Elsewhere

A History Lecture Focusing on Cantor, Dedekind, Gödel, Turing, etc.

• Norman Wildberger's next video in the above series is here. This professor, who has some dissenting views, explicitly tells his class that he does not belive in any of what he is teaching here. I wish, in trying to introduce this, he was more clear on the distinction between provability and truth. You do not get much even on lives here, much less on what minor events in the world induced Gödel to leave Vienna for Princeton.
• Virginie Pérotin on What do we really know about woker co-operatives. Richard Wolff's organization, Democracy at Work, provides an overview.
• Sebastian Gechert et al. have published Measuring capital-labor substitution: the importance of method choices and publication biases.
• Tony Freeth et al. have a report, A model of the cosmos in the ancient greek Antikythera Mechanism. I have written about this curious artifact before.

Three Mistakes Made By Marx

1.0 Introduction

I have previously considered some points on which Marx is vulnerable. In this post, I mention three mistakes. I take the first from Joan Robinson and probably the second point too. I take the third from Ajit Sinha. The second is perhaps the least original.

You can find lots of stupid stuff and nonsense about Marx. Pointing out these particular mistakes is beyond many. To have an opinion about these points, one must read Marx. One might even accept that Marx is mistaken on these points, yet find his general vision correct.

2.0 The Rate of Profits in Volume 1 of Capital

Marx defines the rate of profits in his system of labor values as:

r = s/(c + v)

where c is constant capital, v is variable capital, and s is surplus value. All are measured in value.

Some components of constant capital consist of long-lived machinery. Marx says the value of output is (c + v + s). In this expression, constant capital cannot be the total value of the capital stock. It makes most sense to think of c here as the value of capital used up in a given period. If a machine lasts ten years, it passes approximately 1/10th of its value into the output produced each year. That is, c is a flow.

(This is an approximation. One should resist the urge to think of depreciation as solely a matter of physical deterioration of capital goods.)

But, in the formula for the rate of profits, c is a stock. See appendix below. Marx uses the same variable, c, for a stock and a flow. This, at least, risks confusion.

3.0 The Law of the Tendency of the Rate of Profits to Fall

One can also write the rate of profits in the system of labor values as:

r = (s/v)/((c/v) + 1) = e/(1 + occ)

where e is the rate of exploitation (also known as the rate of surplus value) and occ is the organic composition of capital.

Marx was famously concerned with "the laws of motion" of the capitalist mode of production. In particular, he thought that technical progress, under capitalism, leads to an increase in the organic composition of capital. This supposed trend has been called Marx-biased or capital-using technical change. If the rate of exploitation stays the same, the tendency of the rate of profits to fall follows from the above formula.

(One might note that under these assumptions, a constant rate of exploitation with a constant length of the working day implies that wages consist of more commodities, even if they embody a constant quantity of labor time. This is a difficulty in reading Ricardo. He would refer to this as a case of constant "real" wages, while the overwhelming number of economists these days would say real wages have increased.)

Anyways, I raise the question about technical progress in industries producing capital goods. Even if the physical quantities of capital goods with which laborers work is increased by technical progress, the ratio of the value of those capital goods to labor time need not rise. So I do not see that one must expect the organic composition of capital to increase.

A related problem with the above formula was exposed by the Okishio theorem. To be fair to Marx, his claim about the law of the tendency of the rate of profits to fall is in volume 3, which was assembled out of his notes by Engels and not published in his lifetime. Also, he explicitly notes countervailing tendencies, including the cheapening of means of production by technical change in Department I.

4.0 Extra Profits Made from Innovations

Marx argues that the source of income to property (profit, interest, rent, etc.) is value added by workers not paid out in wages. At any point in time, market prices deviate from prices of production and some make value on alienation, while others lose. So abstract from these deviations and assume prices of production prevail.

Even so, some businesses will be introducing new process of production, in which they can make excess profits. Eventually, one expects these excess profits to be wiped out, as other capitalists adopt these new processes and prices of production vary accordingly.

But suppose innovation becomes a regular business, as it now is in Research and Development departments at many businesses. So innovation becomes a regular source of (fluctuating) profits that is not gained from exploiting the worker.

Appendix: Derivation of Straight-Line Depreciation

Consider a production process that produces g widgets from inputs of m long-lasting machines, a units of commodity A, b units of commodity B, and so on, to k units of commodty K. This production process also requires inputs of l person-years of labor. The machine lasts n years. (This example is from chapter X of Sraffa (1960).)

Consider an annuity that costs p_M dollars now and pays out x dollars at the end of each of n years. The interest rate r that equates the price of the annuity to the present value of the payments is such that:

p_M = x/(1 + r) + x/(1 + r)² + ... + x/(1 + r)ⁿ

Or:

x = p_M r(1 + r)ⁿ/[(1 + r)ⁿ - 1]

For a small positive rate of profits:

(1 + r)ⁿ ≈ 1 + n r

Thus, the annual annuity is approximately:

x ≈ p_M[(1/n) + r]

Buying a machine is like buying an annuity. For this special case, the following equation enters Sraffa's system of price equaitions:

p_M m(δ + r) + (p_Aa + ... + p_K)(1 + r) + lw ≈ p_Gg

where:

δ = 1/n

The rate of profits is charged against the value of the entire stock of capital, not merely the value of the flow of used-up capital goods in a single year.

Perturbations Of Markups In Iron Industry In An Example With Produced Iron, Steel, And Corn

Figure 1: Switch Points Varying With Perturbations In Markup In Iron Industry

I have created an example with three produced commodities and a choice of technique. The three produced commodities are called iron, steel, and corn. Corn is taken to be the numeraire and the only commodity purchased by households for consumption. Markups are assumed to vary among industries, even when prices of production prevail.

I was able to locate various fluke switch points in that example. The fluke switch points partition the space of relative markups, as shown in Figure 2. Consider a 45 degree line in this space sloping upward, with s₃/s₁ equal to s₂/s₁. This corresponds to perturbing the markup in the iron industry, with unchanged markups in the steel and corn industries. Figue 1, at the top of this post shows the location of switch points and the maximum wage, as the markup in the iron industry varies.

