Givens For Two Approaches To The Theory Of Value And Distribution

1.0 Introduction

Broadly speaking, the history of political economy contains two approaches to value and distribution. For purposes of this post, I do not distinguish between classical and Marx's political economy. Institutionalists and those who know about German historical schools, for example, might have a complaint about being ignored.

This post is quite unoriginal. I thought I would just record these properties of two approaches.

2.0 Marginalism

Marginalist economics is about the allocation of given resources among alternatives. In marginalism, the theory of value and distribution is almost co-extensive with economic theory. The givens, for the theory of value and distribution, are:

• Endowments, including distribution of endowments among households.
• Tastes or preferences of each agent.
• Technology.

How to take capital as a given endowment is a difficulty with this approach. It can hardly be taken as a given quantity of value. The theory is supposed to explain prices, including the prices of capital goods. This problem is not just with aggregate theory. It is also a problem with microeconomic theory.

Another approach is to take initial quantities of individual capital goods as given. The neo-walrasian approach abandons the long run and the equalization of the rate of profits among industries. Conceptually, some expectations and plans must have been mistaken before the initial point in time. Yet the theory does not seem to accomodate such mistakes at the given time or into the future. Furthermore, debts and entitlements to future income streams do not seem possible to include among the givens. Disequilibrium processes that change the initial endowments and their distribution do not seem possible to include in the theory either.

3.0 Classical Political Economy

Classical political economics analyzes the conditions needed to ensure the reproduction of society. For the theory of value and distribution, the givens are:

• Technology.
• Requirements for use, which I take as net output.
• Wage or the rate of profits.

The theory of value can be combined with other elements of political economy. The classicals had various theories of wages, combined with demographics. Marx rejected Malthus and developed his theory of the reserve army of labor for similar purposes. The theory is compatible with a rejection of Say's law and enduring unemployment. Many have argued for combining this theory with a long-period interpretation of Keynes' general theory. A theory of growth and the dynamics of technical change can be built upon this theory of value and distribution.

Intensive Rent And The Order Of Rentability

I have thought about what would be the minimal structure of an example that combines extensive and intensive rent. I want to include a commodity produced without land, as well as an agricultural commodity.

This post considers a simpler example. An analysis of extensive rent includes the identification of the order of efficiency and the order of rentability, given the wage or the rate of profits. I take the concept of these orders from Alberto Quadrio Curzio. Can these orders be defined in a model of intensive rent? What would the minimum structure of an example be in which to explore this question? I continue to insist on including an industrial commodity with negligible inputs of land.

I suggest Table 1 provides the technology for such an example. Each column specifies the quantities of labor, iron, wheat, and rye needed to produce a unit output of the commodity produced by the corresponding industry. The table also specifies the quantity of land that must be rented to operate that process. Constant returns to scale are assumed, with the limitation that the endowments of each type of land are givens.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	0	c1,2	c1,3	0	0
Type 2 Land	0	0	0	c2,4	c2,5
Iron	a1,1	a1,2	a1,3	a1,4	a1,5
Wheat	a2,1	a2,2	a2,3	a2,4	a2,5
Rye	a3,1	a3,2	a3,3	a3,4	a3,5

I show each type of land as specialized to produce a different kind of agricultural commodity. I am unsure if that specialization is needed for my point. If not, the table defining the techniques below would contain four more techniques. In each, only one of the processes producing the agricultural commodity would be operated.

As noted, the givens include the amount of each type of land available. Let t₁ be the acres of type 1 land available and t₂ be the acres of type 2 land available

The vector d representing the numeraire has components:

• d₁: The quantity of iron in the numeraire.
• d₂: The quantity of wheat in the numeraire.
• d₃: The quantity of rye in the numeraire.

Let net output y consist of a multiple of the numeraire:

y = c d

Net output is among the givens.

Table 2 specifies the techniques. Non-zero coefficients of production in Table 1 should be such that all three commodities are Sraffa basics in all techniques. Not all techniques are feasible for any level of net output.

Table 2: Is Process Operated By Technique?

Input	Industry
	I	II	III	IV	V
Alpha	Yes	Yes	No	Yes	No
Beta	Yes	Yes	No	No	Yes
Gamma	Yes	No	Yes	Yes	No
Delta	Yes	No	Yes	No	Yes
Epsilon	Yes	Yes	No	Yes	Yes
Zeta	Yes	No	Yes	Yes	Yes
Eta	Yes	Yes	Yes	Yes	No
Theta	Yes	Yes	Yes	No	Yes
Iota	Yes	Yes	Yes	Yes	Yes

One can examine which processes are introduced in the cost-minimizing technques as the level of net output expands. That is, is process II adopted before process III or vice versa? Is process IV or process V operated first? Presumably, the answer to these questions depends on the wage or the rate of profits, whichever variable is taken as given in solving the system of equations for prices of production. This model is a model of intensive rent in that, for example, Epsilon or Zeta is cost minimizing, Type 2 land will be fully farmed and obtain a rent. The scarcity of land is shown by having two processes operating side-by-side on a single type of land. Anyways, the analysis outlined here corresponds to determining the order of efficiency in a model of extensive rent.

