## Monday, April 22, 2024

### A Perverse Switch Point For Neoclassical Economics, Non-Perverse For Austrians

 Figure 1: The Wage-Rate of Profits Frontier
1.0 Introduction

This post completes a demonstration that the economic life of a machine is independent of the capital-intensity of a technique. I here fill in the upper right of a two-by-two table. I have previously filled in the upper-left and lower right entries. And I also have an example for the lower left.

2.0 Technology

Tables 1 and 2 present coefficients of production for processes which can be combined to produce a new output of corn. The example has the structure of one from Salvatore Baldone. Corn is assumed to be the sole consumer good and the numeraire. In the first process, labor works with corn input to produce a machine with a physical life of three years. Labor works with seed corn and machines of the specified age, in each of the remaining processes, to produce a unit of corn and machines a year older. The machine is discarded after being operated in the last process.

 Input Process (I) (II) (III) (IV) Labor 21/50 39/200 117/200 39/100 Corn 12/125 39/100 141/250 117/200 New Machines 0 1 0 0 One-Year Old Machines 0 0 1 0 Two-Year Old Machines 0 0 0 1

 Output Process (I) (II) (III) (IV) Corn 0 1 1 1 New Machines 1 0 0 0 One-Year Old Machines 0 1 0 0 Two-Year Old Machines 0 0 1 0

3.0 Choice of Technique

Three techniques exist for producing a net output for corn. The Alpha, Beta, and Gamma technique differ in the economic life of the machine. In Alpha, the economic life of the machine continues to be one-year, while it remains its full physical life of three years in Gamma. Figure 1, at the top of this post, displays the wage curves for the three techniques. Gamma is cost-minimizing at a low rate of profits. Beta is cost-minimizing at an intermediate rate of profits, and Alpha is cost-minimizing at a high rate of profits. The wage curves for the Alpha and Beta techniques intersect at a low rate of profits within the frontier.

Perhaps this example is clarified by looking at prices. As shown in Figure 2, the price of a new machine is positive for the whole range of feasible rate of profits in the system of equations for each technique. Figure 3 graphs the prices of old machine of each age as a function of the rate of profits. Suppose the machine is operated for its physical life of three years, as under Gamma. The price of a two-year old machine is positive for rates of profits up to a switch point with Beta, at approximately 61 percent. The price of a one-year old machine is positive even for rates of profits slightly higher. Thus, the analysis of the choice of production based on prices confirms that Gamma is cost-minimizing at low rates of profits.

 Figure 2: The Price Of A One-Year Old Machine

 Figure 2: The Prices Of Old Machines

The price of a two-year old machine under Beta is zero. The price of a one-year old machine, when it is operated for two years, is positive between a rate of profits of approximately 24 and of 73 percent. That is, in a range of high rates of profits where Gamma is no longer cost-minimizing, Beta is cost-minimizing in the indicated range. Finally, Alpha is cost-minimizing in the highest range of profits, when the prices of one-year old and two-year old machines is negative under both Beta and Gamma.

If the physical life of a machine were two years, this example would be one of a reswitching. Around the switch point between Alpha and Beta at approximately 73 percent, a lower rate of profits is associated with a decreased value of capital per worker and an increased net output of corn per worker. On the other hand, around this switch point, a lower rate of profits is associated with an increased economic life of the machine.

4.0 Conclusion

This example fills in the upper right of this table. Table 1. The economic life of a machine is independent of the capital intensity of the techniques in which the machine is operated.

## Friday, April 19, 2024

### A Letter From Marx To Engels In 1858 Outlining His Critique Of Political Economy

I have previously repeated some transcriptions from the correspondence Marx and Engels. I tried to concentrate on some formulations from Engels of ideas important in Marxism, Marx stating what is important and novel in Capital, or outlines of the project, particularly concentrating on the so-called transformation problem. This long letter certainly belongs in this collection.

Marx's ideas were fairly well-developed in 1858. Here he has promised Duncker, a publisher, to write what eventually became A Contribution to the Critique of Political Economy. After delay, he only delivered to the publisher the first part of what he was aiming for. I have always found this short book less inspiring, as compared with Capital. Capital does not contain anything on advocates of labor notes, though. The introduction is notable for a short statement of historical materialism, with various aphorism. It went through German (Prussian?) censorship, but I have never seen anybody comment on that. Marx thought his work was scientific, I guess, and thus the censors could not object.

Anyways, we see that Marx only published the first of six parts that he had planned, and only the first subpart of that. Volume 3 of Capital has something on land and rent. And one could argue that he incorporated some of his theory of wages into volume 1. But his developed publications do not contain anything explicit on the state, international trade, and the world market. I think, with his work habits, if Marx promised a few pages, you can expect a long chapter. If he promised a chapter, you can expect a book. Or maybe you will never get anything.

I include the start of this letter for completeness. I did not look up what the first two paragraphs are about, other than to see that the battle of Alma was part of the Crimean war.

2 April 1858

Dear Frederick

The Guardian stories highly AMUSING. A correspondent of the Daily Telegraph (DIRECTLY UNDER PAM'S AUSPICES) writes of the great danger of being 'DEAF' in Paris, and says that all 'DEAF ENGLISHMEN' were being hounded by the police as Allsops. Also that ENGLISHMEN were leaving Paris en masse, partly because of police chicanery, partly for fear of an outbreak. For if the latter were to happen and the Bonapartists be victorious, the John Bulls feared they might be massacred by the MADDENED SOLDIERS, whereat the correspondent himself naively comments that IN SUCH A CASE [HE] SHOULD LIKE TO BE ANYWHERE ELSE BUT IN PARIS. This DESERTION by the Bulls AT THIS MOMENT OF COMMERCIAL DEPRESSION is queering the pitch of the Parisian épicier and householdr, whores, etc. Have you seen that 300 million francs have AVOWEDLY 'DISAPPEARED' FROM THE BUDGET, AND NOBODY KNOWS WHAT HAS BECOME OF THEM? There will, BY AND BY, be further REVELATIONS about Bonapartist FINANCE, and then the asses on the Tribune will realise the wisdom of not having published the very ELABORATED ARTICLES I sent them on the subject six months ago.219 The fellows are asses and anything which is not, in the crudest sense, a 'question of the day' they tend to cast aside as UNINTERESTING, only to go and compile the most egregious rubbish about the selfsame subject as soon as it does become à l'ordre du jour.

Nota bene: in the military clubs here it is being rumoured that EVIDENCE has been discovered among the papers left by Raglan that, 1. at the battle of the Alma he rightly suggested to attack the Russians, not from the direction of the coast, but from the opposite flank, and drive them into the sea; 2. that he proposed to advance on Simferopol after the batde of the Alma; 3. that at Inkerman it was only by dint of the most urgent pleas and MENACES that he extorted from Canrobert the order for Bosquet to hasten to his [Raglan's] assistance. It is further said that, if the boasting on the other side of the Channel were to continue, these PAPERS would be published, providing proof that the French were ever ready TO BETRAY THEIR DEAR ALUES. Indeed, a few HINTS which de Lacy Evans dropped in the HOUSE OF COMMONS seem to indicate something of the kind.

I've been so ill with my bilious complaint this week that I am incapable of thinking, reading, writing or, indeed, of anything SAVE the ARTICLES for the Tribune. These, of course, cannot be allowed to lapse since I must draw on the curs as soon as possible. But my indisposition is disastrous, for I can't begin working on the thing for Duncker until I'm better and my fingers regain their VIGOUR and GRASP.

