I Did Not Invent The Concept Of Fluke Switch Points

I find myself writing about generic switch points, fluke switch points, anomalous switch points, fake switch points, normal switch points, and 'perverse' switch points. I do not seem to have definitions with the precision of those in mathematical analysis.

A switch point is a fluke when any perturbation of some parameters, such as coefficients of production, destroys defining features of the switch point. The concept of a fluke goes back to the 1966 symposium on capital theory in the Quarterly Journal of Economics:

"If, by a fluke more than one switch of technique happened to take place at exactly the same point, the nonzero columns [of the matrix formed by the difference of two Leontief matrices] would be more than one" (Pasinetti 1966: 511).

Other participants recognize this fluke case in which four wage curves intersect at a switch point, with processes replacing one another in two industries:

"'Adjacent" techniques on two sides of a switching point of a switching point will usually differ from each other only with respect to one activity" (Bruno, Burmeister, Sheshinski 1966: 542).

Two wages curves tangent at a switch point is another fluke case. A perturbation leads to either the reswitching of techniques or of one cost-minimizing technique around the rate of profits at which the switch point formerly existed. Other fluke cases arise when a switch point exists at the maximum wage or the maximum rate of profits:

"Cases with multiple roots or cases in which the curves cross only at end points... These ... are cases which one technique can be ignored since it is dominated" (Bruno, Burmeister, Sheshinski 1966: 534).

Pierangelo Garegnani recognizes the possibility of the fluke case with two wage curves tangent:

"The possibility that, at r^* and r^**, the two wage curves touch without intersecting is excluded...” (Garegnani 1966: 567).

Fluke switch points exist in both models of single and joint production. Vienneau (2021) examines fluke switch points in pure fixed capital models, while Vienneau (2022) partitions a parameter space, with fluke switch points, in a model of extensive rent. Vienneau (2024) looks at fluke switch points in a model of non-competitive markets with single production. The characterization of fluke switch points is useful for analyzing structural economic dynamics with a choice of technique.

References

Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80(4): 526-553.

Garegnani, P. 1966. Switching of techniques. Quarterly Journal of Economics 80(4): 554-567

Pasinetti, Luigi L. 1966. Changes in the rate of profit and switches of technique. Quarterly Journal of Economics 80(4): 503-517.

Vienneau, Robert L. 2021. Fluke switch points in pure fixed capital systems. Centro Sraffa working papers n. 48.

Vienneau, Robert L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.

Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.

Surveys Of The Cambridge Capital Controversy

'Neoclassical' economists accepted, in the third quarter of last century, that the theories they teach and apply have no rigorous foundation. They are illogical and incoherent. Why does that not matter? This presents a puzzle.

Many have surveyed or responded to the Cambridge Capital Controversy. Here are some surveys and responses:

• Jack Birner. 2002. Cambridge Controversies in Capital Theory: A Methodological Analysis. Routledge.
• Mark Blaug. 1974. The Cambridge Revolution: Success or Failure?. London: Institute of Economic Affairs. (I stumbled upon this negative review in the History of Political Economy.)
• Christopher Bliss. 1975. Capital Theory and the Distribution of Income. Elsevier North-Holland.
• Edwin Burmeister. 1982. Capital Theory and Dynamics. Cambridge University Press.
• Avi J. Cohen and G. C. Harcourt. 2003. Whatever happened to the Cambridge capital theory controversies. Journal of Economic Perspectives 17(1): 199-214.
• Avinash Dixit. 1977. The accumulation of capital theory. Oxford Economic Papers 29(1): 1-29.
• Roger W. Garrison. 2006. Reflections on reswitching and roundaboutness. In Money and Markets: Essays in Honor of Leland B. Yeager (ed. by Roger Koppl). Routledge.
• G. C. Harcourt. 1969. Some Cambridge controversies in the theory of capital. Journal of Economic Literature 7(2): 369-405.
• G. C. Harcourt. 1972, 2022. Some Cambridge Controversies in the Theory of Capital. Cambridge University Press.
• Daniel Hausman. 1981. Capital, Prices and Profits. Columbia University Press.
• Andrés Lazzarini. 2011. Revisiting the Cambridge Capital Theory Controversies: A Historical and Analytical Study. Pavia University Press..
• Joseph E. Stiglitz. 1974. The Cambridge-Cambridge controversy in the theory of capital: a view from New Haven. Journal of Political Economy 82(4): 893-903.
• Leland B. Yeager. 1979. Capital paradoxes and the concept of waiting. In Time, Uncertainty, and Disequilibrium: Exploration of Austrian Themes (ed. by M. J. Rizzo). Lexington Books.

Harcourt (1972) is my favorite of these surveys - an utterly conventional view. I disagree with much in many of these surveys and responses. But those who have never been exposed to the CCC will learn something from any of them.

To Read Adorno's Minima Moralia Requires Understanding Of Marx

Well-known contributions to philosophy in Europe and America in the twentieth century are often divided into analytical and continental philosophy. Analytical philosophers often state their arguments with formal reasoning and notation, while concentrating on narrow points. How do you know that you have not always used 'green' to mean grue? Continental philosophers provide a more intuitive reasoning and focus on larger issues such as culture. Gender is performative. I take no issue to those who argue that the division is not well-defined. I lean more towards the analytical side, albeit I try to reject logical positivism.

Sometimes, when I read postmodernists - another ill-defined term - I can follow, but I do not retain much. I find intriguing Lukacs' essay on reification, in which he builds on Karl Marx's work on commodity fetishism and vulgar political economy. I have more use for Foucault's post structuralism than Derrida's deconstruction. I find Antonio Gramsci insightful, which is no surprise for somebody building on Piero Sraffa. Herbert Marcuse has a point about instrumental reason in the service of a system that is irrational as a whole. I am never sure when Slavoj Zizek is joking.

But here I want to focus on Theodor Adorno and his 1951 book Minima Moralia: Reflections from Damaged Life. I could not make much out of his book, Negative Dialetics. I have yet to read The Dialetic of Enlightenment.

As I poorly recall, Minima Moralia has a narrative arc, although it takes some work to perceive it. I was surprised at passages that presume an understanding of technical terms in Marx's political economy. I note a few here.

Here Adorno rejects some concepts of a post-capitalist society because they continue commodity fetishism:

"Sur l'Eau. He who asks what is the goal of an emancipated society is given answers such as the fulfillment of human possibilities or the richness of life. Just as the inevitable question is illegitimate, so the repellent assurance of the answer is inevitable, calling to mind the social-democratic ideal of the personality expounded by heavily-bearded Naturalists of the 'nineties, who were out to have a good time. There is tenderness only in the coarsest demand: that no-one shall go hungry any more. Every other seeks to apply to a condition that ought to be determined by human needs, a mode of human conduct adapted to production as an end in itself. Into the wishful image of an uninhibited, vital, creative man has seeped the very fetishism of commodities which in bourgeois society brings with it inhibition, impotence, the sterility of the never-changing. The concept of dynamism, which is the necessary complement of bourgeois 'a-historicity', is raised to an absolute, whereas it ought, as an anthropological reflex of the laws of production, to be itself critically confronted, in an emancipated society, with need..." -- Adorno: 155-156.

