Hayek In The Pure Theory Of Capital

This is more of my commonplace book, based on reading some of Hayek's The Pure Theory of Capital. Hayek rejects Böhm Bawerk's average period of production, a single number summarizing the capital intensity of a structure of production. He still strives to say that more capital-intensive techniques are longer in some non-aggregate sense. I probably will not finish this book. To compare and contrast this book with J. R. Hicks' Value and Capital would be of interest.

Hayek correctly takes issue with the treatment of capital as a homogeneous quantity and the explanation of interest by the supply and demand of capital:

"the attempts to explain interest, by analogy with wages and rent, as the price of the services of some definitely given 'factor' of production, has nearly always led to a tendency to regard capital as a homogeneous substance the 'quantity' of which could be regarded as a 'datum', and which, once it had been properly defined, could be substituted, for purposes of economic analysis, for the fuller description of the concrete elements of which it consisted." (Hayek 1941, p. 5)

and:

"As we shall see, it is more than doubtful whether the discussion of 'capital' in terms of some single magnitude, however defined, was fortunate even for its immediate purpose, i.e. the explanation of interest. And there can be no doubt that for the understanding of the dynamic processes it was disastrous. The problems that are raised by any attempt to analyse the dynamics of production are mainly problems connected with the interrelationships between the different parts of the elaborate structure of productive equipment which man has built to serve his needs." (Hayek 1941, p. 6)

Sraffa, of course, analyzes the structure of productive equipment in a different way.

Hayek is one of the originators of the notion of intertemporal general equilibrium

"An effective discussion of the problems of capital theory must, however, move precisely in that neglected field which deals with general equilibria that are not at the same time stationary states." (Hayek 1941, p. 6)

Hayek goes on to a discussion later built on by Lachmann and paralleling Robinson. At any point, the complex of capital equipment is unsuitable, inherited from the past in which the present was not completely foreseen. In equilibrium, the plans of all agents are adjusted. Equilibrium is not a state that an economy can get into in historical time. Nevertheless, equilibrium analysis has a logical use in a preliminary analysis. Hayek does not want to consider a temporary partial equilibrium, instead of a general intertemporal equilibrium. Furthermore, Hayek puts money aside, including the lending and borrowing of money in most of this preliminary study.

Hayek distinguishes between permanent and non-permanent resources and between producible and non-producible resources.

"The term capital itself, in so far as it is required to describe a particular part of the productive resources, will accordingly be used here to designate the aggregate of those non-permanent resources which can be used only in this indirect manner to contribute to the permanent maintenance of the income at a particular leve1." (Hayek 1941, p. 54, italics in original)

Both producible and non-producible resources can be capital. The latter include something like oil, which can be exhausted by drilling.

In his non-homogeneous concept to capital, Hayek still thinks a connection exists between capital-intensity and time:

"There are two main ways in which the productivity of investment shows itself ... the investment of any group of services of the permanent resources can be combined into one single 'process' of production in two ways. In the one case it is the duration of the actual process of production where the time factor enters, and in the other case is the durability of the product (or of the non-permanent resources used in production)." (Hayek 1941, Ch. 6, p. 65, italics in original)

Hayek was an early adopter of the distinction and terminology of point input, flow input, point output, and flow output:

"it is useful to construct ideal limiting cases which show their peculiarities in the purest form. The first case is best represented if we conceive of a continuous application of input through a period of time, leading to an output all of which matures at a moment of time at the end of the period. This has been described as the 'continuous input – point output' case. The second case is ideally represented if we imagine durable good which is produced at a moment of time and then rendes services continuously over a period of time. This case has correspondingly been described as the 'point input – continuous output' case." (Hayek 1941, p. 66-67, italics in original)

Hayek distinguishes the investment period from the length of the process and the period of production:

"The fundamental fact with which we are concerned is the change in the periods for which particular units of input are invested, that is, the interval between the application of a unit of input and the maturing of the quantity of output due to that input. This interval of time we shall describe as the investment period of that unit of input.” (Hayek 1941, p. 69, italics in original)

Hayek recognizes that an investment period requires a high degree of abstraction to identify and may not always exist.

Hayek is considering the choice of technique, in which many techniques are known for producing a given final output:

"The considerations advanced above - and it is important to remember this throughout the discussion - have nothing to do with technological progress in this sense. On the contrary, they refer to changes under conditions where knowledge is stationary. All that is assumed is that at any moment there are known possible ways of using the available resources which would yield a greater return than those actually adopted, but would not yield this return until a later date, and for this reason are not actually used.