The following is a SHORT OUTLINE OF THE FIRST PART. The whole thing is to be divided into 6 books: 1. On Capital. 2. Landed Property. 3. Wage Labour. 4. State. 5. International Trade. 6. World Market.

1. Capital falls into 4 sections, a) Capital en général. (This is the substance of the first instalment.) b) Competition, or the interaction of many capitals, c) Credit, where capital, as against individual capitals, is shown to be a universal element, d) Share capital as the most perfected form (turning into communism) together with all its contradictions. The transition from capital to landed property is also historical, since landed property in its modern form is a product of the action of capital on feudal, etc., landed property. In the same way, the transition of landed property to wage labour is not only dialectical but historical, since the last product of modern landed property is the general introduction of wage labour, which then appears as the basis of the whole business.

WELL (IT IS DIFFICULT FOR ME TO-DAY TO WRITE), let us now come to the corpus delicti.

I. Capital. First section: Capital in general. (Throughout this section wages are invariably assumed to be at their minimum. Movements in wages themselves and the rise and fall of that minimum will be considered under wage labour. Further, landed property is assumed to be zero, i. e. landed property as a special economic relation is of no relevance as yet. Only by this procedure is it possible to discuss one relation without discussing all the rest.)

1. Value. Simply reduced to the quantity of labour; time as a measure of labour. Use-value—whether regarded subjectively as the USEFULNESS of labour, or objectively as the UTILITY of the product—is shown here simply as the material prerequisite of value, and one which for the present is entirely irrelevant to the formal economic definition. Value as such has no 'substance' other than actual labour. This definition of value, first outlined by Petty and neatly elaborated by Ricardo, is simply bourgeois wealth in its most abstract form. As such, it already presupposes 1. the transcending of indigenous communism (India, etc.), 2. of all undeveloped, pre-bourgeois modes of production which are not in every respect governed by exchange. Although an abstraction, it is an historical abstraction and hence feasible only when grounded on a specific economic development of society. All objections to this definition of value derive either from less developed relations of production or else are based on confused thinking, whereby the more concrete economic definitions from which value has been abstracted (and which may therefore also be seen, on the other hand, as a further development of the same) are upheld as against value in this its abstract, undeveloped form. In view of the uncertainty of messieurs les économistes themselves about the precise relation of this abstraction to later, more concrete forms of bourgeois wealth, these objections were plus ou moins justified.

The contradiction between the general characteristics of value and its material existence in a particular commodity, etc.—these general characteristics being the same as those later appearing in money—gives rise to the category of money.

2. Money.

Some discussion of precious metals as vehicles of the money relation.

a) Money as a measure. A few comments on the ideal measure in Steuart, Attwood, Urquhart; in more comprehensible form among the advocates of labour money (Gray, Bray, etc. An occasional swipe at the Proudhonists). The value of a commodity translated into money is its price. For the moment price appears only in this purely formal distinction between it and value. Thus, in accordance with the general law of value, a specific amount of money merely expresses a specific amount of objectified labour. In so far as money is a measure, the variability of its own value is of no importance.

b) Money as a means of exchange, or simple circulation.

Here we need only consider the simple form of circulation as such. All the other conditions by which it is determined are external to it, and hence will not be considered till later (presuppose more highly developed relations). If the commodity be C and money M then, although simple circulation evinces the two circuits or final points: C—M—M—C and M—C—C—M (this latter constituting the transition to c), the point of departure and the point of return in no way coincide, save by chance. Most of the so-called laws put forward by economists do not consider money circulation within its own confines, but as subsumed under, and determined by, higher movements. All this must be set aside. (Belongs in part to the theory of credit; but also calls for consideration where money appears again, but further defined.) Here, then, money as means of circulation (coin). But likewise as realisation (not simply evanescent) of price. From the simple statement that a commodity, in terms of price, has already been exchanged for money in theory before it is so exchanged in fact, there naturally follows the important economic law that the volume of the circulating medium is determined by prices, not vice versa. (Here, some historical stuff on the polemic concerning this point.) Again it follows that velocity may be a substitute for volume, but that a certain volume is essential to simultaneous acts of exchange in so far as the relation of these themselves is not that of + and —, an equalisation and consideration which will only be touched on at this juncture by way of anticipation. At this point I shall not go further into the development of this section and would only add that the lack of congruence of C—M and M—C is the most abstract and superficial form in which the possibility of crises is expressed. If the law concerning the determination of circulating volume by prices be developed, it will be found that the assumptions made here are by no means applicable to all states of society; hence the fatuity of comparing e.g. the influx of money from Asia into Rome and its effect on prices there tout bonnement with modern commercial relations. On closer examination, the most abstract definitions invariably point to a broader, definite, concrete, historical basis. (OF COURSE, since to the extent that they are definite they have been abstracted therefrom.)

c) Money qua money. This is a development of the formula M—C—C—M. Money, the independent existence of value as opposed to circulation; material existence of abstract wealth. Already manifested in circulation in so far as it appears, not only as a means of circulation, but as realising price. In this capacity c), in which a) and b) appear to be no more than functions, money is the universal commodity of contracts (here the variability of its value acquires importance: value being determined by labour time); it becomes an object of HOARDING. (This would still seem to be an important function in Asia, as formerly in the ancient world and in the Middle Ages GENERALLY. Now persists only in a subordinate capacity within the banking system. In times of crisis money in this form again acquires importance. In this form money considered along with the world-historical DELUSIONS which it engenders, etc. Destructive properties, etc.) As the realisation of all higher forms in which value will appear; definitive forms in which all relations of value are externally concluded. Money, however, once fixed in this form, ceases to be an economic relation which is lost in its material medium, gold and silver. On the other hand, in so far as money comes into circulation and is again exchanged for C, the final process, the consumption of the commodity, again falls outside the economic relation. The principle of self-reproduction is not intrinsic to simple money circulation, which therefore implies something extrinsic to itself. Implicit in money—as the elaboration of its definitions shows—is the postulate capital, i.e. value entering into and maintaining itself in circulation, of which it is at the same time the prerequisite. This transition also historical. The antediluvian form of capital is commercial capital, which always generates money. At the same time the emergence of real capital, either from money or merchant capital, which gains control of production

d) This simple circulation, considered as such—and it constitutes the surface of bourgeois society in which the underlying operations which gave rise to it are obliterated—evinces no distinction between the objects of exchange, save formal and evanescent ones. Here we have the realm of liberty, equality and of property based on 'labour'. Accumulation, as it appears here in the form of HOARDING, is merely greater thrift, etc. On the one hand then, the fatuity of the economic harmonists, modern free traders (Bastiat, Carey, etc.), in upholding this most superficial and most abstract relation of production as their truth, as against the more advanced relations and their antagonisms. Fatuity of the Proudhonists and suchlike socialists, in contrasting the ideas of equality, etc., corresponding to this exchange of equivalents (or presumed AS SUCH), to the inequalities, etc., to which this exchange reverts and from which it emanates. In this sphere, appropriation by labour, the exchange of equivalents, appears as the law of appropriation so that exchange simply returns the same value in another material form. In short, while everything may be 'lovely' here, it will soon come to a sticky end and this as a result of the law of equivalence. For now we come to

3. Capital.

This is really the most important part of the first instalment and one on which I particularly need your opinion. But today I can't go on writing. My bilious trouble makes it difficult for me to ply my pen, and keeping my head bent over the paper makes me dizzy. So FOR NEXT TIME.

Salut.

