Precursors of Piero Sraffa

I want to consider contributions to economics after 1870 that reconsidered classical or Marxist economics, used input-output models and linear algebra, or bear a family resemblance to at least some points in Sraffa's 1960 book.

• Vladimir K. Dmitriev. Used input-output analysis in an interpretation of Ricardo's theory of value.
• Ladislaus Bortkiewicz. Had a simple three-good model, with one basic good, input-output model of prices of production. Sraffa and others argued against aspects of his interpretation of Marx's theory of value.
• Georg von Charasoff. Apparently, around 1909 and 1910, he came up with the concept of "original capital". In a infinite series, much like Sraffa's reduction to dated labor, the capital goods needed more and more indirectly in producing some given net output converge to Sraffa's standard commodity.
• Father Maurice Potron. A fairly conservative Jesuit priest and mathematician, writing in French. I know of him from this collection.
• Wassily Leontief. His empirical work extends from the 1920s. I do not know that those building on his work often cite Sraffa.
• Walter Isard. Applied Leontief input-output analysis to regional or spatial economics in 1951.
• John von Neumann. I am thinking of the 1945-1946 translation of his A Model of General Economic Equilibrium. Kurz and Salvadori read this as a response to Robert Remak.
• Jacob T. Schwartz. A mathematician whose 1961 Lectures on the Mathematical Method in Analytical Economics criticizes neoclassical and Austrian economics. Personally, I found his work not as rigorous as Sraffa's.

I do not know much about many of these authors, but other economists in the post Sraffian tradition have written about them.

Why Does The Labor Theory Of Value Work Empirically As A Theory Of Prices?

Anwar Shaikh On The Transformation Problem

Lots of empirical work shows that prices tend to be proportion to the labor embodied in commodities. My references in this article document this claim. Furthermore, empirical wage-rate of profits curves tend to be close to straight lines. This is not what, say, Sraffa' mathematical economics would lead me to expect. What explains these surprising empirical findings?

Almost 34 minutes in, in the above video, Shaikh makes the above point about the contrast between theory and empirical findings. He concludes with speculation, including with comments on Bertram Schefold's work with input-output matrices formed out of random matrices.

I offer some speculations myself in this post. I do not have much theory to back these suggestions up.

The Leontief matrices obtained from National Income and Product Accounts (NIPAs) are still highly aggregated. The empirical results on the LTV are obtained with matrices that have on the order of, say, 100 industries. One of these industries, if disaggregated, might contain commodities that are produced with a high Organic Composition of Capital (OCC) and a low OCC. Their prices of production would deviate more from labor values than an average combining them both. The extremes would be cancelled out in forming an average.

In my examples of pattern analysis, I also suggest that Sraffa effects could be difficult to see, in that they arise in a transition from one very long run position to another. But I concoct those examples to make a point about possibilities. I do not want to insist on any empirical point here.

Technical progress, despite how I usually model it, is endogenous. If in process of production adopted in some industry, some input is noticeably more expensive than others, managers of firms will seek out and research processes in which that input is reduced or some other cheaper input is substituted for it. Perhaps after a couple of centuries of rapid technical change under these incentives, empirical Leontief input-output matrices will have the properties Schefold highlights for random matrices. I suppose one could confirm this by showing wage-rate of profits curves are closer to affine functions for more highly developed economies. I have done some empirical work along these lines.

Aside: Here is another YouTube video with Anwar Shaikh. He sounds a lot like he accepts Milgate and Eatwell's critique of "imperfectionism". Actually existing capitalism is to be analyzed by a theory that accepts empirical reality, not by deviations from a neoclassical utopia that could never exist in any conceivable world.

Only The Super-Rich Can Save Us!

Neoliberals are hostile to labor unions and every other institution that would allow the vast majority of the population to have some effect on how we are ruled. And they have been so successful that only the super-rich can save us, as the title of a Ralph Nader novel a few years back had it. A couple of recent examples of journalism are about movements of the super-rich:

• Sheelah Kolhatkar writes, in the New Yorker, about Patriotic Millionaires.
• Theodore Schleifer writes, in Vox, about Resource Generation.

I suspect most of the super-rich, however, are vicious, reactionary fools. Apparently, Benjamin Page, Jason Seawright, and Matthew Lacombe provide evidence in their recent book, Billionaires and Stealth Politics. I've read and commented on their previous working paper.

