Extensive Rent And Labor Values

1.0 Introduction

Do scarce natural resources provide additional difficultes for modern reconstructions of classical and Marxian theories of value? After all land can be sold or rented, and labor cannot produce more land. (I put aside Holland.)

This post presents an exposition of the theory of extensive rent, a start on examining possible difficulties. This type of rent provides the least dificulties, as I understand it, for such modern reconstructions. As usual, I present an example, close to the minimal complexity, needed to make my points. The model can obviously be generalized to include many more produced industrial commodities; many more types of agricultural commodities; and many more types of land, each specialized to support the production of one kind of agricultural commodity.

2.0 Technology

Table 1 specifies the technology for this example. Each column defines the coefficients of production for a process. For example, the only iron-producing process requires a_0,1 person-years of labor, a_1,1 tons of iron, and a_2,1 bushels of corn as inputs for every ton iron produced. I assume that each process requires a year to complete and exhibits constant returns to scale. The corn-producing processes each have an upper limit on how much corn they can produce.

Table 1: A Technology

	Iron Industry	Corn Industry
	Process a	Process b	Process c
Labor	a0,1	a0,2	a0,3
Land, Type 1	c1,1 = 0	c1,2 > 0	c1,3 = 0
Land, Type 2	c2,1 = 0	c2,2 = 0	c2,3 > 0
Iron	a1,1	a1,2	a1,3
Corn	a2,1	a2,2	a2,3
OUTPUTS	1 ton iron	1 bushel corn	1 bushel corn

I assume two types of land exist, distinguished by the processes that can be operated on them. A single corn-producing process can be operated on each type of land. Only a certain number of acres of each type of land exists. Each corn-producing process leaves the land unchanged at the end of operating the process. The given quantities of land limit how much corn can be produced. This model cannot accomodate a positive steady-state rate of growth without technical progress.

A full specification for this model should include requirements for use. I assume that the net output must be such that both types of land are farmed, but only one type is fully farmed. Two techniques for production exist, as shown in Table 2. All three processes are operated in each technique, but only one type of land is fully used.

Table 2: Specification of Techniques

Technique	Type 1 Land	Type 2 Land
Alpha	Partially farmed	Fully farmed
Beta	Fully farmed	Partially farmed

3.0 Parameters and Variables

I have already implicitly defined certain parameters above. Table 3 lists certain parameters I use in this model. Table 4 lists variables that I need. Some assumptions are imposed on the matrices A_α and A_β:

• All produced commodities are basic. Iron and corn enter directly or indirectly into the production of both commodities.
• The technology expressed by these matrices is productive. Each matrix satisfies the Hawkins-Simon condition.

Table 3: Selected Parameters

Symbol	Definition
a0, α	Two-element row vector consisting of first two labor coefficients.
a0, β	Two-element row vector consisting of first and third labor coefficients.
Aα	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and second processes.
Aβ	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and third processes.
d	Two-element column vector consisting of iron and corn quantities in the numeraire.

Table 4: Variables

Symbol	Definition
vα	2-element row vector of labor values when type 1 land is free.
vβ	2-element row vector of labor values when type 2 land is free.
p	2-element row vector of prices of unit quantities of iron and corn.
p1	The price of iron, in numeraire units per ton. The first element of p.
p2	The price of corn, in numeraire units per bushel. The second element of p.
rho1	The rent of type 1 land, in numeraire units per acre.
rho2	The rent of type 2 land, in numeraire units per acre.
w	The wage, in numeraire units per person-year.
r	The rate of profits.

4.0 Labor Values

Given the technique in use, how much additional labor would be employed throughout the economy if the net output was such that one additional unit of iron were produced? This is the labor value of iron, and it easily calculated in the theory. The answer to the same question for corn is its labor value.

Suppose type 1 land is free. Then labor values are:

v_α = a_{0, α} (I - A_α)^-1

Labor values, when type 2 land is free, are the corresponding Leontief employment multipliers for the Beta technique. Variations in net output require varying the amount of the land farmed on the type of land that is not fully farmed.

5.0 Prices of Production

With market prices, some operated processes will be obtaining a higher rate of profits than average, and some will be obtaining a lower rate. These variations in the profit rates are perhaps a signal to capitalists that they should disinvest in some industries or processes and increase investment in others. Models of cross-dual dynamics and other models explore these disequilibria.

Prices of production are such that these signals are absent. All operated processes obtain the same rate of profits. I assume profits, rents, and wages are paid out of the surplus product at the end of the year. The following three equations express the condition that all processes obtain the same rate of profits:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

The next equation expresses the condition that the price of the numeraire is unity:

p d = 1

Finally, one of the rents must be zero:

rho₁ rho₂ = 0

The last equation is a defining feature of the theory of extensive rent.

Suppose one of the types of land is rent-free. For deiniteness, let type 1 land be only partially farmed. Then the first four equations are in terms of five variables (p₁, p₂, rho₂, w, r). Just as in the case with only circulating capital, prices of production are specified up to one degree of freedom. In classical political economy, the wage is take as given.

6.0 Choice of Technique

Suppose the wage is non-negative and does not exceed a maximum defined by the technology. The system of equations for prices of production has two solutions. Each solution has the rent on one type of land set to zero. The cost-minimizing technique is the one in which the rent on the other land is positive. If, for a technique, the rent on a type of land is negative, that technique will not be adopted by capitalists. At a switch point, the rents on both types of land are zero.

But the analysis of the choice of technique can be expressed in terms of wage curves. Suppose rents were zero. Consider the first two equations in the system of equations for the prices of production and the equation setting the price of the numeraire to unity. These equations yield a function in which the wage decreases with an increase in the rate of profits. Similarly, the first and third equations yield another decreasing wage curve.

In the case of circulating capital alone, the cost-minimizing technique is found by the wage frontier formed out of the outer envelope of these wage curves. At a given wage, the cost-minimizing technique maximizes the wage.

In this example of extensive rent, the cost-minimizing technique is found by the wage frontier formed out of the inner envelope of the wage curves.

In either case, the appropriate wage frontier shows that a lower rate of profits is associated with a higher wage and vice versa. The maximum wage occurs when the rate of profits is zero. The maximum rate of profits arises when the wage is zero.

7.0 Special Cases

Which land is free and which land pays a rent depends on either the wage or the rate of profits, whichever is taken as exogenous in the system of prices of production. At any rate, a wage frontier exists in which the wage is higher the smaller the rate of profits. This frontier is not the outer frontier of the wage curves for the technique.

Without loss of generality, suppose the Alpha technique is cost-minimizing. Type 1 land is not fully farmed and pays no rent. Then labor values are defined, based on the iron-producing process and the process on type 1 land.

Consider the special case in which a_{0, α} is an eigenvector corresponding to the maximum eigenvector for A_α. Then relative prices of production are equal to relative labor values.

On the other hand, suppose that the numeraire is the standard commodity, as found from a_{0, α} and A_α. Suppose only the standard commodity is produced. In this case, only the process on the rent-free land would be used, in contradiction to the analysis of the choice of technique. And suppose the wage is paid out in the form of the standard commodity. Then the following hold:

• The labor value of gross output is equal to total gross output, evaluated at prices of production.
• The labor value of net output is equal to net output, evaluated at prices of production.
• The labor value of the proportion of the standard commodity paid out in wages is equal to wage goods, evaluated at prices of production.

This special case seems especially forced in the case of extensive rent. Is some reformulation available in which surplus value can be treated as the sum of profits and rent?

I do not address the use of labor values in Marx's account of exploitation, Marx-biased technical change, and so on. The special cases in which the labor theory of value hold make obvious that, for a given technology, a higher rate of profits require a lower wage. And this wage frontier continues to hold in models of extensive rent.

8.0 Conclusion

The inclusion of natural resources, insofar as they can be modeled by extensive rent, does not seem to pose any additional issues for modern formulations of classical and Marxian political economy. It does highlight some issues that arise in models with circulating capital.

Labor values can be calculated for all produced commodities, given the technique in use. They are calculated from the marginal land that receives no rent. But suppose that a choice of technique exists. Then, an analysis at the level of prices of production must be prior to the calculation of labor values. The theory of extensive rent highlights this issue.

As Ricardo and Marx noted, prices of production are generally not proportional to labor values. They are equal in the special case, in which all industries have equal organic compositions of capital, in both models of circulating capital and of extensive rent. In the latter case, the organic composition of capital is found for agriculture from no-rent lands partially farmed.

A commodity of average organic composition is picked out in both models. Total labor values and the labor value of wages are equal to the corresponding aggregates in the system of prices of production when this average commodity is used as numeraire and is produced. These invariants, though, have to restricted to the production of the numeraire with the iron-producing process and the process on no-rent land. It is not clear to me that Marx thought his invariants held in his chapters on rent, given their location towards the end of volume 3 of Capital.

Obviously, these observations on natural resources and rent are just a start. They do seem to match what Ricardo was about in the second chapter of his Principles. The analysis of the choice of technique can be thought of, somewhat, as a critique of Ricardo.

At any rate, prices of production are well-defined in models of extensive rent. And they can be used in an analysis of the choice of technique. As usual, I present the analysis with no mention of utility maximization, preferences, or tastes.