Figure 2: Partitions in the Space of Relative Markups

I make a few points about the graph of the wage at switch points, as the relative markup in the iron industry varies. The markup in the iron industry declines to the right. The partitions between regions 1 and 7 and between regions 9 and 5 are a four-technique pattern of switch points for the Gamma, Delta, Theta, and Eta techniques. Notice that a switch point between the Delta and Gamma techniques exists in both region 1 and in regions 8 and 9. Around this switch point in region 1, a higher wage is associated with the Gamma technique becoming cost-minimizing. Around the corresponding switch points in regions 8 and 9, a higher wage is associated with the Delta technique becoming cost-minimizing. The amount of labor hired throughout the economy for a given technique and given net output is a physical property of the coefficients of production; this measure of labor intensity is independent of wages. In this case, firms want to hire more labor per bushel corn produced net with a higher wage around this switch point in regions 8 and 9. In this sense, the switch point between the Gamma and Delta techniques is 'non-perverse' in region 1 and 'perverse' in regions 8 and 9. (I here call a switch point 'perverse' merely if it does not follow obsolete marginalist dogma.)

I do not think I have previously noted that the areas in which a technique, like Gamma, is cost-minimizing can be disconnected in certain projections, such as Figure 1.

Variation In Switch Points With Markups

Figure 1: Variation of Switch Points with the Markup in the Steel Industry

1.0 Introduction

I want to continue to analyze this example. This example does not do everything I would like with a three-commodity example. Specifically, I do not have a case of triple-switching in the parameter space I explore in this post. That space of relative markups, in a model with n industries, has (n - 1) dimensions. And it is partitioned by (n - 2)-dimensional manifolds, where each manifold corresponds to a fluke switch point. Whether or not reswitching, capital-reversing, or the recurrence of techniques exist depends on relative markups. This post illustrates in an example with three produced commodities.

2.0 Technology

This economy produces a single consumption good, called corn. Corn is also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. So three industries exist. One produces iron, another produces steel, and the last produces corn. Two processes exist in each industry for producing the output of that industry. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce a unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Prices for a Given Technique

I now want to consider prices when firms in each industry are making the going rate of profits in that industry, given the relative markups among industries. To start, I assume that a single process is operated in each industry. In other words, I initially take the technique is given. So the iron industry is characterized by:

• a_0,1: The person-years of labor hired each year, per ton iron produced.
• a_1,1: The tons of iron used as input, per ton iron produced.
• a_2,1: The tons of steel used as input, per ton iron produced.
• a_3,1: The bushels corn used as input, per ton iron produced.

Similiar coefficients of production, with the second subscript varying by industry, characterize the steel and corn industries. They can be read off columns in Table 1 for the the processes comprising the given technique.

Table 3: Variables

Variable	Definition
p1	Price of iron (bushels per ton iron)
p2	Price of steel (bushels per ton steel)
w	Wage (bushels per person-year)
s1	Markup in iron industry
s2	Markup in steel industry
s3	Markup in corn industry
r	Scale factor for rates of profits

To formulate the price equations, for a given technique, I need the variables listed in Table 3. The price equations for the iron, steel, and corn industries are, respectively:

(p₁ a_1,1 + p₂ a_2,1 + a_3,1)(1 + r s₁) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2 + a_3,2)(1 + r s₂) + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3 + a_3,3)(1 + r s₃) + w a_0,3 = 1

The left-hand side of each of these equations shows the costs of producing one physical unit of the output of the corresponding industry. Capital goods are charged with the going rate of profits on them, and wages are paid out at the end of the year. The right-hand side of these equations shows the corresponding revenues. The equations show that revenues cover costs. No extra profits are made in any industry.

4.0 The Choice of Technique

Given markups, the prices equations for a technique can be solved to find the wage, the price of a ton iron, and the price of a ton steel as a function of the scale factor for the rate of profits. Figure 1, at the top of this previous post illustrates wage curves for a specific set of markups in the three industries. The cost-minimizing technique(s) contributes its wage curve to the outer envelope for those wages or scale factors at which it is cost-minimizing. The usual mathematics drawn on in post Sraffian price theory applies even outside of competitive markets, given relative markups among industries.

5.0 Perturbations of Markups

Fluke switch points partition the space of relative markups among industries, as is illustrated in Figure 2. Within each numbered region, the number and sequence of switch points along the wage frontier does not vary, although their specific location does. Table 4 lists the cost-minimizing techniques in each region, in order of an increasing wage. Some partitions exist that are not shown in Figure 2, with corresponding regions not listed in Table 4. Somewhere to the right of Figure 2, there exists a Alpha versus Epsilon pattern over the axis for the scale factor for the rates of profits. Somewhere above, a Beta versus Delta pattern arises over the axis for the scale factor.

Figure 2: Partitions in the Space of Relative Markups

Table 4: Regions

Region	Technique	Notes
1	Beta, Delta, Gamma, Eta	No reswitching, no capital-reversing, no labor-reversing, no process recurrence
2	Beta, Alpha, Gamma, Delta, Gamma, Eta	Reswitching. Capital and labor-reversing for the switch pt. between Gamma and Delta at the lower wage. Process recurrence in the corn industry.
3	Beta, Alpha, Gamma, Eta	No reswitching, no capital-reversing, no labor-reversing, no process recurrence
4	Alpha, Gamma, Delta, Gamma, Eta	Reswitching. Capital and labor-reversing for the switch pt. between Gamma and Delta at the lower wage. Process recurrence in the corn industry.
5	Alpha, Gamma, Eta	No reswitching, no capital-reversing, no labor-reversing, no process recurrence
6	Alpha, Epsilon, Eta	No reswitching, no capital-reversing, no labor-reversing, no process recurrence
7	Beta, Delta, Theta, Eta	No reswitching, no capital-reversing, no labor-reversing, no process recurrence
8	Beta, Alpha, Gamma, Delta, Theta, Eta	Capital and labor-reversing for the switch pt. between Gamma and Delta. Each process recurs a second time in the corn industry.
9	Alpha, Gamma, Delta, Theta, Eta	Capital and labor-reversing for the switch pt. between Gamma and Delta. Process recurrence in the corn industry.

