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, if the managers in the iron industry are able to impose a greater markup over the going rate of profits or fall 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 empirical 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.

Monday, April 26, 2021

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:

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'.

Wednesday, April 21, 2021

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

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 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.

Friday, April 16, 2021

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.

Wednesday, April 14, 2021

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.

Saturday, April 03, 2021

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.

Saturday, March 27, 2021


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.

Saturday, March 20, 2021

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, ..., qS to be the tons steel produced with each of the S processes. Consider the following LP:

Given T, L, and X, choose q1, q2, ..., qS
To maximize Q = q1 + q2 + ... + qS such that
a0(1) q1 + a0(2) q2 + ... + a0(S) qSL
a(1) q1 + a(2) q2 + ... + a(S) qSX
q1 ≥ 0, q2 ≥ 0, ..., qS ≥ 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.

  • 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.

Tuesday, March 16, 2021

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.

Monday, March 08, 2021

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?

Saturday, February 27, 2021

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.

  • 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

Saturday, February 20, 2021

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).
  • 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.

Tuesday, February 02, 2021


  • 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.

Saturday, January 23, 2021

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.

Saturday, January 16, 2021

On The Empirical Verification Of The Cambridge Capital Controversy

1.0 Introduction

My consistent position is that Sraffa and his followers, besides recovering an alternate approach to value and distribution found in classical economics and Marx, demonstrated the logical invalidity of marginalist economics. Empirical results are irrelevant to questions of logical validity.

Wage curves, as constructed from input-output matrices, are rational functions with the numerator and denominator both being some high order polynomial functions. I would have liked to see some more wobbles in those constructed empirically and more examples of reswitching and capital-reversing. Nevertheless, the finding that frontiers are close to linear functions, with only a few switch points, is not consistent with an emphasis on widespread marginal adjustments. It is more consistent with Marx's theory of value and Joan Robinson's understanding of technical change, in which the question of the choice of technique at a given moment in time is, at most, a secondary concern. Schefold's recent work (Schefold 2013, Schefold 2016, Götz and Schefold 2020) with random matrices is of interest here for trying to explain the empirical posts.

I have written about empirical results before. In this post I concentrate on Zambelli (2018) as the most recent, most extensive empirical examination of input-output matrices. See also the comments on Zambelli's work in Götz and Schefold (2020).

2.0 Progress in Empirical Research Work

Increased computer power and more complete consistent national income and product accounts (NIPAs) has supported empirical research. If I recall correctly, Ochoa (1987) looks at wage curves as based on input-output matrices from different times. He looks for pairs that intersect more than onec.

In looking at such a pair, however, many more wage curves are available. One can construct input-output matrices, with one process for each industry, where the processes are not all from one matrix but combine processes among industries from the different matrices. Han and Schefold (2006) take this approach.

But this is not all. One need not limit oneself with processes from pairs of wage curves. One should look at the full range of techniques, where the process for each industry might be from any input-output matrix in your database. Zambelli (2018), in following this approach, uses an algorithm that he and his colleagues cleverly constructed to select the wage curves on the frontier, thereby keeping the combinatorial explosion in this approach somewhat under control.

Ideally, one would like internationally consistent classifications of industries in make and use tables and Leontief input-output matrices that include joint production. If the latter is not available, which it usually not, one needs consistent approximations for single-production. Since make and use tables, and the resulting Leontief input-output tables are typically price data, one needs price indices for industries or commodities. At what level of aggregation do some industries only appear in some tables? Many more questions arise here that are probably beyond me.

3.0 'Perverse' Phenomena

What supposedly 'perverse' phenomena should one look for in techniques formed out of empirical input-output matrices? I suggest instances of the reswitching of techniques, capital reversing, the reverse substition of labor, and the recurrence of processes in individual industries would be of interest. Reswitching on the frontier is sufficient, but not necessary for the occurrence of positive real Wicksell effects. I, like many others, define capital reversing (also known as reverse capital deepening) as equivalent to positive real Wicksell effects. Zambelli (2018), on the other hand, defines capital reversing to arise with positive real or price Wicksell effects.