Among the wide range of possible methods of production known at any one time there will be some which will yield their product after shorter periods of time and some which will not yield it until after longer periods. From among each group of methods involving the same 'amount of waiting' - if we may make provisional use of this vague term - the one that will be chosen will be the one which yields the greatest return from a given investment of factors. But so long as there is any limitation on the 'amount of waiting' for which people are prepared, processes that take more time will evidently not be adopted unless they yield a greater return than those that take less time. (Hayek 1941, p. 72-73)

Hayek uses 'process' synomously with what I usually call a technique. Hayek thinks longer processes, in some sense, support a greater division of labor and often the use of resources that were not used before. Perhaps the consumer good is different but satisfies the same needs.

Here Hayek is once again rejecting an average period of production:

"Before we can pass on to this problem it is necessary to return for a moment to the, difficulty of talking about changes in the length of the process of production. It will probably be fairly obvious by now that as the complete processes of production with which we have to deal become increasingly complex it becomes more and more difficult, and may in some cases be impossible, to say in any general way which of several, alternative processes under consideration is as a whole the shortest or the longest. The total length of time which elapses between the very beginning of the process and the completion of the product may be shorter in one process than in another, and yet by far the greater part of the input used may be applied very early in the first process and very late in the second process. Which of these two processes is to be regarded as the longer? It is impossible to answer this question at the present stage, and there is in fact no general answer to it. It is only mentioned at this point in order to warn the reader against any attempt to provide himself with an answer by introducing some concept of an 'average period' of production. Such a concept, as we shall see, is not only unnecessary but is also highly misleading." (Hayek 1941, p. 76)

The above does not get at reswitching, I do not think. He wants a process in which resources are applied at the start and the end, in contrast to when they are used mostly in the middle.

Somewhere around here, Hayek defines stages of production, much like in his triangles.

I think the following gets at the possibility of reswitching:

"... when we compare two different investment structures, it will not always be possible even to say, on purely technical grounds, which of them involves the greater amount of waiting. At one set of relative values for the different kinds of input and at one rate of interest, the one structure, and at a different set of values or a different rate of interest, the other structure will represent the greater amount of waiting, or will be 'longer' in the sense in which this term has commonly been used." (Hayek 1941, Ch. 11, p. 144)

There is also something suggestive of capital-reversing in Chapter XXI.

"We shall not follow up this point in detail. For the method adopted to give a general picture of the considerations involved is really not adequate for an exhaustive analysis. The reason for this is, of course, as in all other cases where productivity or demand curves for individual factors are regarded as given, that these curves cannot be regarded as simultaneously and independently valid. But to take account of the complicated relations of technological (and psychological) complementarity which are involved, requires another technique which has been evolved quite recently in closely related fields, and which will also have to be used in a more exhaustive investigation of our problems. Here all that we shall mention is that if we were to start from a complete statement of the substitution relationships between all the different resources concerned, all kinds of peculiarities and apparent anomalies in the behaviour of individual factors would appear to be quite consistent with the general tendencies which can be deduced from the cruder type of analysis. It is, for instance, quite possible that while a fall in the rate of interest will create a tendency for the services of most of the permanent factors to be invested for longer periods and for their prices to rise, in the case of some individual factor the effect may well be that it will be invested for shorter periods, or that its price will be lowered, or both." (Hayek 1941, Ch. 21, p. 191-192)

The earliest description of something like reswitching is due to Irving Fisher.

Hayek and other Austrian-school economists are, of course, wrong. In what sense, whether assessed by value measures or physical properties, do more well-chosen, roundabout techniques raise net output? Samuelson (1966) noted that reswitching is not consistent with "the simple tale told by Jevons, Bohm-Bawerk, Wicksell, and other neoclassical writers". Fratini (2019a, 2019b) shows that the adoption of techniques with an increased average period of production, as assessed by financial measures, is compatible with decreased net output per worker. Steedman (2020), in considering techniques associated with the operation of different types of machines, demonstrates that more durable machines can also yield less net output per worker. I know that the economic life of machines do not support Hayek's idea.