Adorno draws on the concept of fetishism in other places. I do not know that the above passage is consistent with Marx and Engels in The German Ideology.

In this next passage he uses the concept of the organic composition of capital to write about how working class consciousness is dimmed:

"Puzzle-picture. Why, despite a historical development that has reached the point of oligarchy, the workers are less and less aware that they are such, can be surmised from a number of observations. While objectively the relation of owners and producers to the productive apparatus grows ever more rigid, subjective class membership becomes all the more fluctuating. This tendency is fostered by economic development itself. The organic composition of capital demands, as has often been noted, control through technical experts rather than through factory owners. The latter were the counterpart, as it were, of living labour, the former correspond to the share of machinery in capital. The quantification of technical processes, however, their dissection into minute operations largely independent of education and experience, makes the expertise of these new-style managers to a large degree illusory, a pretence concealing the privilege of being appointed. That technical development has reached a state which makes every function really open to all - this immanently socialist element in progress has been travestied under late industrialism. Membership of the elite seems attainable to everyone. One only waits to be co-opted... ...Preference goes to those who fit in most exactly...That technical forces might permit a condition free of privileges is accredited by all, even those in the shadow, to the social relations which prevent it. In general, subjective class-membership today shows a mobility that allows the rigidity of the economic order itself to be forgotten..." -- Adorno: 193-194.

And, for the last passage I select, Adorno writes about the law of value and, again, the organic composition of capital:

"Novissimum organum. It has long been demonstrated that wage-labour formed the masses of the modern epoch, indeed created the worker himself. As a general principle the individual is not merely the biological basis, but the reflection of the social process; his conciousness of himself as something in-itself is the iI1usion needed to raise his level of performance, whereas in fact the individuated function in the modern economy as mere agents of the law of value. The inner constitution of the individual, not merely his social role, could be deduced from this. Decisive here, in the present phase, is the category of the organic composition of capital. By this the theory of accumulation meant the 'growth in the mass of the means of production, as compared with the mass of the labour-power that vivifies them'. If the integration of society, particularly in totalitarian states, designates subjects more and more exclusively as partial moments in the network of material production, then the 'alteration of the technical composition of capital' is prolonged within those encompassed, and indeed constituted, by the technological demands of the production process. The organic composition of man is growing. That which determines subjects as means of production and not as living purposes, increases with the proportion of machines to variable capital... Only when the process that begins with the metamorphosis of labour-power into a commodity has permeated men through and through and objectified each of their Impulses as formally commensurable variations of the exchange relationship, is it possible for life to reproduce itself under the prevailing relations of production..." -- Adorno: 228-229.

My understanding of the organic composition of capital is straightforward. I take it to be the ratio of constant capital to variable capital, evaluated either with labor values or with prices of production. I do think about the physical composition of capital goods and of issues associated with depreciation. But I certainly do not go into the cultural effects that Adorno writes about.

Numerical Examples Of The Reswitching Of Techniques In Spatial Economics?

It is my custom to work through, with graphs, example of the reswitching of techniques and other capital-theoretic 'paradoxes' in various models. Sometimes I have created numeric examples, and perturbed them too.

Reswitching examples exist in the literature on spatial, regional, and urban economics. Nowhere have I worked through them.

I think of regional economics as having two traditions for building theoretical models. One is the the Von Thunen model, with a central city and concentric rings of land uses. Transport costs are of importance. Reswitching is manifested by non-adjacent rings being used to produce the same commodities, with other commodities being produced in between. Barnes & Sheppard (1984) have graphs suggesting that they have a concrete numerical example. But they do not present parameter values. Their text suggests that an example can be created based on the example in Metcalfe & Steedman (1979).

Walter Isard invented regional economics in the middle of the twentieth century. This approach has different countries or regions described by individual Leontief matrices. Imports and exports show interactions between the regions.

Pavlik (1990) has a numeric example for this second tradition, reproduced, I think, in section 5.1 of Sheppard & Barnes (1990). They have the production distributed among regions, as I understand it, like in 'gravity models'. And its solution requires an application of an iterative algorithm. I think this implies the eample does not necessarily have a technique that is cost-minimizing across all regions, as opposed to within each region. I suppose it would be good to replicate this example and produce some graphs. (I realize that computers these days provide capabalities that were not also easily available decades ago.)

Perhaps Zaffari & Sbrenna (2024) provide a model on which I should concentrate. This model seems to be in the Isard tradition, with improvements. Their modeling includes transportation costs, the spatial capacity of regions in space, and the endogenity of various variables. I can make various simplifications in developing a concrete numerical example.

Can I work through existing numerical examples in the literature? Can I find numerical values for the parameters in the model in Zaffari & Sbrenna (1984), perhaps simplified, for a reswitching example? I suppose techniques might differ in which regions, processes are run to capacity. Or, perhaps, one region specializes in manufacturing one set of commodities and the other in making manufacturing another set. I do not know how far I will get.

References

• Barnes, Trevor & Eric Sheppard. 1984. Technical choice and reswitching in space economies. Regional Science and Urban Economics 14: 345-362.
• Metcalfe, J. Stanley & Ian Steedman. 1979. Reswitching and primary input use. In Fundamental Issues in Trade Theory (ed. by I. Steedman) New York: St. Martin's Press.
• Pavlik, C. 1990. Technical reswitching: a spatial case. Environment and Planning A 22:1025-1034.
• Sheppard, Eric & Trevor Barnes. 1990. The Capitalist Space Economy: Geographical Analysis after Ricardo, Marx and Sraffa. London: Unwin Hyman.
• Zaffari, Gabriel & Giacomo Sbrenna. (2024) Sraffa goes to space: spatial elements of political economy. Review of Political Economy DOI: 10.1080/09538259.2024.2434532

'Neoclassical' Economists On The Lack Of Foundation For Some 'Neoclassical' Economics

1.0 Introduction

Last century and into this one, 'neoclassical' economists noted the lack of theoretical foundation for certain widely used models in economics. They noted that the interest rate is generally not equal to the marginal product of capital. This post quotes three prominent 'neoclassical' economics, over decades, noting the lack of theoretical foundation for such an equality.

For the purposes of this post, I have little to say about my disagreements with these authors. I will note that Sraffians have something to say about microeconomics too. I also do not want to go into here why empirical work with these unfounded models is almost always a kind of humbug.

2.0 Frank Hahn

Frank Hahn attacks my favorite school of thought. He says:

"Sraffa ... confined himself to the remark that the [missing] equation cannot be one which demands the equality of the marginal product of 'capital' and the rate of profit. ... the neoclassical economist has the same view but his reasons are not those given by Sraffa." -- Frank Hahn (1982) The neo-Ricardians. Cambridge Journal of Economics. 6(4): 362.

