Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Roger Garrison On The Inadequacy Of Hayekian Triangles

Hayek introduced his triangles in his lectures for his book Prices and Production. The first edition was in 1931 and the second in 1935. He attempted a more general treatment of capital theory in his 1941 book, The Pure Theory of Capital. I want to claim that Hayek knew that he could not draw his triangles under these more general assumptions. And that he knew that capital theory needed more development than he was able to give.

Jack Birner knows this, although I do not know a reference off-hand. I want to say that Roger Garrison knows this too, that he presents his diagrams of trianlges in his 2001 book, Time and Money: The Macroeconomics of Capital Structure, mainly for their heuristic and pedagogical value. He says something like this in the following:

"...The widely recognized but rarely understood Hayekian triangle, introduced in his 1931 lectures at the London Schoold of Economics, were subsequently published (in 1931 with a second edition in 1935) as Prices and Production. The triangle, described in the second lecture ... is a heuristic device that gives analytical legs to a theory of business cycles first offered by Ludwig von Mises... Triangles of different shapes provide a convenient but highly stylized way of describing changes in the intertemporal pattern of the economy's capital structure.

In retrospect, we see that the timing of Hayek's invitation to lecture at the London School of Economics takes on a special significance. We learn from the preface of the subsequent book that had the invitation come earlier, he couldn't have delivered those lectures; had it come later, he probably wouldn't have delivered them.

(The invitation) came at a time when I had arrived at a clear view of the outlines of a theory of industrial fluctuations but before I had elaborated it in full detail or even realized all the difficulties which such an elaboration presented. (Hayek, [1935] 1967: vii)

Hayek mentions plans for a more complete exposition and indicates that his capital theory would have to more developed in much greater detail and adapted to the complexities of the real world before it could serve as a satisfactory basis for theorizing about cyclical fluctuations.

A decade after the London lectures, the more complete exposition took form as The Pure Theory of Capital (1941). In this book Hayek fleshed out the earlier formulations and emphasized the centrality of the 'capital problem' in questions about the market's ability to coordinate economic activities over time. The 'pure' in the title meant 'preliminary to the introduction of monetary considerations.' Through some 450 pages in length, the book only achieved only the first half of the original objective. The final sixty pages of the book did contain a 'condensed and sketchy' (p. viii) treatment of the rate of interest in a monetary economy, but the task of retelling the story in Prices and Production in the context of the Pure Theory was put off and ultimately abandoned. The onset of the war was the proximate reason for cutting the project short; Hayek's exhaustion and waning interest in business-cycle issues - and his heightened interest in the broader issues of political philosophy - account for his never returning to the task. In later years, he acknowledged that Austrian capital theory effectively ended with his 1941 book and lamented that no one else has taken up the task that he originally set for himself (Hayek, 1994: 96).

More fully developing the Austrian theory of the business cycle came to be synonymous with writing the follow-on volume to Hayek's Pure Theory. Many a graduate student has imagined himself undertaking this very project, only to abandon the idea even before the enormity of the task was fully comprehended. Thus, while the comparatively simple relationships of capital-free Keynesian theory captured the attention of the economics profession, the inherently complex relationships of Austrian theory languished.

Time and Money is not the sequel to Hayek's Pure Theory. Rather, the ideas and graphical construction in the present volume take the original Hayekian triangle of Prices and Production to be the more appropriate point of departure for creating a capital-based macroeoconomics. The trade-off between simplicity and realism is struck in favour of simplicity. Hayek's triangles allow us to make a graphical statement that there is a capital structure and that its intertemporal profile can change. This statement enables the Austrian theory to make a quantum leap beyond the competing theories that ignore capital altogether or that treat capital as a one-dimensional magnitude." -- Roger Garrison (2001: 10-11).

I think J. R. Hicks' Value and Capital is an attempt to formalize some ideas from Hayek's capital theory. Garrison does not mentioned the possibility that maybe the formalization of a general theory along Austrian lines is impossible.

Anyways, here is an attempt to fairly shorten the quotation:

Hayek ... indicates that his capital theory would have to more developed in much greater detail ... before it could serve as a satisfactory basis for theorizing about cyclical fluctuations... The final sixty pages of [Hayek 1941] did contain a 'condensed and sketchy' (p. viii) treatment of the rate of interest in a monetary economy, but the task of retelling the story in Prices and Production in the context of the Pure Theory was put off and ultimately abandoned... More fully developing the Austrian theory of the business cycle came to be synonymous with writing the follow-on volume to Hayek's Pure Theory... Time and Money is not the sequel to Hayek's Pure Theory. Rather, the ideas and graphical construction in the present volume take the original Hayekian triangle of Prices and Production to be the more appropriate point of departure... -- Roger Garrison (2001: 10-11).