One can use this analysis to consider the effects, on the choice of technique, of perturbations of markups in one industry, given the markups in the other two industries. Figure 1, at the top of this post, plots the maximum wage and the wage at switch points against the relative markup in the steel industry, given that the iron and corn industry are competitive. It corresponds to a horizontal line in Figure 2 at s₃/s₁ = 1. Around the switch point between the Gamma and Delta techniques in regions 8 and 9, a higher wage is associated with more employment per unit of net output of corn in the economy as a whole. Thus, the managers in the steel industry being able to impose a greater markup over the going rate of profits, or falling below the general competitive level, affects the possibilities for labor pressing for greater wages.

Figures 3 and 4 show the effects of perturbations of markups in the corn industry, given competitive markups in the iron and steel industry. They correspond to a vertical line in Figure 2, extending above the area shown, at s₂/s₁ = 1. Here, region 4 demonstrates that variations in markups can bring about or remove the reswitching of techniques.

Figure 3: Variation of Switch Points with the Markup in the Corn Industry (Part 1)

Figure 4: Variation of Switch Points with the Markup in the Corn Industry (Part 2)

One could also consider a ray from the origin in Figure 2 at 45 degrees. This allows one to examine the effects of of perturbations of markups in the iron industry, given competitive markets in the steel and corn industry.

5.0 Conclusion

I think this analysis qualifies this idea:

"It is evident that between the two limits of this maximum rate of profit an immense scale of variation is possible. The fixation of its actual degree is only settled by the continuous struggle between capital and labour, the capitalist constantly tending to reduce wages to their physical minimum, and to extend the working day to its physical maximum, while the working man constantly presses in the opposite direction. The question resolves itself into a question of the respective powers of its combatants." -- Karl Marx, 1865. Value, Price and Profit

Details of the class struggle between capital and labor are altered by the results of conflict among capitalists.

Adam Smith On The Source Of Profits And Rents In The Exploitation Of The Worker

"In that early and rude state of society which precedes both the accumulation of stock and the appropriation of land, the proportion between the quantities of labour necessary for acquiring different objects seems to be the only circumstance which can afford any rule for exchanging them for one another. If among a nation of hunters, for example, it usually costs twice the labour to kill a beaver which it does to kill a deer, one beaver should naturally exchange for or be worth two deer. It is natural that what is usually the produce of two days or two hours labour, should be worth double of what is usually the produce of one day’s or one hour’s labour...

...As soon as stock has accumulated in the hands of particular persons, some of them will naturally employ it in setting to work industrious people, whom they will supply with materials and subsistence, in order to make a profit by the sale of their work, or by what their labour adds to the value of the materials. In exchanging the complete manufacture either for money, for labour, or for other goods, over and above what may be sufficient to pay the price of the materials, and the wages of the workmen, something must be given for the profits of the undertaker of the work who hazards his stock in this adventure. The value which the workmen add to the materials, therefore, resolves itself in this case into two parts, of which the one pays their wages, the other the profits of their employer upon the whole stock of materials and wages which he advanced. He could have no interest to employ them, unless he expected from the sale of their work something more than what was sufficient to replace his stock to him; and he could have no interest to employ a great stock rather than a small one, unless his profits were to bear some proportion to the extent of his stock...

...As soon as the land of any country has all become private property, the landlords, like all other men, love to reap where they never sowed, and demand a rent even for its natural produce. The wood of the forest, the grass of the field, and all the natural fruits of the earth, which, when land was in common, cost the labourer only the trouble of gathering them, come, even to him, to have an additional price fixed upon them. He must then pay for the licence to gather them; and must give up to the landlord a portion of what his labour either collects or produces. This portion, or, what comes to the same thing, the price of this portion, constitutes the rent of land, and in the price of the greater part of commodities makes a third component part." -- Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, Book I, Chapter VI

Smith is saying that the price of a produced commodity, under a system with private property, consists of the sum of the value of the means of production used up in making that commodity and the value added by labor. And that value added by labor is not entirely paid out to workers as wages. Some of it goes to pay profits and rent. This is exploitation of the worker, as many saw it at the time.

I do not see that this claim necessarily depends on any quantitative relationship between the labor embodied in a commodity and "natural prices", where the latter are centers of gravitational attraction for market prices at a given point in time. Does it depend on the macroeconomic invariants Marx erroneously asserted to hold when he came to write on the transformation problem in Volume 3 of capital?

Many of the so-called Ricardian socialists thought of profits as rent imposed on top of prices proportional to labor values. It is this deviation of prices from labor values that, in their view, socialism would abolish. Some had the idea of even then of paying workers in labor vouchers that would be exchangable (in co-operatives?) for goods priced in terms of labor values.

Karl Marx, in many places, famously opposed this view.

References

• W. Paul Cockshott and Allin Cottrell. 1993. Towards a New Socialism
• Anton Menger. 1899. The Right to the Whole Produce of Labor (Trans. by Herbert Foxwell).
• Adam Smith. 1776. An Inquiry into the Nature and Causes of the Wealth of Nations (I go by the Cannan edition).
• Noel W. Thompson. 1984. The People's Science: The Popular Economy of Exploitation and Crisis 1816-34 Cambridge University Press.

Richard Wolff Versus Steven ("Destiny") Bonnell

A Debate on Socialism versus Capitalism

I suppose Destiny entered this debate in good faith. But why did he have to lie about what Wolff was saying? Wolff certainly never repudiated the labor theory of value in his opening remarks. And later Wolff explicitly argued that the Democrats are not socialist. One may not understand or agree with Wolff, but why lie?

Apparently "Destiny" feels he lost badly and has been trying to find somebody less eminent to contrast his ignorance with. (I also stumbled upon a debate last December between "Socialism Done Left" and Victor Magarino.) Maybe it is not too interesting to watch somebody with a twitch stream try to educate himself. I probably have all sorts of disagreements with those here drawing on Cockshott and Cottrell or on Shaikh. But I certainly do not understand this subculture of Twitch and Discord streamers, whatever that means.