Technical Terms: Continuous (flow) input or point input, continuous (flow) output or point output, Intertemporal general equilibrium, Investment period, Period of production, Permanent and non-permanent resources, Producible and non-prodicable resources, Stages of production, Structure of production

Wages, Employment Not Determined By Supply And Demand

1.0 Introduction

I do not think I have presented an introductory example in a while in which an increased wage is associated with firms wanting to employ more labor, given the level of net output. This example is presented as a matter of accounting for a vertically integrated firm.

Exact calculations with rational numbers are tedious in this example. I expect that if anybody bothers to check this, they would use a spreadsheet. As far as I can tell, Microsoft Excel uses double precision floats.

2.0 Technology

The managers of a competitive, vertically-integrated firm for producing corn know of the four production processes listed in Table 1. Corn is a consumption good and also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. The first process produces iron, the second process produces steel, and the last two processes produce corn. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce the specified unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year. The data on technology are taken from a larger example.

Table 1: Technology

Input	Process
	a	d	e	f
Labor	1/3 person-year	7/20 person-year	1 person-year	3/2 person-year
Iron	1/6 ton	1/100 ton	1 ton	0 tons
Steel	1/200 ton	3/10 ton	0 tons	1/4 ton
Corn	1/300 bushel	0 bushel	0 bushels	0 bushels
Output	1 ton iron	1 ton steel	1 bushel corn	1 bushel corn

The managers of the firm have available two techniques for producing corn from inputs of labor, with intermediate inputs being constantly replaced. The iron-producing, steel-producing, and first corn-producing processes are operated in the Gamma technique. The second corn-producing process, as well as the iron and steel-producing processes, are operated in the Delta technique. Iron, steel, and corn all enter, either directly or indirectly, into the production of corn in both techniques. Vertically-integrated firms can also operate a linear combination of the Gamma and Delta technique.

3.0 Quantity Flows

One can consider various levels of operations in each of the processes for each of the technique. I consider two examples of snychronized production, in which inputs of labor simultaneously produce a net output of corn for consumption. A structure of production, consisting of specific capital goods, intervenes between the inputs and output. The labor input reproduces that structure, as well as producing the output.

3.1 Gamma Quantity Flows

Suppose 14,000/11,619 ≈ 1.205 tons iron are produced with the first process, 100/11,619 ≈ 0.0086 tons steel are produced with the second process, and 34,997/34,857 ≈ 1.004 bushels corn are produced with the third process. Then the quantity flows illustrated in Table 2 result. 14,000/11,619 tons iron are used as inputs among the three industries. These inputs are replaced by the output of the iron-producing process. 100/11,619 tons of steel are used as inputs among the three industries, and these inputs are replaced by the output of the steel-producing process. 140/34,857 bushels of corn are used as inputs among the three industries, leaving a net output of one bushel corn. In short, these quantity flows are such that 49,102/34,857 ≈ 1.409 person-years produce one bushel corn net. Obviously, I did not pick a very good set of coefficients for this example to support exact calculations in rational numbers.

Table 2: Vertically-Integrated Production with the Gamma Technique

Input	Process
	a	d	e
Labor	14,000/34,857 person-year	35/11,619 person-year	34,997/34,857 person-year
Iron	7000/34,857 ton	1/11,619 ton	34,997/34,857 ton
Steel	70/11,619 ton	30/11,619 ton	0 tons
Corn	140/34,857 bushel	0 bushel	0 bushels
Output	14,000/11,619 ton iron	100/11,619 ton steel	34,997/34,857 bushel corn

3.2 Delta Quantity Flows

Suppose 100/23,331 ≈ 0.00429 tons iron are produced with the first process, 25,000/69,993 ≈ 0.3572 tons steel are produced with the second process, and 69,994/69,993 ≈ 1.00001 bushels corn are produced with the fourth process. By the same logic as above, these quantity flows are such that 1807/1111 ≈ 1.626 person-years produce one bushel corn net.

Table 3: Vertically-Integrated Production with the Delta Technique

Input	Process
	a	d	f
Labor	100/69,993 person-year	1,250/9,999 person-year	34,997/23,331 person-year
Iron	50/69,993 ton	250/69,993 ton	0 ton
Steel	1/46,662 ton	7,500/69,993 ton	34,997/139,986 tons
Corn	1/69,993 bushel	0 bushel	0 bushels
Output	100/23,331 ton iron	25,000/69,993 ton steel	69,994/69,993 bushel corn

4.0 Prices

Which technique will the managers of the firm choose to adopt? By assumption, they take the price of corn and the wage as given on the consumer and labor markets. For simplicity, assume that price of a bushel corn is unity. That is firms treat the price of the consumer good as numeraire. At the end of the year, firms own a stock of iron, steel, and corn. They sell some of the corn to consumers. They retain the iron, steel, and enough corn to continue production the next year.