Machinery And The Honesty Of David Ricardo

Consider the introduction of new, advanced machinery into a capitalist economy. This will raise productivity and be good for the population as a whole. It will displace workers, at least temporarily, who were previously making the product of the machine with handicraft production or now obsolete machines with lower productivity. But the production of the machines requires workers too. So, ignoring short-run frictions, will the workers not remain as well off?

David Ricardo believed something like this at one point in his life. But he had come to the opposite conclusion when he revised his Principles for the third edition. And he was forthright in saying so. Some of the displaced workers will be more or less permanently unemployed. By the way, this was not a matter of coming to agree with Malthus on a point about effectual demand.

Ricardo's change of mind was not some abstract academic view. This was a time in England shortly after the Luddites were at their peak. The Luddites had been rioting and destroying new machinery being introduced by industrialists. Ricardo's friend, J. R. McCulloch writes to Ricardo, and he immediately saw the potential of these changes (Ricardo, Works, volume 8, pp. 381-386):

Edinburgh 5 June 1821

My dear Sir

I have to apologise for being so long in returning you my best thanks for the valuable present of the third Edition of your great work - I congratulate you on its success - It is the best proof that can be given of the growing attention now paid to this important science; and it must have a powerful influence in furthering the dissemination of sound principles -

At the same time I must say (and I say it with that regret which I ever must feel in differing widely from one to whom I shall always be proud to look up as to my master) that in my humble opinion the Chapter on Machinery in this Edition is a very material deduction from the value of the work... ...Excess of candour has in this instance occasioned your doing a very serious injury to your favourite science - It was certainly proper that you should have renounced your previous opinions the moment you were satisfied of their fallacy; but this may be done in various ways, and I do not think it was at all necessary for you to make a formal recantation - our object never has been and never can be any other than to endeavour to promote the real interests of the science...

However the manner in which you have published your change of opinion is of comparatively little consequence - It is what I consider the extreme erroneousness of the principles to which you have incautiously lent the sanction of your name that has excited my principal regret - It is impossible to fritter away your argument by fencing it about with conditions - If it is good for any thing at all it is conclusive against all employment of machinery - It is not with greater or less gross or net produce that we have the smallest concern in considering this question; but simply whether does machinery produce commodities cheaper or not? If it does not produce them cheaper it will not be erected, and if it does produce them cheaper its erection must be profitable to every class of persons - The example which you have given does not, as far as I can perceive, by any means warrant a single one of the extraordinary conclusions you have drawn from it - You have not said whether the machine worth £7,500 is to last one, ten, or one hundred years -

...Your argument is to be sure hypothetical; but the hypothesis will be thrown aside, and all those who raise a yell against the extension of machinery, and ascribe to it that misery which is a mere necessary consequence of the oppressiveness of taxation, and of the restraints on commerce will fortify themselves by your authority! If your reasoning and that of Mr. Malthus be well founded, the laws against the Luddites are a disgrace to the Statute book -

Let me beg of you to reconsider this subject - A heresy on a mere doctrinal point is of no moment; but really I could not recommend to any of my friends to bestow the least attention on the study of this science, if I was satisfied that it remained yet to be settled whether the reducing of the price of commodities was advantageous or not - Truly if we are not got this length, our disputes about profits and our other remote conclusions ought to afford infinite amusement to the scoffers - But, I, at least, am not in this quandary - I will take my stand with the Mr. Burke of the American war not with the Mr. Burke of the French revolution - with the Mr. Ricardo of the first not of the third edition - Were there nothing else to allege on the subject I should be perfectly satisfied with what I consider the inherent fallacy involved in all the arguments which have been advanced against machinery...

Were I not aware that in all your speculations you are actuated solely by a desire to contribute to the improvement of the science, I should not have presumed to address to you this hasty and ill-digested letter - But I am satisfied that opinions dictated equally by a regard to the interests of the science, and coming from one who is not the least sincere of your admirers, though they may seem erroneous, will claim and meet with your attentive perusal - I am with the greatest regard and esteem

ever faithfully yours

J. R. McCulloch

Those are extracts from a long letter. I have left out many details of the argument.

Ricardo's friendship with Malthus is another testament to his personality. They continually argued that the other was wrong on political economy. Ricardo would lend out his notes on one of Malthus' books (Works, volume 2) to his friends. He did not try to publish them, for they did not make much sense without the text of Malthus' Principles of Political Economy. Malthus explained to Ricardo that he was mistaken, both in person and through a long interchange of letters. It was Malthus' insistence that even in agriculture, no product and its capital advances consist of the same mixture of commodities that induced Ricardo, as I understand it, to take up the labor theory of value.

Anyways, despite these persistent disagreements, Ricardo continued as a life-long friend of Malthus. I do not think I have that temperament.

Edit: Reference as suggested in coments:

• Paul A. Samuelson. 1989. Ricardo was right! Scandinavian Journal of Economics 91(1): 47-62.

Did Marginalism Become Accepted As A Reaction to Marxism?

I take it for granted that marginalism became accepted partly because Marx had used the best in classical political economy in his account of why socialism would and should transcend capitalism. This post presents some who have argued for or asserted the same.

I start by summarizing an argument from Antonia Campus. Campus argues that the marginalists in the 1870s did not have an accepted theory of production, cost, and price. Only in the 1890s did the marginal productivity theory of distribution become accepted. Now that that theory has been demolished, in the 1960s, we are back into the confusion of the 1870s, with every person his own capital theorist.

"With the publication in 1867 of Volume 1 of Capital, Ricardo's theory of distribution and value had in fact reappeared, not in the conciliatory form of J. S. Mill's Principles, but in the dangerous one which had been typical of this theory in the decade following Ricardo's death. According to Böhm-Bawerk, this theory constituted for the Germany of 1884 'the focal point about which attack and defence rally in the war in which the issue is the system under which human society shall be organized'.

On account of the impasse in which the theory of distribution was, and the ensuing chaos in economic theory, there was the danger that Ricardo's theory of distribution - in the most advanced elaboration it had found in Volume I of Marx's Capital - might fill the gap, and become even in Britain the 'focal point' in the struggle for and against the established social order. This danger must have seemed not too abstract, in the climate of Socialist revival of the 1880s, and especially after the foundation of the Social Democratic Foundation in 1881 and the Fabian Society in 1883." -- Antonia Campus. 1987. Notes on cost and price: Malthus and the marginal theory. Political Economy: Studies in the Surplus Approach 3(1): 11.

Here is Luigi Pasinetti saying something along the same lines:

"What turned out to be so devastating was the social impact of [Marx's] writings. The immediate practical effect of Marx's call for a social revolution was to elicit a strong social reaction. The establishment of the Western nations, at the end of the nineteenth century, became scared by Marx's revolutionary call. This by itself explains a lot of the fortune that in academic circles blessed marginalism in the 1870s, whose success was essentially analytical...

...In academic circles, this no doubt represented a radical change, but not in the strict sense of a scientific 'revolution', though some historians of economic thought later hastened to call it so (the 'Marginal Revolution'). Conceptually, it was a 'counter-revolution', an anachronistic achievement, yet a beautiful one, reached with the most sophisticated tools of economic analysis (precisely what the Classical economists had lacked).

At the end of the nineteenth and the beginning of the twentieth century, marginal economic theory led to conclusions which were pleasing to the establishment, especially in terms of a splendid detachment from the hot social issues that were boiling up in the real world, and in terms of arguments that could easily be used for the advocacy of unrestricted laissez-faire policies, supposedly leading, in ideal conditions, to optimal positions..." -- Luigi L. Pasinetti (2007).

Pierangelo Garegnani summarizes some of Sraffa's unpublished notes:

"For the school of 'cost', Sraffa is here referring to Ricardo's 'cost value', influenced, Sraffa says, by the author's 'anti-landlord complex' whereby 'rent not entering cost is disgraced'. The second school, that of 'utility', and with it the conflict between the two, came instead into being, Sraffa continues, when Ricardo’s 'cost' theory was 'taken up by Marx and used as a weapon for the workers'. That provoked by reaction the 'immediate simultaneous success' of the utility-based theory of value of Jevons, Menger and Walras – a theory that, significantly enough was ignored when it made its first appearance in the work of authors such as Dupuit and Gossen, before Marx's work created the need to develop a substitute for labor values." – Pierangelo Garegnani. 2005. On a turning point in Sraffa’s theoretical and interpretative position in the late 1920s. European Journal of the History of Economic Thought 12(3).

Gunnar Myrdal says something similar:

"One point which emerges from our analysis of the classical exchange value and real value theory is that Marx's theory of surplus value is not the result of a 'gross misunderstanding.'...Marx was right in saying that his surplus value theory follows from the classical theory of real value...Moreover, Marx was not the first to draw radical conclusions from it. All pre-Marxist British socialists derived their arguments from Adam Smith and later from Ricardo. Economists did not welcome these inevitable conclusions...He thus touched upon a sore point of economic theory and, probably for this reason, caused so much irritation amongst economists. They often tried not so much to prove him wrong, which would not have been too difficult, as to show that he was an utter fool, a bungler, misguided by those despised German philosophers...The classical theory of value leads inevitably to a rationalist radicalism, if not necessarily in Marx's formulation, at any rate in that direction. For the historian of thought the real puzzle is why the classics did not draw these radical conclusions." -- Gunnar Myrdal, The Political Element in the Development of Economic Theory.