I wish Wolff had been more concrete sometimes. When he was talking about the arguments within the second international that led to the third international splitting off, he could have named Eduard Bernstein and mentioned Rosa Luxemburg's pamphet Reform or Revolution? I would like to hear more about the Portugese communist party. Did they evolve during the Eurocommunism debate in the 1970s? Does providing such details conflict with Wolff's approach of trying to present these ideas in a popular format? It does give those who want to, some keywords on which to search.

I follow Wolff in taking people in the tradition of struggle at their word. For much of the twentieth century, those parties improving the lives of western European countries called themselves socialist. For decades, many have said that the Democratic Party in the USA is not that. Of course, if I take Lenin at his word, those trying to set up a government of a vanguard party implementing central planning are also socialists.

I get that Mondagon has existed for a half-century or more. I suppose Wolff could explain more about where the tradition of worker-managed cooperatives come from. Would he point to pre-Marist utopian socialists? Has he ever commented on Yugoslavia under the communists? I think of dairy products, but, apparently, cooperatives widely exist for residential apartments in some countries today.

Some think that socialist institutions will grow inside capitalist societies until they become dominant. Examining how capitalist institutions grew inside Feudalism, without initially being dominant, is obviously apropos when debating socialism versus capitalism. (Here Wolff talks about how Paul Baran taught him that to consider how a post-capitalist society might grow up under capitalism, one might want to consider how capitalism grew under feudalism.)

In explaining how co-ops can obtain outside investment without ceding control from the workers, Wolff might have mentioned non-voting stock, which many corporations have issued now. Has Wolff ever commented on the possibility of co-ops issuing bonds? Formally, creditors do not control organizations that they lend to, but practically they can. I would also like to hear from Wolff how the law might be changed to encourage co-ops. (This is not, presumably, a matter of prohibiting small proprietors.)

I happen to know that PlayStations are optimized for the mathematics used in demonstrating that the labor theory of value, surprisingly, is empirically valid. Linear algebra is useful for graphics.

Brad DeLong, Noah Smith, And Others On The Cambridge Capital Controversy

Brad DeLong and Noah Smith chat about the Cambridge Capital Controversy on their podcast, Hexapodia is the key insight. Noah says it was only mentioned in one of his classes on macroeconomics, but seems to say he was never formally taught it. Given his summary near the start of this discussion, he has obviously read something about it. Brad thinks the language used by the MIT economists in the 1960s was badly and inaccurately phrased and poorly suited to shed light. He says he had trouble figuring out why anybody would disagree that a single aggregate index could not be rigorous and theoretically justified. So this discussion, if I understand correctly, could be expected to fit well with the name of their podcast. But maybe not.

Brad distinguishes between the ownership of physical capital goods and the physical productivity of those goods. Do they anywhere distinguish between capital as the financial value of assets and capital as a heterogenous odd lot of means of production? Brad notes that 19th century economists brought up the (unconvincing) idea that those providing capital are incurring sacrifices by "waiting".

Noah knows that marginal productivity theory was developed as a theory of "just deserts", not that he accepts this idea. Aggregate models with a single index for the amount of capital are not helpful in figuring out how much business owners should be paid to be consistent with a flourishing society.

At one point, Noah says that somebody had a better index for capital, but he cannot recall who. Brad brings up Christopher Bliss and the slope of the production possibilities frontier for an intertemporal choice over, say, wheat today and wheat a week from now. I aggree Christopher Bliss' response to the CCC is an important marginalist response. But Noah should not have deferred so much at this time. He was trying to remember Edwin Burmeister's work.

I do not think Noah quite gets why the interest rate is generally not equal in equilibrium to the marginal product of capital. He brings up the question here of why did the participants in the CCC not also question an aggregate index for labor. Reswitching, capital reversing, the reverse substitution of labor, and so on can arise in models with heterogenous labor. Do either Brad or Noah distinguish between the price of the services of a capital good for, say, a year and the interest rate? Each kind of labor is measured in a homogeous unit, person-years. What is the analogy with capital supposed to be here?

At one point, Brad explains reswitching. Neither notes that in a comparison of long run positions, a higher wage can be associated with the adoption of a technique in which firms want to employ more labor to produce a given net output.

For me, what the English side showed is that prices of production do not follow the logic of supply and demand. Prices are not indices of relative scarcity. Marshall's principle of substitution does not characterize comparisons of (long-run) equilibrium. Classical political economy had a different approach to value and distribution, and that approach is logically consistent.

In trying to put what should have been the MIT side as strongly as possible, Brad describes the rate of profits "as a control variable" that provides a signal for how to allocate scarce resources. He thinks that by not acknowledging this role, the English side misses something important. Is Brad's position consistent with the above understanding of price theory?

I gather that this podcast is for a popular audience. It is not intended to be a comprehensive academic survey. So one should expect some gaps. They do not bring up Joan Robinson's distinction between historical and logical time or Post Keynesian's ideas on the difference between risk and uncertainty. To be fair, I do not discuss how long run positions can be reached very much myself.

Bill Mitchell now has a two part series on the CCC. Matias Vernengo points out he had an on-topic post in 2012. I find I had a bulleted summary in 2017. This example with three produced commodities is fairly comprehensive. Alexander Douglas has a 2018 Medium post offering an appreciation of Joan Robinson as a philosopher.