Suppose the Iota technique is feasible and cost-minimizing. The solution of the equations for prices of production yields the rent per acre for each type of land. Which type of land obtains the larger rent per acre? Does this order vary with the given wage or rate of profits? Is reswitching of this order possible? Does postulating a stable ratio of the rate of profits among industries change the answers, at least in detail? I suggest that this analysis corresponds to determining the order of rentability in a model of extensive rent.

I assume that the order of rentability varies with distribution and that reswitching of this order is indeed possible. As far as I know, nobody has answered these questions or presented a numerical example. I always think that nothing I say would surprise Betram Schefold, Heinz Kurz, or Neri Salvadori, for example. What I try to do is present concrete examples of their more abstract analyses. My identification of fluke cases, of extending the analysis to markup pricing, and presentation of graphs to aid visualization of the results are my own tweaks, I guess.

Elsewhere

• John Cassidy on Donald Harris in the Mew Yorker.
• Dean Dettloff on The Catholic case for communism in America.
• Enrico Bellino and Gabriel Brondino, Circular vs. one-way production processes: two different views on production and income distribution, in Review of Political Economy. I want to recall this for my work on Hayekian triangles.
• Bob Murphy and David Glasner on the Sraffa-Hayek debate. (They spend to much time on monetary policy before getting to the debate.)
• Leland Yeager and Steve Hanke. 2024. Capital, Interest, and Waiting: Controversies, Puzzles, and New Additions to Capital Theory. I probably will not order this.
• Pierangelo Garegnani. 2024. Capital Theory, the Surplus Approach, and Effective Demand. Appears to now be available.

A 1-Dimensional Diagram For Extensive Rent With Markup Pricing

Figure 1: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture

This post is an elaboration on this past post.

The analysis of the choice of technique varies with perturbations of relative markups in industry and agriculture. Figure 1 depicts this variation, while Figure 2 is an enlargement of the lower range of the relative markup in industry. The heavy solid lines in Figure 1, other than the horizontal line at the top, are switch points on the inner frontier of the wage curves. They divide the space into areas in which the cost-minimizing technique is labeled in bold. The dashed lines are intersections on the outer frontier of the wage curves. The order of fertility varies across dashed lines. The dotted lines are intersections of rent curves. The order of rentability varies across dotted lines.

Figure 2: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture (Enlarged)

Fluke cases partition the graph (Table 1), as shown by the thin vertical lines and the numbering of the resulting regions. To call the intersections of wage curves on the outer frontier 'switch points' is an abuse of language. Quantity flows and the technique do not vary around these intersections. The same types of land are fully farmed, and the type of land that is only partially farmed is farmed to the same extent. The same quantity of iron is produced. Nevertheless, the order of efficiency varies around such intersections. Likewise, to call the intersections of rent curves ‘'switch points' is a similar abuse of language. The order of rentability varies around these intersections.

Table 1: Fluke Switch Points

Regions	Description
Between 1 and 2	Rent curves for Type 1 and 3 lands tangent at switch point.
Between 2 and 3	Switch point for rent curves for Type 1 and 2 lands at wage of zero.
Between 3 and 4	Switch point for wage curves for Beta and Gamma at wage of zero.
Between 4 and 5	Switch point for wage curves for Alpha and Beta at wage of zero.
Between 5 and 6	Three wage curves intersect at single switch point.
Between 6 and 7	Wage curves for Alpha and Beta tangent at switch point.
Between 7 and 8	Switch point for rent curves for Type 1 and 2 lands at wage of zero.

The analysis of the choice of technique is qualitatively invariant in each numbered region. The number and sequence of intersections of wage curves and of rent curves do not vary within a numbered region. Table 2 lists the cost-minimizing technique along the wage frontier, from a wage of zero to the maximum wage, in each numbered region. Variations in the order of fertility and in the order of rentability are also indicated.

Table 2: Variations in the Cost-Minimizing Technique

Region	Range	Technique	Order of Fertility	Order of Rentability
1	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2		Type 3, 1, 2
	w2 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
2	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2			Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
3	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 1, 3, 2
	w1 ≤ w ≤ w2			Type 3, 1, 2
	w2 ≤ w ≤ w3		Type 3, 1, 2
	w3 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
4	0 ≤ w ≤ w1	Gamma	Type 1, 2, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2	Beta	Type 1, 3, 2	Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
5	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2		Type 1, 2, 3
	w2 ≤ w ≤ w3	Beta	Type 1, 3, 2	Type 1, 3, 2
	w3 ≤ w ≤ w4			Type 3, 1, 2
	w4 ≤ w ≤ w5		Type 3, 1, 2
	w5 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
6	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w4		Type 1, 2, 3
	w2 ≤ w ≤ w3		Type 2, 1, 3
	w3 ≤ w ≤ w4			Type 2, 1, 3
	w4 ≤ w ≤ w5	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w5 ≤ w ≤ w6			Type 3, 2, 1
	w6 ≤ w ≤ wα, max		Type 3, 2, 1
7	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2			Type 2, 1, 3
	w2 ≤ w ≤ w3	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w3 ≤ w ≤ w4			Type 3, 2, 1
	w4 ≤ w ≤ wα, max		Type 3, 2, 1
8	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 2, 1, 3
	w1 ≤ w ≤ w2	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w2 ≤ w ≤ w3			Type 3, 2, 1
	w3 ≤ w ≤ wα, max		Type 3, 2, 1

In region 1 at the extreme left in Figure 1, agriculture has overwhelming market power, as compared to industry. Owners of Type 2 land do not obtain a rent except when workers make a high wage. Owners of Type 3 land obtain the highest rent per acre, whatever the order of efficiency. In region 8 at the extreme right, owners of Type 3 land do not obtain a rent when workers make a low wage. In this range, industry has much more market power than agriculture. The possibilities are complicated in between these extremes.