And again:

"The Sraffian picture of neoclassical theory is this. At any moment of time we can observe something physical called the stock of capital (K) as well as the amount of labor (L). There is a concave production function

Y = F(K, L)

where Y is output. In a neoclassical equilibrium all inputs are used and must be paid their marginal products. The latter are known once (K, L) are known. Hence the rate of profit of capital, the real wage and the distribution of income are all known once F(), K and L are known. The concavity of F further implies that the rate of return on capital is non-increasing (generally decreasing) in K. This construction, to be called the parable, Sraffians claim not to be watertight except in the single good economy. In this they are generally correct." -- Frank Hahn (1982: 370)

3.0 Edwin Burmeister

This is from a standard reference work:

"Imposing some set of conditions on the technology T() should be sufficient to ensure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good model often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister (1977, 1979) has proved that a negative real Wicksell effect is a necessary and sufficient condition for an index of capital, k, and a neoclassical aggregate production function F(k) defined across steady-state equilibria such that (i) c = F(k), (ii) r = F’(k), and (iii) F’’(k) < 0. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987). Wicksell effects. The New Palgrave.

That index is Champernowne's chain index measure of capital.

4.0 Emmanuel Farhi

Here is Emmanuel Farhi giving a lecture in 2018 agreeing with the above authors. His history of the CCC is in the first half hour. There is an accompanying paper (working paper version here). Farhi's co-author, David Rezza Baqaee, seems to be pursuing this approach.

5.0 Conclusion

Economics presents a problem for the sociology of 'knowledge'.

Socialism Works In Growing Organic Rice In Latin America

The largest producer of organic rice in Latin America, over the last decade, is the Movimento dos Trabalhadores Rurais Sem Terra (MST), or the Landless Rural Workers Movement, in Brazil. They have 1.5 million members, in 23 of 26 Brazilian states.

Article 5, Section XXIII of Brazil's constitution mandates that land serve a social function. I gather that MST was formed by peasants occupying unused land. They have a radical democratic organization, with some practices that remind me of the 1871 Paris commune. Their base consists of many small settlements working together.

They grow more than rice. Their goals include self-sustaining, self-managed agriculture. They avoid pesticides and use bio-fertilizers. (I guess this is a nice way of talking about manure.) This is a model opposed to a few large business owning huge tracts of land dedicated to production for the export market.

Since I have a vestigial interest in software, I'll mention work by Prof. Celso Alexandre Souza de Alvear and others to develop Sementes, a plugin for web sites designed especially with marketing products from the solidarity economy. There is also Ciranda. If you are going to market organic food, you want the customer to be able to easily access information about ingredients and organizations that grow it. The Arvoredo app (I have not read that paper) "facilitates documentation, monitoring, and evaluation of grassroots environmental governance activities, including tree planting, agroforestry practices, tree nursery construction, and seed collection efforts". Recently, MST has launched Iaraa, an AI tool. The software must be in service of the collective's larger goals.

A corresponding urban organization exists, the Movimento dos Trabalhadores Sem-Teto (MTST) or Movement of Homeless Workers. And MST seems to be networked with other organizations providing models for a post-capitalist society.

As usual, I disclaim much knowledge of these organizations. You can check out an article in Jacobin by Joao Paulo Rodrigues, the write up for the 1991 Right Livelihood Award, or a write up from Grassroots International.

Labor Demand In Corn Industry With The Recurrence Of Truncation

Figure 1: Labor Demand In Corn Industry

1.0 Introduction

This post continues my example of the recurrence of truncation without reswitching. Here I present graphs of the economy-wide demand for capital and for labor, as well as a sectorial demand for labor.

2.0 Technology and Techniques

Of the two industries in the model, one produces machines, and the other produces corn. Machines are fixed capital. Their physical life is two production cycles - that is, two years - in each industry. Corn is circulating capital in each industry and also a consumption good. A bushel corn is also the numeraire. Old machines cannot be transferred between industries. Constant returns to scale (CRS) and the free disposal of machines are assumed . Tables 1 and 2 define specific numeric values for a technology that meets these specifications, with a direct labor input in each process.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 Price Systems and the Choice of Technique

The choice of technique can be analyzed in a model of pure fixed capital by constructing the wage frontier as the outer envelope of the wage curves for each technique. I take a bushel of corn as the numeraire. I assume that wages are paid out of the surplus at the end of the year, not advanced at the beginning. Figure 2 shows the wage curves for the example. Figure 2 shows an enlargement.

Figure 2: Wage Curves

Figure 3: Wage Curves (Enlarged)

The order of the cost-minimizing techniques, with an increasing rate of profits or a decreasing wage, is Alpha, Gamma, Delta, and Beta. This is also in order of increasing labor intensity. I measure labor-intensity by employment, economy-wide, needed to produce a net product of a bushel corn.

This is not a reswitching example. Techniques do not reswitch even off the frontier. The economic life od the machine is truncated in the corn industry for Alpha and Beta. Alpha and Beta are cost-minimizing at the furtherest extremes of the rate of profits, with Gamma and Delta cost-minimizing for middling rates of profits. So this is an example of the recurrence of truncation in the corn industry.

4.0 Economy-Wide Demand Curves for Capital and Labor

The above supports the drawing of economy-wide demand curves for capital and labor. Suppose net output for the economy as a whole is one bushel corn. Figure 4 shows the demand curve for capital. The value of inputs of corn and machines needed as inputs for the cost-minimizing technique are aggregated to obtain the quantity of capital at each point on the demand curve. Switch points are horizontal line segments in the graph. Around each switch point, a lower rate of profits is associated with a demand for a greater quantity of capital. This property is consistent with outdated marginalist theory. It is not a general property.

Figure 4: Economy-Wide Demand Function for Capital

The demand function for capital is not vertical between switch prices. These curves reflect the variation of prices with the rate of profits, given the technique. This variation is known as the price Wicksell effect. Edwin Burmeister champions David Champernowne's chain index measure of capital. This chain index eliminates price Wicksell effects. Given no capital-reversing, this chain index can be used to show that the rate of profits equals the marginal product of capital. This equality plays no role in solving the price system or in analyzing the choice of technique.

You can also draw an economy-wide demand curve for labor (Figure 5). Switch points are again shown as horizontal line segments. Here, a more labor-intensive technique is adopted at a lower wage. This, too, is a general property. But to emphasize the effects of the recurrence of truncation, I want an example that does not contradict obsolete marginalist theory at an aggregate level.

Figure 5: Economy-Wide Demand Function for Labor

5.0 Sectorial Demand Curves for Labor

I now consider the demand for labor in each industry. Figure 6 plots a sectorial demand curve in the machine industry. Prices of production are assumed to prevail at each level of the wage. One new machine is produced gross by the machine industry. Alpha and Gamma both operate the machine for a single year in producing new machines. Thus, the amount of labor employed with the given gross output in the machine industry is the same for Alpha and Gamma, as shown by the single vertical line to the left. Beta and Delta both operate the machine for its full physical life of two years in the machine industry, also, as shown in the graph. The sectorial labor demand function in the machine industry is a downward-sloping step function approximation to the traditional, non-justified story.

Figure 6: Labor Demand In Machine Industry

Figure 1, at the top of this post is the demand function for labor in the corn industry. Alpha and Beta both operate the machine for one year in the corn industry, while Gamma and Delta operate the machine in this industry for the full two years. The switch point between Beta and Delta exhibits the reverse substitution of labor. A higher wage around this switch point is associated with firms wanting to ultimately employ more labor per bushel corn produced gross in the corn industry.