Elsewhere

• Samuel Bowles on how "Marx’s representation of the power relationship between capital and labour in the firm is an essential insight"
• Economics in the Rear-View Mirror on renewal of Paul Sweezy's instructorship.
• Manoucher Parvin, in 1992, asks, "Is teaching neoclassical economics as the science of economics moral?"
• Alexander Douglas on micro-founded macro.
• A review of Fredric Jameson's new book, in Jacobin earlier this month.
• A review of Fredric Jameson's new book, in the September issue of Harper's.

The Production Of Commodities And The Structure Of Production: An Example

Figure 1: Wage Curves for the Two Techniques

1.0 Introduction

Economists following the Austrian school often represent the structure of production with Hayekian triangles (Hayek 1931, Rothbard 1962, Skousen 1990, Garrison 2001, Machaj 2017). Typically, goods of the highest order are produced with unaided labor or other unproduced original resources. (Fillieule (2007), in which a Hayekian triangle is constructed with an infinite stream of unproduced inputs, is an exception.) Rarely, is a model of the production of commodities by means of commodities, as in Sraffa (1960), considered. This post illustrates how to construct a Hayekian triangle with such a model, in the case with circulating capital.

It also illustrates how the use of such triangles to tell the stories that Austrian economists want to tell cannot be sustained.

This post uses one special case to consider another. A model of the production of commodities with only circulating capital and one original factor of production, labor, is used to examine Hayekian triangles, which cannot be rigorously constructed in more general cases (Hayek 1941). This approach is consistent with the research strategy recommended by Harwick (2022). Harwick's argues that capital theory needs to consider exemplary special cases, not begin with the most general case. Here, the most general cases would be general joint production and flow-input, flow-output models. (Hayek 1941 seems to be the source for the classification of inputs as point input or flow input and outputs as point output or flow output.)

Other approaches to capital theory by economists following the Austrian school, such as the use of the financial measure of Duration by Lewin & Cachanosky (2021), are not treated in this post. Fratini (2019) criticizes this financial approach, approach based on reswitching.

2.0 Technology and Net Output

Suppose technology is as characterized by the coefficients of production in Table 1. All techniques are characterized by single production, no fixed capital, and no joint production. In the Alpha technique, the first corn-producing process is operated. The second corn-producing process is operated in the Beta technique. The ale-producing process is operated in both techniques.

Table 1: Regions

Input	Corn Industry		Ale Industry
	Process I	Process II	Process III
Labor	1 person-yr.	275/464 person-yrs.	1 person-yr.
Corn	1/10 kilo-bushels	113/232 kilo-bushels	2 kilo-bushels
Ale	1/40 kilo-liters	1/800 kilo-liters	2/5 kilo-liters
OUTPUTS	1 kilo-bushel	1 kilo-bushel	1 kilo-liter

Each column in the Leontief matrix and corresponding direct labor coefficient defines a production process. Each process exhibits constant returns to scale and requires a year to complete. Each of the produced commodities are available at the end of the year. All commodities enter, either directly or indirectly, into the production of all commodities and the economy is productive. Labor is directly required to operate each process.

This analysis takes the proportions in the net product as given. These proportions are specified by a column vector, as in Table 2. This numeraire is the net product or net output of Sraffa's standard system for the Alpha technique. This special case has implications for the shape of Hayekian triangles, as seen below.

Table 2: The Numeraire

Commodity	Amount
Corn	d1 = (337 - 29 (29)1/2)/455 kilo-bushels
Ale	d2 = (17 + 25 (29)1/2)/1,820 kilo-kiters

The capital goods required as inputs are found by solving the model. The money supply is here assumed to adjust as needed. The construction of a traverse from one equilibrium to another is a difficult question. In a disequilibrium process, one would need to consider disappointed expectations and inconsistent plans, for example. Mainstream economists have a theory of non-steady state equilibrium paths. Rosser (1983) argues that reswitching manifests as a cusp catastrophe in this theory. Gram & Harcourt (2017) note some issues raised by taking the theory of equilibrium paths as an answer to the reswitching controversy. In the theory, any time to get into equilibrium is too long.

