More On Truncation Without Reswitching

Figure 1: Partitions For A Parameter Space

This post extends this and this. I am considering perturbations of coefficients of production in a numerical example of the recurrence of truncation without the reswitching of techniques.

The example consists of two industries that produce new machines and corn, respectively. The physical life of a machine is two years in each industry. Corn acts as circulating capital, as well as the consumption good.

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain forward substitution of labor in the corn industry at the switch point between Alpha and Gamma and reverse labor substitution at the switch point between Beta and Delta. Accordingly, I consider permutations of the coefficients of production in the first row of the input matrix. These coefficients of production specify inputs of circulating capital for each process, when operated at unit level.

Previously, I considered perturbations of the pair of coefficients that specify the inputs of corn needed for each machine when the machine is run in the second year of its life. Today, I consider perturbations in a_1,1 and a_1,3, the coefficients that specify the inputs of corn needed for each machine when the machine is run in the first year of its life.

Figure 1, at the top of this post, partitions a part of the parameter space. In drawing this figure, a_1,2 = 0.635 and a_1,4 = 0.319. Each partition corresponds to a fluke switch point. For the values of a_1,2 and a_1,4 in Table 1 in this post, the parameters corresponding to the fluke switch point in which the four wages curves intersect on the axis for the rate of profits is an economically meaningless point in the second quadrant.

A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. Four of the five partitions correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect.

The sequence of techniques along the wage frontier is invariant in each numbered region in the slice of the parameter space in Figure 1. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Other partitions, depicted in Figure 2, exist up and to the right. One is for a switch point between Alpha and Gamma on the wage axis, and another is for the switch point between Gamma and Delta on the wage axis. Three new regions are associated with these partitions. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,3, not exceeding unity, in the upper left. Delta is cost-minimizing alone in a region to the northeast. Gamma and Delta are cost-minimizing, in that order for an increasing rate of profits, in a region bounded below by region 3 and also by these two additional regions.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma	No switch point.
7	Gamma, Delta	Lower rate of profits associated with truncation in corn industry, greater output per worker.
8	Delta	No switch point.

Figure 2: An Expanded View of Partitions For A Parameter Space

Suppose technical change lowers corn inputs for the first year a machine is operated, in either industry, while leaving all other coefficients of production unchanged. Some trajectories going roughly from the northeast to the southwest in Figures 1 and 2 cross through region 4. Under this specific form of technical progress, the recurrence of truncation can arise and then disappear in secular time. But it need not.

Alternative Textbooks Have Been Available For Teaching Introductory Economics For Decades

"It is true that we cannot, in the time available, teach every- thing that we would like. But why do we pick out for treatment just that selection of topics that is least likely to raise any questions of fundamental importance?" -- Joan Robinson

Over the years, many have proposed textbooks for introductory economics. Some of these are supplementary readings. I include introductions to both micro and macroeconomics:

• Richard Goodwin. 1970. Elementary Economics from the Higher Standpoint. Cambridge University Press.
• Joan Robinson and John Eatwell. 1973. An Introduction to Modern Economics. McGraw-Hill.
• Marc Linder. 1977. Anti-Samuelson (2 volumes). Urizen Books.
• Walsh, Vivian and Harvey Gram. 1980. Classical and Neoclassical Theories of General Equilibrium: Historical Origins and Mathematical Structure. Oxford University Press.
• Morishima, Michio. 1985. The Economics of Industrial Society. Cambridge University Press.
• Yanis Varoufakis. 1998. Foundations of Economics: A Beginner's Companion. Routledge
• Hugh Stretton. 2000. Economics: A New Introduction. Pluto Press.
• Steve Keen. 2002, 2011. Debunking Economics. Zed Books.
.
• Tony Myatt and Rod Hill. 2010. The Economics Anti-Textbook: A Critical Thinker's Guide to Microeconomics. Zed Books.
• Thomas, Alex M. 2021. Macroeconomics: An Introduction. Cambridge University Press.
• Goodwin, Neva et. al. 2024. Essentials of Economics in Context, 2nd edition. Routledge.
• CORE Econ. 2024. The Economy 2.0: Economics for a Changing World.

I have not read a few of those myself. This list is confined to textbooks. Maybe a few are too advanced for introductions. Some are now of historical interest. Thomas' textbook, I guess, is focused on India. I do not include more popular writing. Most I've only dipped into.

Alternative approaches have been available for half a century.

Elsewhere

• Hasan Piker gets endorsed, by Customs and Border Protection (CBP), as being successful at promoting his ideas.
• David Harvey on Sraffa in the New Left Review.
• The most recent episode of Matt Sitman and Sam Adler-Bell's podcast, Know Your Enemy, is on the effects of podcasts on last year's U.S. presidental election.