Conclusion

So much for explaining wages and employment by well-behaved supply and demand functions for labor.

Even without reswitching or capital-reversing, the marginalist textbook stories do not work.

How To Draw Hayekian Triangles In A Model Of The Production Of Commodities

1.0 Introduction

This post revisits an analysis of Hayekian triangles in the context of a circular flow of production. Here I go through the mathematics to show how to construct the "triangle".

2.0 The Technique and Net Output

A technique is specified by a row vector a₀ of direct labor coefficents of a square Leontief input matrix A. Each labor coefficient and corresponding column of the Leontief matrix specify a process to produce one unit of the good produced by that industry. All coefficients are specified in physical units, such as barrel oils per kilowatt. I assume:

• The economy is in a stationary state.
• Constant returns to scale (CRS) prevail.
• All direct labor coefficients are positive.
• Every good enters, either directly or indirectly, into the production of every good.
• The Leontief matrix specifies a productive technique in that a suplus product can be produced.
• Full employment is assumed. More generally, the units in which labor is measured is scaled such that employment is unity.
• Wages are paid at the end of the year, not advanced with the payments for capital goods at the beginning of the year.

These assumptions are stronger than needed.

The proportions of net output are assumed to be specified by a column vector d. This vector is also a numeraire. The level of net output y is specified by the scalar c:

y = c d

This formulation allows for specifying any number of techniques, all with the same numeraire and composition of net output, but at different levels.

3.0 Quantity Flows

The net output vecotr y and the gross output vector q are related as:

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

Total employment is unity:

a₀ q = 1

These equations have a solution. Consumption per worker is:

c = 1/[a₀ (I - A)^-1 d

Gross quantities are:

q = c (I - A)^-1 d = c d + c A d + c A² d + ...

The first term in the infinite expansion on the right-hand side is the net product available at the end of the given year. The second term is the quantities of capital goods being produced in the current year to support the production of the net output in the next year. The third term is the capital goods being produced in the current year to eventually produce the net output two years hence. Note that all of these vectors, of consumption goods, specific capital goods, and so on are heterogeneous.

4.0 Labor and Capital Flows in the Hayekian Triangle

The labor l_i expended in the current year, with previously produced capital goods, to produce the net output, that is, goods of the first order, is defined as:

l₁ = c a₀ d

The labor l_i expended in the current year to produce goods of each of the higher orders is:

l_i = c a₀ A^{i - 1} d, i = 2, 3, ...

The sum of these quantities of labor is unity, that is, the labor force employed in the current year.

The capital goods expended in the current year to produce goods of each order is:

k_i = c Aⁱ d, i = 1, 2, ...

The sum of these quantities of capital goods used in the current year is A q.

5.0 Value Flows in the Hayekian Triangle

Let z_i be the addition in value for each stage in the Hayekian triangle. This is merely the value added by original factors of production, properly time discounted for each stage:

z_i = w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

The notation reflects the interdependence of the wage 𝑤(𝑟) and the interest rate 𝑟 in a stationary state.

For goods of first order, the length of this step in the Hayekian triangle is:

z₁ + z₂ + ... = c p(r) d

where p(r) is a row vector of prices. For goods of the second order, the length of the step in the Hayekian triangle is:

z₂ + z₃ + ... = c p(r) d - z₁

For goods of the third order, the length of the step in the Hayekian triangle is:

z₃ + z₄ + ... = c p(r) d - (z₁ + z₂)

These steps can be continued. This completes one derivation of the lengths of the steps in a Hayekian triangle.

The length of the ith step can also be expressed as:

z_i + z_{i + 1} + ... = p(r) k_i (1 + r)ⁱ + w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

With a couple of substitutions and factoring, the above becomes:

z_i + z_{i + 1} + ... = c [p(r) A (1 + r) + w(r) a₀] A^{i - 1} d (1 + r)^{i - 1}

Or:

z_i + z_{i + 1} + ... = c p(r) A^{i - 1} d (1 + r)^{i - 1}, i = 1, 2, ...

The Hayekian triangle, with an infinite number of steps, has now been derived, in two ways, from the circulating capital case of a model of the production of commodities by means of commodities.

6.0 Conclusion

The above derivations assume knowledge of the solutions of the price system for the technique. A more complete exposition would present that solution. It would also show that the Hayekian triangle approaches one constructed with a geometric series, as the order of goods increases. The composition of capital goods approaches that of Sraffa's standard system.

On A Transition To Socialism

I have just started David Schweickart's After Capitalism. I have yet to get to his exposition of how transition to economic democracy might be achieved.

To my mind, a transition to Schweickart's preferred system could start with a government fund acquiring enough equity in corporates to prescribe certain organization decisions. As I recall, the United States government bailed out General Motors during the 2008 worldwide depression. These were loans; they could have taken equity. The Norwegian government accumulates a Sovereign Wealth Fund from North Sea oil. The Australian government has superanuation funds.

Once a government has enough ownership of some corporations, they are not required to maintain them as hierarchical, authoritarian institutions. They could, say, get rid of the board of directors. An elected workers' council could appoint management.

The government fund can still own the capital. The firm pays a fee for renting the capital. I once worked for a wholl owned subsidiary, a capitalist firm owned by another capitalist firm. We had to pay such a fee to the owning company.

Schweickart has the revenue obtained by such fees used to create an investment fund. Investment will be directed not solely by profitability. He has the investment fund being paid out on a per capita basis, as a first approximation. Some of the investments are national in scope. The remainder goes to regions on a per capita basis, as a first approximation. This works well with a federal system, like Australia, Canada, and the USA. Public needs would have some influence. For example, some investment might be deliberately directed to less developed regions of the country or to update old technology with more green, ecological technology. New worker-directed firms can be created, and current firms can invest in new processes and new products.

I can see how some of these fees might also pay for, say, head start, pension funds, and so on. Not everybody is in the work force. Schweickart recommends a job guarantee program. He imagines current social programs would continue with funding, I guess, by current tax systems. This is in parallel with new investment fund.

The workers, however, would be the residual claimant under economic democracy. They might not even be paid wages. Profit calculations would differ. Maybe profits are broken down and distributed every week, every two weeks, or every month.

Under Schweickart's economic democracy, the means of production are not privately owned but rented from the public. Workers do not sell their labor power for a wage. Products are produced for buying and selling on markets. As now, large islands exist where the maket is replaced by internal planning. Now those are privately owned corporations. Under this new structure, these firms would be governed by the workers themselves.

We could start transitioning to such a system today.

But I do not see where the political will comes from to do this. On the other hand, I have seen changes in my lifetime that I did not think possible: the fall of the Berlin Wall, the end of apartheid in South Africa, the fall of fascism in Chile, a genuine democracy created for a while in the Philippines.

On The Incoherence Of Austrian Business Cycle Theory

I thought I would try to summarize again some objections to the Austrian school.