The numeric example is constructed on the basis of the following assumptions, in addition to those already given for characterizing the available technology:

• Stationary states are compared.
• Only one original factor of production, labor, is used.
• One person-year is employed. More generally, full employment is assumed, with a given size of the labor force.
• Wages are paid at the end of the year, not advanced with the payments for capital goods at the beginning of the year.

Hayekian triangles cannot be constructed under the more general assumptions of Hayek (1941).

3.0 Price Systems

Prices and distribution are specified with one degree of freedom, by a given technique. Prices must satisfy the following equations, given the technique, for the same rate of profits to be obtained in both industries:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_{0, 1} = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + w a_{0, 2} = p₂

The following equation follows from the definition of the numeraire.

p₁ d₁ + p₂ d₂ = 1

Adam Smith called these prices 'natural prices'. David Ricardo called them 'prices of production'. Alfred Marshall called them 'normal prices'. At any rate, they define the appropriate price system for an economy in a synchronized state, as in this thought experiment.

The model in this article is open. Distribution, including the wage and the interest rate, are not determined. Dumenil & Levy (1985) present a model in which workers consume their entire wage and the interest rate varies with shifts in the capitalists' utility functions. Marglin (1984) also presents a model with intertemporal utility maximization, as well as non-neoclassical ways of closing the model. Closures with intertemporal utility maximization can explain how the interest rate varies. Multiple equilibria can arise in such models.

Figure 1 shows the wage curves for the two techniques. The choice of the numeraire results in the wage curve for the Alpha technique being a straight line, that is, an affine function. Look at the intercepts on the abscissa. The maximum rate of profits for the Beta technique exceeds the maximum for the Alpha technique. On the other hand, the intercepts on the ordinate show that net output per worker for the Beta technique exceeds the net output per worker for the Alpha technique. If the curvature for the wage curve for the Beta technique were concave to the origin, these properties of the intercepts would be reversed in a reswitching example.

The analysis of the choice of technique, in the circulating capital case, does not rely on the construction of the so-called factor price frontier. Milano (2024) notes that this is a misleading name for the wage frontier in Figure 1. Rather, this analysis has an algorithmic justification in the analysis of prices of production (Bharadwaj 1970, Woods 1990, Bidard 1990, Vienneau 2005 and 2017). Bidard & Klimovsky (2004) that the analysis based on the wage frontier does not necessarily extend to general joint production. Specific analyses have been developed for special cases of joint production, notably in the case of fixed capital and of extensive and intensive rent (Pasinetti 1980).

Figure 2 illustrates another analysis of the cost-minimizing technique. The left-hand side is constructed with prices for the Alpha technique, while the right-hand side illustrates prices for the Beta technique. One can see that greater profits are obtained, at low and high rates of profits, by adopting the second corn-producing process at Alpha prices. Likewise greater profits are obtained, at intermediate rates of profits, by adopting the first corn-producing process at Beta prices. Thus, the cost-minimizing technique, at a given interest rate, is the technique contributing its wage curve to the outer frontier at that interest rate.

Figure 2: Extra Profits for Price Systems

This example can be used to explore two parables, what one might call a neoclassical and a Austrian story:

• A lower interest rate incentivizes managers of firms to adopt a more capital-intensive technique. With more capital per worker, the steady-state net output per worker is higher.
• A lower time preference results in a willingness to defer consumption and a lower interest rate. This lower time preference is seen in the support of a longer structure of production and less consumption per worker.

The second switch point demonstrates that the first story is ill-founded. The construction of Hayekian triangles below demonstrates the second story is ill-founded. The example does demonstrate that the second story is not refuted merely by objections to the first.

4.0 Perverse Hayekian Triangles at the First Switch Point

In a previous post, I showed how to find the labor used to produce net output, in the current and each future year, when production is synchronized in a given year.

The increment in each step, for a given technique, is the value added by the original factors of production in that step. That is, an increment is the product of the wage and the labor for that order of goods, properly costed up with interest charges.

Figures 3 and 4 display the Hayekian triangles at the first switch point. The ordinate is the order of goods, as in Menger (187?). The abscissa is the value of the goods of that order that ultimately go into producing consumption goods, that is, goods of the first order. The length of the first step is the intersection of the corresponding wage curve with the ordinate in Figure 1. The increments in the steps are the values added by original, unproduced factors of production. In this case, labor is the only original factor of production.