Your

K. M.

There is a lot in this letter. In much of Capital, Marx takes wages as at a given level, but not always. The given level does not have to be a physical subsistence. We see the order of exposition is value, money, capital. The simple circulation of commodities, with money as an intermediary, provides the possibility of crises. Marx asserts the theory of endogenous money. Much is given here about presenting his ideas as in opposition to the Ricardian socialists. Marx gestures that his account of exploitation is consistent with justice in exchange.

## Saturday, April 13, 2024

### The Fundamental Sraffian Theorem

1.0 Introduction

I have been reading Robin Hahnel. Hahnel argues even more strongly than Steedman did that labor values are redundant. And he argues for the importance of the fundamental Sraffian theorem. I think this may be Hahnel's coinage. Anyways' this is my working my way through some of what I think he is saying.

Hahnel has some interesting things to say, not discussed here, about analyzing environmental concerns in a Sraffian framework. I ignore the chapter in Hahnel (2017) on the moral critique of capitalism. Following Eatwell (2019) and others, I hold that mainstream economists do not have a theory of value and distribution, anyways.

2.0 The Setting

Suppose a capitalist economy is observed at a given point in time. n commodities are being produced, each by a separate industry. Suppose the technique in use can be characterized by a row vector a0 and a n x n square matrix A. Let the column vector d denote the quantities of each commodity paid to the workers for a unit of labor.

The jth element of a0 is the amount of labor directly employed in the jth industry in producing one unit of a commodity output from that industry. "We suppose labour to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have previously been reduced to equivalent differences in quantity so that each unit of labour receives the same wage…" - Piero Sraffa (1960).

The jth column of A is the goods used up in producing one unit of a commodity output. For example, suppose iron is produced by the first industry and steel is produced by the second industry. a1,2 is then the kilotons of iron needed to produce a kiloton of steel. Assume that every good enters directly or indirectly into the production of each commodity. Iron enters indirectly into the production of tractors if steel enters directly into the tractor industry. Assume a surplus product, also known as a net output, exists. That is:

0 < λPF(A) < 1

where λPF(A) is the dominant eigenvalue of the matrix A. The dominant eigenvalue is also known as the Perron-Frobenius root.

3.0 Prices of Production

Suppose the wage purchases the specified bundle of commodities. And also assume the wage is advanced. One can define the input-output matrix with wage goods included:

A+ = A + d a0

I think that Sraffa treats the input-output matrix as A+ in chapter 1 of his book.

The system of equations for prices of production is:

p A+ (1 + r) = p

where p is a row vector, and r is the rate of profits. One can show that, given a surplus product, not including wage goods, a solution exists.

Fundamental Sraffian Theorem: The rate of profits, r, in the system of prices of production is positive if and only if:

0 < λPF(A+) < 1

In fact, the rate of profits is:

r = 1/λPF(A+) - 1

Under these assumptions, the price of each produced commodity is positive with the above rate of profits. And this economically meaningful solution is unique, up to the specification of a numeraire.

4.0 Increased Surplus Profit

Suppose one or more of the elements of A+ decrease. Then 1 - λPF(A+), which is strictly positive, increases. The surplus product that capitalists capture is increased by decreased components of the real wage and by decreased commodity inputs into production.

Suppose that the real wage is given and that an innovation results in a new technique, B, being available. This technique might have increased coefficients and decreases in other coefficients, as compared to A. It might even have a new column or delete a column for an industry that is not used to directly produce a wage good. This new technique is adopted at the given wage if and only if:

1 - λPF(B+) > 1 - λPF(A+)

Suppose further that:

1 - λPF(B) < 1 - λPF(A)

Then the maximum rate of profits, at a wage of zero, decreases. Suppose no reswitching exists. I think this is what is meant by Capital-Using, Labor-Saving technical change. This is also known as Marx-biased technical change. Marx's law of the tendency of the rate of profits to fall, presented in volue 3 of Capital, is not justified by this analysis.

5.0 Quantity Flows

This framework can also be used to examine the rate of growth. Suppose employment, at an instant of time, is unity:

L = a0 q = 1

where q isthe column vector of gross outputs. In this formulation, employment increases at the rate of growth.

Let consumption out of the surplus product be in the composition of the column vector e, and let c be the level of such consumption. It is most coherent to take this consumption as not made by the workers:

We could hardly imagine that, when the workers had a surplus to spend on beef. their physical need for wheat was unchanged. -- Robinson (1961)

So prices of production associated with this treatment of qunatity flows are as above.

Let the column vector j represent investment goods. These are part of the surplus product. Then the column vector q of gross outputs satisfies the following equation:

q = A+ q + c e + j

The above is extremely general. I now consider a steady-state rate of growth. Assume constant returns to scale in every industry. The vector of investment goods is in the same proportion as existing capital goods:

j = g A+ q

Here I present a derivation, since I typed this out to check myself. Substituting the specification of investment goods and after some algebraic manipulations, one has:

c e = [I - (1 + g)A+] q

Assuming the rate of growth is less than the maximum, one has:

c [I - (1 + g)A+]-1e = q

Premultiplying by the row vector of labor coefficients, one has:

c a0 [I - (1 + g)A+]-1e = a0 q = 1

The solution of the system of equations for quantity flows is:

c = 1/{a0 [I - (1 + g)A+]-1e}

The maximum rate of growth is:

gmax = 1/λPF(A+) - 1

The level of consumption out of the surplus product is lower, the higher the rate of growth and vice versa. One can also consider the impact on the rate of growth of changes in the elements of the matrix A+. I believe one can prove the following:

Theorem: The steady state rate of growth, g is higher if:

• Consumption out of the surplus product, where the surplus product does not include wages, is lower.
• Necessary wages are lower.
• The dominant eigenvalue, λPF(A), of the input-output matrix is lower.

The theorem highlights dilemmas in development economics. One does not want to obtain a higher rate of growth by lowering wages for those who are already pressed. It does not help for foreign aid to end up in luxury consumption either. In chosing the technique out of a range of possibilities, one would like the one that maximizes the rate of growth. Unless the rate of growth equals the rate of profits, that is, consumption out of the surplus product does not occur, the cost-minimizing technique is unlikely to be efficient in this sense.

6.0 Conclusion

The theory of value and distribution has a family resemblance to modern formulations of classical and Marxian political economy. Labor values are not discussed. It is focused on prices of production. Yet, with its consideration of dynamic changes in dominant eigenvalues, it seems to be consistent with an analysis of the formal and real subsumption of labor to capital. The formulation in this post can easily be generalized in various ways, Hahnel emphasizes inputs from nature and mentions the theory's consistency with homogeneous labor inputs. The analysis of growth should include technical change. I am interested in fixed capital. Some issues arise with general joint production, but the model is open in any case.

References
• John Eatwell. 2019. 'Cost of production' and the theory of the rate of profit. Contributions to Political Economy.
• Robin Hahnel. 2017. Radical Political Economy: Sraffa Versus Marx. Routledge.
• Joan Robinson. 1961. Prelude to a critique of economic theory. Oxford Economic Papers, New Series. 13 (1): 53-58.

## Saturday, April 06, 2024

### Keynes And Robinson On Supposed Self-Regulating Markets

Here is John Maynard Keynes in 1926:

"Let us clear from the ground the metaphysical or general principles upon which, from time to time, laissez-faire has been founded. It is not true that individuals possess a prescriptive 'natural liberty' in their economic activities. There is no 'compact' conferring perpetual rights on those who Have or on those who Acquire. The world is not so governed from above that private and social interest always coincide. It is not so managed here below that in practice they coincide. It is not a correct deduction from the Principles of Economics that enlightened self-interest always operates in the public interest. Nor is it true that self-interest generally is enlightened; more often individuals acting separately to promote their own ends are too ignorant or too weak to attain even these. Experience does not show that individuals, when they make up a social unit, are always less clear-sighted than when they act separately.