Austrian Business Cycle Theory (ABCT) focuses on the consequences of the monetary authority setting the monetary interest rate below the natural rate of interest. Following Knut Wicksell somewhat, the theory argues that capitalist entrepreneurs will lengthen production processes. Since these decisions do not synchronize with household consumption and savings decisions, the artificial boom is unsustainable. A bust is the result.

This theory is built on mistaken capital theory. When Sraffa spanked Hayek, he deliberately put capital theory aside.

What would it mean for a production technique to be more capital-intensive? To examine the (il)logic of this approach, I make various simplifying assumptions.

Accordingly, consider vertically-integrated firms. They produce a given commodity or, rather, a basket of commodities, in fixed proportions. The only input is homogenous labor. All means of production, tools, intermediate goods are produced and used internally.

Under these assumptions, one can talk about (net) output per person-year. Productivity is well-defined. I start by postulating that with a more capital-intensive technique, workers are more productive.

It turns out that a lower interest rate does not induce managers of firms to adopt more capital-intensive techniques. Numerical examples illustrating this point have been available in the literature since the 1960s. And they are accepted by all sides. "The interesting point, however, is the perversity, not the duplicity." -- Robinson and Naqvi (1967).

I have now demonstrated that marginalist capital theory, including the Austrian variant, is invalid. Given typical assumptions, the traditional stories about 'capital' markets do not follow. But what about all that stuff Austrian school economists say about the structure of production?

They are wrong there, too. First, I consider aggregate measures of the period of production. Bohm-Bawerk's measure assumes simple interest, not compound interest. The counterexamples mentioned above demonstrate it is invalid to conclude cost-minimizing entrepreneurs will lengthen the period of production when they anticipate lower interest rates.

Nicolas Cachanosky and Peter Lewin have recently proposed a financial measure of Duration. By this measure, the technique chosen at a lower interest rate, around a switch point, has a larger Duration. But a larger Duration is associated with lower productivity in the counter-examples.

You could also consider how long capitalists will choose to run given machinery. Machines last for more than one production cycle, and capitalists must choose to set their economic life. Surely, a lower interest rate will provide incentives to capitalists to increase their economic life. Well, no. A longer economic life of a machine can be associated with a less capital-intensive technique in the sense that the productivity of labor is decreased. I happen to know that the recurrence of the period of truncation is possible without the reswitching of techniques.

And then there are Hayekian triangles. They do not work either. Hayekian triangles, as Roger Garrison notes, are heuristic pictures, useful for pedagogic purposes. Hayek unsuccessfully tried to put them on rigorous foundations in Prices and Production. But it all fell apart. He even discovered capital-reversing, in some sense. You can find more recent statements with these triangles, but nothing that addresses the difficulties that Hayek found, much less anything that surmounts them.

Jevons Versus Marshal On Ricardo

I have been pointing out that much teaching in most universities and high schools in economics is propaganda. Mistakes that were exposed more than half a century ago continue to be taught. Alternatives have been available, at varying levels, in textbooks for decades.

Does an alternative, building on classical political economy and Marx, exist? Assertions on this topic go back more than a century.

A (bad) way of reading classical political economy is that its proponents were struggling towards developing the one true system, that of marginalist economists. With this incorrect view of continuity, you might say incorrectly that they overemphasized supply. Their theories were corrected by developing theories of utility and demand.

A better reading recognizes that they had their own approach. Jevons held this view, although he was wrong about which approach was better:

"When at length a true system of Economics comes to be established, it will be seen that that able but wrong-headed man, David Ricardo, shunted the car of Economic science on to a wrong line - a line, however, on which it was further urged towards confusion by his equally able and, wrong-headed admirer, John Stuart Mill." -- William Stanley Jevons, The Theory of Political Economy, Preface to the Second Edition, p. li.

Marshall, on the other hand, was an early progenitor of a supposedly generous reading that blurs the distinctiveness of the classical theory of value and distribution:

"1... [Ricardo's] book makes no pretence to be systematic. He was with difficulty induced to publish it; and if in writing it he had in view any readers at all, they were chiefly those statesmen and business men with whom he associated. So he purposely omitted many things which were necessary for the logical completeness of his argument, but which they would regard as obvious. And further, as he told Malthus in the following October, he was 'but a poor master of language.' His exposition is as confused as his thought is profound; he uses words in artificial senses which he does not explain, and to which he does not adhere; and he changes from one hypothesis to another without giving notice.

If then we seek to understand him rightly, we must interpret him generously, more generously than he himself interpreted Adam Smith. When his words are ambiguous, we must give them that interpretation which other passages in his writings indicate that he would have wished us to give them. If we do this with the desire to ascertain what he really meant, his doctrines, though very far from complete, are free from many of the errors that are commonly attributed to them...

...Again, in a profound, though very incomplete, discussion of the difference between 'Value and Riches' he seems to be feeling his way towards the distinction between marginal and total utility. For by Riches he means total utility, and he seems to be always on the point of stating that value corresponds to the increment of riches which results from that part of the commodity which it is only just worth the while of purchasers to buy; and that when the supply runs short, whether temporarily in consequence of a passing accident, or permanently in consequence of an increase in cost of production, there is a rise in that marginal increment of riches which is measured by value, at the same time that there is a diminution in the aggregate riches, the total utility, derived from the commodity. Throughout the whole discussion he is trying to say, though (being ignorant of the terse language of the differential calculus) he did not get hold of the right words in which to say it neatly, that marginal utility is raised and total utility is lessened by any check to supply.

2. But while not thinking that he had much to say that was of great importance on the subject of utility, he believed that the connection between cost of production and value was imperfectly understood; and that erroneous views on this subject were likely to lead the country astray in practical problems of taxation and finance; and so he addressed himself specially to this subject. But here also he made short cuts." -- Alfred Marshall, Principles of Economics, Appendix I

Marshall is wrong here. For example, Ricardo describes riches as a collection of commodities. They were not measured along a single scale, whatever measurement level you might think that scale obtains. Even less could his labor values be said to have been marginal utilities.

Samuel Hollander is the greatest exponent in my lifetime of the view of continuity in the development of theories of value and distribution. Even he, though, recognizes that Marx had reasons for his reading of Ricardo, but I forget where.

Supply And Demand Breaking Down Half A Century Ago: The Sonnenschein-Mantel-Debreu Theorem

"[M]ainstream economists [divide] into effective 'castes', with only a tiny but exalted subset of the profession undertaking the detailed mathematical work needed to discover the weaknesses in the theory. The vast majority of economists believe that this high caste, the mathematical economists, did their work properly, and proved that the theory is internally consistent. The caste has indeed done its work properly, but it has proved precisely the opposite: that the theory is consistent only under the most restrictive and specious of assumptions." - Steve Keen, Debunking Economics

1.0 Introduction

Economists like to tell stories about supply and demand, in which a higher price of a good signals that it is more scarce and encourages agents to substitute other goods for the more scarce good. Mainstream economists have known for more than half a century that these stories have no justification in the most rigorous versions of their theory. Their stories are ad hoc and arbitrary.

I have summarized the Cambridge Capital Controversy before. Here I concentrate on the Sonnenschein-Mantel-Debreu (SMD) theorem.