I know of W. J. Ashley from Sraffa's unpublished notes:

The marginal conception of value which this generation owes to Jevons and Menger was clearly enough expounded by Longfield in 1833, but it passed unregarded... It is evident that their inattention was due, not to dissatisfaction with what men like Longfield offered them, but to satisfaction with the apparently sufficient formulae they had already mastered...

...Meanwhile ... the dissemination of the teachings of the so-called 'scientific' socialists - of Lassalle's 'Iron Law of Wages,' and Marx's 'Surplus Value' - disposed conservatively minded thinkers to re-examine that Ricardian teaching to which the Socialists, with so much show of reason, were in the habit of appealing." -- W. J. Ashley (1907).

The idea that marginalism became accepted partly as a reaction to Marxism is an established take from those who have examined the question over more than a century. You do not even need to be a radical to believe it.

Other takes emphasize a reaction to Henry George, Mirowski's account of physics envy, and the extension of Ricardo's theory of rent to all so-called factors of production.

Ian Steedman edited Socialism and Marginalism in Economics: 1870-1930 in 1995. The essays in this book are intelligent, informed, and much more nuanced than this post. Some socialists adopted marginalism. George Bernard Shaw was convinced by Philip Wicksteed,\ and remained a Fabian. Apparently, in Denmark it was not even a controversy. Wicksell was a radical in Sweden, advocating birth control. But on economics he wasn't very socialist. Vladimir Dmitriev and Ladislaus Bortkiewicz interpreted Ricardo and Marx with linear production models, a tradition continued by Robert Remak, John Von Neumann, and Wassily Leontief.

Adam Smith, David Ricardo, And The Labor Theory Of Value

1.0 Introduction

I resolutely am not commenting on unhappy current events.

Smith and Ricardo thought a (simple) LTV was not applicable to capitalism. Prices do not tend to or orbit around labor values. At least that is their claim.

Ricardo had more to say about the LTV.

This argument is not new. Smith confined the LTV to a supposed "early and rude state of society which precedes both the accumulation of stock and the appropriation on land" (WoN, book 1, chapter 6; see also book 1, chapter 8). Ricardo thought this was sloppy reasoning. The LTV does not become nonapplicable merely because of the accumulation of capital and of the division of society into capitalists and workers (Principles, 3rd edition, chapter 1, section III).

2.0 Technology

A simple model of circulating capital can be used to make Ricardo's point. Let a₀ be a row vector of direct labor coefficients. Let A be the Leontief input-output matrix. An element of a₀ and the corresponding column of A specify the labor time and the capital goods needed to operate a process to produce one unit of the output of that industry. The technology satisfies the following common assumptions:

• Some labor is needed to operate every process in each industry.
• Constant returns to scale prevail.
• Each commodity enters, directly or indirectly, into the production of every commodity. Iron, for example enters indirectly into the production of automobiles if iron is needed to produce steel and steel is needed to produce cars.
• The technology is productive. For some level of operation of the processes for each industry, some commodities are left over after reproducing the capital goods used in producing them.

Now for the unusual special case. Let λ be the largest eigenvalue of the Leontief matrix. This eigenvalue is also known as the Perron-Frobenius root of the Leontief matrix. Assume that the vector of direct labor coefficients is a corresponding left-hand eigenvector:

a₀ A = λ a₀ (Display 1)

3.0 Labor Values

Let v be the row vector of labor values. By definition, labor values satisfy the system of equations in Display 2:

v A + a₀ = v (Disp. 2)

The total labor to produce a commodity is the sum of the labor values of the capital goods used in that industry and the direct labor coefficient.

Under the special case assumption, labor values are a multiple of direct labor coefficients:

v = (1/(1 - λ)) a₀ (Disp. 3)

One can check this solution by merely plugging it into the solution in Display 2:

(1/(1 - λ)) a₀ A + a₀ = (λ/(1 - λ)) a₀ + a₀ = (1/(1 - λ)) a₀ (Disp. 4)

Since the above is a one-line proof, I thought I would include it. Labor values are also an eigenvector of the Leontief matrix.

4.0 Prices

Under the usual assumptions, the row vector p of prices satisfies the system of equations in Display 5:

p A (1 + r) + w a₀ = p (Disp. 5)

The scalar w is the wage, and r is the rate of profits.

Let R be the maximum rate of profits, obtained when the wage is zero, and the workers live on air. For the special case, the solution to the price system is quite simple:

R = (1/λ) - 1 = (1 - λ)/λ (Disp. 6)

r = R (1 - w) (Disp. 7)

p = v (Disp. 8)

One can check this solution by plugging it into the system of equations in Display 5.

So Ricardo was correct. The LTV could apply to capitalism under a special case. The failure of the LTV to apply in general is because those special case conditions cannot be expected to arise.

5.0 Conclusion

Ricardo had a point. As any ent would tell you, Smith was too hasty. The simple conflict between the wage and the rate of profits in Display 7 applies more generally than the above special case. The LTV, even when it is not valid, points to theories of the returns to capital.

I suppose reconstructing arguments between Ricardo and Smith with modern economics may conflict with how some historians do history. Even saying that Smith was analyzing capitalism could be considered an anachronism.

Adam Smith On A Labor Theory Of Value

The following are the first three paragraphs of the introduction to the Wealth of Nations:

"The annual labor of every nation is the fund which originally supplies it with all the necessaries and conveniencies of life which it annually consumes, and which consist always either in the immediate produce of that labour, or in what is purchased with that produce from other nations.

According, therefore, as this produce, or what is purchased with it, bears a greater or smaller proportion to the number of those who are to consume it, the nation will be better or worse supplied with all the necessaries and conveniencies for which it has occasion.

But this proportion must in every nation be regulated by two different circumstances: first, by the skill, dexterity, and judgment with which its labour is generally applied; and, secondly, by the proportion between the number of those who are employed in useful labour, and that of those who are not so employed. Whatever be the soil, climate, or extent of territory of any particular nation, the abundance or scantiness of its annual supply must, in that particular situation, depend upon those two circumstances." -- Adam Smith

When people talk about Adam Smith having a labor theory of value, they are not normally referencing the above. They are usually thinking of his few pages on a supposed "early and rude state of society which precedes both the accumulation of stock and the appropriation on land (book 1, chapter 6; see also book 1, chapter 8). Or they are thinking of his use of labor commanded as a measure of welfare (book 1, chapter 5).

But consider the above quotation. One can break down the annual labor of a nation in several ways. One can look at the proportion of the labor of the country which is needed to produce the necessaries for the entire labour employed. The remaining labour produces commodities that make up profits, interest, rent, payments to unproductive laborers, and so on.

This surplus can be consumed as necessaries and conveniences of life. One might call the latter luxuries. Or it might be used for accumulation. If the size of the labor force is to grow and the consumption of the individual worker is not to decrease, some of it must be used for accumulation. Smith thought that as the market increased, so would the division of labor. In a virtuous cycle, a greater proportion would be available for accumulation.

So in the very beginning of his most well-known book, Adam Smith points to the question of the size, distribution, and use of the surplus.

A Derivation Of Prices Of Production With Linear Programming

1.0 Introduction

This post illustrates a derivation of prices of production, based on certain properties of duality theory as applied to linear programming. I strive to be more concise and elementary than previous expositions. This exposition is based on John Roemer's Reproducible Solution (Analytical Foundations of Marxian Economic Theory, Cambridge University Press, 1981).

You will find no utility maximization or supply and demand functions below. I have no need for such hypotheses. Nevertheless, one can read this derivation as consistent with marginalism.

2.0 Technology and Endowments

Two commodities, iron and corn, are produced in this example. Managers of firms know a technology consisting of the processes defined in Table 1. Each column shows the inputs and outputs for a process operated at a unit level. All processes take a year to complete and provide their output at the end of the year. Each process exhibits constant returns to scale (CRS). For convenience, assume all coefficients of production defined in the table are positive. The inputs to production are totally used up by operating these processes.

Table 1: The Technology

INPUTS	Processes
	Iron Industry		Corn Industry
	a	b	c	d
Labor	a0,1(a)	a0,1(b)	a0,2(c)	a0,2(d)
Iron	a1,1(a)	a1,1(b)	a1,2(c)	a1,2(d)
Corn	a2,1(a)	a2,1(b)	a2,2(c)	a2,2(d)
OUTPUT	1 ton iron	1 ton iron	1 bushel corn	1 bushel corn

The endowments of iron and corn in the firm's inventory at the start of the year are also given parameters. Table 2 lists the remaining variables in this post. Presumably, the endowments are from production during the previous year. They are unlikely to be in the proportions needed to continue production. For example, if the managers of a firm decide to specialize in producing corn, they will have no endowments of iron.