We cannot, therefore, settle on abstract grounds, but must handle on its merits in detail, what Burke termed 'one of the finest problems in legislation, namely, to determine what the State ought to take upon itself to direct by the public wisdom, and what it ought to leave, with as little interference as possible, to individual exertion.' We have to discriminate between what Bentham, in his forgotten but useful nomenclature, used to term Agenda and NonAgenda, and to do this without Bentham's prior presumption that interference is, at the same time, 'generally needless' and 'generally pernicious.' Perhaps the chief task of Economists at this hour is to distinguish afresh the Agenda of Government from the Non-Agenda; and the companion task of Politics is to devise forms of Government within a Democracy which shall be capable of accomplishing the Agenda. -- John Maynard Keynes, The end of laissez faire. In Essays in Pursuasion, 1931

Here is Joan Robinson in 1962, from a later reprint:

"It is possible to defend our economic system on the ground that, patched up with Keynesian correctives, it is, as he put it, the 'best in sight'. Or at any rate that it not too bad, and change is painful. In short, that our system is the best system that we have got.

Or it is possible to take the tough-minded line that Schumpeter derived from Marx. The system is cruel, unjust, turbulent. but it does deliver the goods, and, damn it all, it's the goods that you want.

Or, conceding its defects, to defend it on political grounds - that democracy as we know it could not have grown up under any other system and cannot survive without it.

What is not possible, at this time of day, is to defend it, in the neo-classical style, as a delicate self-regulating mechanism, that has only to be left to itself to produce the greatest satisfaction for all.

But none of the alternative defenses really sound very well. Nowadays, to support the status quo, the best course is just to leave all these awkwards problems alone." -- Joan Robinson. 1964. Economic Philosophy. Pelican: p. 130.

## Thursday, April 04, 2024

### A History Of Production Processes In Volume 1 Of Capital

1.0 Introduction

I have written many posts on formal results related to my favorite interpretations of the theory of value and distribution in Marx's Capital. But Marx's work is not solely about formalism. One aspect of volume 1 is a history of production processes up to Marx's day. Much opportunity exists to build on this history. Some have done this in works I have not read much of. I have been reading Soren Mau, and many years ago I read much of Charles Babbage's On the Economy of Machinery and Manufactures. Ian Wright, too, has had something to say about the impersonal force of capital at the level of a totality. I found Harry Cleaver's study guide useful in writing this post, even though I read Capital more analytically than politically.

This post provides a brief overview of aspects of Capital I have not mentioned before. Much more can be be found in volume 1. For instance, I say nothing about business cycles here.

2.0 Primitive Accumulation

Marx does not proceed in chronological order. He concludes volume 1 with a section on primitive accumulation. He describes the enclosure movement, which led to many peasants becoming vagabonds and beggars. And he describes strict laws against such. This leads to a labor force of workers with the double freedom of being free to sell their labor power and of being free from ownership of the means of production. I do not recall much about the transformation of natural economies, in overseas colonies, to capitalist economies. For example, the imposition of money taxes on colonial subjects constrained those subjects to seek waged labor so as to obtain the needed money.

3.0 (Re)producing the Presuppositions of Capitalism

But Marx starts his exposition, more or less, with a system in which production units are coordinated by selling commodities on markets for money. His book is about, from one perspective, how capitalism produces and reproduces its own presuppositions on an expanded scale.

According to Marx, capitalism initially takes over existing processes. He calls this formal subsumption. In the putting-out system, also called the domestic system, a merchant provides handicraft workers in their homes with raw materials to work up. The merchant collects their products and sells them on the market. Even though the workers are physically isolated, this system is the beginning of the aggregation of workers into a labor force.

Chapter 13 describes co-operation, and chapter 14 describes what Marx calls manufacture. Many handicraft workers are brought together under one roof. They might all be executing the same tasks, each transforming raw materials, through several steps, into a single commodity. Or they might be producing different products, perhaps with some vertical integration. The outputs of some workers are semi-finished products used as inputs by other workers.

In chapter 10, Marx writes about struggles over the length of the working day. In the formalism, absolute surplus value is increased if workers can be made to work longer, while requiring the same length of time needed to produce the commodities that they purchase from their wages. Some of this chapter seems to be still relevant in the United States today, where many low-pay workers have more than one job. Can their employers send them home and then call them back in the same day? Do the working days for these jobs sometimes add up to 24 hours? Computer and communication technology allow your employer to call on you at any time.

Marx has an interesting story about how mill owners might have found it in their own interest for laws to be passed regulating hours and working conditions. Each mill owner in Manchester would like to hire healthy workers, overwork them, and then discard them. But if all mill owners are doing this, where are healthy workers to be found? This story could be modeled with game theory, I guess.

Once so many workers are gathered in one place, the production processes which they operate can be 'rationalized'. This redivision and reallocation of tasks is advanced along with the introduction of machinery, as Marx discusses in chapter 15, Machinery and Modern Industry.

When I was young, the Rochester Museum and Science Center had a diorama - probably on the same floor as this one - showing the inside of a mill. A wheel was turned by a waterfall, and a system of gears and pulleys delivered the power to several rows of machines. For Marx, a machine is combination of tools, such as a lever, an inclined plane, a screw, etc., where the motive power might not be a human. The motive power could be water or steam, for example.

The result of this introduction of machinery and the re-organization of tasks is what Marx calls the real subsumption of capital. The pace at which laborers work is regulated by the machine. Automation is such that workers serve the machine, not vice versa. Laborers can no longer competitively duplicate their activities outside of employment bhy capitalists.

The increase in productivity brought about by real subsumption results in the increase of relative surplus value. The length of the working day required to produce the commodities purchased by wages decreases. Even with the same length of the working day and the same basket of wage goods, surplus labor time increases. The increase in relative surplus value is even compatible with a somewhat decreased working day and a somewhat larger basket of wage goods. This increase in relative surplus value is about forces operating behind the back of capitalists. They do not fund innovation so as to decrease the necessary labor embodied in wage goods.

4.0 Extensions and Conclusion

This history can be continued. Peter Drucker is of some importance in the development of the idea of management as a profession. Taylorism, named after Frederick Taylor, the author of Principles of Scientific Management, provides analytical tools for breaking tasks down further and regimenting them. Operations Research (OR) and Command, Control, Communications, and Information (C3I) are some disciplines building on the work of Babbage. For reasons of Information Assurance (IA), even relatively privileged computer programmers do not own the means of production; in the modern corporation, almost insurmountable hoops prevent you from hooking your own devices up to the corporate intranet or sharing files between your work and personnal computer. The history extends to office work and service jobs; it is not confined to factory work.

The result of this history is a division within the workshop and other entities organized in private firms, as well as among many productive enterprises in many industries scattered over a country or many countries. For the most part, no one worker oversees the entire production of a commodity. Rather, each becomes a detail worker. I tend to assume that a Leontief input-output matrix shows the interactions of all industries, but this structure is the result of a historical process. Furthermore, every capitalist is constrained by competition. The domination of capital is not solely about the relationship between individual employers and employees. It is also about an overarching dominating logic. By the way, I know some who take pride, probably rightfully, in their work in this setting.

The question for the socialist is whether this increased productivity can be maintained, with an increased population and developed social relationships among all, while somehow abolishing this domination. Can something like this division of labor be made transparent? If so, how do we get there?