Figure 3: Hayekian Triangle at First Switch Point

Figure 4: Hayekian Triangle at First Switch Point for Goods of Higher Order

Notice that the step for goods of first order is longer for the Beta technique. For goods of order 42 and higher, the step for the Alpha technique is higher. Since Beta is cost-minimizing at a lower interest rate around this interest rate, a lower interest rate is associated with the adoption of a shorter structure of production to produce more goods of the first order. This effect is perverse from the perspective of the Austrian story that Hayek developed his triangles to illustrate.

One can look at the ratio of successive increments of step sizes in these Hayekian triangles (Figure 5). This ratio is a constant for the Alpha technique. Because of the composition of the consumption good, this ratio does not vary with the order of goods. This ratio, for the Beta technique, approaches a constant as the order of goods increases without bound. Under the assumptions in which a Hayekian triangle can be constructed from a model of the production of commodities, this ratio always approaches a limiting value. This limit increases with the interest rate for which the triangle is constructed. It is smaller for a larger maximum interest rate. Since the intersection of the wage curves with the abscissa in Figure 1 is larger for Beta, the structure of production for Alpha is ultimately longer in Figure 4.

Figure 5: Ratios of Value Added at First Switch Point

5.0 Hayekian Triangles at the Second Switch Point

One can also consider Hayekian triangles at the second switch point (Figure 6). This switch point is perverse from the perspective of neoclassical theory. Around this switch point, the Alpha technique is cost-minimizing at a lower switch point. The Alpha technique has a longer structure of production and produces less goods of the first order. This pivoting of the structure of production is just the story that Hayek invented his triangles to tell.

Figure 6: Hayekian Triangle at Second Switch Point

"While the controversy about public works was developing, Professor Robbins sent to Vienna for a member of the Austrian school to provide a counter attraction to Keynes. I very well remember Hayek's visit to Cambridge on his way to the London School. He expounded his theory and covered a black board with his triangles. The whole argument, as we could see later, consisted in confusing the current rate of investment with the total stock of capital goods, but we could not make it out at the time. The general tendency seemed to be to show that the slump was caused by [excessive] consumption. R. F. Kahn, who was at that time involved in explaining that the multiplier guaranteed that saving equals investment, asked in a puzzled tone, 'Is it your view that if I went out tomorrow and bought a new overcoat, that would increase unemployment?' 'Yes,' said Hayek, 'but,' pointing to his triangles on the board, 'it would take a very long mathematical argument to explain why.'" -- Joan Robinson. 1972. The second crisis of economic theory.

Conclusion

The above has analyzed a reswitching example, with two techniques, in a model of the production of commodities by means of commodities. The switch point that is normal, from a mainstream neoclassical perspective, has perverse Hayekian triangles. The switch point that is perverse from a mainstream perspective has Hayekian triangles consistent with the Austrian story about how an increased time preference rotates the triangle to lengthen the structure of production. If the wage curve for Beta were concave to the origin, on the other hand, the first, normal switch point would also be normal from an Austrian perspective, and the second switch point would be perverse from both a neoclassical and Austrian perspective. But what would happen if only one or three switch points existed? Hayekian triangles cannot both be made rigorous and sustain an Austrian story on capital theory.

On the other hand, this post has shown that Austrian economists are correct to claim to have some perspectives distinct from mainstream economists. A refutation of tradional neoclassical capital theory is not, by itself, sufficient to refute all varieties of Austrian capital theory. Furthermore, the analysis in this post has illustrated that, in a model of the production of commodities, the ratio of step sizes in Hayekian triangles approaches a constant (as in Fillieule 2007), dependent on the interest rate for which the triangles are drawn and the maximum interest rate.

This post has drawn no conclusions about monetary theory or macroeconomics. Economists from the Austrian school often draw on capital theory for such purposes. Capital theory can hardly be radically modified without further consequences. Considering such consequences, however, has been on the agenda since Hayek (1941).

Adam Smith On A Labor Theory Of Value

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

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

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

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

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

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

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

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

William Baumol On Marx

This is more for my commonplace book. The first two quotations are part of a symposium with Morishima and Samuelson:

"This paper will suggest that the meaning of the relationship between values and prices described in Capital has been widely misunderstood. Commentators as eminent as Mrs. Robinson and Professor Samuelson have sought in the transformation discussion issues which Karl Marx never meant it to contain. Writers on 'the transformation problem' since L. Bortkiewicz have focussed on an issue that is largely peripheral; and others like E. B6hm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless lne reads into them an interpretation different from that which Marx repeatedly emphasized.