Region 5 includes the competitive case, in which s₁ = 1/2. As previously noted, this region exhibits a reswitching of the order of efficiency. Region 2 exhibits a reswitching of the order of rentability. I find worth emphasizing the ranges of the wage in each region in which the orders of efficiency and rentability differ. Such a range exists for each numbered region. Landlords with less fertile land can receive more rent per acre than those with more fertile land. Under capitalism, you are not rewarded, in general, for your contributions or for the contributions of the resources that you own to production. This disconnect applies both to competitive and non-competitive markets.

The analysis of the choice of technique with extensive rent is more complicated than such analysis in a model with only circulating capital:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari (2010).

The quantity and price systems are interconnected. Assumptions on the level of net output are required to determine which techniques are feasible. All three types of land need to be farmed, at least partially, to satisfy the requirements for use in the numeric example.

Introducing relative market power among industries further complicates the analysis of the choice of technique. Extensive rent and which land is marginal, that is, receives no rent, depend on the wage and the relative market power of industry and agriculture. Changes in market power can have similar qualitative effects, such as the introduction of the reswitching of the order of rentability, as structural economic dynamics resulting from technical innovation.

Sraffa's Publications

The following is a list of publications I expect to see in the first volume of Sraffa's collected works:

• Monetary inflation in Italy during and after the war (This is Sraffa's dissertation).
• The current situation of Italian banks. Manchester Guardian. 1922.
• The bank crises in Italy. Economic Journal. 1922.
• On the relations between cost and quantity produced. Annali di economia 1925. (I think an English translation is in the Cambridge Journal of Economics.)
• The laws of returns under competitive conditions. Economic Journal. 1926.
• Increasing returns and the representative firm. Economic Journal. 1930. (This is a symposium with Robertson and Shove.)
• An alleged correction of Ricardo. Quarterly Journal of Economics. (with L. Einaudi).
• Dr. Hayek on money and capital. Economic Journal. 1932.
• Money and Capital: A Rejoinder. Economic Journal. 1932. (This is a rejoinder to Hayek's reply.)
• Malthus on public works. Economic Journal. 1955.
• The Production of Commodities by Means of Commodities. Cambridge University Press. 1960.
• Production of commodities: A comment. Economic Journal. 1962. (This is a response to Harrod's review.)

I expect the three articles in Italian above will be published with Italian and English on facing pages. A couple of letters to editors might be in the first volume. I do not know what I expect for Sraffa's editorial comments in Ricardo's Collected Works or for Garegnani's notes. The second volume should contain Sraffa's lectures in the 1920s, especially those on value.

As I try to recall what was on my crashed hard drive, I'll probably have more lists like this.

Extensive Rent With Markup Pricing: An Example

Figure 1: Choice of Technique with Extensive Rent and Competitive Markets

1.0 Introduction

I am making some slow progress on recreating past posts. For variation, I here take the wage as given.

If I write this up more formally, I intend not to try to relate it to Marx's concept of absolute rent. The point is to illustrate the following comment with long-lasting variations in market power between industry and agriculture:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari. Rent, Resources, Technologies, Springer,2010: 34.

This story illustrates how the owner of Type II land is affected by the distribution between wages and profits, depending on the degree of market power between industry and agriculture. Under competitive markets in the example, this owner receives a rent, no matter what the distribution between wage and profits. With a large enough relative markup in agriculture, as compared with industry, they receive zero rents for some distributions between wages and profits. I also illustrate two fluke cases for relative markups.

2.0 Technology and Endowments

Table 1 presents the technology for the example. The second column shows the inputs of labor, iron, and corn needed to produce a ton of iron. The remaining three columns to the right are the coefficients of production for processes to produce corn. A unit level of operation of a process in agriculture produces a bushel corn and requires an input of one of three types of land, as shown. Constant returns to scale prevail, although the level of operation of the processes producing corn is limited by the available acreage.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV
Labor	1 person-yr.	9/10 person-yrs.	6/10 person-yr.	29/50 person-yr.
Type 1 Land	0	1 acre	0	0
Type 2 Land	0	0	49/50 acre	0
Type 3 Land	0	0	0	2/5 acre
Iron	9/20 ton	1/40 ton	3/2000 ton	29/500 ton
Corn	2 bushels	1/10 bushel	9/20 bushel	13/100 bushel

In the three processes for producing corn, process III requires more labor per acre of land than process II, and process IV requires even more. Output per acre of land also increases across these three processes. Process III requires less seed corn per acre than process II, and process IV requires even less. Given these contrasts, processes II, III, and IV cannot be ranked by physical efficiency alone. Iron inputs per acre do not even vary monotonically among processes II, III, and IV, further illustrating the need for prices to rank lands by efficiency.

I take the numeraire as a commodity basket consisting of 1 ton iron and 1 bushel corn.