## Tuesday, March 26, 2024

### Perverse Switch Point For Austrian Economics

 Figure 1: The Wage-Rate of Profits Frontier

This post continues a series of posts demonstrating that the change in the economic life of a machine at a switch point is independent of the change of the capital intensity of the technique at a switch point. I here fill in the lower left in a a two-by-two table.

The wage curves above are for the an example with the same structure as in the previous post in this series.This is a 'one-good' model. The manager of firms know three processes to produce a widget, also known as a machine. In the first process, labor and a new widget are used to produce new widgets and a one-year old widget. In the second process, labor and a one-year old widget are used to produce new widgets and a two-year old widget. In the last process, labor and a two-year old widget are used to produce new widgets. The output coefficients, b1,2 and b1,3, specify how many new widgets are produced by the second and third widgets. New widgets are the only commodities that can be consumed. The numeraire is a new widget.

The choice of technique arises because managers of firms can choose different economic lifetimes for the machine. Free disposal is assumed.

One can create a system of equations for the quantity flows for each technique. The net output is one new widget. The solution shows at what level each process is operated. It also shows how many widgets of each age are advanced as capital goods and how much labor is employed, per net output.

One can also create a system of equations for prices of production for each technique. Wages are paid out of the surplus, and the same rate of profits is obtained in operating each process. Figure 1 shows the wage curves for the example, for the specified parameters, b1,2 and b1,3. The cost-minimizing technique is on the outer frontier.

Around the single switch point, the value of capital per worker is higher at a lower rate of profits. This is a negative real Wicksell effect, a formalization of the notion of capital deepening. A greater output per worker is associated with a lower rate of profits. So, as far as real Wicksell effects go and unlike the previous one, this example conforms to obsolete marginalist teaching.

But consider the economic life of a machine. Around the switch point, a lower rate of profits is associated with a shorter economic life of a machine. And operating the machine for a shorter time results in a greater net output per worker. A choice of a shorter life of a machine can be associated with either more or less output per worker.

The machine operates at non-constant efficiency in the example. Whether the efficiency is increasing or decreasing over various years cannot be defined, in general, in models with multiple inputs for the machine. Furthermore, the change in the properties of the analysis of the choice of technique do not need to arise from perturbing parameters for processes in which the machine is operated. These changes can arise from perturbing other processes in multicommodity models.

 Figure 2: A Parameter Space for the Example

Perturbing the parameters b1,2 and b1,3 cannot create an example with a switch point falling into the upper right of my two-by-two table (Figure 2). Thus, I need to consider either a model with a different structure or perturbing additional parameters in ths one-good model.

## Saturday, March 23, 2024

### Keynes And Marx

1.0 Introduction

John Maynard Keynes had some amusing jibes against Marx, but does not provide any substantial argument against the theory in Marx's Capital. In fact, one can draw parallelisms between elements of Keynes' and Marx's theories. This post provides a brief start on justifications for these assertions.

2.0 Jibes

Keynes' most explicit and most well-known statement about Marx is probably this:

"How can I accept a doctrine which sets up as its bible, above and beyond criticism, an obsolete economic textbook which I know to be not only scientifically erroneous but without interest or application for the modern world? How can I adopt a creed which, preferring the mud to the fish, exalts the boorish proletariat above the bourgeois and the intelligentsia vho, with whatever faults, are the quality in life and surely carry the seeds of all human advancement? Even if we need a religion, how can we find it in the turbid rubbish of the Red bookshops? It is hard for an educated, decent, intelligent son of Western Europe to find his ideals here, unless he has first suffered some strange and horrid process of conversion which has changed all his values." - Keynes, 1931

This is from his essay, 'A short view of Russia'. One might also want to look at the essays, 'Am I a Liberal?' and 'Liberalism and Labour'. These three were all republished in Essays in Pursuasion. The latter two are more about party politics in Britain in the 1920s.

3.0 The General Theory, Including Drafts

Keynes distinguished, in drafts of the General Theory, between three types of economies. He designated the first type as a barter of co-operative economy. The second type is a neutral entrepreneur economy or a neutral economy, for short. This type is summarized by Marx's symbols C-M-C and acts like the first type. Agents exchange commodities for commodities, with money acting as an intermediary, but having no independent effect on the course of events. A monetary theory of production, though, deals with a money-wage economy, also known as an entrepreneur economy. This type is characterized with Marx's symbols M-C-M'. Money matters in all runs. I rely on secondary literature which I do not reference below.

Another parallelism between Keynes and Marx is in the choice of units:

"In dealing with the theory of employment I propose, therefore, to make use of only two fundamental units of quantity, namely, quantities of money-value and quantities of employment. The first of these is strictly homogeneous, and the second can be made so. For, in so far as different grades and kinds of labour and salaried assistance enjoy a more or less fixed relative remuneration, the quantity of employment can be sufficiently defined for our purpose by taking an hour's employment of ordinary labour as our unit and weighting an hour's employment of special labour in proportion to its remuneration; i.e. an hour of special labour remunerated at double ordinary rates will count as two units. We shall call the unit in which the quantity of employment is measured the labour-unit; and the money-wage of a labour-unit we shall call the wage-unit." -- Keynes. 1936.

The above seems close to Marx's theory of value.

4.0 After The General Theory

I pick two examples of others drawing on parallels between Keynes and Marx. Perhaps the above section heading is unfair to Kalecki. When he visited Cambridge, the Keynesians found that he was in on all their jokes. Kalecki divides the economy into a sector producing consumer goods and a sector producing investment goods. He was aware of Rosa Luxemburg, and Luxemburg built on Marx's schemes of simple and expanded reproduction.

Joan Robinson (1953) makes the point about units of measurement that I note above. She had a lot more to say about Marx.

5.0 Conclusion

Many have had lots more to say about the relationship of the ideas of Keynes and Marx. One question is whether Marx was wrong to claim that the development of capitalism will lead to its own demise, perhaps in the depth of a great depression. Does not Keynes provide the political tools to keep capitalism going indefinitely? Some Marxists rejected Keynesianismn on these grounds.

Another question is about long-run theory. Keynes' General Theory is not confined to the short run. No need exists for the labor market to eventually clear. Sraffa provides a theory of prices consistent with a non-clearing labor market, and he takes output as given. Should there not be a theory that integrates the approaches of Keynes and Sraffa? But this has been on the agenda for decades.

Lots more can be said.

References

## Tuesday, March 19, 2024

### Traditional And 'Perverse' Switch Points For Austrian And Neoclassical Economics

 Figure 1: The Wage-Rate of Profits Frontier
1.0 Introduction

This is one in a series of posts demonstrating that the change in the economic life of a machine at a switch point is independent of the change of the capital intensity of the technique at a switch point. I want to illustrate each entry in a two-by-two table in a previous post. The example in this post has two switch points. One fits the traditional Austrian and neoclassical stories, as in the entry in the upper-left of the table. The other switch point is 'perverse' in both ways, as in the entry in the lower right.

2.0 Technology

The example is of a 'one-good' economy, in which the produced commodity has a physical lifetime of three years when used in production. When the commodity is newly produced, it can also be used by households as a consumption good. I make the usual assumption of constant returns to scale. Each column of Table 1 shows the inputs for a production process operated at unit level. The corresponding columns show the outputs for each production process, again operated at a unit level. The efficiency of this machine varies over its lifetime.