Interpretation of the intentions of the writings of the dead is always a questionable undertaking, particularly since defunct authors cannot defend themselves. Yet there are some cases in which a careful rereading of the pertinent writings indicates that the author did speak for himself and spoke very clearly-the trouble in such cases seems to be that somethinv aboit the onriinal nresentation prevents most readers, even some very careful ones, from seeing what the writer intended.

A notable case in point is D. Ricardo's discussion of the labor theory of value. It is hard to understand how a careful reader of any edition of the Principles can overlook Ricardo's recognition of the role of the quantity and the durability of capital in the determination of price. The labor theory is explicitly proposed as a remarkably good approximation to the determination of competitive price. But, ultimately, Ricardo holds to a cost of production theory of pricing, not to a pure labor theory. Yet until Stigler's fine article on the subject (1958), in which this is documented beyond any shadow of a doubt, virtually any text was prepared to ascribe to Ricardo the purest of labor theories, and even J. H. Hollander and E. Cannan (see G. J. Stigler for references) suggested that Ricardo retreated grudgingly under fire to the cost of production model of the third edition. Only a few commentators, notably A. Marshall, J. Viner, and P. Sraffa, saw Ricardo's analysis for what it so plainly was from the first edition on.

I emphasize this case for two reasons; first, because I will try to show that the correct interpretation of Marx' intentions is equally evident, and second, because I will suggest that the false analogy between Ricardo's and Marx' value theories may help to explain our misunderstanding of the latter...

...In Ricardo, the labor theory of value was meant as a good approximation to a full explanation of the determination of prices. However Marx probably never intended to produce such an approximation and it certainly was not his intention when he wrote about the transformation problem; yet that objective, or something close to it, is often attributed to Marx.

I will provide evidence that Marx did not intend his transformation analysis to show how prices can be deduced from values. Marx was well aware that market prices do not have to be deduced from values (nor, for that matter, values from prices). Rather, the two sets of magnitudes which are derived more or less independently were recognized by Marx to differ in a substantial and a systematic manner. A subsidiary purpose of the transformation calculation was to determine the nature of these deviations. But this objective and, indeed, any explanation of pricing as an end in itself, was of very little consequence to Marx, for the primary transformation was not from values into prices but, as Marx and Engels repeatedly emphasize, from surplus values into the non-labor income categories that are recognized by 'vulgar economists,' i.e., profits, interest, and rent." -- William J. Baumol. 1974. The transformation of values: What Marx 'really' meant (an interpretation). Journal of Economic Literature 12(1): 51-62.

"2. I am surprised that 'On the question of whether [Marx's] purpose was successful in some sense or another [Samuelson] can find only a few relevant paragraphs in Baumol's text.' I am surprised because, so far as I know, there is no such paragraph. The only objective of my paper was to determine what Marx had set out to accomplish and how Marx believed he had accomplished his objectives, because I don't think it is appropriate to criticize anyone until we are sure we are criticizing what he actually said, not what we suspect he might have said, or should have said, or someone else says he might have said...

...4. Professor Samuelson proposes his peace terms, which require me to admit that for an explanation of 'actual wage-profits distribution,' presumably as for an explanation of actual pricing of commodities, 'the Volume I analysis is indeed a detour.' So much I admit readily and without reservations, and I contend Marx would readily have admitted it too, for in fact he did so repeatedly. Actual prices and actual wages, profits, rents and interest payments clearly were to him explainable by the classical mechanism, which is what he admittedly took over in Volume III. Marx never claimed, in fact he specifically denied, that one gets better numbers for any of these magnitudes from a Volume I than from a Volume III analysis.

Thus, for his part, all that Professor Samuelson has to do to end the disagreement between us is to admit that Marx himself was not particularly interested in the determination of these magnitudes, which he considered a surface manifestation and were important to him only because he believed them to conceal the underlying social production relationships...

...One final comment. Obviously, Ricardo is not easy reading, and our predecessors did often hold a multiplicity of views among which they themselves were not always able to distinguish, and such problems are brushed aside far too often in writings But there are some cases, albeit rare, where an author has said clearly and repeatedly, 'I do mean A, I do not mean B,' yet many people have refused to listen. Ricardo did repeatedly say that his was a cost of production model. He did say, at length, in every edition of the Principles that quantity and durability of capital make a difference to value. I suggest to any interested reader that he treat himself to a reexamination of pages 3043 (from the third edition) and 52-66 (from the first edition) of Volume I of Piero Sraffa [5, 1951] to see whether he can come away disagreeing with Jacob Viner's (humorous?) conclusion that 'Ricardo's actual words show that from the first he held that the relative values of commodities are always partly dependent on the relative amounts of fixed capital employed in their production' (Viner's italics) [6, 1930]." -- William J. Baumol. 1974. The fundamental Marxian theorem: A reply to Samuelson: Comment. Journal of Economic Literature 12(1): 74-75.