Paul Krugman Ignorant Of The Cambridge Capital Controversy

Zach Carter has an appreciation of Joan Robinson's work on imperfect competition, with a bit about the role of Cambridge circus in helping Keynes write the General Theory. Paul Krugman, gatekeeper, reacts:

"Nice appreciaton of Joan Robinson, although no mention of her later role. Sad to say, as a student I mainly encountered her through the 'Cambridge capital controversy', a huge intellectual muddle. Somehow Robinson and others managed to convince themselves that the moral legitimancy of capitalism rested on the existence of a well-defined measure of 'capital' that had a well-defined marginal product. What followed was a tortured debate that illuminated nothing much, and eventually just faded away. Oh well. But Zach Carter is right: we value thinkers for their best work, not their detours, and Robinson made a huge contribution." -- Paul Krugman, 25 April 2021

In my work trying to extend the CCC, I usually jump into the middle. I probably have a summary years ago for the beginner, but I cannot find such. Quickly looking, I find these posts:

• Quotations from some experts on the CCC
• A sophisticate neoclassical response to the CCC
• Why do mainstream economists not make more out of the CCC?
• Fields impacted by the CCC
• Catalog of neoclassical responses to the CCC
• Textbook recommendations (I have longer lists in some other posts, I think.)

Somebody coming here from Twitter who does not pay attention to academic economics might not find these too helpful. I write hardly anything at all about the 'moral legitimancy of capitalism'.

Two Four-Technique Patterns With Markup Pricing

Figure 1: Wage Curves for the Example

1.0 Introduction

The Cambridge Capital Controversy (CCC) applies to models both of competitive industries and of non-competitive industries. Around a switch point exhibiting capital-reversing, a higher wage is associated with greater employment per unit of net output produced. It is not merely a question of what technology is available. It is also a matter of the power of firms among industries to extract value from their workers, their upstream suppliers, and their downstream buyers. And of the countervailing power to resist such exercise of power. Can you cite literature in Industrial Organization (IO) that explicitly recognizes the logical consequences of the CCC for IO?

This post presents an example of a fluke switch point, or rather two fluke switch points, in a global pattern. It is a case in which capital-reversing just begin to arise.

2.0 Technology

I have used an example with this structure before. This economy produces a single consumption good, called corn. Corn is also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. So three industries exist. One produces iron, another produces steel, and the last produces corn. Two processes exist in each industry for producing the output of that industry. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce a unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Prices and the Choice of Technique

I consider prices of production in which:

• Labor is advanced and is paid out of the surplus product at the end of the year.
• Firms in differenct industries are able to enforce barriers to entry, with stable ratios of profits among industries.
• A bushel of corn is the numeraire.

Let s₁ r be the rate of profits in the iron industry, s₂ r be the rate of profits in the steel industry, and s₁ r be the rate of profits in the corn industry. I consider the case with s₁ set to unity, and the ratio of the other two markups to this as noted in the figures in this post. Maybe I should go back to calling r the scale factor for the rate of profits.

Under these assumptions, a system of three equations can be set out for the three techniques. The variables in these equations are the rate of profits r, the wage, the price of iron, and the price of steel. They can be solved with one degree of freedom remaining open. Figure 1, at the top of this post, shows the wage curves for each technique. The wage curves for the Zeta and Theta techniques, at least, are more curved than is typically found in the empirical literature.

One can also find the price of iron and steel, as shown in Figures 2 and 3 below. At a switch point, the wage and prices are the same for all techniques on the frontier.

Figure 2: The Price of Iron in the Example

Figure 3: The Price of Steel in the Example

At the first switch point, managers of firms in the iron and corn industry do not care which of the two processes in their industry they operate. Contrawise, at the other switch point, managers of firms in steel industry and the corn industry do not care.

4.0 Conclusion

This combination of fluke switch points is an intersection of two one-dimensional manifolds in the two-dimensional parameter space formed by the relative markups, s₂/s₁ and s₃/s₁. Each manifold characterizes one of the two four-technique fluke switch points. I do not think I have yet constructed an example of manifolds partitioning such a two-dimensional space of relative markups.

Fluke Switch Points in Pure Fixed Capital Systems

I have a working paper at the Centro Sraffa.

Abstract: This article considers structural economic dynamics, in models with fixed capital and a choice of technique, of the production of commodities. Fluke switch points are described and cataloged. For fluke switch points, parameter perturbations create a qualitative change in how the choice of technique varies with distribution. Techniques are presented for visualizing partitions of parameter spaces such that the analysis of the choice of technique does not vary within each region. Implications are drawn about the choice of the truncation of the operation of (or the economic life of) machines and about the adoption of roundabout techniques.

Algebraic Geometry

An Introduction to Algebraic Geometry

I have been looking for fluke switch points in certain parameter spaces of coefficients for polynomial equations. Bertram Schefold has pointed out to me that I may want to look into algebraic geometry. This may be beyond me. I consider what I have been doing as exploratory mathematics, and I have been relying on numerical algorithms. I started with thinking that there is a parallel to bifurcation theory. But Barkley Rosser convinced me that I should not use that terminology without an explicit dynamic system, presumably of market prices. These two threads on Math Overflow suggest I might want to look at Bertrametti et al. Lectures on Curves, Surfaces and Projective Varieties. I need a physical book for this, I think, not just a PDF.

Flummery From Robert A. Heinlein

He had been droning along about 'value,' comparing the Marxist theory with the orthodox 'use' theory. Mr. Dubois had said, 'Of course, the Marxian definition of value is ridiculous. All the work one cares to add will not turn a mud pie into an apple tart; it remains a mud pie, value zero. By corollary, unskillful work can easily subtract value; an untalented cook can turn wholesome dough and fresh green apples, valuable already, into an inedible mess, value zero. Conversely, a great chef can fashion of those same materials a confection of greater value than a commonplace apple tart, with no more effort than an ordinary cook uses to prepare an ordinary sweet.'

'These kitchen illustrations demolish the Marxian theory of value — the fallacy from which the entire magnificent fraud of communism derives — and to illustrate the truth of the common-sense definition as measured in terms of use.'

Dubois had waved his stump at us. 'Nevertheless — wake up, back there! — nevertheless the disheveled old mystic of Das Kapital, turgid, tortured, confused, and neurotic, unscientific, illogical, this pompous fraud Karl Marx, nevertheless had a glimmering of a very important truth. If he had possessed an analytical mind, he might have formulated the first adequate definition of value... and this planet might have been saved endless grief.'

-- Robert A. Heinlein, Starship Troopers

I think this sufficient demonstration that Heinlein's ignorant character is attacking a straw person.