 Input Process (I) (II) (III) Labor 30 180 39/2 New Widgets 1 0 0 One-Year Old Widgets 0 1 0 Two-Year Old Widgets 0 0 1

 Output Process (I) (II) (III) New Widgets 3 b1, 2 b1, 3 One-Year Old Widgets 1 0 0 Two-Year Old Widgets 0 1 0

The economic life of a machine may be less than its physical life. I assume free disposal. Three techniques can be defined:

• Alpha: The machine is discarded after one year.
• Beta: The machine is discarded after two years.
• Gamma: The machine is discarded after three years.
3.0 The Choice of Technique

A system of equations defines prices of production for each technique. For example, the following two equations characterize the Beta technique:

(1 + r) + 30 w = 3 + p2
p2 (1 + r) + 180 w = b1, 2

Wages are assumed to be paid out of the product. A common rate of profits is charged on the prices of the capital goods advanced. The revenues recorded on the right-hand side are as appropriate for joint production. The price of a two-year old machines do not appear in the equations. Under Beta, two-year old machines are discarded, and their price is zero.

Given an externally specified rate of profits, the system of equations for each system can be solved for the wage and the prices of the old machines that are used in the production processes for that technique. Figure 1, at the top of this post, graphs the wage curves for the three techniques, for the values of b1, 2 and b1, 2 examined in this post. The cost-minimizing technique at any rate of profits is the technique that contributes its wage curve to the outer frontier. At a (non-fluke) switch point, two techniques are cost-minimizing. This example is a reswitching example.

 Figure 2: The Prices of Old Machines

The prices of old machines provide an equivalent method of analyzing the choice of technique with fixed capital. For each technique, the price of a new machine is unity. Figure 2 graphs the price of old machines. The left panel shows the price of a one-year old machine for the Beta and Gamma techniques, the two techniques in which such machines are operated. The right panel shows the price of a two-year old machine for the Gamma technique. The Gamma technique is only cost-minimizing in the range up to the maximum rate of profits when the price of old machines is positive or zero. That is Gamma is cost minimizing outside the two indicated switch points.

Consider the range for the rate of profits between the switch point. Gamma is not cost-minimizing here, and the economic life of a machine is truncated from its physical life. Beta is not cost-minizing in this range either, for the price of a one-year old machine is negative, under the system of equations for Beta prices. Consequently, the Alpha technique is cost-minimizing between the indicated switch points. (I thought a bit about how to draw a flowchart for a market algorithm for this example.) The analysis of the choice of technique with the construction of the outer wage frontier yields the same results as an analysis of truncation based on negative prices for old machines.

Two switch points exist on the outer envelope of the wage curves. For the first switch point, Alpha is preferred at a slightly higher rate of profits, and Gamma is preferred at a slightly lower rate of profits. That is, a lower rate of profits around the switch point results in the operation of the machine for a longer economic life of a machine. It also results in a cot-minimizing technique requires a greater value of capital per worker, and an increase in output per head.

All of this is reversed at the second switch point. A lower rate of profits around this switch point is associated with a shorter economic life of the machine, a smaller value of capital per worker, and a decrease in output per head.

4.0 Conclusion

This post has filled in two entries in a two-by-two table. In these entries, either output per head and the economic life of a machine rise with a higher wage around a switch point. Or, in contrast to traditional marginalist and Austrian theory, they both fall with a higher wage around a switch point. This example is not yet sufficient to demonstrate that the economic life of a machine is independent of measures of capital-intensity, as used in mainstream marginalist economics.

## Saturday, March 16, 2024

### Elsewhere

• Matt McManus on Thomas Sowell.
• A review of Adam Shatz's biography of Frantz Fanon.
• Nathan Robinson interviews Kohei Saito on degrowth.
• I have not read Bob Rowthorn on neo-ricardianism in decades. I wish NLR made PDFs of old articles freely available.

## Monday, March 11, 2024

### To Do: Perverse Switch Points And The Economic Life Of A Machine

 Traditional Marginalist Story 'Perverse' Marginalist Story Traditional Austrian Story Negative real Wicksell effect, greater net output per worker Positive real Wicksell effect, smaller net output per worker Longer economic life of machine Longer economic life of machine 'Perverse' Austrian Story Negative real Wicksell effect, greater net output per worker Positive real Wicksell effect, smaller net output per worker Shorter economic life of machine Shorter economic life of machine

I have been thinking about perturbations of coefficients in a model of fixed capital. This research can be redirected to find examples to fill in the above two-by-two table. Under obsolete marginalist teaching, a lower rate of profits encourages firms to addopt more capital-intensive techniques. At least two measures of capital intensity are available. Burmeister champions a measure of real Wicksel effects. Böhm-Bawerk championed a measure of the period of production which I am identifying, in this context, with the economic life of a machine.

The upper-left entry is the only one that conforms to the traditional story with both measures. I want to show that all four entries are possible. By perturbing an example from Salvadore Badone, I can fill in three of the entries, all but the bottom right. By perturbing an example of a 'one good' model, I can fill in that square and repeat two others. I also have an example from Bertram Schefold. I'd like to find a single example with perturbations that can fill in all four squares.

I want to recall that this work complements the corn-tractor model from Ian Steedman. Around each switch point, a different type of tractor is produced in Steedman's model, unlike in these examples. Each tractor works at constant efficiency, while I allow efficiency to vary. We both look at variations of the economic lives of machines. And this analysis is examining an issue independent of capital-intensity, as usually argued about in the Cambridge Capital Controvery. Demonstrating this independence is rather the point of filling in the above table.

I need a survey of analyses of fixed capital that does not end with a pure fixed capital model. Or, at least, I need to summarize a paper from Biao Huang. Perhaps I can avoid such a survey by just citing a model of pure fixed capital for existence but otherwise de-emphasize it. My goal is to be as terse as possible, with illustrations.

I also need to say something about why economists of the Austrian school should care. It seems to me that such economists often say that they have long ago developed their theory where it no longer relies on aggregate measures or physical measures of capital-intensity. I want to assert that they have not succeeded and still implicitly rely on the intuition from previous theory. Saverio Fratini makes a similiar case. It seems to me that I just need to note the existence of these claims and argue that the economic life of machines is one aspect of the Austrian theory of capital.

A difficulty arises of where to publish this. My previous version was rejected from Metroeconomica. Their editors, reviewers, and readers are unlikely to be astonished by these claims. On the other hand, some editors and authors of mainstream journals would claim they have long ago moved to and then transcended abstract models which this sort of work does not address. Yet they continually have a non-articulated background intuition inconsistent with the theory of prices of production. Fabio Petri has long argued along these lines.

## Wednesday, March 06, 2024

### New Interpretations Of Marx

This post is basically complaining that I cannot keep up.

I think I am fairly informed on Karl Marx. I do not read German, and I have not even read some early works. My area of concentration is reading Capital as a work of mathematical economics, which cuts against the subtitle and, maybe, de-emphasizes a break with classical, especially, Ricardian political economy.

More generally, I thought Marx generally praises the tremendous increase of productivity brought about by the bourgeoisie. He downplays the accompanying environmental degradation. Imperialism extends capitalism into non-European colonies. Marx deplores the violence, but thinks rationalization of such societies is progress.

As I understand it, some of the literature below challenges these ideas. This is partly because of the current context. But it is also because of new texts brought into circulation by the second attempt at a Marx-Engels-Gesamtausgabe (MEGA2). David Ryazanov led the first attempt at MEGA. Stalin first dismissed him to internal exile, then killed him in one of his purges. I have not read him, but I gather Musto draws on Marx's reflections from visiting Algiers on a doctor's recommendation. Anderson, I guess, draws on journalistic writings. These two offers re-evaluate what Marx has to say about colonialism. I am currently reading Soren Mau. The instruments and violence and intellectual hegemony of those presently the interests of capitalists as universal interests help maintain the reproduction of capitalism. Mau looks at a third means for such reproduction.