Baumol returned to Marx in 1983:

"I find few things as discouraging as the persistent attribution of positions to a writer whose works contain repeated, categorical, indeed emotional, denuciations of those views. Marx's views on wages are a prime example. Both vulgar Marxists and vulgar opponents of Marx have propounded two associated myths: that he believed wages under capitalism are inevitably driven near some physical subsistence level, and that he considered this to constitute of robbery of the workers and a major evil of capitalism. Yet Marx and Engels tell us aggain and again, sometimes in the most intemperate language, that these views are the very opposite of theirs. These observations, incidentally, are hardly new discoveries..." -- William J. Baumol. (1983). Marx and the iron law of wages. American Economic Review 73(2): 303-308

The Production Of Commodities And The Structure Of Production

Many of my examples illustrate simple structures of production for models in which commodities are produced with commodities. Economists following the Austrian school often illustrate the structure of production with Hayekian triangles. Accordingly, this post illustrates a Hayekian triangle with a model in which commodities are produced out of commodities. I consider the case in which only circulating capital exists. This post is a rewrite of this one.

The following are taken as given for the technique in use:

• A: The nxn Leontief input-output matrix in physical terms. Assume all commodities are basic and the economy is productive.
• a₀: The n-element row vector of direct labor coefficients.
• d: An n-element column vector that is in the proportions in which commodities are consumed.

Define:

denom = a₀(I - A)^-1d

From the given data, one can find the quantities of labor-time in the first column below.

Distribution of Work in a Given Year

Labor Time	Purpose
a0d/denom	To produce (1/denom) d, a basket of commodities for consumption at the end of the current year.
a0A d/denom	To produce capital goods to be used to produce the (1/denom) d basket of commodities for consumption at the end of the next year.
a0A2 d/denom	To produce capital goods to be used to produce capital goods to produce the (1/denom) d basket of commodities for consumption at the end of two years hence.
...	...

The first column can be summed:

(1/denom) a₀(I + A + A² + ...)d = a₀(I - A)^-1d/denom = 1 person-year

So the above table shows a decomposition, per person-year, of employment in a given year. It is a Hayekian triangle. Because it is constructed from a model of the production of commodities, the elements go on forever. No last year exists in the future for which capital goods are currently being produced.

The capital goods, Aⁿ d, approach the ratios of Sraffa's standard commodity. Consequently, the ratio of labor inputs approaches a constant, related to the maximum rate of profits.

Hayekian triangles are not necessarily set out with a single physical measure of an unproduced input at each stage. The value of capital goods per worker varies because of three effects:

• Composition effect: a different mixture of capital goods is used for different rates of growth.
• Price Wicksell effect: the capital goods are re-evaluated at different prices with a different interest rate.
• Real Wicksell effect: the capital goods vary with the technique, and the cost-minimizing technique varies with the interest rate.

The wage-rate of frontier is useful for visualizing these effects. They do not go away just because one represents the structure of production as a Hayekian triangle. The sort of regularities that Machaj (2017), for example, assumes with variations in the interest rate lack logical foundation.

I want to that the price of the commodities consumed at the end of the year is:

[w(r)/denom] [a₀d + a₀A d (1 + r) + a₀A² d (1 + r)² + ... ]

The terms in the above are useful in representing a Hayekian triangle in price terms.

A model of the production of commodities by means of commodities with general joint production cannot necessarily be represented to a series of inputs of dated labor inputs. I go back and forth on my intution in the case of pure fixed capital.

References

• Renaud Fillieule. 2007. A formal model in Hayekian macroeconomics: the proportional goods-in-process structure of production. Quarterly Journal of Austrian Economics 10: 193-208.
• Harris, Donald J. 1973. Capital, distribution and the aggregate production function. American Economic Review 63(1): 110 - 113.
• Machaj, Mateusz. 2017. Money, Interest, and the Structure of Production: Resolving Some Puzzles in the Theory of Capital. Lanham: Lexington Books.