Elsewhere

The Banach-Tarski Paradox

• The Hahn-Banach theorem is related to how mainstream economics model perfect competition. The video above is mind-bending math.
• The Mountain Goat blog.
• A profile of some economists at Berkeley, some of who I have read when they collaborated with Thomas Piketty or A. Dube. I have read deLong, as well, of course.
• If you cannot name 'capitalism', 'neoliberalism', or 'neoclassical economics', it is difficult to criticize them. Here is a popular account about right-wingers crying about researchers daring to use such terms.
• I conclude with a recent talk, below, by Yanis Varoufakis on the need for pluralism in economics. He argues, at least (I still have more to watch):
• Economic theory can be performative, counter-performative, or reflexive (without using those terms).
• Time, money, debt, and interest rates do not appear in the models in the textbook.
• Markets can exist without a society being capitalist.
• Once you have learned all these models that have nothing to say about capitalism, you might possibly say something intelligent.

Yanis Varoufakis: From an economics without capitalism to markets without capitalism.

The Production Function In A Discrete Technology

Figure 1: Isoquants For The Production Function

1.0 Introduction

I often assume a discrete technology in my demonstrations that what many mainstream economists teach is mostly incoherent balderdash. Some incompetents have told me that such well-established results "just show that the particular production functions that you have chosen don't work. This is not a generic result." So, for my amusement, I will go through a simple example here to explain how any continuously differentiable production function can be approximated arbitrarily closely by the production function for a discrete technology.

By the way as far as I know, capital-reversing is consistent with continuously differentiable production functions. Does the late Emmanuel Farhi's work on this theme make progress on Wolfgang Eichert's work? I sometimes worry that a serious exploration by a mainstream economist of the Cambridge capital controversy would lead to psychological depression.

2.0 The Model

I consider a single sector of an economy where, say, Q tons steel are manufactured from inputs of labor and iron. The managers of firm know of S processes for producing steel, where each process is characterized by an ordered pair of coefficients of production. That is, the technology for making steel, T, is defined as:

T = { (a₀(s), a(s)) | s = 1, 2, ..., S }

In the sth process, the services of at least a₀(s) person-years of labor and a(s) tons iron must be applied for every ton steel produced.

With only two inputs, I can assume that labor coefficients are ordered to be increasing:

0 < a₀(1) < a₀(2) < ... < a₀(S)

And that iron coefficients are decreasing:

a(1) > a(2) > ... > a(S) > 0

With this specification of technology, one can formulate a linear program (LP). Let L be the person-years of labor available to this firm, and let X be the tons of iron available. Define q₁, q₂, ..., q_S to be the tons steel produced with each of the S processes. Consider the following LP:

Given T, L, and X, choose q₁, q₂, ..., q_S

To maximize Q = q₁ + q₂ + ... + q_S such that

a₀(1) q₁ + a₀(2) q₂ + ... + a₀(S) q_S ≤ L

a(1) q₁ + a(2) q₂ + ... + a(S) q_S ≤ X

q₁ ≥ 0, q₂ ≥ 0, ..., q_S ≥ 0

Let the solution of this LP be:

Q = F(L, X)

Then F is the production function for steel production.

3.0 Selected Properties of a Production Function

A production function as defined above exhibits constant returns to scale (CRS). Figure 1, at the top of this post, displays isoquants for a particular technology with S equal to four. Any point in the interior of the line seqment between (a₀(1) Q, a(1) Q) and (a₀(2) Q, a(2) Q), for example, is a switch point. The extremes are non-switching points, where only one process in the technology is operated.

Figure 2 graphs the output of steel as a function of the labor input, given a specified quantity of iron available for input. The physical marginal product is shown below. The marginal product is non-increasing. The horizontal steps are non-switching points, and the vertical jumps occur at switch points.

Figure 2: The Marginal Product Of Labor

In this example, labor and iron inputs are treated formally the same. So, as Figure 3 shows, the graph of the output of steel as a function of the iron input, and of iron's physical marginal product, look qualitatively the same as the output of steel as a function of the labor input.

Figure 3: The Marginal Product Of Iron

Linear programming, as I understand it, is not taught as introductory mathematics. On the other hand, one can explain the above graphs without knowledge of calculus. Are there still recent introductory textbooks for microeconomics with graphs like the above?

4.0 Conclusions

One can generalize the above to consider a production function for more than two inputs. The processes will not be ordered as above, and isoquants would be graphed in a higher dimensional space. Another generalization would consider multiple production funtions, one for each sector, with given prices for the produced outputs. Given endowments for inputs, also known as 'factors of production', the dual problem assigns shadow prices to the inputs. Also, endowments are not given in long-period models.

References

• Eichert, Wolfgang. 2014a. Long-period positions in multi-sectoral Cobb-Douglas economies. Metroeconomica 65 (1): 136-153.
• Eichert, Wolfgang. 2014b. Technological Change in Multi-Sectoral Economies: Theoretical Change in Multi-Sectoral Economies. Doctoral thesis, University of Graz.
• Pasinetti, Luigi L. 1977. Lectures on the Theory of Production. New York: Columbia University Press.

Private Truths, Public Lies In Mainstream Economics?

I sometimes wonder if most mainstream economists think that most of what they were taught, teach, and research are some combination of false, incoherent, and useless for understanding actually existing capitalism. But they go along out of some sense of professionalism and a belief that their colleagues do not share their views. That is many privately think they are a minority of one, but publically espouse the orthodoxy.

As far as professionalism goes, I suppose some believe that those who go on from their microeconomics class, for example, are expected to have been exposed to certain material. I would hope that some question the ethics of not letting the students know that they are being taught one approach, named marginalism, and other approaches exist.

Maybe one of these days, I will read Timur Kuran's book.