And we also now have available a new translation of the first volume of Capital. From Heinrich, I learned that the structure of the first chapter was quite different in the first edition. Some turns of phrase, such as, "Moneybags must be so lucky", come from the Moore and Aveling's english translation.

Anyways, here are some recent works on Marx:

I think a tendency exists to treat capital as something like an emergent, over-arching subject. One can see this in writing from Ian P. Wright. Philip Mirowski argues markets are computing automata, and computers are often taken as models of the mind these days. Another book I want to consider reading is Benjamin Labatut, 2023, The Maniac, Penguin Random House. This is a novelization of the life of Johnny Von Neumann.

## Saturday, March 02, 2024

### Labor Values And Invariants

1.0 Introduction

This post is an attempt to work through some linear algebra that some have used to understand Karl Marx's Capital. I have recently explained how, in a simple model, prices of production are equal to labor values if the organic composition of capital does not vary among industries. That special case is the setting of volume 1.

In capitalism, workers rent themselves out to their employers. They work longer, under the dominion of capital, than needed to produce the commodities which they purchase with their wages. Marx explains the returns to ownership (profits, interest, rent, etc.) by the distinction between the use value and the exchange value of labor power.

This post removes the special case assumption. It considers certain relationships between the system of labor values and the system of prices of production. These relationships are highlighted towards the start of volume 3. I ignore Hegel, on his head or otherwise.

2.0 Quantity Flows

Suppose a capitalist economy is observed at a given point in time. n commodities are being produced, each by a separate industry. Suppose the technique in use can be characterized by a row vector a0 and a n x n square matrix A.

The jth element of a0 is the amount of labor directly employed in the jth industry in producing one unit of a commodity output from that industry. "We suppose labour to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have previously been reduced to equivalent differences in quantity so that each unit of labour receives the same wage…" - Piero Sraffa (1960).

The jth column of A is the goods used up in producing one unit of a commodity output. For example, suppose iron is produced by the first industry and steel is produced by the second industry. a1,2 is then the kilotons of iron needed to produce a kiloton of steel. Assume that every good enters directly or indirectly into the production of each commodity. Iron enters indirectly into the production of tractors if steel enters directly into the tractor industry. Assume a surplus product, also known as a net output, exists.

Let y be the column vector of net outputs and q the column vector of gross outputs, both in physical terms. In Leontief's work, y is taken as given. Gross outputs and net outputs are related as:

y = q - A q

Or:

q = (I - A)-1 y

The labor force needed to produce this net product is:

a0 q = a0 (I - A)-1 y = 1

I have taken units in which labor is measured to be such that this labor force is unity. Employment is such that the net output is produced, the capital goods in producing the net output are reproduced, the capital goods used in producing those capital goods are reproduced, and so on.

3.0 Labor Values

Let ej be the jth column of the identity matrix. The labor force needed to produce this net output is:

vj = a0 (I - A)-1 ej

That is, the (direct and indirect) labor needed to produce a net output of one unit of the jth commodity is vj. The row vector of labor values is:

v = a0 (I - A)-1

4.0 Prices of Production

At any time, market prices are such that different industries are making different rates of profits. Under competitive conditions, without barriers to entry in the various industries, a kind of leveling process is going on.

One can imagine a vector of prices such that this leveling process is already completed with the observed technique and wage. Let p be that row vector of prices of production, with all industries obtaining the same rate of profits.

I need an assumption about the composition of commodities purchased from the wage, w, since I want to explore the labor value embodied in the wage. Accordingly, assume that the wage is a proportion of the final product. The wage ranges from zero to unity, inclusive. The physical composition of the wage is w y. Wages are advanced. Define:

A*(w) = A + w y a0

I gather the vector operation at the end of the above expression is the outer product. Prices of production satisfy the equation in the following display:

p A*(w) (1 + r) = p

where r is the rate of profits. That is, p is a price vector consistent with the observed technique and wage.

By the Perron-Frobenius theorem, the eigenvalue of A*(w) with the maximum modulus is real, positive, and does not exceed unity. The corresponding rate of profits is non-negative. The eigenvector consists of all positive entries. Thus a solution exists for the above equation. Furthermore, the wage and the rate of profits are related by a decreasing function. The maximum wage occurs at a rate of profits of zero. The maximum rate of profits is finite and occurs at a wage of zero.

Prices of production have been found up to a scaling factor. They are generally not proportional to labor values, as Ricardo and Marx knew.

5.0 Invariants

The scale for prices of production can be fixed by specifying a numeraire. Consider, instead, the imposition of an identity between the system of labor values and the system of prices of production.

5.1 Case 1: Total Gross Output

The labor value of gross output is equal to the price of gross output if and only if:

v q = p q

Imposing the above condition fixes the scale for prices.

5.2 Case 2: Total Net Output

Alternatively, the labor value of net output is equal to the price of net output if and only if:

v y = p y = 1

I have taken advantage above of the scaling of units of labor time. This invariant is my favorite of the three invariants considered here.

5.3 Case 3: The Rate of Profits

The labor value of advanced capital is v A*(w) q, while its price is p A*(w) q. The labor value of profits is:

(1 - w) v y = (1 - w)

The rate of profits does not differ between the system of labor values and the system of prices if and only if:

(1 - w)/[v A*(w) q] = (1 - w) p y/[p A*(w) q]

6.0 Concluding Observation

The above post has defined three invariants, each equating a sum or ratio of labor values to the corresponding sum or ratio in the system of prices of production. Only one invariant can generally hold, though, in the given model. This has led to quite a bit of literature arguing that one of these or other invariants is central to Marx's argument.

Some have another approach. They adopt another model in which all three invariants hold. In fact, more than one such model has been developed.

An approach I find of interest looks at a special composition of final output. Whatever the composition of the final output, one can iterate by looking at the composition of the capital goods used in producing that final output. A number of iterations leads to a composite commodity of close to the output of something like Marx's industry of average organic composition of capital.

Or one can retain an interest in how labor is allocated among industries, while exploring prices of production with an arbitrary numeraire. The fundamental theorem of Marxism holds in this setting. Must one draw quantitative relationships between the system of labor values and the system of prices of production?

Others might want to explore the historical and empirical evolution of the parameters of the model in the post and related models.

Reference

## Saturday, February 24, 2024

### Utility Maximization A Tautology?

Economists proved over half a century ago that certain stories are unfounded in the theory. For example, one might think that if some workers are involuntarily unemployed, a drop in real wages would lead to a tendency for the labor market to clear. The Cambridge Capital Controversy revealed some difficulties. In response, some economists turned to the Arrow-Debrue-McKenzie model of intertemporal equilibria in which it is not clear that one could even talk about such concepts. The Mantel-Sonnenschein-Debreu theorem shows that this model lacks empirical content. Utility theory provides a closure for some models. Formally, one can demonstrate the existence of equilibria under certain assumptions. But existence does not get one very far.