Table 2: Parameters and Variables

Additional Parameters
ω1	Endowment of iron (in tons) for the firm.
ω2	Endowment of corn (in bushels) for the firm.
Parameters taken as given by managers of the firm
p	Price of iron (in bushels per ton).
w	The wage (in bushels per person-year).
Decision Variables
q1(a)	Quantity of iron (in tons) produced by the first process.
q1(b)	Quantity of iron (in tons) produced by the second process.
q2(c)	Quantity of corn (in bushels) produced by the third process.
q2(d)	Quantity of corn (in bushels) produced by the fourth process.
r	The rate of profits.

3.0 The Primal Linear Program

Managers of firms choose the quantities to produce with each process to maximize the increment z in value, subject to the constraint that they can buy the needed inputs at the start of the year out of the revenue obtained by selling their endowment. The objective function for the primal linear program is:

z = {p - [p a_1,1(a) + a_2,1(a) + w a_0,1(a)]} q₁(a)

+ {p - [p a_1,1(b) + a_2,1(b) + w a_0,1(b)]} q₁(b)

+ {1 - [p a_1,2(c) + a_2,2(c) + w a_0,2(c)]} q₂(c)

+ {1 - [p a_1,2(d) + a_2,2(d) + w a_0,2(d)]} q₂(d)

The quantities in the square brackets above are the costs of operating each process at a unit level. A bushel corn is taken as numeraire. The quantities in the squiggly brackets are the net revenues (also known as accounting profits) of operating each process at a unit level. Scaling these net revenues by the level of operation for each process results in the total accounting profit for the firm.

The constraints are:

[p a_1,1(a) + a_2,1(a)] q₁(a)

+ [p a_1,1(b) + a_2,1(b)] q₁(b)

+ [p a_1,2(c) + a_2,2(c)] q₂(c)

+ [p a_1,2(d) + a_2,2(d)] q₂(d) ≤ p ω₁ + ω₂

q₁(a) ≥ 0, q₁(b) ≥ 0, q₂(c) ≥ 0, q₂(d) ≥ 0

The statement of the constraints is based on the assumption that wages are paid at the end of the year, not advanced at the start.

4.0 The Dual Linear Program

The above linear program has a dual. In the dual, the rate of profits r is chosen to minimize the charge y on endowments:

y = (p ω₁ + ω₂) r

Such that:

[p a_1,1(a) + a_2,1(a)](1 + r) + w a_0,1(a) ≥ p

[p a_1,1(b) + a_2,1(b)](1 + r) + w a_0,1(b) ≥ p

[p a_1,2(c) + a_2,2(c)](1 + r) + w a_0,2(c) ≥ 1

[p a_1,2(d) + a_2,2(d)](1 + r) + w a_0,2(d) ≥ 1

r ≥ 0

Each constraint in the dual specifies that the revenues obtained from operating a process at the unit level do not exceed the costs, where costs include a charge for the going rate of profits. In other words, no super-normal profits can be obtained.

5.0 Some Observations About Duality

The value of the objective functions are equal in the solutions to the primal and dual LPs. In other words, the increment in value obtained by the decisions of the manager of a firm is charged to the value of the endowment.

Suppose the solution of the primal LP results in some process being operated at a positive level. Then the corresponding constraint in the dual LP is met with equality in its solution. Likewise, if a constraint in the dual is met with inequality, then that process will not be operated in the dual.

If the rate of profits in the solution to the dual is positive, then the constraint in the primal LP will be met with equality. That is, the whole value of the endowment will be used for further production.

6.0 Prices of Production

I introduce a final assumption. The solution to these LPs must be such that the economy can continue. In the context of this exposition, some firms must produce iron, and some must produce corn. Thus, one of the first two constraints in the dual LP must be met with equality. One of next two constraints must also be met with equality.

Consider the case when only one of the processes for producing iron is operated, and the same is true of the processes for producing corn. The dual LP yields a system of two equations in three variables: the price of iron, the wage, and the rate of profits. This system specifies prices of production.

This formulation solves for the choice of the technique, as well as prices of production. It can be generalized to allow for the production of many more commodities and many more processes for producing each commodity. A generalization can allow for heterogeneous labor. Another generalization allows for the production and use of fixed capital, that is, machines that last for many years. For a given wage, prices and the rate of profits drop out of the equations for prices of production for the chosen technique. These prices do not support the parables often told in introductory economics classes with supply and demand. For example, unemployment cannot necessarily be eliminated by lowering the wage and encouraging firms to thereby hire more labor.

7.0 Conclusion

The above illustrates some elements of a theory of value. This is neither a labor theory of value, nor Marx's theory of value. The theory is focused on production and has implications about how labor is allocated among industries, a central concern of Karl Marx.

Precursors Of The Modern Revival Of Classical Political Economy

A revolution occurred in price theory about two-thirds of a century ago. Several scholars independently developed components of this revolution. This post merely lists selected literature. I have previously tried to briefly describe why some of these authors are precursors.

• Ladislaus von Bortkiewicz (1907) On the correction of Marx's fundamental theoretical construction i the third volume of Capital. Translated and reprinted by Sweezy.
• David G. Champernowne (1945-1946) A note on J. V. Neumann's article on "A model of economic equilibrium". Review of Economic Studies 13 (1): 10-18.
• Georg von Charasoff (2010) Das System des Marxismus: Darstellung und Kritik. Berlin: H. Bondy.
• V. K. Dmitriev (1974) Economic Essays on Value, Competition, Utility. English Trans.
• Walter Isard (1951) Interregional and regional input-output analysis: A model of a space economy. Review of Economics and Statistics 33 (4): 318-328.
• Wassily Leontief (1928). The economy as a circluar flow.
• Maurice Potron (2010) The Analysis of Linear Economic Systems: Father Maurice Potron’s Pioneering Works (ed. by Christian Bidard and Guido Erreygers). Routledge.
• Jacob Schwartz (1961). Lectures on the Mathematical Method in Economics. New York: Gordon & Breach.
• Piero Sraffa (1960) Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory Cambridge
• John Von Neumann (1945-1946) A model of economic equilibrium. Review of Economic Studies 13 (1): 1-9.

Some Assertions Of Marx And Some Remarks On The Labor Theory Of Value

1.0 Introduction

I have been reading fools in other parts of the Internet. Hence this post.

2.0 Assertions

Marx says the following (I am least sure of 6):

1. Both sides to an exchange gain. (Capital, volume 1, chapter 5)
2. Nobody, neither consumers, nor workers, nor investors, nor the managers of firms, make decisions on the grounds of the labor time embodied in commodities. (Capital, volume 1, chapter 1, section4)
3. Surplus value (dividends, interest, rent, etc.), in an ideal competitive capitalist economy, is not stolen from the worker. (Capital, volume 1, chapter 7, section 2)
4. Relative prices do not tend towards relative surplus values. (Capital, volume 1, chapter 5, last footnote; Capital, volume 3, part II)
5. After the revolution, workers will not receive equal wages. (Critique of the Gotha Program)
6. After the revolution, the planning authority should not necessarily set prices equal to labor values. (Critique of the Gotha Program, Poverty of Philosophy)?
7. Labor is not the source of all wealth.

3.0 On The Labor Theory Of Value

Consider a capitalist economy. At the start of a year, say, firms own certain plants and goods in process in their inventories. During the year, laborers working with this plant and inputs for energy, lubricants, semi-finished goods, and so on, work up these goods to produce an immense quantity of commodities, the gross product for the year. Firms, who own this gross product, buy and sell some of these commodities among themselves or retain some to replace used up plant and inventories. The renainder is the net product. This net product consist of goods in the form that can be used to expand the plant, including the capital goods needed to work with this plant, and the necessaries and conveniencies of life.

This net product, at a certain level of abstraction, can be seen as divided into wages and income for the owners. The overall rate of profits is the ratio of the net product to the sum of the capital goods used up in production and wages, if one treats wages as advanced.

The above is the perspective of classical political economists such as Adam Smith and David Ricardo. But the above summary is of an approach that needs better definition. I have defined the rate of profits as a ratio between heterogenous quantities, a nonsense quantity. One needs prices or some other way of forming a single number out of the large mish mashes of commodities.

Following Sraffa, assume that a uniform rate of profits is made in each line of production. Then, still at a very abstract level, one can set up a system of equations for prices of production. "...the distribution of the surplus must be determined through the same mechanism and at the same time as are the price the price of commodities" (Sraffa). And this system of equations can be solved.

Ricardo and Marx did not have available the concept of an eigenvalue. They had a conjecture. The ratio defining the rate of profits can be found by evaluating large aggregates of commodities with the labor embodied in each commodity. They knew and said that there was some error here. Nevertheless, this approach attempts "to penetrate the inner physiology of bourgeois society," as Marx put it. Ricardo correctly identified the trade-off between proportional wages and the rate of profits, a trade-off later rediscovered by Nicholas Georgescu-Roegen and taken up by Paul Samuelson under the moniker of the 'factor price frontier'.

Ricardo and Marx both focused on a good of average capital-intensity, in some sense. Ricardo sought a measure of value that somehow did not vary with distribution or improvements in technology. If the ratio of a price of production to the price of such a commodity varied with distribution, that variation was supposed to be caused by the invidual commodity, not by stretching or shrinking of the yardstick. One of Ricardo's insight was that a higher wage could be accompanied by a lower price of production of some commodities.