If General Equilibrium Theory (GET) were to have empirical implications, it would restrict what was possible for market behavior. It turns out that, however, supply and demand functions can have almost any shape. No reason exists, in the theory, for equilibria to be unique or stable. As Andreu Mas Colell and his co-authors put it, anything goes.

I rely more on Alan Kirman's presentations than the original papers for the SMD theorem.

2.0 General Equilibrium Theory (GET)

Leon Walras invented GET and set out its canonical problems: the existence of an equilibrium, its uniqueness, and its stability. For the latter, he invented the tatonnement process, an auction in which no transactions are allowed until prices are found in which demand and supplies are equal. The Arrow-Debreu-McKenzie model is the current canonical statement of GET. For purposes of this post, you can consider a pure exchange economy.

Supply and demand are functions. For example, the quantity demanded and supplied of butter are depicted as functions of its price. The difference between demand and supply is an excess demand function.

Expressing the supply and demand of butter as only a function of its price seems inadequate. Should the demand not also depend on the price of margarine? If the price of bread fell and consumers consumed more bread, would not their demand for butter also rise? Would not the supply of bread, and thus the demand for butter, be impacted by decisions of farmers between growing wheat and producing crops for ethanol?

GET attempts to model all these interactions. Households, in a competitive pure exchange economy, are assumed to start with given endowments, with a certain basket of goods. They also are assumed to have preferences among these goods and to face given prices. The households decide how much of each good in their endowment to sell on the market and how much more to buy. In the jargon, they maximize their utility subject to a wealth constraint.

So for any set of prices, the model describes the difference, for each household, between the quantity demanded on the market of each good and their endowment of each good. This is the household's excess demand function. Under certain general and non-restrictive assumptions, individual excess demand functions have certain supposedly intuitive properties. I think the demonstration that demand functions slope down, if substitution effects dominate income effects, applies to the analysis of a household's maximization problem.

Aggregate or market excess demand functions are found by summing over all households. (Aggregate demand, in this sense, is not the aggregate demands in macroeconomics. They are specified for each of thousands of goods, not somehow summed over all goods.) Suppose the market excess demand for some good was positive at some price vector. Then the households would be trying to buy more of that good than exists. This is a disequilibrium.

An equilibrium exists when the prices are such that utility-maximization decisions of the households are mutually consistent. No good exists in which the households want to buy more than the aggregate endowment of that good.

3,0 Characterization of Market Excess Demand Functions

Arrow & Debreu and McKenzie proved that, under fairly general conditions, an equilibrium exists. I am unsure if the first welfare theorem, from GET, is the theoretical justification for claims that an unregulated capitalism can be efficient. Debreu always denied this interpretation, as I understand it. Debreu (1959) provides no attempt to describe how an equilibrium can be achieved. This remains an unsolved problem (see Fisher 1983).

Almost any functions can be excess demand functions. The restrictions are that the functions be continuous, homogeneous of degree zero, and satisfy Walras' law. Also, we only consider the functions bounded an arbitrarily small distance away from zero. That is, the behavior of the function when all prices are zero is not considered.

Homogenity here means only relative, not absolute prices matter. It does not matter if prices are denominated in dimes or dollars, euros or yuan.

Walras law states that if the excess demand for some good is positive, at disequilibrium prices, then some other markets have excess supplies. The disequilibria cancel out, in some sense.

The conclusion is that GET has no empirical implications at the level of markets.

4.0 Failed Attempts at Workarounds

Market excess demand functions can inherit nice properties on individual excess demand functions if all individuals are identical and have homothetic preferences. The latter implies that Engel curves are linear functions. Your relative demands for different goods, for say, chicken or lobster, does not depend on your income.

These assumptions were typical of macroeconomists for a long time after the so-called rational expectations revolution. They talk a lot about micro foundations, but their models lack them. They could not accommodate individuals with different tastes or with tastes that varied in some way with income.

Kirman may have been sensitized to the importance of the SMD theorem by his attempts, with co-authors to relax these assumptions. What happens if individuals have homothetic preferences, but individuals vary among themselves in their preferences? The same class of functions can still be excess demand functions, with the above extremely limited constraints. How about if individuals have identical preferences, but they are not necessarily homothetic? This does not help. Nor does it help to include production.

5.0 Conclusion

Kirman suggests, as I understand it, that part of the problem is that individuals interact in the model only through markets. Maybe some sort of norms or fashions shape preferences to provide some sort of coordination. Or maybe economists should consider broad classes of households as having common preferences. This type of approach is like that of the classical political economists who assumed, for example, that workers consume all their income (they do not have much), capitalists save, and landlords indulge in spending on luxuries.

References

• Gerard Debreu (1959) Theory of Value: An Axiomatic Analysis of Economic Equilibrium.
• Gerard Debreu (1974) Excess demand functions. Journal of Mathematical Economics 1(1): 15-21.
• Franklin M. Fisher (1983) Disequilibrium Foundations of Equilibrium Economics.
• Alan Kirman (1989) The intrinsic limits of modern economic theory: the emperor has no clothes. Economic Journal 99(395): 126-139.
• Alan Kirman (1992). Whom or what does the representative individual represent? Journal of Economic Perspectives 6(2): 117-136.
• Alan Kirman and K. J. Koch (1986) Market excess demand in exchange economies with identical preferences and collinear endowments. Review of Economic Studies 53(3): 457-463.
• Rolf R. Mantel (1974) On the characterization of excess demand functions. Journal of Economic Theory 7(3): 348-353.
• Hugo Sonnenschein (1972) Market excess demand functions. Econometrica 40(3): 549-563.
• Leon Walras(1874/1877) Elements of Pure Economics.

The Centre Of The Solving Subsystem In A Model With Fixed Capital And Scarce Land

1.0 Introduction

This post revisits my example with fixed capital and two types of land. It presents, by means of an example, the concept of the centre of a solving subsystem. Quadrio Curzio & Pellizzari (2010) introduce the solving subsystem in models of rent so as to first solve the price equations without rent. Schefold (1989) introduces the centre of the price system for a pure fixed capital model to, following Sraffa, initially eliminate the prices of old machines from price equations. As far as I know, nobody has combined these concepts before.

The concept of a solving subsystem clarifies how a switch point can lie along a single wage curve. A system of equations for prices is associated with each technique. Each operated process contributes an equation equating revenues and costs. The revenues can include the prices of joint products, and costs include a charge for the rate of profits on advanced capital goods. A last equation specifies the value of the numeraire as unity. In models of extensive rent, a subsystem can be formed from the processes that characterize industrial processes, with no inputs from land, and processes run on land that are not scarce. The resulting subsystem, with the equation for the numeraire concatenated, can be solved, given the rate of profits, for the wage and the prices of produced commodities. In models of intensive rent, the solving subsystem includes the equations for industrial processes and a linear combination of the equations for the processes that operate on one type of land to the limits of its endowment. As Sraffa (1960) explains, a variable for rent is eliminated by this linear combination. In the case of extensive rent, with no joint production otherwise, the solving subsystem also applies to a model of single production. In any case, the solution to the solving subsystem can then be used to find rents. The example in this post, extends the concept of a solving subsystem to a case with extensive rent and fixed capital. I do not know if the concept of a solving subsystem can usefully apply to joint production more generally

The centre of a pure fixed capital system (Schefold 1989) helps solve the price system of a pure fixed capital system. Joint utilization of machines does not exist in any process in a model of pure fixed capital. Old machines are not consumer goods. In the example, a single commodity is a consumption good and acts as numeraire. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete. Nice properties of single production systems generalize to such cases of fixed capital. In particular, the "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang, 2019). The cost-minimizing technique can be determined by the construction of the wage frontier. These properties are not retained in the combination of pure fixed capital with scarce land. The centre still helps solve the price system.