In a consistent accounting scheme, the price of iron and steel are such that:

• The same (accounting) rate of profits is obtained in all operated processes.
• The cost of the inputs, per bushel corn produced gross, for the corn-producing process not operated for a technique does not fall below that for the operated process.

The first condition specifies prices of intermediate goods and the rate of profits the accountants register. The second condition states that no pure economic profits can be obtained. Under these conditions, the managers of the firm can price their capital stock at the end of any year.

4.1 Prices at a Low Wage

Suppose the wage is w = 19,296/352,547 ≈ 0.05473 bushels per person-year. The accountants set the price of iron at p₁ = 6,860/27,119 ≈ 0.2530 bushels per ton iron and the price of steel at p₂ = 76,454 ≈ 2.819 bushels per ton steel. Table 4 shows the cost per unit output for each process and the resulting rate of profits obtained by operating each process. In constructing the tables for price systems, wages are assumed to be advanced. Under these assumptions, the rate of profits is 9/4, that is 225 percent, in each process comprising the Gamma technique. A lower rate of profits is obtained in the remaining corn-producing process, and it will not be operated. This is a consistent accounting system for the vertically-integrated firm, given the wage.

Table 4: Costs and the Rate of Profits at a Low Wage

Process	Cost	Rate of Profits
a	(1/6)p1 + (1/200)p2 + (1/300) + (1/3)w = 27,440/352,547	225 percent
d	(1/100)p1 + (3/10)p2 + (7/20)w = 305,816/352,547	225 percent
e	p1 + w = 2,308/7,501	225 percent
f	(1/4)p2 + (3/2)w = 554,839/705,094	≈ 27.1 percent

4.2 Prices at a Higher Wage with the Original Technique

Now suppose the wage is higher, namely w = 1,332/5,197 ≈ 0.2563 bushels per person-year. Consider prices of p₁ ≈ 0.2622 bushels per ton iron and p₂ ≈ 0.4167 bushels per ton steel. Table 5 shows cost accounting for these prices.

Table 5: Costs and the Rate of Profits at a High Wage (Incomplete)

Process	Cost	Rate of Profits
a	0.141 Bushels per ton iron	85.9 percent
d	0.2241 Bushels per ton steel	85.9 percent
e	0.5379 Bushels per bushel	85.9 percent
f	0.5178 Bushels per bushel	93.1 percent

Notice the same rate of profits is obtained in operating the first three processes. But the cost of producing a bushel corn with the last process is lower than in producing corn with process e. A larger rate of profits is obtained in operating that process. The managers of the firm will realize that their accounting implies that the Delta technique should be operated. If this firm were not vertically integrated and iron and steel were purchased on the market, a market algorithm would also lead to the Delta technique being adopted at this wage.

4.3 Prices at the Higher Wage with the Cost-Minimizing Technique

Continue to consider a wage of w = 1,332/5,197 ≈ 0.2563 bushels per person-year. The accountants report prices of p₁ = 1,420/5,197 ≈ 0.2732 bushels per ton iron and p₂ = 2,402/5,197 ≈ 0.4622 bushels per ton steel. Table 6 shows costs per unit output for the five processes under these prices.

Table 6: Costs and the Rate of Profits at a High Wage

Process	Cost	Rate of Profits
a	710/5,197	100 percent
d	1,201/5,197	100 percent
e	2,752/5,197	≈ 88.8 percent
f	1/2	100 percent

With this set of prices, the Delta technique is operated, and a rate of profits of 100 percent is obtained. The cost of operating the first corn-producing process exceeds the cost of operating the corn-producing process in the Delta technique. With a higher wage, the managers of a cost-minimizing firm will choose to operate a corn-producing process that requires more labor per bushel corn produced gross. (3/2 person-years is greater than 1 person-year.) More labor will also be hired per bushel corn produced net.

5.0 Conclusion

Table 7 summarizes these calculations. The ultimate result of a higher wage in the range considered is the adoption of a more labor-intensive technique. If this firm continues to produce the same level of net output and maximizes profits, its managers will want to employ more workers at the higher of the two wages considered. So much for the theory that, given competitive markets, wages and employment are determined by the interaction of well-behaved supply and demand curves on the labor market.