The given data includes the land available and the requirements for use. These are such that all three type of land must be at least partially farmed. The given data are in principle observable at a single moment in time. Different types of land are distinguished by how corn is grown on them. No need exists to imagine marginal adjustments

Three techniques, Alpha, Beta, and Gamma, can feasibly satisfy requirements for use. In all three techniques, all four processes are operated. One of the types of land is not fully cultivated in each technique (Table 2). The choice of technique is based on cost-minimization or profit maximization.

Table 2: Techniques of Production

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Partially farmed	Fully farmed	Fully farmed
Beta	Fully farmed	Partially farmed	Fully farmed
Gamma	Fully farmed	Fully farmed	Partially farmed

3.0 Prices of Production

Prices of production are here defined for a given ratio of markups in the industrial and agriculture sectors. The rate of profits in the process producing iron is s₁ r, while the rate of profits in each of the three processes producing corn is s₂ r. I call r the scale factor for the rate of profits. Prices of production satisfy the following system of equations:

(p₁ a_1,1 + p₂ a_2,1)(1 + s₁ r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + s₂ r) + ρ₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + s₂ r) + ρ₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + s₂ r) + ρ₃ c_3,4 + w a_0,4 = p₂

I am assuming that wages and rents are paid out of the surplus at the end of the period of production. The relative market power of industry over agriculture, or vice versa, is expressed by the ratio s₁/s₂. When this ratio is unity, the equations characterize a competitive capitalist economy. Today, I experiment with assuming markups lie on a simplex:

s₁ + s₂ = 1

Each of the processes in the technology contributes an equation to the system of equations defining prices of production. The rate of profits is calculated on the value of the capital goods advanced. Rent and wages are paid out of the surplus. Four equations are defined in terms of seven variables, the prices of iron and corn, the rents per acre on each of the three types of land, the wage, and the scale factor for the rate of profits. The coefficients of production and the markups s₁ and s₂ are taken as given.

Specifying the numeraire removes one degree of freedom:

p₁ + p₂ = 1

All rents must be non-negative and one type of land must pay no rent. This is the type of land not fully cultivated. The following equation specifies that the rent on at least one type of land is zero:

ρ₁ ρ₂ ρ₃ = 0

This constraint removes another degree of freedom. The system of equations for prices of production has one degree of freedom for this model of markup pricing with extensive rent. This degree of freedom can be expressed as a function showing how the wage decreases with the scale factor for the rate of profits, or vice versa. With the wage somehow specified, the rate of profits in industry and agriculture, the price of iron, and rent on the scarce land are determined.

4.0 The Choice of Technique: An Example with Competitive Markets

Figure 1 illustrates wage curves when s₁ and s₂ are 1/2. Markets are competitive. The cost-minimizing technique corresponds to the wage curve on the inner frontier. Figure 2 illustrates the corresponding rent curves.

Figure 2: Order of Rentability with Competitive Markets

Consider a wage of zero or just barely positive. Beta is cost-minimizing. If the requirements for use were small enough that they could be satisfied by only farming Type 2 land, a technique with same wage curve as Beta would be cost-minimizing. No land would pay rent, and the scale fator for the rate of profits would be as shown on the right-most wage curve. With somewhat greater requirements for use, Type 1 land would be taken into cultivation, and the scale factor would be as shown on the wage curve for Alpha. Type 2 would be fully cultivated and pay a rent. Finally, with the originally postulated requirements for use, Type 3 land is cultivated. In the range for the smallest wage, the order of fertility, from most fertile to least fertile land, is Type 2, Type 1, Type 3. The order of fertility is also known as the order of efficiency.

The order of efficiency, for a wage of zero and competitive markets, also orders techniques in decreasing order of the maximum rate of growth. Since land of each type is limited in quantity, the maximum rate of growth must decline, without innovatin.

This is an example of the reswitching of the order of efficiency. When Gamma is cost-minimizing, the order of efficiency varies from Type 2, Type 1, Type 3 lands to Type 1, Type 2, Type 3 lands and back. The inner frontier shows the cost-minimizing technique for this example. The intersections of the wage curves on the outer frontier show variations in the order of efficiency. The wage curve for the Beta technique is never on the inner frontier, and Beta is never cost-minimizing.

The intersections for the rent curves occur at wages different from those for which wage curves intersect. When Gamma is cost-minimizing, the order of rentability varies from Type 1, Type 2, Type 3 to Type 2, Type 1, Type 3. The order of efficiency first deviates from the order of rentability, then matches at a higher wage, deviates at an even higher wage, and matches again at a still higher wage. Whether or not the orders of efficiency and rentability match also varies with the wage in the range where Alpha is cost-minimizing and Type 1 land pays no rent.

5.0 A Three-Technique Fluke Switch Point

Suppose the relative markup in industry is lower. Figures 3 and 4 show the wage and rent curves for a fluke case, in which all three rent curves intersect at a single point. Under Gamma, the order of efficiency, at a wage of zero, starts at Type 2, 1, and 3 lands. The order of rentability, being Type 1, 2, and 3 lands, differs. At a higher wage, the order of efficiency changes, and then matches the order of rentability. When Alpha is cost-minimizing, the orders of efficiency and rentability match. They are both Type 3, 2, and 1 lands.