I tend, in pointing out the invalidity of marginalist economics, to de-emphasize any concern with the direction of price Wicksell effects. As I understand it, the direction of price Wicksell effects is dependent on the selection of the numeraire. Also, I am aware of Burmeister's championing of Champernowne's chain index for capital. On the other hand, Baldone (1984) suggest this defense of mainstream economics fails. Fratini (2010) has an example with a continuous variation of techniques along the wage frontier and in which negative price Wicksell effects swamp positive real Wicksell effects, which I guess is a propos here.

4.0 Wage Frontiers and Aggregate Production Functions

I have been talking about wage frontiers and wage curves above. One can construct the aggregate production 'function', given the analysis of the choice of technique. In this analysis, one takes net output as of a given physical composition. It is convenient to take net output as the numeraire. The composition of capital goods varies at switch points, and their prices vary between switch points. At one point, though, Zambelli considers variations in the composition of capital goods between switch points, as I understand it. I relegate an explanation of what he is doing here to an appendix.

5.0 Conclusion

Zambelli (2018) is impressive empirical work. The failure of so-called neoclassical theory in 60 percent of the cases examined, as I understand it results, from a concentration on price Wicksell effects, which would not disconcert, for example, Burmeister. I also have difficulties with how Zambelli relates the aggregate production function to a problem of minimizing the value of aggregate capital.

Appendix: The Construction of a Microeconomic Production Function

I illustrate the construction of a production funcition as the solution of a maximization problem. A more general presentation would start with netput vectors and assume convexity. I briefly glanced at the appendix to chapter VI in Pasinetti (1977) in writing this.

For concreteness, suppose the managers of a firm have given quantities, x1, x2, and x3, of three resources and know of four fixed-coefficient processes for producing a single commodity. The coefficicients of production for these four processes are:

(a.j)T = (a1, j, a2, j, a3, j), j = 1, 2, 3, 4.

Let qi, i = 1, 2, 3, 4, be the decision variables denoting how much output is produced with each process. Consider the linear following linear program (LP). Maximize output y:

y = q1 + q2 + q3 + q4

such that:

a1, 1 q1 + a1, 2 q2 + a1, 3 q3 + a1, 4 q4x1
a2, 1 q1 + a2, 2 q2 + a2, 3 q3 + a2, 4 q4x2
a3, 1 q1 + a3, 2 q2 + a3, 3 q3 + a3, 4 q4x3
qi ≥ 0, i = 1, 2, 3, 4 = 1, 2, 3, 4.

The constraints express the condition that no more of a resource (also known as a factor of production) can be used than is given. Every process must be operated at a non-negative level. Let f express the solution of this LP as a function of factors of production:

y = f(x1, x2, x3)

This is a discrete version of the production function for a given commodity. It has properties commonly assumed in marginalist economics. It exhibits constant returns to scale (CRS) and non-increasing marginal products. If one wanted to construct a production function differentiable everywhere, one could assume an uncountably infinite set of production processes.

I might as well write down the dual problem. It is to choose factor prices w1, w2, w3 to minimize:

w1 x1 + p2 x2 + p3 x3

such that:

a1, 1 w1 + a2, 1 w2 + a3, 1 w3 ≥ 1
a1, 2 w1 + a2, 2 w2 + a3, 2 w3 ≥ 1
a1, 3 w1 + a2, 3 w2 + a3, 3 w3 ≥ 1
a1, 4 w1 + a2, 4 w2 + a3, 4 w3 ≥ 1
w1 ≥ 0, w2 ≥ 0, w3 ≥ 0

For a solution of these two LPs, the values of their objective functions are equal. Factor prices are such that output is completely distributed among the owners of the resources whose services are used in producing the given commodity. If a constraint in the dual is met with inequality, the corresponding decision variable in the primal LP is set to zero. That process is not operated. If a constraint in the primal LP is met with an inequality, that resource is in excess supply and its price is zero. Even though you see no derivatives above, this is an exposition of an aspect of the theory of marginal productivity.