Bushwa From Jeffrey Clemens In The Journal of Economic Perspectives

"The labor supply curve slopes upward, reflecting differences in workers’ reservation wages (as driven by outside opportunities related, perhaps, to leisure, home production, and economic assistance that can be received while out of work). The labor demand curve slopes downward, tracing out the relationship between the quantity of labor employed and the marginal revenue product of that labor. This, in turn, reflects the assumption of a constant price (due, perhaps, to a perfectly competitive market for the firm’s output) and a production function in which, holding capital and technology fixed, labor has diminishing marginal productivity.

In a perfectly competitive labor market, a freely set wage will adjust to equilibrate supply and demand..."

-- Jeffrey Clemens. 2021. How do firms respond to minimum wage increases? Understanding the relevance of non-employment margins. Journal of Economic Perspectives (Winter): 51 - 72.

Clemens then considers shifts in demand and supply curves for labor with changes in prices due to market power in final goods, changes in benefits, and other aspects of jobs. He never notes his framework is balderdash. Empirical evidence, which Clemens cites, cannot make up for his basic incoherence. Mayhaps, Clemens could read Fabio Petri's textbook when it is published.

I do not expect to read articles about the speed of the ether in absolute space in physics journals. Nor do I expect to read about the weight of phlogiston in chemistry journals. Why does the American Economic Association publish articles that make astrologists look good?

Vienneau (2005) Is A Necessary Resource For Arguments About A Minimum Wage

Maybe, perhaps, that is a bit hyperbolic. But it has been known for at least half a century that, even in competitive markets, wages and employment cannot be explained by the interaction of well-behaved supply and demand curves for labor. If you do not want to read me, check out, for example, Garegnani (1970) or Opocher and Steedman (2015). Shove (1933) illustrates how far awareness of the difficulties go. White (2001) is a demonstration that I am not the only one to draw practical conclusions from the theory.

Cohort after cohort, generation after generation, in the supposedly best schools promulgate falsehoods, ignorance, and incoherent nonsense.

References

• Garegnani, Pierangelo. 1970. Heterogeneous capital, the production function and the theory of distribution. Review of Economic Studies 37(3): 407-436.
• Opocher, Arrigo and Ian Steedman. 2015. Full Industry Equilibrium: A Theory of the Industrial Long Run Cambridge: Cambridge University Press.
• Shove, G. F. 1933. Review of The Theory of Wages. Economic Journal (Sep.)
• Vienneau, Robert L. 2005. On labour demand and equilibria of the firm. Manchester School 73(5): 612-619.
• White, Graham. 2001. The poverty of conventional economic wisdom and the search for alternative economic and social policies Austrlian Review of Public Affairs

Neoclassical Economists Being Wrong

What is neoclassical economics? (This post draws on something I wrote on Usenet more than a decade ago.) I believe I might have introduced this list of three key assumptions, as noted by Roy Weintraub, into the wikipedia article on the topic:

• People have rational preferences between outcomes that can be identified and associated with values.
• Individuals maximize utility and firms maximize profits.
• People act independently on the basis of full and relevant information.

One should recognize that neoclassical economics is associated with mathematical formalism. So neoclassical economists speaking among the clergy would prefer the language of topology and the algebra of relations for stating their assumptions.

The point of neoclassical economics is to build a theory on those assumptions which emphasizes equilibrium, characterizes economics as the allocation of scarce resources, and justifies supply and demand reasoning. Neoclassical economists wanted to argue:

• Equilibrium prices are scarcity indices
• Marshall's principle of substitution is generally applicable

Neoclassical economists are unable to state assumptions that justify such reasoning. Weintraub's assumptions, suitably formalized, don't succeed. They do not succeed because one can construct examples with these assumptions in which the negation of neoclassical claims hold. I and others have done this. This is a matter of logic.

Just to show you that others characterize neoclassical economics in the same way as I do:

"The [Demand-and-Supply-based Equilibrium] theory visualizes the economy as an aggregate of atomistic individuals (producers and consumers) making their decisions autonomously, with no interference from the influence of 'externalities'. Relative prices and quantities are determined simultaneously in equilibrium as an outcome of the interplay of 'forces of demand and supply', generated by the optimizing behavior of individuals subject to their resource constraints. A certain symmetry characterizes the behaviour of producers and consumers. Each producer, given the technological possibilities, chooses the profit-maximizing activities and outputs, at the going prices; each consumer, given his budget constraints and scales of preferences, maximizes satisfaction at the going prices. It is through the operation of the 'fundamental' and 'universal' principle of substitution that individuals adjust their chosen quantities in response to the parametrically given prices...

Further, the notion of 'change' in the DSE theory gets restrictively predetermined by the theory in the following ways. First, all changes in quantities within the system are seen as the outcome of the ever-active principle of substitution. Thus the changes are primarily in relative quantities involving allocational variations. The role of prices as a scarce-resource allocator, given the resources, dominates the theory as contrasted with the resource-creational role of prices in classical theory... Secondly, all changes are explained as induced by changes in relative prices and operate through the decisions of individuals who are only 'quantity adjusters'; that is, all influences affecting quantities have to be necessarily mediated through relative prices or changes on the market and are outcomes of the atomistic responses of individuals. The relative prices acquire the all-powerful role of resource-allocation and the 'market' becomes the 'arena' of action." -- Krishna Bharadwaj (1989).

Here are a couple of examples of the incorrect reasoning to which I object:

"It is indeed the great contribution of the Pure Logic of Choice that it has demonstrated conclusively that even such a single mind could solve this kind of problem only by constructing and constantly using rates of equivalence (or 'values' or 'marginal rates of substitution'), that is, by attaching to each kind of scarce resource a numerical index which cannot be derived from any property possessed by that particular thing, but which reflects, or in which is condensed, its significance in view of the whole means-end structure...