Figure 3: Choice of Technique for Three-Technique Fluke Case

Figure 4: Order of Rentability for Three-Technique Fluke Case

I suppose the most interesting aspect of this example, other than its fluke property, is that the switch point for the order of efficiency at the low wage w₁ would not exist, in some sense, if the scale factor was taken as given. The scale factor for the rate of profits for this switch point exceeds the maximum scale factor for the Gamma technique.

6.0 A Fluke Switch Point with a Variation in the Order of Efficiency at a Wage of Zero

I next want to consider an even lower markup in industry, for another fluke case. A switch point for the order of efficiency exists in Figure 5 on the axis for the scale factor for the rate of profits. Figure 6 shows the corresponding rent curves.

Figure 5: Choice of Technique for Fluke Case for Order of Efficiency

Figure 6: Order of Rentability for Fluke Case for Order of Efficiency

When Gamma is cost-minimizing at a positive wage, the orders of efficiency and of rentability are Type 1, 2, and 3 lands.

When Beta is cost-minimizing, the orders of efficiency and rentability initially match, with a order of Type 1, 3, and 2 lands. At a higher wage, the order of rentability changes, so they do not match. At an even higher wage, the order of efficiency changes, so the orders once again match, with Beta cost-minimizing.

Alpha is cost-minimizing at the highest range of feasible wages. The orders of efficiency and rentability match, with an order of Type 3, 2, and 1 lands.

7.0 An Example with Higher Market Power in Agriculture

Finally, I want to consider a non-fluke case, with an even lower markup in industry. Figures 7 and 8 show the wage curves and rent curves here. The comments about the orders of efficiency and rentability in the previous section apply, with the exception that the switch point at a wage of zero and disappeared below the axis for the scale factor for the rate of profits.

Figure 7: Choice of Technique for Example with Market Power for Agriculture

Figure 8: Order of Rentability for Example with Market Power for Agriculture

I like that the orders of efficiency and rentability totally reverse, from the lowest to the highest wage.

More qualitative changes in the analyhsis of the choice of technique, including fluke cases, arise for even lower relative markups in industry. But I want to stop here, where the reswitching of the order of efficiency has vanished and Type 2 land obtains no rent for some intermediate range of the wage.

8.0 Conclusion

One can see that the owners of Type 2 land care about competitive forces among capitalists, and the struggle between the workers and the capitalists. If capitalists in agriculture are able to impose lasting barriers to entry, as compared to capitalists in industry, the struggle over the wage can leave them with no rent at all. Furthermore, the specific level of rents per acre, for all types of land, varies at specific wages, depending on the struggle among capitalists. The order of efficiency can vary from the order of lands from the highest rate of growth to the lowest, even at a wage of zero, depending on the struggle among capitalists. I can expand the above write-up to include one of my one-dimension diagrams.

I draw heavily on the work of Alberto Quadrio Curzio in this post. He looked deeply into the theory of rent in post-Sraffian price theory. The book, Resources, Production, and Structural Dynamics (edited by Mauro L. Baranzini, Claudia Rotondi, and Roberto Scazzieri) is dedicated to him. It contains a lot that differs from this kind of formal analysis, considering institutions for fostering innovation to overcome resources scarcities and so on.

Marginalism As A Reaction To Marx

This is another post for my commonplace book. It is extraordinary difficult to get a rational explanation for the marginal revolution. Trying to imitate nineteenth-century physics seems to be part of it.

"Marx implicitly assumes the the whole of social reproduction is mediated through the exchange of commodities, including the reproduction of labor power, that is, the reproduction of people themselves. We can view the labor that produces what productive workers consume as the labor necessary for the reproduction of society and the labor that capitalists appropriate in the form of surplus value as the surplus labor time of society, in the sense that only the necessary labor time would be required to enable reproduction of people and productive facilities on the same scale. Thus the wage-labor mechanism allows capitalists as a class to appropriate the surplus labor time of the society without giving the workers as a class any equivalent.

A situation in which one person gives another something for which the giver receives no equivalent is commonly called exploitation. Because this is exactly the situation in capitalist production, Marx argues that, from the point of view of the labor theory of value, the source of surplus value lies in the exploitation of labor.

If you do not accept the postulate that labor produces the whole value added, you will not see much basis for the claim that wage-labor is exploitative. I think this is the main reason that the labor theory of value has fallen into disrepute among orthodox economists. To avoid the characterization of capitalist social relations as exploitative requires the construction of some other theory of value that makes the wage seem to be a complete social equivalent for the labor that workers actually perform." -- Duncan Foley, Understanding Capital: Marx's Economic Theory, Harvard University Press, 1986: 38-39.

Oscar Lange is an example of a marginalist economist who was also a socialist.

Books After Marx

Here is a list of books by Marxists that have stood the test of time. I am being impressionistic and probably idiosyncratic. I am more interested in analysis of what is than political organizing. What should be added? Removed?