All the parameters and variables in the primal LP are in physical units (for example, bushels, tons, person-years). It does not make much sense to me in an aggregate production function, with output and arguments in price terms, to maximize the value of output for a given value of capital or to minimize the value of capital for a given value of output. Nevertheless, that is what Zambelli does in Section 5.3 of his paper. I suppose he wanted to present a comprehensive empirical exploration of aggregate neoclassical theory, taking its illogic as given.

  • Baldone, Salvatore. 1984. From surrogate to pseudo production functions. Cambridge Journal of Economics 8: 271-288.
  • Burmeister, E. 1980. Capital Theory and Dynamics. Cambridge: Cambridge University Press
  • Fratini, Saverio M. 2010. Reswitching and decreasing demand for capital. Metroeconomica 61 (4): 676-682.
  • Han, Zonghie and Bertram Schefold. 2006. An empirical investigation of paradoxes: reswitching and reverse capital deepening in capital theory. Cambridge Journal of Economics 30: 737-765.
  • Kersting, Götz and Bertram Schefold. 2020. Best techniques leave little room for substitution: a new critique of the production function. Centro Sraffa Working Paper n. 47.
  • Ochoa, E. M. 1987. Is reswitching empirically relevant? US wage-profit-rate frontiers, 1947-1972. Economic Forum 16: 45-67.
  • Pasinetti, Luigi L. 1977. Lectures on the Theory of Production New York: Columbia University Press.
  • Schefold Bertram. 2013. Approximate surrogate production functions. Cambridge Journal of Economics 37 (5): 1161-1184.
  • Schefold Bertram. 2016. Profits equal surplus value on average and the significance of this result for the Marxian theory of accumulation.. Cambridge Journal of Economics 40 (1): 165-199.
  • Zambelli, Stefano. 2018. The aggregate production function is NOT neoclassical. Cambridge Journal of Economics 42: 383-426.

Wednesday, January 13, 2021

Books To Make You More Muddled

I have not read all of these, and you might think I am being unfair with this post title. If you want critiques of post modernism, try the Amin and Eagleton referenced at the end of this post. Sokal, after the cited book, participated in interesting colloquia with those who were scholars of what he was attacking and mocking. If you want to see how little I know about this area, you can look at my posts on Gramsci, Foucault, Wittgenstein, or Zizek.

  • Gellner, Ernest. 1959. Words and Things: A Critical Account of Linguistic Philosophy and a Study in Ideology. London: Gollantz.
  • Gross, Paul R. and Norman Levitt. 1998. Higher Superstition: The Academic Left and its Quarrels with Science. Baltimore: John Hopkins Press.
  • Hicks, Stephen R. C. 2004. Explaining Postmodernism: Skepticism and Socialism from Rousseau to Foucault. New Berlin: Scholarly Publishing.
  • Pluckrose, Helen and James A. Lindsay. 2020. Cynical Theories: How Activist Scholarship Made Everything about Race, Gender, and Identity - and Why This Harms Everybody. Pitchstone Publishing.
  • Sokal, Alan and Jean Bricmont. 1998. Fashionable Nonsense: Postmodern Intellectuals Abuse of Science. New York: Picador USA.

Saturday, January 09, 2021

John Roemer's Reproducible Solution

Can I adapt Roemer's work, suitably taking into account later work by D'Agata and Zambelli, to found this approach to markup pricing? As a start, I here quote Roemer on a reproducible solution (RS), before he takes into account unequal rates of profits and a choice of technique. Given the role of endowments, is this a neoclassical approach, like Hahn's 1984 CJE paper? Even so, is it a valid justification for Sraffa's price equations? Notice there are no subscripts for time below.

"There are N capitalists; the νth one is endowed with a vector of produced commodity endowments ων ... Capitalist ν starts with capital ων, which he seeks to turn in more wealth at the highest rate of return. Thus the program of capitalist ν is
Facing prices p, to
choose xν0 to
max (p - (p A + L)) xν
s.t. (p A + L) xνp ων
(The constraint says that the inputs costs can be covered by current capital.) Let us call Aν(p) the set of solution vectors to this program." -- Roemer (1981: 18-19, I made changes for typesetting mathematics).