2.0 Technology, Endowments, Final Demand

Tables 1 and 2 specify the technology. This technology extends an example of fixed capital from Baldone (1974). Labor uses circulating capital to manufacture a machine in process I. The machine has a physical life of three years. Labor uses circulating capital and the machine to produce corn on type 1 land in processes II, III, and IV. The machine is operated on type 2 land in processes V, VI, and VII. A process that produces corn jointly produces a machine one year older than the machine used as input, up to its physical life. One hundred acres of each type of land are assumed to exist. Final demand is for 87 bushels corn, a level that ensures one or the other type of land is scarce. The numeraire is a bushel of corn.

Table 1: Inputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Labor	a0,1 = 0.4	a0,2 = 0.2	a0,3 = 0.6	a0,4 = 0.4	a0,5 = 0.23	a0,6 = 0.59	a0,7 = 0.39
Type 1 Land	0	c1,2 = 1	c1,3 = 1	c1,4 = 1	0	0	0
Type 2 Land	0	0	0	0	c2,5 = 1	c2,6 = 1	c2,7 = 1
Corn	a1,1 = 0.1	a1,2 = 0.4	a1,3 = 0.578	a1,4 = 0.6	a1,5 = 0.39	a1,6 = 0.59	a1,7 = 0.61
New Machines	0	1	0	0	1	0	0
Type 1 1-Yr. Old Machines	0	0	1	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	0	1	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	0	1	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	0	1

Table 2: Outputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Corn	0	b1,2 = 1	b1,3 = 1	b1,4 = 1	b1,5 = 1	b1,6 = 1	b1,7 = 1
New Machines	1	0	0	0	0	0	0
Type 1 1-Yr. Old Machines	0	1	0	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	1	0	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	1	0	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	1	0

3.0 Techniques

Tables 3, 4, and 5 specify the techniques that may be chosen with this technology. Alpha, Beta, and Gamma differ in the economic life of the machine on non-scarce, type 1 land. No processes are operated on type 2 land. Under Delta, Epsilon, and Zeta, on the other hand, type 1 land is not farmed at all, and the economic life of the machine varies among the techniques in the processes operated on type 2 land. The remaining techniques fully cultivate one or the other type of land and require rent to be paid to landlords

Table 3: Techniques of Production with Non-Scarce Land

Technique	Processes	Type 1 Land	Type 2 Land
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, II, III, IV	Partially farmed	Fallow
Delta	I, V	Fallow	Partially farmed
Epsilon	I, V, VI	Fallow	Partially farmed
Zeta	I, V, VI, VII	Fallow	Partially farmed

Table 4: Techniques of Production with Type 1 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Eta	I, II, V	Fully farmed	Partially farmed
Theta	I, II, III, V	Fully farmed	Partially farmed
Iota	I, II, III, IV, V	Fully farmed	Partially farmed
Kappa	I, II, V, VI	Fully farmed	Partially farmed
Lambda	I, II, III, V, VI	Fully farmed	Partially farmed
Mu	I, II, III, IV, V, VI	Fully farmed	Partially farmed
Nu	I, II, V, VI, VII	Fully farmed	Partially farmed
Xi	I, II, III, V, VI, VII	Fully farmed	Partially farmed
Omicron	I, II, III, IV, V, VI, VII	Fully farmed	Partially farmed

Table 5: Techniques of Production with Type 2 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Pi	I, II, V	Partially farmed	Fully farmed
Rho	I, II, III, V	Partially farmed	Fully farmed
Sigma	I, II, III, IV, V	Partially farmed	Fully farmed
Tau	I, II, V, VI	Partially farmed	Fully farmed
Upsilon	I, II, III, V, VI	Partially farmed	Fully farmed
Phi	I, II, III, IV, V, VI	Partially farmed	Fully farmed
Chi	I, II, V, VI, VII	Partially farmed	Fully farmed
Psi	I, II, III, V, VI, VII	Partially farmed	Fully farmed
Omega	I, II, III, IV, V, VI, VII	Partially farmed	Fully farmed

Under techniques Eta through Omicron, type 1 land is fully farmed and pays rent. Under Eta, Theta, and Iota, the machine is operated for only one year on type 2 land and then discarded. The techniques differ on the economic life of the machine on type 1 land. Under Kappa, Lambda, and Mu, the machine is operated for two years on type 2 land, while it is operated for its full physical life of three years under Nu, Xi, and Omicron. Under Pi through Omega, type 2 land is scarce and pays rent. Each technique between Eta and Omicron corresponds to a technique between Pi and Omega in which the same processes are operated. The economic life of the two types of machines are the same in these corresponding techniques. The scale at which the processes are run varies so as to vary which type of land is fully farmed.

4.0 The Price System for Omicron

I consider the price equations for Omicron to illustrate the concepts of the solving subsystem and of the centre. All seven processes are operated under Omicron, and type 1 land is scarce. The following seven displays, in obvious notation, specify the price system for Omicron:

a_1,1(1 + r) + w a_0,1 = p₀

(a_1,2 + p₀)(1 + r) + rho₁ c_1,2 + w a_0,2 = b_1,2 + p_1,1

(a_1,3 + p_1,1)(1 + r) + rho₁ c_1,3 + w a_0,3 = b_1,3 + p_1,2

(a_1,4 + p_1,2)(1 + r) + rho₁ c_1,4 + w a_0,4 = b_1,4

(a_1,5 + p₀)(1 + r) + w a_0,5 = b_1,5 + p_2,1

(a_1,6 + p_2,1)(1 + r) + w a_0,6 = b_1,6 + p_2,2

(a_1,7 + p_2,2)(1 + r) + w a_0,7 = b_1,7

Revenues for operating each process at a unit level are shown on the right-hand side of these equations. Revenues for the first process are obtained by selling new machines. Revenues for the second process result from products of both corn and a type 1 one-year old machine. That type 1 machine, in turn, enters into the advanced costs of the third process, and so on. Type 1 land obtains a rent, and type 2 land is free.

The first equation and the last three of the seven constitute the solving subsystem for Omicron. Given the rate of profits, the solving subsystem specifies the wage, the price of a new machine, and the prices of one-year old and two-year old machines when operated on free type 2 land. The remaining three equations can then be used to find the rent on type 1 land and the prices of one-year old and two-year old machines when operated on type 1 land. The solving subsystem for Omicron is also the solving subsystem for Zeta, Nu, and Xi. In all these techniques, the machine is run for its full physical life of three years on free type 2 land.