Fundamentally, in a system in which the knowledge of the relevant facts is dispersed among many people, prices can act to co-ordinate the separate actions of different people in the same way as subjective values help the individual to co-ordinate the parts of a plan. It is worth contemplating for a moment a very simple and commonplace instance of the action of the price system to see what precisely it accomplishes. Assume that somewhere in the world a new opportunity for the use of some raw material, say, tin has arisen, or that one of the sources of tin has been eliminated. It does not matter for our purpose and it is significant that it does not matter which of these two causes has made tin more scarce. All that the users of tin now need to know is that some of the tin they used to consume is now more profitably employed elsewhere and that, in consequence, they must economize tin. There is no need for the great majority of them even to know where the more urgent need has arisen, or in favor of what other needs they ought to husband the supply... The whole acts as one market, not because any of its members survey the whole field, but because their limited individual fields of vision sufficiently overlap so that through many intermediaries the relevant information is communicated to all. The mere fact that there is one price for any commodity ­ or rather the local prices are connected in a manner determined by the cost of transport, etc. - brings about the solution which (it is just conceptually possible) might have been arrived at by one single mind possessing all the information which is in fact dispersed among all the people involved in the process." -- F. A. Hayek (1945).

"Let us then suppose that... there is a strike on the part of one group of workers, say the plasterers, or that there is some other disturbance to the supply of plasterers' labour... The rise in plasterers' wages would be checked if it were possible either to avoid the use of plaster, or to get the work done tolerably well and at a moderate price by people outside the plasterers' trade: the tyranny, which one factor of production of a commodity might in some cases exercise over the other factors through the action of derived demand, is tempered by the principle of substitution." -- Alfred Marshall (1920).

Hayek and Marshall were writing before it was known that the assumptions of neoclassical economics could not justify their reasoning.

Here is an ignorant or dishonest neoclassical economist perpetuating ignorance to another generation:

"Suppose the number of carpenters suddenly increases, due to the immigration of thousands of new carpenters from Mexico. Both before and after the change, carpenters receive their marginal revenue product... But the wage after the migration is lower than the wage before. Since the supply of carpenters is higher than before, the equilibrium wage is lower.

...an increase in the supply of an input I own drives down its price (and marginal revenue product) and so decreases my income. The same is true for an increase in the supply of an input that is a close substitute for an input I own. If I happen to own an oil well, I will regard someone else's discovery of a new field of natural gas--or a process for producing power by thermonuclear fusion--as bad news." -- David D. Friedman (1990).

David cannot state his assumptions. Here is a quote from a refereed paper:

"This note considers a linear programming (LP) formulation of the theory of the firm. A neoclassical non-increasing labour demand function is derived from the solution of the LP. It is argued that only a small number of points on this curve, one or two in the examples provided, are equilibria of the firm. Equilibria are characterized by decisions of the managers of the firms that allow the same decisions to be made in successive periods. Hence, one can explain the quantity of labour that firms desire to hire either by a traditional neoclassical labour demand function or by an analysis of equilibria of the firm, but generally not both. Explaining wages and employment by well-behaved supply and demand functions for labour is of doubtful logic." -- R. L. Vienneau (2005).

References

• Krishna Bharadwaj. 1989. Themes in Value and Distribution: Classical Theory Reappraised London: Unwin-Hyman.
• David D. Friedman. 1990. Price Theory: An Intermediate Text 2nd Edition.
• F. A. Hayek. 1945. "The use of knowledge in society. American Economic Review 35 (5): 519-530.
• Alfred Marshall. 1920. Principles of Economics: An Introductory Volume 8th edition.
• E. Roy Weintraub. 2007. Neoclassical economics. The Concise Encyclopedia of Economics.
• Robert L. Vienneau. 2005. On labour demand and equilibria of the Firm. Manchester School 73 (5): 612-619.

Elsewhere

• A start (see links on the left) of advice to a student interested in how to introduce an evolutionary approach into economics.
• A podcast severely critical of mainstream economics.
• A substack post of an avowed creed for neoliberals. I don't think the author has studied the scholarly literature on the topic.

These links do not present a hopeful picture.

Update 13 February 2021:

• Some obituaries of Michael Perelman.
• David Pakman on socialism, social democracy, and its difference with democratic socialism. I do not think I have been consistent on the distinction. According to his Wikipedia entry, Pakman studied economics and communications as an undergrad at UMass Amherst.
• Recent talks by Steve Keen. Sometime in the 1980s, I too discovered if I wanted to learn about economic theory, I would be better off in a library than reading mainstream textbooks.

Greg Mankiw Should Try To Make A Honest Living

A Discussion On The Best (or Worst) Of Mankiw

Peter Bofinger has expanded a series of tweets to point out some stuff that is just wrong in Mankiw's introductory textbook. The video above is a virtual panel discussion in which Mankiw graciously pretends to respond to Bofinger. Rüdiger Bachmann and Anna Reisch also participate. I do not know the host, Thomas Fricke. Questions from the audience are fielded towards the end. I concentrate on Sascha Buetzer (1:07:45) below. Other questions are from Janina Urban (1:18:41) and Thomas Kopp (1:21:29). I am probably missing something.

I wonder whether Anna Reisch knows about Adolph Lowe's political economics. I know of this through his 1965 book On Economic Knowledge. I gather Lowe thought it was the task of economists to say whether a given end state is internally consistent and to explain how it could be reached.

I want to point out some hypocrisy from Mankiw. He does not even bother arguing that Bofinger has pointed out confusion and nonsense in his textbook. He says that he sees his job as presenting the consensus of mainstream economics, not his own theories. He tries to minimize the imposition of his own idiosyncrasies. Now Buetzer is, if I hear correctly, the senior advisor to the German director to the International Monetary Fund (IMF). Buetzer offers the difficult proposition that "textbooks should strive to be factually correct." And one of his points is that, "all modern empirical evidence ... point to ... there is no equity-efficiency tradeoff from moderate levels of redistribution, but rather the opposite." This is the "mainstream in mainstream institutions". Does Mankiw say he will then update his textbook in the next edition to reflect the mainstream view? Of course not. He starts presenting his own idiosyncratic reasons for rejecting empirical evidence.

Economists should strive not to teach falsehoods and nonsense and not to promote the teaching of falsehoods and nonsense. Maybe Mankiw is correct that if he discarded from his textbooks stuff that is, at best, just wrong, his textbooks would not sell as well. That is no justification for retaining balderdash. Although Mankiw may disagree, he is not entitled to an income from textbooks.