Roemer defines a RS:

"Definition 1.1: A price vector p is a reproducible solution for the economy {A, L; b; ω1, ..., ωN} if:
  • For all ν, there exists xν in Aν(p), such that (profit maximization)
  • x = Σ xν and xA x + (L x) b (reproducibility)
  • p b = 1 (subsistence wage)
  • A x + (L x) ≤ ω = Σ ων (feasibility)
We shall also refer to the entire set {p, x1, ..., xN} as a reproducible solution." -- Roemer (1981: 19-20, with for math).

A RS can only exist if the elements of the endowment vector are in certain proportions:

"Theorem 1.2: Let the model {A, L, b} be given with A productive and indecomposable, and the rate of exploitation e > 0. Let {p, x1, ..., xN} be a nontrivial RS. (i.e., Σ xν = x0). Then the vector of prices p is the E[qual] P[rofit] R[ate] vector p*. Furthermore, a RS exists if and only if omega is an element of C*, where C* is a particular convex cone in [the space of n-dimensional real vectors] containing the balanced growth path of {A, L, b}. (C* is specified precisely below.)" -- Roemer (1981: 20, with changes for math).

Even though endoments are taken as given in defining the firm's LP, endowments are endogenous in the sense that they must lie close to those on a balanced growth path. I like to have labor advanced and wages paid out of the surplus, instead of vice versa as above. The above does not allow for a choice of technique. Roemer has at least some of this in later chapters.

  • John E. Roemer. 1981. Analytical Foundations of Marxian Economic Theory. Cambridge University Press.

Thursday, January 07, 2021

23 February 1981: King Juan Carlos Becomes A Spanish National Hero

I only know about this at the level of a Wikipedia article. Or maybe a short newspaper article. Some of you doubtlessly know more.

Some Spanish military officers, pining for the certainty of a fascist authortarian state, assaulted the Congress of Deputies in 1981. They held the deputies hostage. Some showed real physical courage. The prime minister and deputy prime minister refused to sit down when ordered so, despite having guns pointed at them.

I'd like to conclude that, despite this failed coup attempt, Spain is a thriving democracy today. But I think political parties today are addressing problems more connected with austerity after 2008 than with nostalgia for Franco. I conclude with a couple references about violence in politics.

  • Hannah Arendt. 1969. On violence. In Crises of the Republic New York: Harcourt Brace Jovanovich.
  • Georges Sorel. 1950. Reflections on Violence (Trans. by T. E. Hulme) London: Collier-Macmillan.

Saturday, January 02, 2021

The Tractor-Corn Model: A Start

1.0 Introduction

In my ROBE article, I consider fluke switch points arising from perturbations of coefficients of production in the Samuelson-Gargenani model, but in the case with only circulating capital. An obvious generalization is to consider fixed capital. This generalization is simplified by restricting oneself to the case in which machines operate with constant efficiency. Steedman (2020) analyzes this case, and this post is a start on working through elements of the corn-tractor model he leaves as homework. I do not know how far I will go in rewriting my paper for this case.

2.0 Technology for a Technique

In the model, corn is produced by labor working working with a specified type of tractor. And that type of tractor is itself produced by labor working with that type of tractor.

Each type of tractor defines a technique, where a technique is specified by six parameters:

  • a: The number of tractors (of a given age) whose services are used for a year in producing a new tractor.
  • b: The person-years of labor needed to work with tractors (of a given age) to produce a new tractor.
  • n: The number of years a tractor lasts when used in producing new tractors.
  • α: The number of tractors (of a given age) whose services are used for a year in producing a bushel of corn.
  • β: The person-years of labor needed to work with tractors (of a given age) to produce corn.
  • ν: The number of years a tractor lasts when used in producing corn.