• Frederick Engels, Anti-Dühring, 1877. I think German comrades learned Marxism during the second international more from this thick tome. I recommend other works for introductions these days.
• Vladimir Lenin, What is to be Done?, 1902. Lenin lays out a strategy and defines a vanguard party. And the Bolsheviks are in power at the end of 1917.
• Rosa Luxemburg, The Accumulation of Capital, 1913. Luxemburg argues that capitalism needs a less advanced sector (or maybe military purchases from the state) to provide demand. Growth paths can be defined by Marx's scheme for expanded reproduction, but why would capitalists make these invevestments?
• Rudolf Hilferding, Finance Capital, 1910. I have not read this one. Hilferding recognizes that joint stock companies and financial institutions have changed capitalism from the era of small business.
• Nikolai Bukharin, Economic Theory of the Leisure Class, 1919. Extends the approach of Marx's Theories of Surplus Value to analyze works of the marginal revolution. Where does Bukharin have the time for scholarly work?
• Antonio Gramsci, Selections from the Prison Notebooks, 1971. Originally written in Mussolini's prisons. Argues that in advanced societies, communists must first change civil society before obtaining state power.
• Paul A. Baran and Paul Sweezy, Monopoly Capital, 1966. How should Marx's analysis be updated for the world of modern corporations? The editors of Monthly Review have ideas.
• Piero Sraffa, The Production of Commodities by Means of Commodities, 1960. Minimalist, as in modern art. I think many have still not absorbed this.
• Guy Debord, The Society of the Spectacle, 1967. This is more Marxist than I expected. Builds on the idea of commodity fetishism. I could learn more about the situationists in Paris in May 1968.

But I do not want to get into current events.

Profits Not Explained By Merit, Increased Risk, Increased Ability To Compete, Etc.

This post is for my commonplace book. I have been reading Enrico Bellino and Gabriel Brondino's "Circular vs. one-way production processes: two different views on production and income distribution" (Review of Political Economy, 2024).

Bellino and Brondino draw a distinction between a circular model of production and a one-way model. In the latter, a finite, dated series of non-produced inputs (for example, labor) results in the production of the output. This dichotomoy is distinct from the distinction between a theory focusing on the reproduction of society and a theory focusing on the allocation of scarce resources.

Bellino and Brondino argue that, if subsistence wages enter into the costs of production, the one-way model becomes a circular model of production.

The one-way model supports the illusion that returns to capital can come from markups over costs. The circular model demonstrates that profits come from the ability of a surplus product to be generated in production. It is not a matter of prices.

"Finally, when production is circular, profits depend first and foremost on the ability of the economic system to generate a surplus, that is, the possibility to produce commodities such as corn, cloth, wood, steel and chips to a greater degree than required, and secondly on the capitalists' power to appropriate a part of it (if workers also get a share of the surplus) or all of it (if workers earn only the subsistence wage). Profits thus emerge in the sphere of production and in the relative bargaining strength of capitalists in fixing real wages. They are not the merit of a particular input, firm, or industry. The profits of an individual are a claim on the surplus, a claim which is independent of the risk taken, the ability to compete, or the sector in which it produces. These qualities may explain why one capitalist can get more than others, but not why there are profits in the economic system. In a circular system, the generation of surplus is a necessary condition for the existence of profits. [Footnote]

[Footnote:] Saying that profits depend on the economy's capacity to generate a surplus does not deny the possibility that, at an individual level, a capitalist may attempt to increase their profits by raising prices. However, such increases can only be temporary under competitive conditions. The daring capitalist will eventually lose market share since they will attract other competitors, making their rate of profit converge to the general one..." -- Bellino and Brondino

I have a question about the history of economic thought. Ricardo had examples of the one-way model of production, for example, in his first chapter showing why a simple labor theory of value. The Ricardian socialists, as I understand it, often explained profits by capitalists being able to exert force to markup prices over costs. Does Bellino and Brondino's argument illustrate a superiority of Marx over the Ricardian socialists?

Another Hayekian Triangle Not Supporting The Austrian School

Figure 1: Hayekian Triangles for The Two Techniques

1.0 Introduction

This post is a variation on this one.

2.0 Technology and Net Output

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

Table 1: Regions

Input	Corn Industry		Ale Industry
	Process I	Process II	Process III
Labor	1 person-yr.	275/464 person-yrs.	1 person-yr.
Corn	1/10 kilo-bushels	113/232 kilo-bushels	2 kilo-bushels
Ale	1/40 kilo-liters	1/200 kilo-liters	2/5 kilo-liters
OUTPUTS	1 kilo-bushel	1 kilo-bushel	1 kilo-liter

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

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

Table 2: The Numeraire

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

3.0 The Choice Of Technique And Hayekian Triangles

The usual analysis of prices and production and the choice of technique yields the wage curves in Figure 2 below. The Beta technique is cost-minimizing for a low interest rate. The Alpha technique is cost-minimizing for a higher interest rate.

Figure 2: Wage Curves for the Two Techniques

Figure 1, at the top of this post, shows the Hayekian triangles at the single switch point. Around this switch point, a lower interest rate does extend the structure of production. But it does not require more savings to achieve that extension.

4.0 Conclusion

The above has constructed Hayekian triangles not consistent with Austrian business cycle theory. A coordinated state does not necessarily rotate the Hayekian triangle to have a longer structure of production with less consumption (more savings).

Paul Kengor's The Devil and Karl Marx

Paul Kengor's The Devil and Karl Marx: Communism's Long March of Death, Deception, and Infiltration is a stupid and ignorant book. Kengor is particularly keen to chronicle atheist attacks from Marx and communists on religion.