Sraffa identified a 'standard economy' built into the quantity flows for his system of equations. In the standard system, which has the same level of employment as the actual system, capital goods, gross output, and net output are all composed of commodity baskets in which the quantities of commodities enter in the same proportion. Suppose the net product of the standard system is adopted as the numeraire for the wage, which, therefore, ranges from zero to unity. Then one can calculate the rate of profits as a ratio of homogeneous quantities, without ever evaluating commodities at prices.

This rate of profits is not a price phenomenon insofar as it does not depend of relative prices. The general rate of profits is unchanged by evaluating commodities at market prices, at prices of production with a uniform rate of profits, at prices of production with varying markups among industies refracting market power, at labor values, or at energy values. It just does not matter.

This perspective on the labor theory of value, which I mainly take from Pierangelo Garegnani, presents it as a technical solution to a technical problem in trying to understand capitalist economies. Fratini and Ravagnani has some interesting work on the standard system. Whatever socialogical insights one may take from Marx's early writings on alienation or Lukacs writings on reification does not seem to have much to do with the matter.

Garegnani notes how little understanding some critics of Marx had of what they were attempting and failing to criticize:

"We find instead no mention whatsoever, in Böhm-Bawerk, of the correct proposition that the rate of profit is determined when the real wage is specified, or of the ensuing inverse relation between the wage and the profit rate which the labour theory of value had allowed Ricardo and Marx to establish, overcoming the deficiencies of Smith’s determination of profits..." (Garegnani 2018)

and

Two observations seem sufficient to indicate how Böhm-Bawerk coped with Marx’s Volume III prices of production, without essentially changing his conclusions about Marx’s work. The first is that Marx’s idea that the rate of profit and, therefore, the prices of production originate from a redistribution of aggregate surplus value is broken down by Böhm-Bawerk into a 'premise' and no less than four distinct 'arguments', thus rendering Marx's reasoning practically incomprehensible. The second observation is that the crux of Böhm-Bawerk's argument, contained in ten out of the essay's 110 pages, shows some awareness of the true difficulties besetting Marx's theory of prices of production, centring around the use of 'values' rather than 'prices of production' in the price equations themselves. [footnote:] See Böhm-Bawerk’s criticism of what he reconstructs as Marx's 'fourth argument' on labour values in Böhm Bawerk (1896). In the course of his criticism, Böhm-Bawerk notices that the determination of aggregate surplus value cannot ignore the fact that wage goods can be sold at 'prices' of production which deviate from 'values', the same point, we may recall, made by Marx in his sketch of a theory of prices." (Garegnani 2018)

References

• Saverio M. Fratini and Fabio Ravagnani, Sraffa and the 'slogans not used'.
• Pierangelo Garegnani. 2018. On the labour theory of value in Marx and in the Marxist tradition Review of Political Economy

'The' Labor Theory of Value

1.0 Introduction

This post argues that there is more than one labor theory of value.

2.0 The Labor Theory of Property

John Locke argued that what one mixes one labor with, one has a right to own. One could read Marx's Capital as a reductio ad absurdum of this labor theory of property. I disagree with this reading.

3.0 Labor Commanded as a Theory of Welfare

Given a unit of money - one dollar or one british pound - the labor commanded by that money is the amount of person-years of labor you can hire with that money. This quantity of the labor is the reciprocal of the wage. The following quotation from Adam Smith, I assert, is NOT a statement of a theory of value:

"The real price of everything, what everything really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What everything is really worth to the man who has acquired it, and who wants to dispose of it or exchange it for something else, is the toil and trouble which it can save to himself, and which it can impose upon other people. What is bought with money or with goods is purchased by labour as much as what we acquire by the toil of our own body. That money or those goods indeed save us this toil. They contain the value of a certain quantity of labour which we exchange for what is supposed at the time to contain the value of an equal quantity. Labour was the first price, the original purchase-money that was paid for all things. It was not by gold or by silver, but by labour, that all the wealth of the world was originally purchased; and its value, to those who possess it, and who want to exchange it for some new productions, is precisely equal to the quantity of labour which it can enable them to purchase or command." -- Wealth of Nations, Book 1, Chapter 5: Of the real and nominal price of commodities, or their price in labour, and their price in money.

Smith is interested in a measure of wealth. The money price varies too much for this purpose. I do not recall why Smith rejects the price of corn, that is, the price for the most common food of the people. I think he finds it also quite variable.

4.0 Labor Embodied as Natural Prices

Adam Smith, however, does has a labor theory of value. He states:

"In that early and rude state of society which precedes both the accumulation of stock and the appropriation of land, the proportion between the quantities of labour necessary for acquiring different objects seems to be the only circumstance which can afford any rule for exchanging them for one another. If among a nation of hunters, for example, it usually costs twice the labour to kill a beaver which it does to kill a deer, one beaver should naturally exchange for or be worth two deer. It is natural that what is usually the produce of two days or two hours labour, should be worth double of what is usually the produce of one day's or one hour's labour." -- Wealth of Nations, Book 1, Chapter 6: Of the component parts of the price of commodities.

This labor theory of value only holds for an imaginary pre-historic society with no private property. Under capitalism, the labor commanded by a commodity exceeds the labor embodied in it, the latter being what is most commonly meant by the labor value of a commodity. One can read Smith as asserting that the profits on stock and the rent of land result from the exploitation of workers. Profits and rent are the result of value added by workers not paid out in wages.

5.0 The Ricardian Socialists

John Francis Bray, John Gray, Charles Hall, Thomas Hodgskin, 'Piercy Ravenstone', William Thompson, and others are known as the Ricardian socialists. Thompson (2002) argues that they drew more on Smith than Ricardo. Generally, they read Smith as explaining the returns to capital and land as a matter of exploitation and took the labor theory of value as a normative theory about what prices should be.

Accordingly, they advocated reforms to get rid of parasitical returns to capital and land. Some suggested the founding of banks that would issue labor notes. I think this idea was advocated not only so that the worker would get the full value of their product, but had something to do with abolishing the instability seen in economic crashes and business cycles. Some actually founded banks and volutary communities to implement these ideas. Nearby here, Ithaca has HOURs as labor notes. I have seen stores that say they accept them, but have never seen one.

Karl Marx rejected the claim that workers had a right to the whole product of their labor. This was a point that Marx raised against Ferdinand Lasalle, as well as against the Ricardian socialists.

6.0 Ricardo and a Simple Labor Theory of Value

Ricardo presented a simple labor theory of value as a descriptive theory of prices in a capitalist economy. He takes the distinction between market prices and 'natural' prices from Smith, ultimately, I guess, from William Petty. Market prices tend to and bob around natural prices, also known as prices of production. Suppose the whole distribution over time of the labor inputs used to manufacture two commodities is the same. Then the ratio of their prices of production will, indeed, be the ratio of their labor values.

But Ricardo immediately considers, in later parts of the first chapter of his Principles what happens when, say, the ratio of fixed capital and circulating capital varies among the methods with which commodities are produced. And he realizes that a simple labor theory of value cannot be exactly true. This is why Ricardo is sometimes described as having a 93 percent labor theory of value. He does not think land presents a fatal complication. One can calculate embodied labor values on marginal land that pays no rent. He searches for an 'invariable standard' which he ultimately decides is best approximated by a commodity of average capital intensity.

James Mill, Jane Marcet, John Ramsay McCulloch, Harriet Martineau, Robert Torrens, and J. S. Mill followed, popularized, and built on Ricardo Some have claimed that such followers of Ricardo had a confused understanding of Ricardo's theory of value.

7.0 Marx, a Simple Labor Theory of Value in Volume 1, and the Volume 3 Invariants

Marx was appreciative of Ricardo. But Marx says that Ricardo does not examine the value-form. Ricardo, according to Marx, does not analyze the preconditions of capitalism, for example, how it is that workers are available to be hired for money, how their labor-power is available on the market as a commodity.

Not only did Marx criticize Ricardo for ignoring the distinction between labor-power and labor. The labor value of labor is a meaningless phrase. Marx also argued that Ricardo was not able to abstract interest on money, profits on capital, and rent on land into the generalization of surplus value. Ricardo has a distinction between fixed and circulating capital, but cannot see Marx's distinction between constant and variable capital. So Ricardo has no notion of the organic composition of capital. Ricardo's theory of value and distribution, like Marx's does not need the imposition of Say's law. According to King (1983) precursors for all these notions can be found in the works of the Ricardian socialists.

Marx, in volume 1 of Capital, considers how labor must be distributed among industries to sustain production in a commodity-producing economy. He focuses on how surplus value is generated when labor-power is a commodity bought and sold on a labor market. For these purposes, he adopts a simple labor theory of value. Market prices tend towards ratios in the proportion of embodied labor values.

In volume 3, Marx depicts value as generated as in volume 1, but redistributed among industries. Prices of production are such that the same rate of profits is obtained in all industries. He asserts the rate of profits is the same in the value system and in the system of prices of production. Furthermore, total profits is supposedly equal to total surplus value, and total values are equal to total prices. These invariants hold in the production of a commodity of average organic composition of capital, in some sense.

In later chapters in Volume 3, Marx proceeds to an even lower level of abstraction. He considers different types of rent and financial capital.

