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. a_1,2 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 < λ_PF(A) < 1

where λ_PF(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 < λ_PF(A⁺) < 1

In fact, the rate of profits is:

r = 1/λ_PF(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 Product

Suppose one or more of the elements of A⁺ decrease. Then 1 - λ_PF(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 - λ_PF(B⁺) > 1 - λ_PF(A⁺)

Suppose further that:

1 - λ_PF(B) < 1 - λ_PF(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:

g_max = 1/λ_PF(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, λ_PF(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.

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.

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?

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, b_1,2 and b_1,3, 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, b_1,2 and b_1,3. 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 b_1,2 and b_1,3 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.

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

• Michal Kalecki, 1943. Political aspects of full employment, Political Quarterly.
• John Maynard Keynes, 1931. Essays in Persuasion.
• John Maynard Keynes, 1936. The General Theory of Employment, Interest and Money. Harcourt, Brace and Company.
• Joan Robinson, 1953. An open letter from a Keynesian to a Marxist.

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.

Table 1: Inputs for The Technology

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

Table 2: Outputs for The Technology

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 = b_{1, 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 b_{1, 2} and b_{1, 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.

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.

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

Table 1: Lower Rate of Profits Around A Switch Point

	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.

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:

• William Clare Roberts, 2016, Marx's Inferno: The Political Theory of Capital, Princeton University Press.
• Kevin B. Anderson, 2016, Marx at the Margins: On Nationalism, Ethnicity, and Non-Western Socities, University of Chicago Press.
• Marcello Musto, 2020, The Last Years of Karl Marx: An Intellectual Biography, Stanford University Press.
• Michael Heinrich, 2021, How to Read Marx's Capital: Commentary and Explanations on the Beginning Chapters, Monthly Review Press.
• Soren Mau, 2023, Mute Compulsion: A Marxist Theory of the Economic Power of Capital, Verso.
• Kohei Saito, 2023, Marx in the Anthrocene: Towards the Idea of Degrowth Communism, Cambridge University Press.
• Adrian Johnston, 2024, Infinite Greed: Inhuman Selfishness of Capital, Columbia University Press.
• Karl Marx, 2024, Capital: Critique of Political Economy: Volume 1, Edited and translated by Paul Reitter, Edited by Paul North, Preface by Wendy Brown, Afterword by William Clare Roberts, Princeton University Press.

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.

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. a_1,2 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 e_j be the jth column of the identity matrix. The labor force needed to produce this net output is:

v_j = a₀ (I - A)^-1 e_j

That is, the (direct and indirect) labor needed to produce a net output of one unit of the jth commodity is v_j. 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

• Francis Seton. 1957. The 'Transformation Problem'. Review of Economic Studies 24 (3): 149-160.

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.

Elsewhere

• Solidarity Federation
• Francisco Nunes-Pereira and Mário Graça Moura. On the survival of a flawed theory of capital: mainstream economics and the Cambridge capital controversies, Cambridge Journal of Economics. Concludes mainstream economists continue to use flawed aggregate production functions because of an instrumentalist methodology and conformism.
• . Colin Rogers. Whitehead’s fallacy of misplaced concreteness and the unfortunate uselessness of all monetary-macro theory micro-founded on Walrasian-Pareto general equilibrium theory, Cambridge Journal of Economics. Argues that neo-Walrasian general equilibrium theory cannot support monetary theory, despite the practices of mainstream economists.
• Theo Santini and Ricardo Azevedo Araujo. Vertical integration, technical progress and structural change,Cambridge Journal of Economics. Builds on the theory of vertical integration to develop measures of technical progress.
• The Levy Institute podcast. John Harvey talks to James Galbraith in the first episode.
• Reading Capital with comrades, a podcast. I'm not sure lectures work in this format.
• Pitchfork Economics, a podcast.
• An interview with Nick Hanauer and his new co-written book.

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. a_1,2 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 e_j be the jth column of the identity matrix. The labor force needed to produce this net output is:

v_j = a₀ (I - A)^-1 e_j

That is, the (direct and indirect) labor needed to produce a net output of one unit of the jth commodity is v_j. 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.

Orwell Remembers Revolution

Pablo Picasso's Guernica

This post has long quotations, as is typical of a commonplace book.

Where did socialism work? In Barcelona, Spain, and, more generally Aragon and Catalonia, from August 1936 to April 1937. This was the anarchist version, and was resisted by all governments, including the Soviet Union.