Juvenile Literary Works from Marx

Kengor starts with a poem that Marx wrote when he was 19. Marx envisions himself, I guess, as a ferocious violin player, inspired by the devil. Apparently, Marx was a fan of Goethe. Kengor never uses the phrase 'sturm und drang'. He goes on about some other poems and a play of Marx's. Kengor thinks that Marxists unjustly ignore these early works. They are in the first volume, though, of the Marx-Engels Collected Works.

Some Bits and Pieces from the Lives of Marx and Engels

The second section, after this prelude, is about Marx and Engels. It has little to say about Marx's ideas. He calls Marxism utopian, seeking a heaven on earth. He says nothing about distinction between scientific and utopian socialism.

Kengor, in this part of the book has a lot to say about Marx's life. When discussing his movement among countries, he does not even mention that nations kept on kicking him out. Kengor hops around in chronology. He is big on Marx not earning a living while studying and agitating. Also, Marx had poor bathing habits.

He brings up Bruno Bauer several times. But he does not mention that The Holy Family mocks Bauer. When bringing up Mikhail Bakunin, Kengor does mention the later falling out of Bakunin and Marx. Does Marx's intolerance in splits in the First International have something to do with Lenin's behavior? With Stalin?

Inasmuch as Kengor even discusses Marx's work, he stops at 1848, with the Communist Manifesto. The quotation about religion as the "opiate of the people" is from Marx's Critique of Hegel's Philosophy of Right, from 1944. Marx's argument is for changing material conditions so that religion will no longer be needed as "the sigh of the oppressed creature". This idea is not inconsistent with arguments for atheism. Such idealist, superstructural arguments are beside the point, though. The context of this well-known phrase from Marx is over Kengor's head.

Here is all Kengor has to say about Capital:

"Marx had wasted over two decades writing Das Kapital, a long, ridiculous tome, a waste of money as well as time. He had initially received a three hundred dollar advance for the book, but extended over twenty-three years of drawn-out writing, it equated to a little over a dozen dollars a year.

The Bolshevik War on Religion

Kengor does not connect the ideas of Marx to Lenin and events in the Soviet Union. At a more personal level, I do not know that Marx was responsible for what happened to his children after his death, including the suicides of two daughters. I had not known Lenin spoke at Paul Lafarge's funeral. Kengor brings up Bukharin several times, but does not mention that he was a victim of Stalin's show trials. I do not think Engels not getting married to the women that he slept with is on the same moral plane as the tortures in communist prisons in eastern and central Europe.

Excerpts from the House Un-American Activities Committee

I do not know why I should care about testimony in the 1940s and 1950s to the House Unamerican Activities Committee. I was under the impression that a united front and a popular front are different concepts. The former is all non-facist forces, while the latter is all leftist forces. Anyways, Kengor does not discuss this distinction or even why it is a proplem for the Communist Party to work with Christians, and specifically Catholics, for specific reasons. Yes, the Communist Party was atheist. Kengor barely mentions the relationship of Catholics to Franco, and only in the context of what communists say. He only mentions Hitler in the context of turns in the communist party line with the Hitler-Stalin pact and then the German invasion of the Soviet Union. He has nothing to say about how this has anything to do with Marx's ideas. And he doesn't really discuss communist sectarian activities in taking over front groups. I am not surprised that communists were sometimes deceitful.

Various Intellectuals

The fifth part consists of two chapters. Kengor writes a bit about the life of the occulist Aleister Crowley, the New York Times's reporter Walter Duranty (who believed what the Soviets told him in the 1930s), the gay activist Harry Hay, Wilhelm Reich, Walter Benjamin, and Kate Millett. Why this selection? Perhaps because snippets of their lives are suitable for ad hominem. Kengor is fairly candid that Crowley does not have much to do with Marxism or communism, for example.

We have another time Kengor says he does not know what he is talking about:

"Trying to discern the inane and impentrable ideas of the men of the Frankfurt School is a soul-crushing exercise in futility. One must spend years scouring pages and footnotes of thick volumes (mostly in untranslated German) trying to arrive at a vague flickering of understanding at what in the devil's name these madmen were thinking about. It would be bad enough if this venture was simple a waste of one's time - especially given the sacrifice of more edifying reading - but what is worse is the strain and toxicity to the intellect and the soul. One is struck again and again at how some Godless intellectuals (especially German ones) can descend into such rank intellectual vacuity, ambiguity, and downright stupidity..."

If I thought Kengor was a person of good-will, I might emphasize here.

Conclusion

The single chapter in the conclusion section neither summarizes the book nor follows from what comes before. We get complaints about Obama's totalitarianism, gay marriage, the Frankfurt school, critical theory, Gramsci, relativism, and cultural Marxism. We also get echos of Fulton Sheen and Pope Pius X's 1907 encyclical on modernism. Apparently, Kengor has never heard of post-modernism. Kengor does not even know what he has covered in the book:

"In this book, we have looked at key figures of the Frankfurt School, including Georg Lukacs, Walter Benjamin, The Sexual Revolution's Wilhelm Reich, and (among others) Herbert Marcuse..."

No, he has not discussed Lukacs or Marcuse, even to the extent of saying something about some moments in their lives.

Some Overall Comments

The book has lots of redundancy. Kengor distinguishes between socialism and communism. I do not see why the book should go through the same quotations from the same papal encyclicals again to see that the Catholic Church opposed both communism and socialism.