Perturbation Of A Model Of The Recurrence Of Truncation

Figure 1: A Partition of Part of the Parameter Space

1.0 Introduction

This post is a continuation of the example in a previous post. That example is of the recurrence of truncation without reswitching. I here consider perturbations of selected coefficients of production and of relative markups in the two industries.

2.0 Perturbation of Efficiency of Circulating Capital for Old Machines

The first industry in the example produces machines, and the second produces corn. Machines constitute fixed capital, and corn functions as circulating capital, as well as a consumption good.

Two processes are available in each industry. The second process uses an old machine to produce the output of that industry, whether a new machine or corn. Old machines are specific to the industry in which they were (jointly) produced. The choice of technique, in this model of pure fixed capital, is equivalent to choosing to truncate the economic life of a machine in either industry. The four possible techniques are defined, along with the technology, in the previous post.

Consider perturbations of a_1,2 and a_1,4. The first coefficient of production is the amount of corn input, as circulating capital, needed to operate an old machine in the machine industry. The second coefficient of production is the input of corn needed to operate an old machine in the corn industry.

Figure 1, at the top of this post depicts a partition of the resulting parameter space. The variation of the choice of technique with distribution is invariant in each numbered region. Table 1 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.

The diagram yields the following results:

• Region 1: If old machines are inefficient enough, then the economic life of machine is one year.
• Region 5: With improvement in the efficiency of old machines in the machine industry, machines are operated for two years in the machine industry at large rates of profits.
• Region 2: With improvement in the efficiency of old machines in the corn industry, machines are operated for two years in the corn industry at large rates of profits
• Region 4: Recurrence of the truncation of machines in the corn industry occurs at a specific range of these coefficient of production.
• Region 3: With sufficient improvement in the efficiency of old machines, no possibility arises of the economic life of machine being operated for two years only in the machine industry, whatever the distribution of income.

All these regions are around a quintuple fluke switch point. The partition between regions 1 and 5 occurs for parameters for which managers of firms are indifferent, when the wage is zero, about the economic life of a machine in producing new machines. The partition between regions 1 and 2 occurs when managers of firms are indifferent, also at a wage of zero, about the economic life of a machine in producing corn. The intersection of these two partitions must also be an intersection of the other three partitions.

3.0 Perturbation of Relative Markups

Now suppose the technology is fixed, as in the post. Let s₁ r be the rate of profits in the machine industry, and s₂ r the rate of profits in the corn industry. As a normalization condition, I assume the sum of the relative markups is unity:

s₁ + s₂ = 1

Figure 2 displays the effects on the choice of technique of perturbations of persistent relative markups.

Figure 2: Effects on The Choice of Technique of Perturbations of Relative Markups

High enough market power for the corn industry, as compared to market power in the machine industry, can eliminate the possibility of extending the economic life of the the machine in the corn industry. This is seen in region 5, to the left on the graph. Persistent high enough market power for the machine industry, can also eliminate the recurrence of the truncation of machines in the corn industry. This is seen in region 3 to the right on the above graph.

4.0 Conclusion

The choice of technique varies with variations in distribution between wages and profits. Both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. Technical progress, however, has the benefit of increasing productivity.

Steve Keen On His Breakaway From Nonsense In Marginalist Economics

Steve Keen appeared on the 30 April epsiode of 1Dime Radio, a podcast. Towards the start, he tells a story:

"My change actually came from a very technical piece of economics because my first-year lecturer is a still a good friend these days, Professor Frank Stillwell - back in those days, he was Doctor Frank Stillwell – explained what's called the theory of the second-best in the first-year lectures. And this is something you normally learn in a third or fourth-year honors course or master's or PhD qualifying. And by the time you’ve got to that level, most people were being so saturated with neoclassical thinking that they would have just regarded it as, 'Oh, that’s a nice little curiosity', and they forget about it. What it showed was that = I think it got the originators the Nobel prize at one point – they show that if you were two steps away from what's regarded as perfection by neoclassical economics, then moving one towards it, and not the whole two, would actually make social welfare worse. And the example that Frank used was if you have wage negotiations, the ideal according to neoclassical theory is that you have workers who bargain on their own personnel right and firms who, again, bargain for employees on an individual basis – no collusion within labour or capital.