The right staged their coup against the Republic in July 1936. Orwell went to Spain in December, and he wanted to kill fascists. He describes it as almost happenstance that he ended up in the militia under the Party Of Marxist Unity (POUM). This party contained, among others, followers of Trotsky and did not follow Stalin's line. Orwell describes how he found Barcelona:

"This was in late December 1936, less than seven months ago as I write, and yet it is a period that has already receded into enormous distance. Later events have obliterated it much more completely than they have obliterated 1935, or 1905, for that matter. I had come to Spain with some notion of writing newspaper articles, but I had joined the militia almost immediately, because at that time and in that atmosphere it seemed the only conceivable thing to do. The Anarchists were still in virtual control of Catalonia and the revolution was still in full swing. To anyone who had been there since the beginning it probably seemed even in December or January that the revolutionary period was ending; but when one came straight from England the aspect of Barcelona was something startling and overwhelming. It was the first time that I had ever been in a town where the working class was in the saddle. Practically every building of any size had been seized by the workers and was draped with red flags or with the red and black flag of the Anarchists; every wall was scrawled with the hammer and sickle and with the initials of the revolutionary parties; almost every church had been gutted and its images burnt. Churches here and there were being systematically demolished by gangs of workmen. Every shop and café had an inscription saying that it had been collectivized; even the bootblacks had been collectivized and their boxes painted red and black. Waiters and shop-walkers looked you in the face and treated you as an equal. Servile and even ceremonial forms of speech had temporarily disappeared. Nobody said 'Senior' or 'Don' or even 'Usted'; everyone called everyone else 'Comrade' and 'Thou', and said 'Salud!' instead of 'Buenos dias'. Tipping was forbidden by law; almost my first experience was receiving a lecture from a hotel manager for trying to tip a lift-boy. There were no private motor-cars, they had all been commandeered, and all the trams and taxis and much of the other transport were painted red and black. The revolutionary posters were everywhere, flaming from the walls in clean reds and blues that made the few remaining advertisements look like daubs of mud. Down the Ramblas, the wide central artery of the town where crowds of people streamed constantly to and fro, the loudspeakers were bellowing revolutionary songs all day and far into the night. And it was the aspect of the crowds that was the queerest thing of all. In outward appearance it was a town in which the wealthy classes had practically ceased to exist. Except for a small number of women and foreigners there were no 'well-dressed' people at all. Practically everyone wore rough working-class clothes, or blue overalls, or some variant of the militia uniform. All this was queer and moving. There was much in it that I did not understand, in some ways I did not even like it, but I recognized it immediately as a state of affairs worth fighting for. Also I believed that things were as they appeared, that this was really a workers' State and that the entire bourgeoisie had either fled, been killed, or voluntarily come over to the workers' side; I did not realize that great numbers of well-to-do bourgeois were simply lying low and disguising themselves as proletarians for the time being." (Chapter I)

Orwell finds equality within the militia, despite ranks and orders:

"At this time and until much later the Catalan militias were still on the same basis as they had been at the beginning of the war. In the early days of Franco's revolt the militias had been hurriedly raised by the various trade unions and political parties; each was essentially a political organization, owing allegiance to its party as much as to the central Government. When the Popular Army, which was a 'non-political' army organized on more or less ordinary lines, was raised at the beginning of 1937, the party militias were theoretically incorporated in it. But for a long time the only changes that occurred were on paper; the new Popular Army troops did not reach the Aragon front in any numbers till June, and until that time the militia-system remained unchanged. The essential point of the system was social equality between officers and men. Everyone from general to private drew the same pay, ate the same food, wore the same clothes, and mingled on terms of complete equality. If you wanted to slap the general commanding the division on the back and ask him for a cigarette, you could do so, and no one thought it curious. In theory at any rate each militia was a democracy and not a hierarchy. It was understood that orders had to be obeyed, but it was also understood that when you gave an order you gave it as comrade to comrade and not as superior to inferior. There were officers and N.C.O.S. but there was no military rank in the ordinary sense; no titles, no badges, no heel-clicking and saluting. They had attempted to produce within the militias a sort of temporary working model of the classless society. Of course there was no perfect equality, but there was a nearer approach to it than I had ever seen or than I would have thought conceivable in time of war.

