Friday, May 28, 2021


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

Saturday, May 22, 2021

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 pM 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:

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


x = pM 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:

xpM[(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:

pM m(δ + r) + (pAa + ... + pK)(1 + r) + lwpGg


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.

Tuesday, May 18, 2021

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.

Tuesday, May 11, 2021

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

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
Alphaa, c, e
Betaa, c, f
Gammaa, d, e
Deltaa, d, f
Epsilonb, c, e
Zetab, c, f
Etab, d, e
Thetab, 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:

  • a0,1: The person-years of labor hired each year, per ton iron produced.
  • a1,1: The tons of iron used as input, per ton iron produced.
  • a2,1: The tons of steel used as input, per ton iron produced.
  • a3,1: 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
p1Price of iron (bushels per ton iron)
p2Price of steel (bushels per ton steel)
wWage (bushels per person-year)
s1Markup in iron industry
s2Markup in steel industry
s3Markup in corn industry
rScale 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 a1,1 + p2 a2,1 + a3,1)(1 + r s1) + w a0,1 = p1
(p1 a1,2 + p2 a2,2 + a3,2)(1 + r s2) + w a0,2 = p2
(p1 a1,3 + p2 a2,3 + a3,3)(1 + r s3) + w a0,3 = 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
1Beta, Delta, Gamma, EtaNo reswitching, no capital-reversing, no labor-reversing, no process recurrence
2Beta, Alpha, Gamma, Delta, Gamma, EtaReswitching. Capital and labor-reversing for the switch pt. between Gamma and Delta at the lower wage. Process recurrence in the corn industry.
3Beta, Alpha, Gamma, EtaNo reswitching, no capital-reversing, no labor-reversing, no process recurrence
4Alpha, Gamma, Delta, Gamma, EtaReswitching. Capital and labor-reversing for the switch pt. between Gamma and Delta at the lower wage. Process recurrence in the corn industry.
5Alpha, Gamma, EtaNo reswitching, no capital-reversing, no labor-reversing, no process recurrence
6Alpha, Epsilon, EtaNo reswitching, no capital-reversing, no labor-reversing, no process recurrence
7Beta, Delta, Theta, EtaNo reswitching, no capital-reversing, no labor-reversing, no process recurrence
8Beta, Alpha, Gamma, Delta, Theta, EtaCapital and labor-reversing for the switch pt. between Gamma and Delta. Each process recurs a second time in the corn industry.
9Alpha, Gamma, Delta, Theta, EtaCapital 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.

Friday, May 07, 2021

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.

  • 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.

Tuesday, May 04, 2021

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.

Saturday, May 01, 2021

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.