But the real world is you’ve got trade unions on one side and employer associations on the other. So in the neoclassical view, you get equilibrium where the worker gets paid their marginal product. That's the ideal. When you allow that there's both trade unions and monopolies (employer associations), you move to another point where it's indeterminate what the wage will be and it’s a bargaining point which might make the firms better off or the workers better off compared to the ideal. But if you abolish one or the other, either get rid of the trade union or get rid of the employer association, the outcome is necessarily lower social welfare than the previous case where had both the trade union and the employer association.

And I fell for the conventional argument. I accepted all the idea of supply and demand and equilibrium and so on. And then to have if pointed out that if you take into account the reality there’s plenty of distortions from what's called the perfectly competitive ideal, then getting rid of them sequentially will make things worse. I thought, there's got to be something wrong with the theory if you can simply demolish it so easily. So I checked my textbook. There was no mention of the theory of the second-best there

I then went down to the economics department library, which is in the same building as where the lectures were, the Merewether Building at Sydney University. And I went looking for the journal papers. I found the original. And then I was horrified that this is not covered in the textbooks.

So I went to the journals again looking for the most recent journal papers. And I found one by Paul Samuelson which was called – first of all I found a journal paper by a Marxist and that surprised me. That was in the Cambridge Journal of Economics by Bhaduri. And I was amazed that a Marxist got into a journal. That surprised me. But then I read Samuelson, a paper called 'A summing up'. And he basically conceded defeat in a debate over the definition of capital which I did even know was happening. But it was actually taking place between 1960, when Piero Sraffa published A Production of Commodities by Means of Commodities, through to the – probably petered out in the late sixties. No mention of it, Samuelson conceded defeat in that paper, but you read his textbook which I had at the time – no recognition of the dispute there either. So I thought I'm being lied to by my textbooks.

And I stopped reading the textbooks. I read them anyway for reference, obviously. But I go and take a look at the journal papers and seeing what's being said in the journals. And the gap between what I was being taught versus the journals wasn't a case that I was getting the simplified version in the textbooks and the sophisticated stuff is in the journals. I was seeing completely contrary results for absolutely fundamental arguments in the textbooks. And I just thought these textbooks are mendacious. Whether they know it or not, they're lying about the nature of economics.

So that was my breakaway point and I’ve never looked back. So that's why I regard economics as unscientific in the extreme, because there have been so many anomalies and so many logical disproofs, and so many empirical failings, this theory should not even be around anymore. It should be like phlogiston in chemistry. But it still dominates economics today. And they are so bloody arrogant about it. That's the other terrifying thing. They are so sure that they’ve got the right answers to everything when history and logical analysis shows that they've got the wrong answers to everything." -- Steve Keen (My transcription)

Richard Lipsey and Kelvin Lancaster published 'The general thory of second best' in 1956-1957, in the Review of Economic Studies. Neither won the Nobel prize. Lancaster won the John Bates Clark medal, which is very prestigious.

Keen does not go into this, but I think a distinction exists between the theory of the second best and the results of the Cambridge capital controversy (CCC). John Eatwell is good on this disticntion. The theory of the second best is one of a number of imperfections and frictions, like transaction costs, information asymmetries, principal agent problems, externalities, search costs, and incomplete contracts. Underlying these imperfections and frictions is an ideal theory. But the CCC shows that this ideal theory is incoherent. Maybe I am too firm on this distinction. If you are clear on all these imperfections and frictions, you know that the ideal is unattainable anyways, whatever policy the government adopts. Talk of government non-intervention in the economy is incoherent.

I think Keen is conflating Amrit Bhaduri's 1969 Economic Journal article, 'On the significance of recent controversies on capital theory: a Marxian view' with later articles.

Paul Samuelson did modify the tenth edition of his textbook. But later editions are befuddling.

I might as well say something about how I developed my views, keeping in mind that I have never been an academic economist. Sometime in the 1980s, I came across a reference to Robinson as the 'British Galbraith'. I had always like Galbraith, who I thought of as a popular writer and advocate of liberalism. So I looked up Joan Robinson's writing. I came across much about Sraffa and the Cambridge capital controversy. I ended up reading some of the same journal articles as Keen. I do not know that you should trust my self-depiction, but perhaps I continued looking for some, any response that defends what is in the textbooks for intermediate microeconomics. I could see that the question was not merely whether aggregate production functions, in macroeconomics, are a useful simplification.

Those defending marginalist economics in the CCC do not end up supporting the view in the textbooks. Capital is not a factor of production. If follows that interest is not a payment for the services of a factor of production. The aggregate production function is theoretically unfounded. Marginal productivity is not a theory of distribution. Equations relating payments to marginal products are merely part of the formulation of a system of general equilibrium. No theoretical foundation exists for well-behaved supply and demand functions in, say, labor markets. Maybe I am wrong, but I see it as very difficult to defend mainstream academic teaching as well-informed and honest.