But I admit that at first sight the state of affairs at the front horrified me. How on earth could the war be won by an army of this type? It was what everyone was saying at the time, and though it was true it was also unreasonable. For in the circumstances the militias could not have been much better than they were. A modern mechanized army does not spring up out of the ground, and if the Government had waited until it had trained troops at its disposal, Franco would never have been resisted. Later it became the fashion to decry the militias, and therefore to pretend that the faults which were due to lack of training and weapons were the result of the equalitarian system. Actually, a newly raised draft of militia was an undisciplined mob not because the officers called the private 'Comrade' but because raw troops are always an undisciplined mob. In practice the democratic 'revolutionary' type of discipline is more reliable than might be expected. In a workers’ army discipline is theoretically voluntary. It is based on class-loyalty, whereas the discipline of a bourgeois conscript army is based ultimately on fear. (The Popular Army that replaced the militias was midway between the two types.) In the militias the bullying and abuse that go on in an ordinary army would never have been tolerated for a moment. The normal military punishments existed, but they were only invoked for very serious offences. When a man refused to obey an order you did not immediately get him punished; you first appealed to him in the name of comradeship. Cynical people with no experience of handling men will say instantly that this would never 'work', but as a matter of fact it does 'work' in the long run. The discipline of even the worst drafts of militia visibly improved as time went on. In January the job of keeping a dozen raw recruits up to the mark almost turned my hair grey. In January the job of keeping a dozen raw recruits up to the mark almost turned my hair grey. In May for a short while I was acting-lieutenant in command of about thirty men, English and Spanish. We had all been under fire for months, and I never had the slightest difficulty in getting an order obeyed or in getting men to volunteer for a dangerous job. 'Revolutionary' discipline depends on political consciousness — on an understanding of why orders must be obeyed; it takes time to diffuse this, but it also takes time to drill a man into an automaton on the barrack-square. The journalists who sneered at the militia-system seldom remembered that the militias had to hold the line while the Popular Army was training in the rear. And it is a tribute to the strength of 'revolutionary' discipline that the militias stayed in the field-at all. For until about June 1937 there was nothing to keep them there, except class loyalty. Individual deserters could be shot - were shot, occasionally - but if a thousand men had decided to walk out of the line together there was no force to stop them. A conscript army in the same circumstances - with its battle-police removed - would have melted away. Yet the militias held the line, though God knows they won very few victories, and even individual desertions were not common. In four or five months in the P.O.U.M. militia I only heard of four men deserting, and two of those were fairly certainly spies who had enlisted to obtain information. At the beginning the apparent chaos, the general lack of training, the fact that you often had to argue for five minutes before you could get an order obeyed, appalled and infuriated me. I had British Army ideas, and certainly the Spanish militias were very unlike the British Army. But considering the circumstances they were better troops than one had any right to expect." (Chapter III)

When Orwell gets leave in April 1937, he finds Barcelona is back to normal, but with some omnious overturns. He reflects on his time in the POUM militia:

"The essential point is that all this time I had been isolated - for at the front one was almost completely isolated from the outside world: even of what was happening in Barcelona one had only a dim conception - among people who could roughly but not too inaccurately be described as revolutionaries. This was the result of the militia-system, which on the Aragon front was not radically altered till about June 1937. The workers' militias, based on the trade unions and each composed of people of approximately the same political opinions, had the effect of canalizing into one place all the most revolutionary sentiment in the country. I had dropped more or less by chance into the only community of any size in Western Europe where political consciousness and disbelief in capitalism were more normal than their opposites. Up here in Aragon one was among tens of thousands of people, mainly though not entirely of working-class origin, all living at the same level and mingling on terms of equality. In theory it was perfect equality, and even in practice it was not far from it. There is a sense in which it would be true to say that one was experiencing a foretaste of Socialism, by which I mean that the prevailing mental atmosphere was that of Socialism. Many of the normal motives of civilized life - snobbishness, money-grubbing, fear of the boss, etc. - had simply ceased to exist. The ordinary class-division of society had disappeared to an extent that is almost unthinkable in the money-tainted air of England; there was no one there except the peasants and ourselves, and no one owned anyone else as his master. Of course such a state of affairs could not last. It was simply a temporary and local phase in an enormous game that is being played over the whole surface of the earth. But it lasted long enough to have its effect upon anyone who experienced it. However much one cursed at the time, one realized afterwards that one had been in contact with something strange and valuable. One had been in a community where hope was more normal than apathy or cynicism, where the word 'comrade' stood for comradeship and not, as in most countries, for humbug. One had breathed the air of equality. I am well aware that it is now the fashion to deny that Socialism has anything to do with equality. In every country in the world a huge tribe of party-hacks and sleek little professors are busy 'proving' that Socialism means no more than a planned state-capitalism with the grab-motive left intact. But fortunately there also exists a vision of Socialism quite different from this. The thing that attracts ordinary men to Socialism and makes them willing to risk their skins for it, the 'mystique' of Socialism, is the idea of equality; to the vast majority of people Socialism means a classless society, or it means nothing at all. And it was here that those few months in the militia were valuable to me. For the Spanish militias, while they lasted, were a sort of microcosm of a classless society. In that community where no one was on the make, where there was a shortage of everything but no privilege and no boot-licking, one got, perhaps, a crude forecast of what the opening stages of Socialism might be like. And, after all, instead of disillusioning me it deeply attracted me. The effect was to make my desire to see Socialism established much more actual than it had been before. Partly, perhaps, this was due to the good luck of being among Spaniards, who, with their innate decency and their ever-present Anarchist tinge, would make even the opening stages of Socialism tolerable if they had the chance." (Chapter VIII)