The notation is Steedman's, borrowed from J. R. Hicks. I see that if I keep this notation, I will have to drop my usual practice, in honor of Joan Robinson, of using lowercase Greek letters to refer to a technique.

Consider, for a technique, the (n + ν)-element row vector of labor coefficients a0, the (n + ν) x (n + ν) matrix A of input coefficients, and the (n + ν) x (n + ν) matrix B of output coefficients. This vector and these matrices have a block structure:

a0 =bb (uT)1,n - 2bββ (uT)1,ν - 2β

A = 001,n - 20001,ν - 20
a01,n - 20α01,ν - 20
0n - 2, 1a In - 2,n - 20n - 2, 10n - 2,10n - 2,ν - 20n - 2,1
001,n - 2a001,ν - 20
0ν - 2,10ν - 2,n - 20ν - 2,10ν - 2,1α Iν - 2,ν - 20ν - 2,1
001,n - 20001,ν - 2α

B = 001,n - 201(uT)1,ν - 21
1(uT)1,n - 21001,ν - 20
a01,n - 20001,ν - 20
0n - 2,1a In - 2,n - 20n - 2, 10n - 2, 10n - 2,ν - 20n - 2, 1
001,n - 20α01,ν - 20
0ν - 2,10ν - 2,n - 20ν - 2,10ν - 2,1α Iν - 2,ν - 20ν - 2,1

Obviously, HTML defeated me here. I is the identity matrix, and u is a column unit vector.

Each element of a0 and each column of A and B correspond to a process of production. The first n columns constitute the tractor sector, and the remaining ν columns are the corn sector. I assume constant returns to scale and that each process requires a year to complete. a0, j is the person-years of labor that enters the jth process per unit-level of operations. The jth column of A is the inputs consumed by the process, and the jth column of B is the outputs. The first row index is for corn. The first row of A is zero, since corn is not used as an input in any process. The second row index is for new tractors. The remaining row indices are for old tractors. Once a tractor is used in tbe production of tractors, it can no longer be used in producing corn. Likewise, a tractor used in the corn sector cannot be transferred to the tractor sector.

3.0 An Annuity

Consider an annuity cn(r) bought for a dollar at the start of a year. This annuity pays out the sum cn(r) at the end of the first year, at the end of the second year, and so on through the end of the nth year. This arrangement implicitly specifies an interest rate r which equates the cost and the present value of the payments:

1 = cn(r)/(1 + r) + cn(r)/[(1 + r)2] + ... + cn(r)/[(1 + r)n]

A bit of algebra reveals that the payments for the annuity are given by the following formula:

cn(r) = r (1 + r)n/[(1 + r)n - 1]

The limit as the interest rate approaches zero can be found by L'Hôpital's rule. It is:

cn(0) = 1/n

I need these formulas below.

4.0 The Quantity System

Now I want to consider a steady state in which the economy grows at a uniform rate of 100 g percent. Let the column vector q specify the level of operation of each process. I postulate that q has the following form:

qT = [q1, q1/(1 + g), ..., q1/(1 + g)n - 1, q2, q2/(1 + g), ..., q2/(1 + g)ν - 1]

where q1 and q2 are variables to be determined. Let e1 be the first column of the identity matrix. Consumption in a steady-state is c(g) e1, where:

c(g) e1 = [B - (1 + g) A] q

Expanding the first element of the column vectors on both sides, one gets:

c(g) = (1 + g) q2/cν(g)

The second element yields:

0 = {[(1 + g)/cn(g)] - (1 + g) a} q1 - (1 + g) α q2


0 = [1 - a cn(g)] q1 - α cn(g) q2

Given g, the above is a linear equation in q1 and q2. New tractors do not enter into consumption. Quantity flows are specified such that one person-year of labor is employed:

a0 q = 1


(1 + g) b q1/cn(g) + (1 + g) β q2/cν(g) = 1


b cν(g) q1 + β cn(g) q2 = cn(g) cν(g)/(1 + g)

A linear system of two equations in two unknowns, given the rate of growth, has now been derived.