Recurrence Of Truncation Without Reswitching

Figure 1: Wage Curves In The Example

1.0 Introduction

I have presented this example before. This example is another case of exploring or demonstrating code written for Matlab or Octave.

The structure of the example is the minimum multi-industry example with circulating and fixed capital in all industries and in which the choice of technique is to select the economic life of a machine.

The recurrence of truncation is like the recurrence of a process in single production. As far as I know, no numeric example exists in the literature of the recurrence of truncation without reswitching. This example might have been surprising if I were writing half a century ago. Its possibility is obvious in the work of Bertram Shefold, Heinz Kurz & Neri Salvadori, Ian Steedman, and others. Although reswitching and capital-reversing do not arise in the example, the reverse substitution of labor does.

2.0 Technology and Techniques

Two industries exist in the example. One industry produces machines, and the other industry produces corn. Corn is a consumption good, the good for circulating capital, and the numeraire. Machines are fixed capital. Each machine has a physical life of two years. Old machines cannot be transferred between industries. I assume constant returns to scale (CRS) and the free disposal of old machines. Labor is advanced and paid out of the surplus of corn.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs, at a unit level of operation, consist of 1/10 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.

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

The machines operate an non-constant efficiency in both industries. An old machine, in the machine industry, is used to produce more new machines than a new machine. The inputs of labor services and corn increase with the age of the machine. In the corn industry, an ole machine is used to produce less corn than a new machine. The input of labor services decrease and the corn input increases with the age of the machine.

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
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

3.0 Price Systems and the Cost-Minizing Technique

The economic life of a machine is chosen to minimize cost. A system of equations for prices is associated with each technique. This system can be solved. In the solution, the wage is a function of the rate of profit. Each price of a produced commodity is also a function of the rate of profits.

Figure 1 shows the wage curves, for the four techniques in the example. The cost-minimizing technique at each wage or rate of profits is the technique with its wage curve on the outer frontier. The cost-minimizing techniques are indicated on the figure. Maybe I should experiment with perturbing parameters to see if I can get a more visually obvious graph. Figure 2 shows an enlargement, emphasizing rates of profits around the switch point between Gamma and Delta.

Figure 2: Wage Curves In The Example (Enlarged)

At any rate, the cost-minimizing techniques, in order of an increasing rate of profits, are Alpha, Gamma, Delta, and Beta. Each pair of techniques at a switch point on the frontier differs in one process. A switch point in which the economic life of a machine differs in both industries would be a fluke case. No fluke switch points exist in this example, without perturbing some coefficients of production.

4.0 Prices of Old Machines

Identifying when prices of old machines are negative provides another method of analyzing the choice of technique in models of pure fixed capital. A negative price indicates that the economic life of a machine should be shortened. The machine should be truncated and discarded.

Figure 3 plots the price of old machines in the machine industry, for the two techniques in which old machines are operated in this industry. The switch points, at which the price of an old machine is zero, are indicated. As can be seen in Figure 2, the switch point between Alpha and Beta is not on the outer frontier.

For rates of profits less than that at the switch point between Gamma and Delta, the price of an old machine in the machine industry is negative for the Delta price system. If the Delta technique were in operation, prices would signal that machines in the Delta industry should be truncated. This trunction results in the Gamma tecnique being adopted.

Figure 3: The Price of an Old Machine in Machine Production

Figure 4 plots the price of old machines in the corn industry. Old machines are operated in this industry only for Beta and Delta. Since the price of these old machines are negative, in the Gamma price system, for rates of profits less than the rate at which the price is zero, the machine is truncated at these rates and the Alpha technique is adopted. Likewise, at rates of profits greater than the rate at which the price of this machine is zero, in the Delta system, the machine is truncated and the Beta technique is cost-minimizing at these rates.

Figure 4: The Price of an Old Machine in Corn Production

This analysis of prices of old machines has re-justified the analysis of the choice of technique in Section 4.

5.0 Extra Profits in Extending the Economic Life of Machines

A third method of examining the choice of technique is available.

Under Alpha and Gamma, the machine is truncated in the machine industry. The price of an old machine in the machine industry is zero under those price systems. Figure 5 shows extra profits, for each technique, available in operating the machine for a second year. if the life of this type of machine is extended under Gamma, the Delta technique is adopted. Extra profits are available in so extending the life of the machine at any rate of profits greater than at the switch point between Gamma and Delta. Gamma cannot be cost-minimizing in this range.