8.0 Conclusion

These theories present opportunities for confusion and muddle, for talking past one another. Does Smith have a labor theory of value? Ricardo? Marx? Is a labor theory of value supposed to be a measure of wealth, a descriptive theory for prices in a capitalist economy, or a theory of economic planning in a post-capitalist society? And then, of course, some are ignorant and of bad faith. And those who have their own, but different, interpretations of the transformation problem, can get into honest argument.

References

• J. E. King. 1983. Utopian or scientific? A reconsideration of the Ricardian Socialists. History of Political Economy 15(3): 345-373.
• Anton Menger. 1899. The Right to the Whole Produce of Labour: The Origin and Development of the Theory of Labour's Claim to the Whole Product of Industry.
• Noel W. Thompson. 2002. The People's Science: The Popular Political Economy of Exploitation and Crisis 1816-34. Cambridge University Press.

Translation Between The Language Of Classical Economists And Marginalists

Lately, when trying to write up my results I use terminology from classical political economy. The table below maps some terms from classical political economy to terminology for marginalists.

Terminology

Classical	Marginalist
Use value	Utility
Supply	Quantity supplied
Demand	Quantity demanded
(Normal) profits	Interest
Extra profits	(Pure) economic profits
Supernormal profits
Market prices	Short run prices
Natural prices	Long run prices
Prices of production

I probably am leaving some important mapping out. Nuances, at least, exist to distinguish between entries in rows in the tables. And maybe it depends on which classical or marginalist economist you read, and which of their works. I think the first row is the most questionable.

A Short History

William Petty began classical political economy in the 17th century. Classical economics was developed through the work of the physiocrats and such writers as Adam Smith, David Ricardo, and Karl Marx. Marx was also a critic.

About a century and a half ago, economists mistakenly accepted the marginal revolution. Jevons, Menger, and Walras had precursors, but they were regarded as cranks. Marx, however, posed a political problem. Some might mention Henry George here, or maybe even Silvio Gesell. Better have an imitation of physics than talk about the ideas of those opposed to capitalists.

Lionel Robbins made clear in the 1930s, when the theory plainly did not apply, that marginalist economics is about the allocation of scarce resources. Land and labor can be taken as given at a moment in time, but capital cannot.

About half a century ago, economists came to recognize that they were mistaken. By the way, this mistake was also noted in the Heckscher–Ohlin-Samuelson (HOS) model of international trade. The demonstration that marginalism is fundamentally wrong was pushed by Joan Robinson and Piero Sraffa.

So some economists returned to elaborating classical economics. Even in the marginal interregnum, some, such as Leontief and Von Neumann, elaborated classical themes.

But most economists have been spinning in a widening gyre, ignoring the incoherence of their teaching. If a sufficient political movement develops outside of academic economics, maybe more economists will return to serious work. If so, they may even find elements to repurpose in this half century of dissolution and confusion.

Three Quotations: Rousseau, Adam Smith, Engels

Here is Jean Jacques Rousseau:

"...whether those who command are necessarily better than those who obey, and if strength of body or of mind, wisdom or virtue are always found in particular individuals, in proportion to their power or wealth: a question fit perhaps to be discussed by slaves in the hearing of their masters, but highly unbecoming to reasonable and free men in search of the truth." -- Jean Jacques Rousseau, Discourse on Inequality (1755).

Early developers of political economy were not slavish. Here is Adam Smith explaining that returns to capital and land are the result of value added by labor not paid out in wages.

"In the early and rude state of society... the whole produce of labour belongs to the labourer; and the quantity of labour commonly employed in acquiring or producing any commodity, is the only circumstance which can regulate the quantity of labour which it ought commonly to purchase, command, or exchange for.

As soon as stock has accumulated in the hands of particular persons, some of them will naturally employ it in setting to work industrious people, whom they will supply with materials and subsistence, in order to make a profit by the sale of their work, or by what their labour adds to the value of the materials. In exchanging the complete manufacture either for money, for labour, or for other goods, over and above what may be sufficient to pay the price of the materials, and the wages of the workmen, something must be given for the profits of the undertaker of the work, who hazards his stock in this adventure. The value which the workmen add to the materials, therefore, resolves itself in this case into two parts, of which the one pays their wages, the other the profits of their employer upon the whole stock of materials and wages which he advanced. He could have no interest to employ them, unless he expected from the sale of their work something more than what was sufficient to replace his stock to him; and he could have no interest to employ a great stock rather than a small one, unless his profits were to bear some proportion to the extent of his stock...

...As soon as the land of any country has all become private property, the landlords, like all other men, love to reap where they never sowed, and demand a rent even for its natural produce. The wood of the forest, the grass of the field, and all the natural fruits of the earth, which, when land was in common, cost the labourer only the trouble of gathering them, come, even to him, to have an additional price fixed upon them. He must then pay for the licence to gather them, and must give up to the landlord a portion of what his labour either collects or produces. This portion, or, what comes to the same thing, the price of this portion, constitutes the rent of land, and in the price of the greater part of commodities, makes a third component part." -- Adam Smith, The Wealth of Nations, Book I, Chapter VI (1776).

Many read the above as an account of how labor is exploited under capitalism. I find something similar in an essay a young Engels wrote before his life-long partnership with Marx:

"We have seen that capital and labour are initially identical; we see further from the explanations of the economist himself that, in the process of production, capital, the result of labour, is immediately transformed again into the substratum, into the material of labour; and that therefore the momentarily postulated separation of capital from labour is immediately superseded hy the unity of both. And yet the economist separates capital from labour, and yet clings to the division without giving any other recognition to their unity than by his definition of capital as "stored-up labour". The split between capital and labour resulting from private property is nothing but the inner dichotomy of labour corresponding to this divided condition and arising out of it. And after this separation is accomplished, capital is divided once more into the original capital and profit-the increment of capital, which it receives in the process of production; although in practice profit is immediately lumped together with capital and set into motion with it. Indeed, even profit is in its turn split into interest and profit proper. In the case of interest, the absurdity of these splits is carried to the extreme. The immorality of lending at interest, of receiving without working, merely for making a loan, though already implied in private property, is only too obvious, and has long ago been recognised for what it is by unprejudiced popular consciousness, which in such matters is usually right. All these subtle splits and divisions stem from the original separation of capital from labour and from the culmination of this separation- the division of mankind into capitalists and workers-a division which daily becomes ever more acute, and which, as we shall see, is bound to deepen. This separation, however, like the separation already considered of land from capital and labour, is in the final analysis an impossible separation. What share land, capital and labour each have in any particular product cannot be determined. The three magnitudes are incommensurable. The land produces the raw material, but not without capital and labour. Capital presupposes land and labour. And labour presupposes at least land, and usually also capital. The functions of these three elements are completely different, and are not to be measured by a fourth common standard. Therefore, when it comes to dividing the proceeds among the three elements under existing conditions, there is no inherent standard; it is an entirely alien and with regard to them fortuitous standard that decides— competition, the cunning right of the stronger. Rent implies competition; profit on capital is solely determined by competition; and the position with regard to wages we shall see presently." -- Friedrich Engels, Outlines of a Critique of Political Economy (1844).

I think one can discard the conjectural history and moral overtones of the above and freely play in the mathematics of Leontief matrices. But many academic economists nowadays are imposing mind-forged manacles onto another generation.

Marx On The Transformation Problem In 1847

This is the start of Section 5, "Strikes and combinations of workers", in the second chapter of The Poverty of Philosophy:

"'Every upward movement in wages can have no other effect than a rise in the price of corn, wine, etc., that is, the effect of a dearth. For what are wages? They are the cost price of corn, etc.; they are the integrant price of everything. We may go even further: wages are the proportion of the elements composing wealth and consumed reproductively every day by the mass of the workers. Now, to double wages ... is to attribute to each one of the producers a greater share than his product, which is contradictory and if the rise extends only to a small number of industries, it brings a general disturbance in exchange; in a word, a dearth.... It is impossible, I declare, for strikes followed by an increase in wages not to culminate in a general rise in prices: this is as certain as that two and two make four.' (Proudhon, Vol. I, pp. 110 and 111)

We deny all these assertions, except that two and two make four.

In the first place, there is no general rise in prices. If the price of everything doubles at the same time as wages, there is no change in price, the only change is in terms.

Then again, a general rise in wages can never produce a more or less general rise in the price of goods Actually, if every industry employed the same number of workers in relation to fixed capital or to the instruments used, a general rise in wages would produce a general fall in profits and the current price of goods would undergo no alteration.

But as the relation of manual labour to fixed capital is not the same in different industries, all the industries which employ a relatively greater mass of capital and fewer workers, will be forced sooner or later to lower the price of their goods. In the opposite case, in which the price of their goods is not lowered, their profit will rise above the common rate of profits. Machines are not wage-earners. Therefore, the general rise in wages will affect less those industries, which, compared with the others, employ more machines than workers. But as competition always tends to level the rate of profits, those profits which rise above the average rate cannot but be transitory. Thus, apart from a few fluctuations, a general rise in wages will lead, not as M. Proudhon says, to a general increase in prices, but to a partial fall - that is a fall in the current price of the goods that are made chiefly with the help of machines.

