marx‘s theory

of value.

2. The simplicity of the method demonstrates that there are no inconsistencies in Marx‘s

theory, and that the difficulties found to solve the ―transformation problem‖ reside only in

the lack of a clear way to state it mathematically.

3. To illustrate the latter, a comparison with many other methods of transformation shows

that all they are roughly the same.

9 Those units result from Meek’s postulate of invariance, which cannot be interpreted easily for this purpose,

as can be in Bortkiewicz’s case.

25

Concerning the more philosophical implications of the transformation problem:

4. This analysis does not demonstrate by itself the utility of the theory of value in regard to

the calculation of prices. As is argued by neoricardians, it is possible to go directly from the

physical magnitudes to the calculation of prices (Sraffa, Steedman).

5. This analysis does not refute the objections of Samuelson (1957, 1967, 1970, 1971, 1973,

1974), according to which a "transformation" of values into prices can be reduced to a

simple process of elimination of values and replacement by prices, prices obtained

regardless of values. Moreover, this criticism is demonstrated right, regardless of the

conclusions the author derives from it.

6. It is our opinion that the power of Marx‘s theory of value and surplus-value resides in the

way they can directly represent the mechanisms of exploitation, of technical progress and

their relationship in accumulation. A proven method to relate these more abstract

categories to the observable ones, like prices and profit, is essential to any further

development of the theory. This is the only way the theory can prove useful.

We like to think at this problem with analogy to Ptolemaic and Copernican models of solar

system. The first takes account of direct observations, concrete phenomena, but is awkward

and difficult for calculations. The second is based on abstraction, not directly observed

movement, but is quite simple an easy to calculate based on it. So, its prediction power is

enormously better.

Now we know that Copernicus model is not ―right‖, in opposition to a ―wrong‖ one from

Ptolomeus, because there is no absolute framework of reference (relativity). But we surely

know how useful the Copernican model was to Kepler, Newton and so on.

7. References

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Interpretation), Journal of Economic Literature, vol 12, pp 51—62.

Badhuri, A (1969) On the Significance of Recent Controversies on Capital Theory: A Marxian

View, Economic Journal, vol 79, pp 532—9.

Böhm-Bawerk, E. (1896) Una Contradicción No Resuelta en el Sistema Económico Marxista,

Estudios Públicos, N° 10, 1983, pp 167—211.

Bortkiewicz, L. von (1907) On the Correction of Marx‘s Fundamental Theoretical Construction

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