The system is easily solved:

q1 = α cn(g) cν(g) /{[β + αbcν(g) - aβcn(g)](1 + g)}

q2 = [1 - acn(g)] cν(g)/{[β + αbcν(g) - aβcn(g)](1 + g)}

Consumption per worker (in units of bushels corn per person-year) is:

c(g) = [1 - acn(g)]/[β + αbcν(g) - aβcn(g)]

In a comparison of steady states, consumption per worker is higher if the rate of growth is lower. The dependence of the denominator on the rate of growth vanishes under the special case in which:

a cn(g)/b = α cν(g)/β

Somehow, the above says that the organic composition of capital does not vary between the tractor and the corn sectors. The tradeoff, however, between consumption per worker and the rate of growth is still not linear. The maximum rate of growth, G, is the smallest non-negative real solution to:

0 = 1 - a cn(G)

Consumption per worker in a stationary state is:

c(0) = [n - a]ν/[nνβ + αbn - aβν]

One might use the above to discuss the capital-intensity of a technique. If the technique with one type of tractor is more capital-intensive than the technique with another type, one would expect c(0) to be higher with the first type.

5.0 The Price System

I now consider prices. Let p be a row vector of prices, w the wage, and r the rate of profits. In matrix form, the price equations are:

p A (1 + r) + w a0 = p B

A bushel corn is the numeraire:

p e1 = 1

The above consists of a system of (n + ν + 1) equations for (n + ν + 2) variables. The system has one degree of freedom. Labor is advanced, and wages are paid out of the surplus at the end of the year. A tractor of each age and history has a seperate price.

I now rewrite the price equations for the first time. The price of a bushel cotn is unity, and p represents thd price of a new machine. pm,j is the price of a j-year old tractor in the tractor sector. pc,j is the price of a j-year old tractor in the corn sector. The n equations for the tractor sector are:

p a (1 + r) + w b = p + pm,1 a

pm,1 a (1 + r) + w b = p + pm,2 a


pm,n - 1 a (1 + r) + w b = p

The ν equations for the corn sector are:

p α (1 + r) + w β = 1 + pc,1 α

pc,1 α (1 + r) + w β = 1 + pc,2 a


pc,ν - 1 α (1 + r) + w β = 1

Consider the equations for the machine sector. Multiply the first equation by (1 + r)n - 1, the second equation by (1 + r)n - 2, and so on, until the last equation is multiplied by (1 + r)0.Sum these equations:

p a (1 + r)n + w b [1 + (1 + r) + ... + (1 + r)n - 1] = p [1 + (1 + r) + ... + (1 + r)n - 1]

The prices for old tractors appear on both sides of successive equations with the same coefficient and drop out. A similiar procedure for the corn sector yields:

p α (1 + r)ν + w β [1 + (1 + r) + ... + (1 + r)ν - 1] = [1 + (1 + r) + ... + (1 + r)ν - 1]

So far, this procedure works if tractors do not have constant efficiency. The next step requires that, though. The price equations become:

p a cn(r) + w b = p

p α cν(r) + w β = 1

Fpr both the quantity and the price system, a set of (n + ν) equations is reduced to two equations in which (quantities or prices) of old tractors do not enter. The charge for a tractor is that of an annuity that pays out for each year of the tractor's life.

The price system is easily solved. The price of a new tractor is:

p = b/[β + αbcν(r) - aβcn(r)]

Under the special case of equal organic compositions of capital, the ratio of a price of a new tractor to a bushel corn is the ratio of direct labor inputs. Presumably, prices are also proportional to labor values in this special case. The wage curve is:

w = [1 - acn(r)]/[β + αbcν(r) - aβcn(r)]

I have already discussed the wage curve under the guise of the tradeoff between consumption per worker and the rate of growth. The maximum rate of profits R is identical to the maximum rate of growth G.

6.0 Conclusion

Steedman (2020) avoids writing about almost all of the above or leaves it as an exercise for the reader. Basically, I have derived Steedman's first five numbered equations. Some of this is in Chapter 10 of Sraffa (1960).

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