The rise and fall of profits and wages expresses merely the proportion in which capitalists and workers share in the product of a day's work, without influencing in most instances the price of the product. But that 'strikes followed by an increase in wages culminate in a general rise in prices, in a dearth even' - those are notions which can blossom only in the brain of a poet who has not been understood." -- Karl Marx

You can see why after this, Marx and Proudhon were no longer drinking buddies. Anyways, Marx here considers a rise in wages. Echoing Ricardo, Marx argues that some prices drop and others rise. So even though the labor embodied in commodities does not alter, relative prices of production vary because of a variation of wages. I take this to be a statement of the so-called transformation problem.

An early statement of Marx's solution can be found in his 2 August 1862 letter to Engels.

I see in the above an idea Marx takes over from Ricardo. Think of the yearly net output of a capitalist economy as produced by the labor employed during that year. The 'real wage', in Ricardo's terminology, is the proportion of that labor that goes to produce the commodities purchased and thereby consumed by the workers. Suppose productivity increases, and total employment does not change. The workers can thereby obtain a greater quantity of 'necessaries and luxuries', while the real wage declines, depending on how the results of this increased productivity are divided among the classes making up society. The current usage among mainstream economists of the term 'real wage' is an obstacle to reading Ricardo, if one is not careful.

By the way, in a comment on one of my posts on the Temporal Single System Interpretation, Hedlund recommends Robert A. Bryer's "Marx's accounting solution to the 'transformation problem'". This is a chapter in his book Accounting for Value in Marx's Capital: The Invisible Hand

On Sraffian Subsytems And Labor Values

1.0 Introduction

I recently stumbled across McKiernan (2017), an Austrian response to Sraffa's book. This is a weird working paper. He works through Sraffa, with apparently no knowledge of all the textbooks explaining the book. McKiernan correctly notes that Sraffa provides little context about his points. And the mathematics is not always explicit. Naturally, McKiernan makes mistakes. If he ever revisits this, I think he would want to break it up into several papers. (Fabra (1991) is the strangest published response to Sraffians of which I know.)

I only want to focus on one point, Sraffian subsytems. But before doing that, let me at least point out something insightful in McKiernan's paper. He points out one of what I later call a fluke case:

However, Sraffa makes no note of cases in which a scarcity of land is offset by using a more expensive method of production, and that employment meets present desire for the product exactly when applied to all available land, so that only one method is used. In this case, two variables, p_corn and ρ, correspond to one equation. (p. 22)

I think I could also construct a fluke case with intensive rent, which is closer to his point.

2.0 A Criticism Of Subsystems

McKieran mistakenly asserts:

But consider the pricing of a commodity that were not produced in surplus, as could happen in these models with a purely infrastructural commodity. A sub-system for this commodity would have a net production of 0, but these models have all presumed a need for active renewal, so there would be an expenditure of labor. If returns to scale were co-incident, so that the sub-system might be embodied, that sub-system standing alone would produce a wage of 0, but wages are presumed to be shared across sub-systems (else Sraffa’s argument falls apart). Hence, it appears that prices of commodities of this infrastructural sort must be whatever one makes of division by 0. (p. 8)

(If McKieran ever revisits this, I wish he would give a more explicit and formal definition of co-incident returns. I think I get his point, but I do not think I have ever seen this notion in the economics literature.)

Anyways, if the net output of a commodity is zero, the decomposition of the given quantity flows into Sraffian subsytems, with nothing left over, will result in no labor being directed towards producing a net output of that commodity. So one rather gets a quotient of zero divided by zero, not a division of a positive quantity of zero. Nevertheless, one can still find a Sraffian subsystem for producing that commodity.

3.0 A Decomposition of One Set of Quantity Flows

I take an example from my FAQ on the Labor Theory of Value. Consider an economy with the observed quantity flows shown in Table 1. In this little model economy, wheat, iron, and labor are used to produce a net output of 500 quarters of wheat and 8 tons iron. From the postulated observations, one cannot tell whether iron is somehow consumed or whether this economy is undergoing steady growth. Given, say, the real wage, one can calculate prices of production. But many questions are not adressed here.

Table 1: Given Quantity Flows

Inputs		Output
74 qr. wheat & 37 t. iron & 592 worker.	->	592 qr. wheat
18 qr. wheat & 3 t. iron & 48 workers	->	48 t. iron

I find Leontief coefficients of production useful. Table 2 results from dividing the first of Table 1 by 592 qr. wheat and the second row by 48 t. iron.

Table 2: Leontief Coefficients

Inputs		Output
1/8 qr. & 1/16 t. & 1 workers	->	1 qr. wheat
3/8 qr. & 1/16 t. & 1 workers	->	1 t. iron

Suppose these coefficients of production are used to calculate inputs when gross outputs of are approximately 588.2 quarters wheat and 39.2 tons iron. Table 3 results. The net output of the wheat subsytem is 500 quarters wheat, produced from an input of 32000/51 workers. That is 64/51 ≈ 1.25 workers are embodied in each quarter of wheat.

Table 3: Wheat Subsystem

Inputs		Output
1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers		10000/17 qr. wheat
250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers		2000/51 t. iron

Table 4 shows the iron subsystem. In this subsystem, 640/51 ≈ 12.5 workers produce a net output of 8 tons iron. In other words, 80/51 ≈ 1.57 workers are embodied in each ton iron.

Table 4: Iron Subsystem

Inputs		Output
8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers		64/17 qr. wheat
56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers		448/51 t. iron

The quantity flows shown in Tables 3 and 4 add up to the quantity flows in Table 1. In these subsystems are thought of as operating side-by-side, total quantity flows are as in the observed economy. No assumptions on returns to scale are needed for this decomposition into subsystems.

4.0 A Decomposition of Another Set of Quantity Flows

I now want to consider another set of quantity flows that might be observed. Suppose these quantity flows are as in Table 5. The net output of this economy consists of 500 quarters wheat. For ease of calculation, I have used the same coefficients of production as in Table 2. They very well could be different since gross outputs vary from those in Table 1. If coefficients of production vary, so do returns to scale. Anyways, in this example, the net output of iron is zero. But even so, one can find here a subsystem for producing iron.

Table 5: Another Set of Quantity Flows

Inputs		Output
1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers		10000/17 qr. wheat
250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers		2000/51 t. iron

Consider the quantity flows shown in Table 6. The net output is 500 quarters wheat and -8 tons iron. This is not a Sraffian subsystem. It is unbalanced. But these quantity flows are constructed from the same coefficients of production as manifested in Table 5.

Table 6: Unbalanced Quantity Flows

Inputs		Output
1242/17 ≈ 73.1 qr. & 621/17 ≈ 36.5 t. & 9936/17 ≈ 584.5 workers		9936/17 qr. wheat
194/17 ≈ 11.4 qr. & 97/51 ≈ 1.9 t. & 1552/51 ≈ 30.4 workers		1552/51 t. iron

Now consider what quantity flows result from subtracting those in Table 6 from those in Table 5. Table 7 results. This is the same subsystem for producing iron as shown in Table 4. Even though the net output of iron in the observed quantity flows in Table 5 is zero, one can still find in them an iron subsystem.

Table 7: The Iron Subsystem Again

Inputs		Output
8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers		64/17 qr. wheat
56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers		448/51 t. iron

5.0 Results

No assumptions on returns to scale are made in analytically decomposing the observed quantity flows into subsystems.

Given observed physical quantity flows, one can ask how much more labor would be employed if net output of the economy was increased by a small quantity of a specified commodity. In other words, each commodity has an employment multiplier, to use the jargon of Leontief analysis. If non-constant returns to scale do not prevail, the error in this calculation will become more pronounced as the specified quantity increases. An insight behind the differential calculus is that, for continuous functions, a small enough variation has an approximately linear effect.

Reference

• Pail Fabra. 1991. Capitalism versus Anti-Capitalism: The Triumph of Ricardian over Marxist Political Economy. Rowman and Littlefield.
• Daniel Kian McKiernan. December 22, 2017. The begged questions and confusions in Mr. Sraffa's theory of price.

Value And Distribution

"It is the whole process of production that must be called 'human labour', and thus causes all products and all values. Marx and Ricardo used 'labour' in two different senses: the above and that of one of the factors of production ('hours of labour' or 'quantity of labour' has a meaning only in the latter sense). It is by confusing the two senses that they got mixed up and said that value is proportional to quantity of labour (in second sense) whereas they ought to have said that it is due to human labour (in first sense: a non-measurable quantity or not a quantity at all)." -- Piero Sraffa (D3/12/11:64, as quoted by Kurz and Salvadori)

"I shall begin by giving a short 'estratto' of what I believe is the essence of the classical theories of value, i.e. of those which incluce W. Petty, Cantillon, Physiocrats, A. Smith, Ricardo and Marx. This is not the theory of any one of them. I state it of course, not in their own words, but in modern terminology, and it will be useful when we proceed to understand their portata (delivery capacity) from the view of our present inquiry. It will be a sort of 'frame', a machine into which to fit their own statements in a homogeneous pattern, so as to be able to find what is common in them and what is the difference with the later theories." -- Piero Sraffa (D3/12/4:12 as quoted by Pasinetti and by Kurz and Salvadori and by )

1.0 Introduction

This post outlines a coherent theory to build on in trying to understand capitalist economies. I thought originally of trying to explain the unoriginal points in this post without any algebra. I wish I could be more terse. But I find that too difficult. No reason exists that all uses of this theory should be fully formalized.