My purpose of this post is to note that some saw utility theory as a useless tautology at the time of the marginal revolution:

"It is interesting, in this connection, that the earliest critics saw in the theory of marginal utility what we have called a behaviourist theory of choice ... and used exactly the same arguments against it which will be used below against this latter version. Thus [John] Cairnes wrote about Jevon's theory: 'What does it really amount to? In my apprehension to this, and no more - that value depends upon utility, and that utility is whatever affects value. In other words, the name "utility" is given to the aggregate of unknown conditions which determine the phenomenon, and then the phenomenon is stated to depend upon what this name stands for.' Jevon's theory was believed to say no more than this: 'that value was determined by the conditions which determine it - an announcement, the importance of which, even though presented under the form of abstruse mathematical symbols, I must own myself unable to discern'. Some Leading Principles of Political Economy, 1874, p. 15.

[John] Ingram took the same view in A History of Political Economy, 1888, ed. by Ely, 1915, p. 228 and passim. Cairnes, Ingram, and other early critics of marginal utility had, however, directed their criticism also against the mathematical method generally, and the discussion went soon into other channels. The marginalists met the criticism by claiming to be proponents of logical and mathematical method and their tautological psychology thus escaped its well-deserved criticism." -- Gunnar Myrdal (1953) The Political Element in the Development of Economic Theory (trans. by Paul Streeten, Routledge & Kegan Paul, p. 231.

Obviously, Cairnes and Ingram could not have known about results demonstrated a century later. Utility theory manages simultaneously to not say anything about market phenomena, to not be good armchair theorizing, and to be empirically false at the level of the individual.

## Monday, February 19, 2024

### Two Special Cases For The Labor Theory Of Value

1.0 Introduction

A simple labor theory of value holds in two special cases.

1. The rate of profits in the system of prices of production is zero.
2. The vector of direct labor coefficients is an eigenvector of the Leontief input-output matrix corresponding to the maximum eigenvalue.

I do not know if I've worked through this alone before. A more rigorous approach would prove the uniqueness of the solution.

2.0 The Setting

Suppose a capitalist economy is observed at a given point in time. n commodities are being produced, each by a separate industry. Suppose the technique in use can be characterized by a row vector a0 and a n x n square matrix A.

The jth element of a0 is the amount of labor directly employed in the jth industry in producing one unit of a commodity output from that industry. "We suppose labour to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have previously been reduced to equivalent differences in quantity so that each unit of labour receives the same wage…" - Piero Sraffa (1960). I guess the idea is that relative wages are more or less stable.

The jth column of A is the goods used up in producing one unit of a commodity output. For example, suppose iron is produced by the first industry and steel is produced by the second industry. a1,2 is then the kilotons of iron needed to produce a kiloton of steel. Assume that every good enters directly or indirectly into the production of each commodity. Iron enters indirectly into the production of tractors if steel enters directly into the tractor industry. Assume a surplus product, also known as a net output, exists.

2.1 Quantity Flows

Let y be the column vector of net outputs and q the column vector of gross outputs, both in physical terms. In Leontief's work, y is taken as given. Gross outputs and net outputs are related as:

y = q - A q

Or:

q = (I - A)-1 y

The labor force needed to produce this net product is:

L = a0 q = a0 (I - A)-1 y

One might as well take units in which labor is measured to be such that this labor force is unity. Employment is such that the net output is produced, the capital goods in producing the net output are reproduced, the capital goods used in producing those capital goods are reproduced, and so on.

2.2 Labor Values

Let ej be the jth column of the identity matrix. The labor force needed to produce this net output is:

vj = a0 (I - A)-1 ej

That is, the (direct and indirect) labor needed to produce a net output of one unit of the jth commodity is vj. The row vector of labor values is:

v = a0 (I - A)-1

(I could put an aside here about geometric series and an infinite sum of labor time, assuming the current technology was used forever in the past.)

The employment needed to produce a given net output is the sum of the labor values of the individual commodities in net output, v y. One can think of this post as showing one way of decomposing the observed net output and employed workers. With this way of thinking, no assumptions have been made about returns to scale.

Labor values support one way of doing accounting in models like this. One could ask about how much employment would have decreased or increased if final demand had been decreased or increased by some specified quantities of specified commodities.

2.3 Prices of Production

Take y as numeraire. At any time, market prices are such that different industries are making different rates of profits. Under competitive conditions, without barriers to entry in the various industries, a kind of leveling process is going on.

One can imagine a vector of prices such that this leveling process is already completed with the observed technique and wage. Let p be that row vector of prices of production, with all industries obtaining the same rate of profits:

p A (1 + r) + w a0 = p

where r is the rate of profits and w the wage. That is, p is a price vector consistent with the observed technique and wage. Since y is numeraire, one has:

p y = 1

The point is to show that prices of production are labor values in special cases.

3.0 The First Special Case: No Profits

Assume that the rate of profits is zero. The claim is that prices of production are labor values.

First, consider the equation for the numeraire:

v y = a0 (I - A)-1 y = a0 q

By assumption, the amount of labor employed is one unit. So using labor values for prices satisfies the equation for the numeraire. Furthermore, if the rate of profits is zero, the wage is unity. (One might do a bit of algebra here.)

I want to show:

v A + a0 = v

But this is true if and only if:

a0 = v (I - A)

Or:

a0 (I - A)-1 = v

But this is the definition of labor values. So if the rate of profits is zero, prices of production are labor values.

4.0 The Second Special Case: Equal Organic Compositions Of Capital

Suppose that:

a0 A = λ a0

where λ is the eigenvalue with the maximum modulus. By the Perron-Frobenius theorem, this eigenvalue is positive and less than unity. All of the elements of the vector of direct labor coefficients are positive.

Under this special case, the solution to the price equations is:

p = v

and:

r = R (1 - w)

where:

R = (1 - λ)/λ

Suppose:

v = (1/(1 - λ)) a0

By the definition of labor values:

v (I - A) = a0

Or:

(1/(1 - λ)) (a0 - a0 A) = a0

Using the special case assumptions, one has:

(1/(1 - λ)) (a0 - λ a0) = a0

Thus, in this special case, labor values are directly proportional to direct labor coefficients.

I want to show:

v A(1 + ((1 - λ)/λ)(1 - w)) + w a0 = v

Or:

(1/(1 - λ)) a0 A((1/λ) - ((1 - λ)/λ)w) + w a0 = (1/(1 - λ)) a0

Or:

(λ/(1 - λ)) a0((1/λ) - ((1 - λ)/λ)w) + w a0 = (1/(1 - λ)) a0

But the left-hand side is simply (1/(1 - λ)) a. So labor values are prices of production in the special case. Furthermore, prices of production do not vary with distribution in this case.

5.0 Conclusion

Suppose the organic composition of capital does not vary among industries. That is, the vector of direct labor coefficients is the specified eigenvector of the Leontief input-output matrix. So prices of production associated with the observed technique and net output are labor values. How does capital obtain profits in this special case? This is Marx's question in the first volume of Capital.

Objections to the lack of realism of this special case and to the conditions needed to define prices of production are not on point. If you have a theory explaining returns to capital, it should apply in this special case. The question, I gather, is more salient if you think there is something fair about commodities being priced at labor values.

The answer cannot be entrepreneurship, since the returns to entrepreneurship are a non-equilibrium phenomenon. For half a century, economists have known that the answer cannot be supply and demand of capital. For that answer, one must have a unit in which capital can be measured independently of prices. I suppose one can create a self-consistent model with intertemporal utility maximization by households, including households whose income is entirely from returns to ownership. But the mechanics of how such a model works disagree with traditional notions of substitution and scarcity.

A valid answer, it seems to me, must invoke some concept of power. This answer need not be exactly Marx's. The Post Keynesian theory of growth, in which large corporations set the rate of growth, might be part of an answer applicable in some times and places.