Figure 5: Extra Profits in the Machine Industry

The machine is truncated in the corn industry for Alpha and Beta. Figure 6 shows extra profits in the corn-industry, for all techniques, in operating the machine for a second year. Extra profits cannot be obtained for Alpha up to the switch point between Alpha and Gamma. Likewise, extra profits are not available for Beta, in extending the life of the machine in corn-production, for rates of profits greater than at the switch point between Beta and Delta. This method of analyzing the choice of technique, not surprisingly, yields the same result as the other two.

Figure 6: Extra Profits in the Machine Industry

6.0 Recap

The above has illustrated three equivalent methods of analyzing the choice of technique for a pure fixed capital model. Table 4 summarizes the results for this numerical example. The bounds on the ranges of the rates of profits are approximate. Matlab has a funtion, roots(), that returns the (possibly complex) zeros for a polynomial of any degree. I use this function in finding the intersections of wage curves in this example.

Table 4: Cost-Minimizing Techniques

Range	Technique	Truncation
0 ≤ r ≤ 70.21%	Alpha	Machines truncated in both industries.
70.21% ≤ r ≤ 71.19%	Gamma	Machines truncated in machine-production.
71.19% ≤ r ≤ 87.5%	Delta	Machines operated at full physical life in both industries.
87.5% ≤ r ≤ 122.8%	Beta	Machines truncated in corn-production.

At any rate, the machine is truncated in corn-production when both the Alpha and the Beta technique are cost-minimizing. The truncation of the machine in corn-production recurs, being part of the cost-minimizing technique at extremes of low and high rates of profits. This is not, however, an example of the reswitching of techniques.

Negative real Wicksell effects occur at all four switch points. Around each switch point, a lower rate of profits and higher wage is associated with a greater net output of corn per person-year. At the switch point between Alpha and Gamma, truncation in the corn industry is a switch to a more capital-intensive technique. Likewise, at the switch point Gamma and Delta, truncation in the machine industry is a switch to a more capital-intensive technique. As usual, these results disagree with Austrian capital theory and the ideas of economists of this school about roundaboutness.

Around the switch point between Alpha and Gamma, a lower rate of profits or higher wage is associated with truncation in the corn industry and a greater gross output of corn per person-year hired in the corn industry. Around the switch point between Delta and Beta, contrawise, a lower rate of profits or higher wage is associated with the extension of the economic life of the machine in the corn industry and a decrease in the gross output of corn per person-year hired in the corn industry. This second switch point is a manifestation of the reverse substitution of labor, one of those 'perverse' phenomena found in the Cambridge capital controversy.

Menger's Principles Is Obsolete

Carl Menger has a theory of consumer demand, in his Principles of Economics. This theory, one expression of utility theory, is ejected or ignored by other marginalist economists. Bohm-Bawerk is an exception. He also puts forth this theory. For those who want to read something shorter, I recommend William Smart's 1891 An Introduction to the Theory of Value. Heinz Kurz has recently written about Menger.

I take current theory to be revealed preference theory, which was developed by Paul A. Samuelson. Gerard Debreu's 1959 Theory of Value: An Axiomatic Analysis of Economic Equilibrium is canonical. in the theory, each consumer has a preference relation over a space of goods. Suppose all goods can be enumerated. Debreu has No. 2 Red Winter Wheat as an example of one good. Suppose a consumer is presented with vectors of n goods, where n is the number of goods available. Each vector specifies the quantity of each good available. The consumer is assumed to be able to tell, for each pair of vectors, whether they prefer the first to the second, they prefer the second to the first, or they are indifferent between them. Given certain assumptions on preferences, a utility can be assigned to each vector. This utility has some of the properties of numbers. You may not have the mathematics to understand some expositions of this theory, and other expositions exist, for example, in terms of choice functions.

Menger, by contrast, looks at one good at a time. He has a couple of chapters on the theory of the good. In his chapter on value, he classifies wants or needs into different classes. For example, food might be a class. A good, say, water, might go into several classes. You can drink water, use it to water your lawn, or use it to fill a swimming pool. These might be three different classes. The consumer has ranks, in each class, of satisfactions or utilities. The first gallon of water, in the drinking class, might have a rank of 10, while each successive gallon has a lower rank. When the consumer obtains a new gallon of water, they must look at the next satisfaction to be obtained, with the given distribution of existing goods among the classes. The consumer will then allocate this next gallon among these uses accordingly.

None of the structure in Menger's theory survives in modern economics. I think even Kelvin Lancaster's1966 New approach to consumer theory is something different.

Other aspects of Menger’s book are also obsolete. But I want to only focus on one aspect at a time.