Here is something about the atomsphere:

"But besides all this there was the startling change in the social atmosphere - a thing difficult to conceive unless you have actually experienced it. When I first reached Barcelona I had thought it a town where class distinctions and great differences of wealth hardly existed. Certainly that was what it looked like. 'Smart' clothes were an abnormality, nobody cringed or took tips, waiters and flower-women and bootblacks looked you in the eye and called you 'comrade'. I had not grasped that this was mainly a mixture of hope and camouflage. The working class believed in a revolution that had been begun but never consolidated, and the bourgeoisie were scared and temporarily disguising themselves as workers. In the first months of revolution there must have been many thousands of people who deliberately put on overalls and shouted revolutionary slogans as a way of saving their skins. Now things were returning to normal." (Chapter IX)

And then on 3 May 1937, the Assault and Civil Guards attacked the anarchists in Barcelona, starting at the Telephone Exchange which the Anarchists ran. The militia for the POUM was, more or less, on the side of the anarchists. The communists were on the side of suppressing the workers' revolution. The May days changed Orwell's plans and how you got along with others:

"In the hotel the horrible atmosphere of suspicion and hostility had grown worse now that the fighting was over. In the face of the accusations that were being flung about it was impossible to remain neutral. The posts were working again, the foreign Communist papers were beginning to arrive, and their accounts of the fighting were not only violently partisan but, of course, wildly inaccurate as to facts. I think some of the Communists on the spot, who had seen what was actually happening, were dismayed by the interpretation that was being put upon events, but naturally they had to stick to their own side. Our Communist friend approached me once again and asked me whether I would not transfer into the International Column.

I was rather surprised. 'Your papers are saying I'm a Fascist,' I said. 'Surely I should be politically suspect, coming from the P.O.U.M.'

'Oh, that doesn't matter. After all, you were only acting under orders.'

I had to tell him that after this affair I could not join any Communist-controlled unit. Sooner or later it might mean being used against the Spanish working class. One could not tell when this kind of thing would break out again, and if I had to use my rifle at all in such an affair I would use it on the side of the working class and not against them. He was very decent about it. But from now on the whole atmosphere was changed. You could not, as before, 'agree to differ' and have drinks with a man who was supposedly your political opponent. There were some ugly wrangles in the hotel lounge. Meanwhile the jails were already full and overflowing. After the fighting was over the Anarchists had, of course, released their prisoners, but the Civil Guards had not released theirs, and most of them were thrown into prison and kept there without trial, in many cases for months on end. As usual, completely innocent people were being arrested owing to police bungling." (Chapter X)

Orwell goes back to the front, is shot, and goes to the hospital in Barcelona. He is lucky to get out of Spain without, at least, being arrested by the Republican government.

I want to recall that chapters V and X1 are Orwell's attempt to give a historical overview, that is, they are about politics, no longer a memoirist's eyewitness testimony.

Variation On An Example From Schefold

Figure 1: Variation in the Economic Life of a Machine with Technical Progress

This post varies the coefficients of production in an example from Bertram Schefold. I wanted to have 'nice' fractions at a time of zero. Qualitatively, this looks like a previous post.

Reviewers for a recent rejection of an article with another fixed capital example objected to this type of model. I need to relate technical progress to a well-known type (Harrod-neutral, Marx-biased, or whatever) or produce some evidence that this sort of modeling is reasonable.

Table 1 presents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.

Table 1: Coefficients of Production for The Technology

Input	Process
	(I)	(II)	(III)
Labor	(7/25) e- σ t	3 e- φ t	(14/5) e- φ t
Corn	(4/25) e- σ t	(4/25) e- φ t	(2/3) e- φ t
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

The managers of firms need not run the machine for two years. They could discard the machine after only one year. (I assume free disposal.) I call Alpha the technique in which the machine is run for one year, and Beta the technique in which the machine is run for its full physical year of two years. The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative. Prices are found as prices of production, with an external specification of the wage or the rate of profits.