The prices of old type 2 machines can be eliminated from the solving subsystem for Omicron. Multiply both sides of the second equation of the solving subsystem by (1 + r)²:

(a_1,5 + p₀)(1 + r)³ + w a_0,5(1 + r)² = b(1 + r)²_1,5 + p_2,1(1 + r)²

Multiply both sides of the third equation of the solving subsystem by (1 + r):

(a_1,6 + p_2,1)(1 + r)² + w a_0,6(1 + r) = b_1,6(1 + r) + p_2,2(1 + r)

Add these two equations and the last equation of the solving subsystem:

where the row vector and matrix in this system of equations is as follows:

The ordered pair consisting of this row vector and matrix is the centre (Schefold 1989) for the solving subsystem for Omicron. Given the rate of profits, this system of matrix equations can be solved for the wage and the price of a new machine. This price system has the form of a price system for a circulating capital model, with the exception of the dependence of the Leontief input matrix and the vector of direct labor coefficients on the rate of profits. Unlike in the model of circulating capital, the wage curve derived from the centre of a pure fixed capital system can slope up for part of its range. The wage frontier of a pure fixed capital system, however, decreases throughout its length (Baldone 1974, Varri 1974).

The prices of old type 1 machines can be similarly eliminated from the full price system for Omicron.

5.0 Conclusion and Questions

The above illustrates the centre of a solving subsystem. In the example, the solving subsystem shows that a system of seven equations for a price system can be decomposed such that a system of four equations is solved first. And the centre of the solving subsystem shows that that system of four equations can be further decomposed so that a system of two equations is solved first.

Perhaps the centre of a solving subsystem can be used to address a theoretical question. Is the wage frontier always decreasing in a model combining fixed capital and rent? Can the wage frontier sometimes slope up?

In a model of extensive rent, the wage frontier is not the outer envelope of the wage curves for the technique. But it is always decreasing. Each wage curve is found from a solving subsystem. And the solving subsystem is from a related circulating capital model. So the wage curves inherit the properties of circulating capital models. The wage frontier is formed from the wage curves of the cost-minimizing techniques and always is decreasing.

In a pure fixed capital model, the wage frontier is the outer envelope of the wage curves for the techniques and is always decreasing. Individual wage curves can be increasing, but the ranges of the rate of profits at which they are increasing is never on the frontier.

I suspect the wage frontier for a model combining extensive rent and fixed capital can be increasing over some range of the rate of profits. This suspicion should be validated by constructing a numerical example. On the other had, if the wage frontier is alwys decreasing in such a model, that should be capable of a proof. And such a proof, if it exists, will probably use the concept of the centre of a solving subsystem.

References

• Baldone, S. (1974), Il capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIV(1): 45-106. Trans. in Pasinetti (1980).
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Pasinetti, L.L. 1980. (ed.), Essays on the Theory of Joint Production, New York, Columbia University Press.
• Quadrio Curzio, Alberto. 1980. Rent, income distribution, and orders of efficiency and rentability (in Pasinetti 1980).
• Quadrio Curzio, Alberto and Fausta Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer.
• Schefold, Bertram. 1989. Mr. Sraffa on Joint Production and other Essays, London, Unwin-Hyman.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.
• Varri, P. 1974. Prezzi, saggio del profitto e durata del capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIX(1): 5-44. Trans. in Pasinetti (1980).

Old Papers On Rent And One New One

This post annotates some papers that I want to remind myself of.

Montani (1975) references Quadrio Curzio (in Italian), defines the order of fertility and rentability, notes that they are different, and has something like the reswitching of the order of fertility. He does not have the reswitching of the order of rentability. He treats both extensive and intensive rent, but does not combine them. He notes the wage frontier can slope up under intensive rent. I have to read more closely to see if he already has multiple cost-minimizing techniques. I am under the impression that D'Agata first notice this possibility.

Montet (1979) criticizes Metcalfe and Steedman in that their perversities are more general than they know. Land provides another degree of freedom. They have a wage, rent, rate of profits frontier. I generally do not set equations for natural resources out this way. I once set out an example with heterogeneous labor, relabeling 'land' as 'skilled labor'.

Gibson & McLeod (1983) look at extensive, intensive, and external intensive rent. They go into difficulties of defining basics in joint production. One definition is about the decomposability of matrices and the other is about the rank of some sort of block matrix. They define quasi-basics for the latter. D’Agata has some sort of objection to this. They have interchanges in both the CJE and the RRPE.

Erreygers (1995) considers joint production. Toward the end of his paper, he shows how extensive rent fits into this framework. He wants to avoid setting out another equation in the quantity system to constrain levels of operations of processes from requiring more land to be farmed than exist. And rents should be part of the price vector in the price system, not seperate variables. Kurz & Salvadori (1995) show how to define certain block structured matrices to achieve this end. I think Erreygers may have created this approach.

Ianni (2026) is about international trade, not rent. The theory of intensive and extensive rent can show why most lands are specialized, so the theory may have implications for the theory of international trade. Also, my way of analyzing the choice of technique with long-lasting and given ratios of the rate of profits among industries may have implications for trade. Different countries may be modeled as having different rates of profits.

References

• Erreygers, Guido. 1995. On the uniqueness of cost-minimizing techniques. The Manchester School 63: 145-166
• Gibson, Bill and Darryl McLeod. 1983. Non-produced means of production in Sraffa's system: basics, non-basics and quasi-basics. Cambridge Journal of Economics 7: 141-150.
• Ianni, Guido. 2026. Class struggle and the international division of surplus: The distributive surface in the open economy. Metroeconomica
• Montani, Guido. 1975. Scarce natural resources and income distribution. Metroeconomica 27(1): 68=101.
• Montet, C. 1979. Reswitching and primary input use: a comment. Economic Journal 89 (355): 642-647.

Factor Demand Curves For An Example With Fixed Capital And Rent

Figure 1: Demand Curve for Labor

I have created and worked through an example in which a machine with a physical life of three years can be used in producing an agricultural commodity on one of two types of land.

My example is one of capital-reversing. It occurs to me that I have not plotted the demand for so-called factors of production in this example. Accordingly, Figure 1 plots the wage against the employment firms want to offer, given final demand. Switch points are horizontal line segments in this graph. Around the 'perverse' switch point, a higher wage is associated with firms wanting to employ more workers.

Given final demand and the rate of profits, a price system is defined for each technique. I can add up the value of the capital goods that must exist at the start of the year to produce the given final demand. Prices of production are used to aggregate heterogeneous goods. Figure 2 shows the demand for capital, in some sense. Here, too, the 'perverse' switch point is indicated for a step function approximation for an increasing demand curve. The value of capital varies between switch points because of price Wicksell effects.

Figure 2: Demand Curve for Capital

A model with both fixed capital and the rent of natural resources is a step towards realism if you want. It is also a step beyond what can be found from empirical Leontief matrices, as I understand it. Still, wages and employment, for example, cannot be explained in the long run by the interactions of well-behaved supply and demand functions in the labor market.