This post contains an implicit criticism of other theories based on what is not here. The data are objective, and the approach is not that of methodological individualism. I do not draw any well-behaved supply and demand functions. Nor do I assume that the labor market, for example, clears. I do not calculate any marginal quantities. Unobservable preferences are not referred to, and no utility functions are maximized. I do not see why in extending this approach one needs to refer to such fictions.

2.0 Givens From A View From Outer Space

Consider a capitalist economy during a single production cycle, which I take as a year. One can observe the following:

• A column vector, q, of the gross output of produced commodities. Each commodity is measured in physical units (tons, kilowatts, etc.), and is the output of some process of production.
• The row vector, a₀, of labor coefficients of the production processes. The coefficient a_0,j is in physical units (for example, labor units per ton), where labor units are such that total employment over the year is unity. Different concrete activities are weighted by relative wages to reduce labor units to a common abstract labor unit. The product (a_0,j q_j) is the proportion of total employment allocated to the jth industry.
• The square Leontief input-output matrix, A. Each element a_i,j is in physical units (for example, kilograms per ton). The quantity (a_i,j q_j) is the physical units of the ith commodity used in producing the (gross) output of the jth process. The Leontief matrix is assumed to characterize a viable economy that can operate year after year at the same level. For simplicity, assume that each commodity enters directly or indirectly into the production of all commodities. That is, all commodities are Sraffian basic commodities.
• The wage, w, where the wage is the proportion of the price of the net output obtained by the workers.
• The column vector, d, of the commodities on which the wage is spent.

In this formulation, each industry produces a single commodity, and a single process is operated in each industry. Since the data is from a snapshot of the whole economy, in a sense, no assumptions are made on returns to scale. Each industry is operating at whatever scale is observed, with observed inputs being obtained from other industries.

Certain variables are implicit in these definitions of the givens. Net output is related to the the given technique and gross output:

y = q - A q = (I - A)q

where y is the column vector of net output, and I is the appropriately sized identity matrix. The net output is that part of gross output that can be consumed, while leaving available the reproduction of the capital goods used up in producing it so as to continue production in the next year at the same level and with the same technique. One can invert the above relation to find gross output from net output:

q = (I - A)^-1 y

The assumptions ensure that the Leontief inverse exists.

The unit of measurement for labor is chosen such that total employment, at the observed level of gross outputs, is unity:

a₀ q = a₀ (I - A)^-1 y = 1

The numeraire is the net output:

p y = 1

where p is a row vector of prices. The total wage is equal to the price of the commodities on which wages are spent:

w a₀ q = p d

For what it is worth, the physical composition of wages by industry is d a₀.

3.0 Social Relations Between Workers

The data reflect a certain allocation of labor over the industries comprising the economy. A notion of vertical integration is needed to think about how each commodity can be produced in a sustainable way.

For example, suppose a consumer buys one additional automobile of a specified make. To leave the production apparatus unchanged, workers will have to produce an automobile in Michigan, tires outside of Akron, steel from iron ore in Pittsburgh, and so on, with transport workers ensuring everything gets shipped appropriately. (Actually, this example probably reflects my understanding of the structure of production many decades ago.) This can all be expressed in algebra:

v_j = a₀ (I - A)^-1 e_j

where e_j is the jthe column of the identity matrix. The integrated labor units per unit of the jth commodity, v_j, is both an employment multiplier and the labor embodied in the commodity. I have explained it above as a matter of laborers working in parallel in the given year. But one can also think of it as a matter of notionally summing some labor in this year and over past years under the assumption that the observed technique has been used forever in the past at this scale.

One can also apply this algebra to certain collections of commodities. The labor embodied in commodities purchased out of wages, V, is:

V = a₀ (I - A)^-1 d

The labor embodied in the remaining commodities, S, which are produced by the workers but in the control of those owning the means of production are:

S = 1 - V = 1 - a₀ (I - A)^-1 d

The labor embodied in capital goods, C, is:

C = a₀ (I - A)^-1 A q

Certain ratios of these labor values have often been expressed. For example, the ratio, e, of the labor spent during the year to produce commodities not paid to workers to the labor spent to produce the commodities purchased out of wages is:

e = S/V

If one assumes wages are totally consumed, the above ratio has something to do with the maximum possible rate of growth. The ratio, occ, of the labor embodied in capital goods to the labor expended in the year is:

occ = C/(V + S)

4.0 Prices As Relations Between Things

Those making decisions about which processes to operate and at what levels have done so with certain expectations about being able to sell the produced commodities at certain prices. They also have expectations about costs. Observed market prices are almost certainly such that some of these expectations are being disappointed, and some will need to alter their plans. Such alterations include changing the scale at which certain processes are operated in some industries, disinvesting in some industries and investing in other industries, and replacing processes of production (perhaps with newly discovered processes) in various industries.

As a heroic simplification, consider what prices must be to be consistent with the data. Such prices of production are based on the assumption that the allocation of the labor force seen in the data is socially necessary. Perhaps, if the data were to be repeated year after year, market prices would circulate around prices of production or approach them in a 'gravitational' process.

Assuming competitive markets, in some sense, prices of production satisfy the following equation:

p A(1 + r) + w a₀ = p

where r is the rate of profits. In this equation, wages are paid out of the product at the end of the year.

As a matter of mathematics, the assumptions in this model of circulating capital are sufficient to determine prices of production and the rate of profits. They still would be sufficient if one allowed for fixed capital and for land in a model of extensive rent. Likewise, one could allow for limitations in competition by assuming known ratios of the rate of profits among industries. Certain issues arise in determining prices of production and the rate of profits in models of intensive rent and of general joint production. More than one solution may arise in some cases.

These extensions to joint production require the modeling of the choice of technique. I am not sure how I can do this analysis without assumptions on returns to scale. I do see that I can restrict distribution such that the observed technique is cost-minimizing at the observed scale.

It is easier to solve for the wage as a function of the rate of profits:

w = 1/{a₀ [I - (1 + r)A]^-1 y}

The above is a monotone decreasing function. The maximum wage of unity occurs when the rate of of profits is zero. The maximum rate of profits occurs at a wage of zero, and that maximum is 1/occ. Prices of production are:

p = a₀ [I - (1 + r)A]^-1/{a₀ [I - (1 + r)A]^-1 y}

4.1 An Aside On Vertical Integration

I repeat the above equation for prices of production:

p A(1 + r) + w a₀ = p

This can be rewritten as:

p Ar + w a₀ = p(I - A)

Or:

p A(I - A)^-1r + w a₀(I - A)^-1 = p

Or:

p Hr + w v = p

where I am going to call H the vertically integrated Leontief matrix. (Pasinetti probably has another name.) The row vector v is a vector of labor values. The solution of this system of equations is the same as above. When the rate of profits is zero, prices of production are equal to labor values.

5.0 Conclusion

The above has outlined a logical theory for describing a capitalist economy. The starting data are, in principle, observable and close to what can be obtained from the National Income and Product Accounts (NIPA).

This data allow one to examine how the labor force is allocated in a sustainable economy. The proportions of workers that are (re)producing certain aggregates (wage goods and the remainder) or embodied in certain aggregates (the capital goods used up in production) are noted. Presumably, individual commodities may be produced with extremes of labor-intensive methods, but these differences could come close to averaging out in the aggregate. One might want to extend wage goods to include, for example, commodities consumed by, for example, retired workers, students, the unemployed, and those unable to participate in the labor force. One might also want to look at other aggregates such as luxuries spent by those obtaining income produced by the workers but not paid out as (generalized) wage goods and investment goods used in expanding the economy.

These aggregates can also be evaluated with prices. I have drawn a few connections between prices of production and the labor embodied in these aggregates. The maximum rate of profits is the multiplicative inverse of the organic composition of capital. In formulating a system of equations for prices of production in vertically integrated terms, I find labor values useful. Otherwise, this post has drawn no connection between prices of production and the labor embodied in these aggregates. Can one pass easily between prices of production and labor values? The mathematics of eigenvalues and eigenvectors is useful for exploring the theory behind this question. Whatever you think of the answer, including difficulties arising with joint production and of empirical results, seems to be independent of the validity of anything in the above post.

I have talked about some of the conditions needed to sustain the operation of a capitalist economy, while only looking at the data for a single production period. Presumably, the gross output for the next year will be at a different level and mix. Such a change in scale in operation can be expected to alter the vector of labor coefficients and the Leontief input-output matrix.

Non-produced commodities that are available only as a finite stock that are used up in production (for example, oil, ores, and certain minerals) have been abstracted from. I have also ignored physical limitations imposed by bounds on throughput and sinks. The latter issue can be explored by the theory of joint production where one does not impose the assumption of free disposal.

Half a century ago, some economists demonstrated that economists a century and a half ago were mistaken. Jevons, Menger, and Walras made fundamental mistakes that cannot be fixed and are built into the work since then that builds on them. The political economy that had been developed in the century before them provides an alternative that is worth updating and building on.