The references are odd. For example, if I see a discussion of Darkness at Noon, I expect a reference to Koestler's novel, not a book by a historian. I would hope for more than extracts from Wikipedia and from the "About" section from certain web sites.

It fails at the level of the individual sentence. Consider, "Sheen noted that whereas Karl Marx called religion 'the sigh of the oppressed creature, Sheen saw communism as the sigh of the oppressed creature." Is Sheen noting how Sheen saw communism? At one point, Kengor has Manning Johnson testifying to the HUAC that "this was 'the extension of the hand of friendship and cooperation to the church, while in the other hand holding a dagger to drive through the heart of the church.'" But then, "the outstretched hand concealed a knife." Which metaphor do you want?

Adam Smith, David Ricardo, And The Labor Theory Of Value

1.0 Introduction

I resolutely am not commenting on unhappy current events.

Smith and Ricardo thought a (simple) LTV was not applicable to capitalism. Prices do not tend to or orbit around labor values. At least that is their claim.

Ricardo had more to say about the LTV.

This argument is not new. Smith confined the LTV to a supposed "early and rude state of society which precedes both the accumulation of stock and the appropriation on land" (WoN, book 1, chapter 6; see also book 1, chapter 8). Ricardo thought this was sloppy reasoning. The LTV does not become nonapplicable merely because of the accumulation of capital and of the division of society into capitalists and workers (Principles, 3rd edition, chapter 1, section III).

2.0 Technology

A simple model of circulating capital can be used to make Ricardo's point. Let a₀ be a row vector of direct labor coefficients. Let A be the Leontief input-output matrix. An element of a₀ and the corresponding column of A specify the labor time and the capital goods needed to operate a process to produce one unit of the output of that industry. The technology satisfies the following common assumptions:

• Some labor is needed to operate every process in each industry.
• Constant returns to scale prevail.
• Each commodity enters, directly or indirectly, into the production of every commodity. Iron, for example enters indirectly into the production of automobiles if iron is needed to produce steel and steel is needed to produce cars.
• The technology is productive. For some level of operation of the processes for each industry, some commodities are left over after reproducing the capital goods used in producing them.

Now for the unusual special case. Let λ be the largest eigenvalue of the Leontief matrix. This eigenvalue is also known as the Perron-Frobenius root of the Leontief matrix. Assume that the vector of direct labor coefficients is a corresponding left-hand eigenvector:

a₀ A = λ a₀ (Display 1)

3.0 Labor Values

Let v be the row vector of labor values. By definition, labor values satisfy the system of equations in Display 2:

v A + a₀ = v (Disp. 2)

The total labor to produce a commodity is the sum of the labor values of the capital goods used in that industry and the direct labor coefficient.

Under the special case assumption, labor values are a multiple of direct labor coefficients:

v = (1/(1 - λ)) a₀ (Disp. 3)

One can check this solution by merely plugging it into the solution in Display 2:

(1/(1 - λ)) a₀ A + a₀ = (λ/(1 - λ)) a₀ + a₀ = (1/(1 - λ)) a₀ (Disp. 4)

Since the above is a one-line proof, I thought I would include it. Labor values are also an eigenvector of the Leontief matrix.

4.0 Prices

Under the usual assumptions, the row vector p of prices satisfies the system of equations in Display 5:

p A (1 + r) + w a₀ = p (Disp. 5)

The scalar w is the wage, and r is the rate of profits.

Let R be the maximum rate of profits, obtained when the wage is zero, and the workers live on air. For the special case, the solution to the price system is quite simple:

R = (1/λ) - 1 = (1 - λ)/λ (Disp. 6)

r = R (1 - w) (Disp. 7)

p = v (Disp. 8)

One can check this solution by plugging it into the system of equations in Display 5.

So Ricardo was correct. The LTV could apply to capitalism under a special case. The failure of the LTV to apply in general is because those special case conditions cannot be expected to arise.

5.0 Conclusion

Ricardo had a point. As any ent would tell you, Smith was too hasty. The simple conflict between the wage and the rate of profits in Display 7 applies more generally than the above special case. The LTV, even when it is not valid, points to theories of the returns to capital.

I suppose reconstructing arguments between Ricardo and Smith with modern economics may conflict with how some historians do history. Even saying that Smith was analyzing capitalism could be considered an anachronism.

Roger Garrison On The Inadequacy Of Hayekian Triangles

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

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

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

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

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

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

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

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

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

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

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

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

Elsewhere

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

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

Figure 1: Wage Curves for the Two Techniques

1.0 Introduction

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

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

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

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

2.0 Technology and Net Output

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

Table 1: Regions

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

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

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

Table 2: The Numeraire

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

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

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

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

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

3.0 Price Systems

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

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

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

The following equation follows from the definition of the numeraire.

p₁ d₁ + p₂ d₂ = 1

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

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

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

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

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

Figure 2: Extra Profits for Price Systems

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

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

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

4.0 Perverse Hayekian Triangles at the First Switch Point

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

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

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

Figure 3: Hayekian Triangle at First Switch Point

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

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

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

Figure 5: Ratios of Value Added at First Switch Point

5.0 Hayekian Triangles at the Second Switch Point

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

Figure 6: Hayekian Triangle at Second Switch Point

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

Conclusion

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

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

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