Table 7: A More Labor-Intensive Technique at a Higher Wage

Wage	Technique	Labor Intensity
0.05473 bushels per person-year	Gamma	1.409 person-years per bushel
0.2563 bushels per person-year	Delta	1.626 person-years per bushel

This example can be generalized in many ways. Different types of labor can be introduced. More intermediate produced capital goods can be included. Any number of processes can be available for producing each good, including an uncountable infinity. The use of fixed capital introduces more complications. The introductory marginalist textbook story about wages and employment in competitive markets is without foundation.

Why do so many economists teach nonsense?

Alienation And Commodity Fetishism

Marx, in Theories of Surplus Value, quotes James Stuart writing about 'profit upon alienation'. When one sells a good one owns, one has alienated it from oneself. In the typical work relation under capitalism, the managers of firms, that is, the representatives of capitalists, sell the product or services that workers produce. Workers do not own the goods they produce, for they have previously sold their labor-power. That is, they have agreed that their product is not owned by themselves, and neither is their labor.

But alienation means something more in Marx's Paris Manuscripts. The work of Ludwig Feuerbach is important background here. (By the way, this post is based mostly on secondary and tertiary literature, I forget which. I have not read most of the Marx and Engels' works cited here in some time.)

Feuerbach's criticism of Christianity is of some importance for Marx's concept of alienation. According to Feurbach, God's human-like qualities of knowledge, wisdom, and power are projections of extensions of human qualities. We impose them on God. This result of the projection of human qualities is then taken as ruling over us.

Capital goods are themselves the result of human labor. But, in capitalism, the alienated worker is then ruled by these products of human labor. In both cases, human do not usually understand that they are ruled by an entity created collectively by themselves.

The word 'alienation' never appears in the three volumes of Capital as I understand it. Instead, Marx writes about commodity fetishism. I guess the continuity I am claiming is debated among scholars of Marx. But the following sounds like alienation, as developed from the ideas of Feuerbach, to me:

"A commodity is therefore a mysterious thing, simply because in it the social character of men's labour appears to them as an objective character stamped upon the product of that labour; because the relation of the producers to the sum total of their own labour is presented to them as a social relation, existing not between themselves, but between the products of their labour. This is the reason why the products of labour become commodities, social things whose qualities are at the same time perceptible and imperceptible by the senses... with commodities... existence of the things qua commodities, and the value relation between the products of labour which stamps them as commodities, have absolutely no connection with their physical properties and with the material relations arising therefrom. There it is a definite social relation between men, that assumes, in their eyes, the fantastic form of a relation between things. In order, therefore, to find an analogy, we must have recourse to the mist-enveloped regions of the religious world. In that world the productions of the human brain appear as independent beings endowed with life, and entering into relation both with one another and the human race. So it is in the world of commodities with the products of men's hands. This I call the Fetishism which attaches itself to the products of labour, so soon as they are produced as commodities, and which is therefore inseparable from the production of commodities.

The religious analogy suggests, perhaps, that this supposed relation between things appears as a independent being. That is, capital is an entity with seeming agency and its own logic. Marx uses such metaphors as zombies, vampires, and table-turning for this spectral creature. "Capital comes dripping from head to foot, from every pore, with blood and dirt." In a longer exposition, I would also want to go the formal and real subsumption to capital.

It may seem odd to talk about capital as a creature with agency. Many human beings, following certain protocols, can implement a Turing machine, unbeknowest to themselves. The analogy of markets to a computer is harly novel. More prosaically, every manager or firm owner who justifies layoffs; the closure of factories, stores, and offices; or the abandoment of one line of business and the start up of another is appealing to some such logics of capital. Whatever you may think of such people, they are not wrong in claiming to be ruled by an overarching entity.

Capital is not the only supra-personel entity, discussed among Marxists, that seems to have agency. Hardt and Negri's Empire also seems to be such an entity.

I think that Marx condemns capitalism, inasmuch as he does, because he objects to the rule of capital as a supra-personel entity. This creature is constructed by human beings, and we should be able to rule ourselves without such illusions. The rule of this creature hardly seems consistent with a society in which "the free development of each is the condition for the free development of all."

Precursors Of The Modern Revival Of Classical Political Economy

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

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