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.