Figure 1 above and Table 2 below illustrate the analysis of the choice of technique. Until time reaches the pattern over the axis for the rate of profits, it is cost-minimizing to operate the machine for only one year. In Region 2, the machine is operated for two years when wages are low, and for one year when wages are higher. Region 3 is an example of reswitching. Eventually, it is cost-minimizing to operate the machine for two years, for all feasible wages.

Table 2: The Wage Frontier in Selected Regions in Parameter Space

Region	Switch Points	Cost-Minimizing Techniques
1	None	Alpha
2	One	Beta, Alpha
3	Two	Beta, Alpha, Beta
4	None	Beta
5	One	Alpha, Beta

Figure 2 partitions the parameter space such that the characteristics of the variations in the choice of technique technique do not vary within each region. Each region is bounded by thick (non-dotted) lines. Suppose that technical progress in producing and using machines is steady. That is, σ and φ have some fixed values. The dotted line in Figure 2 illustrates a path in logical time for such a thought experiment. The 45 degree line corresponds to the case in which σ and φ are equal, as in Figure 1. The numbering of the regions in Figures 1 and 2 and in Table 2 correspond.

Figure 2: A Parameter Space

Stepping through this example does not provide any qualitative differences from previous posts. Reswitching in models of fixed capital can be manifested as the return of a period of truncation. Around a switch point, a lower wage need not be associated with greater employment for a given net output. And so on.

Bertrand Russell On Bolshevism In 1920

Many to the left of liberals, that is, socialists, communists, and anarchists of various stripes, were opposed to the Soviet Union since its founding. Another example is Bertrand Russell. This is from the preface to his The Practice and Theory of Bolshevism, first printed in November 1920:

"The Russian Revolution is one of the great heroic events of the world's history. It is natural to compare it to the French Revolution, but it is in fact something of even more importance. It does more to change daily life and the structure of society: it also does more to change men's beliefs. The difference is exemplified by the difference between Marx and Rousseau: the latter sentimental and soft, appealing to emotion, obliterating sharp outlines; the former systematic like Hegel, full of hard intellectual content, appealing to historic necessity and the technical development of industry, suggesting a view of human beings as puppets in the grip of omnipotent material forces. Bolshevism combines the characteristics of the French Revolution with those of the rise of Islam; and the result is something radically new, which can only be understood by a patient and passionate effort of imagination.

Before entering upon any detail, I wish to state, as clearly and unambiguously as I can, my own attitude towards this new thing.

By far the most important aspect of the Russian Revolution is as an attempt to realize Communism. I believe that Communism is necessary to the world, and I believe that the heroism of Russia has fired men's hopes in a way which was essential to the realization of Communism in the future. Regarded as a splendid attempt, without which ultimate success would have been very improbable, Bolshevism deserves the gratitude and admiration of all the progressive part of mankind.

But the method by which Moscow aims at establishing Communism is a pioneer method, rough and dangerous, too heroic to count the cost of the opposition it arouses. I do not believe that by this method a stable or desirable form of Communism can be established. Three issues seem to me possible from the present situation. The first is the ultimate defeat of Bolshevism by the forces of capitalism. The second is the victory of the Bolshevists accompanied by a complete loss of their ideals and a régime of Napoleonic imperialism. The third is a prolonged world-war, in which civilization will go under, and all its manifestations (including Communism) will be forgotten.

It is because I do not believe that the methods of the Third International can lead to the desired goal that I have thought it worth while to point out what seem to me undesirable features in the present state of Russia. I think there are lessons to be learnt which must be learnt if the world is ever to achieve what is desired by those in the West who have sympathy with the original aims of the Bolsheviks. I do not think these lessons can be learnt except by facing frankly and fully whatever elements of failure there are in Russia. I think these elements of failure are less attributable to faults of detail than to an impatient philosophy, which aims at creating a new world without sufficient preparation in the opinions and feelings of ordinary men and women.

But although I do not believe that Communism can be realized immediately by the spread of Bolshevism, I do believe that, if Bolshevism falls, it will have contributed a legend and a heroic attempt without which ultimate success might never have come. A fundamental economic reconstruction, bringing with it very far-reaching changes in ways of thinking and feeling, in philosophy and art and private relations, seems absolutely necessary if industrialism is to become the servant of man instead of his master. In all this, I am at one with the Bolsheviks; politically, I criticize them only when their methods seem to involve a departure from their own ideals." -- Bertrand Russell

I have not got very far into this short book itself. Some of the above sounds to me a bit like Richard Wolff. The Soviet Union was an experiment, conducted at great human cost. We should learn from its successes and failures. We can and will do better.