In the United States and elsewhere, we had a progressive movement reacting to the terrible effects and excesses of the "roaring twenties" of a century ago. Of course, there was a fascist movement, too, that resulted in global war.

In the United States, prominent celebrities such as Henry Ford and Charles Lindbergh supported fascism. The super-rich did not step back. The business plot was an attempt by millionaires to stage a coup against Franklin Delano Roosevelt. They tried to get Major General Smedly Butler to act as a figurehead. I know of him for saying, war is a racket. I do not know if this falls in the politics of the super-rich, but I only recently learned about the Christian Front, a fascist organization inspired by the radio demagogue Father Coughlin. In 1940, their office in New York City was raided by the FBI for trying to overthrow the government. Seventeen members was arrested, but their prosecution was unsuccessful. (Caveat: I have not read the books and literature linked to in this paragraph.)

I think we need a better material basis than the well wishes and work of the super-rich to bring about hopeful change.

Towards the Derivation of the Cambridge Equation with Expanded Reproduction and Markup Pricing

I have a new working paper.

Abstract: Does the Cambridge equation, in which the rate of profits in a steady state is equal to the quotient of the rate of growth and the savings rate out of profits, hold in an economy with widespread non-competitive markets? This article presents a multiple-good model of markup pricing in an attempt to answer this question. A balance equation is derived. Given competitive conditions, this model can be used to derive the Cambridge equation. The Cambridge equation also holds in a special case of markup pricing, with one capital good and many consumption goods being produced. No definite conclusions are reached in the general case.

The Factor Price Frontier In The Space Of Factor Rental Prices

Figure 1: Real Factor Price Frontier

1.0 Introduction

Carlo Milana has proposed a new way of visualizing the choice of technique, including in the case of reswitching. This way of describing what he has done is not neccessarily how he thinks of it. In this post, I describe his approach with a reswitching example, in a model of the production of commodities by means of commodities.

2.0 Technology

Table 1 shows the coefficients of production for this example. Coefficients of production specify inputs per unit output. Each process takes a year to complete. Inputs are totally used up in the production of the outputs. (This example is taken from one of my papers.)

Table 1: Coefficients of Production for The Technology

Input	Steel Industry		Corn Industry
	Alpha	Beta
Labor	1	275/464	1 Person-Yr
Steel	1/10	113/232	2 Tons
Corn	1/40	0	(2/5) Bushels

Two techniques of production arise in this example. The Alpha technique consists of the Alpha process for producing steel and the corn-producing process. Both steel and corn are basic commodities, in the sense of Sraffa, for the Alpha technique. The Beta technique consists of the Beta process for producing steel and the corn-producing process. Only steel is a Sraffa-basic commodity for the Beta process. Suppose, however, corn is the only consumption good in this example. Then in the Beta technique, as with the Alpha technique, both steel and corn will be (re)produced for both techniques.

3.0 Prices of Production

If the Alpha technique is in use in a long-period position, prices satisfy the following two equations:

((1/10) p_α,1 + (1/40) p_α,2)(1 + r) + w_α = p_α,1

(2 p_α,1 + (2/5) p_α,2)(1 + r) + w_α = p_α,2

Prices are spot prices. The services of produced inputs are paid for at the start of the year, while wages are paid out of the surplus at the end of the year.

The corresponding equations for prices for the Beta technique are:

((113/232) p_β,1)(1 + r) + (275/464) w_β = p_β,1

(2 p_β,1 + (2/5) p_β,2)(1 + r) + w_β = p_β,2

At this point, I take a bushel corn as the numeraire. One can solve the Alpha system of equations, for example, to find (w_α/p_α,1) as a function of the interest rate. This is the wage curve for the Alpha technique and is shown below. The wage curve for the Beta technique is also graphed. The outer envelope of these curves, called the wage frontier, shows which technique is cost-minimizing at any given interest rate. Both techniques are cost-minimizing at the switch points, which arise for interest rates of 20 percent and 80 percent. Between the switch points, the Alpha technique is cost-minimizing. Outside the switch points, the Beta technique is cost minimizing.

Figure 2: Wage Curves and the Wage Frontier

4.0 Rental Prices for Factor Inputs

In marginalism, the choice of technique is often analyzed in terms of rental prices for factors of production. One can think of the example in terms of three factors: labor, steel, and corn. Steel and corn are capital goods.

Since a choice of production processes arises in the steel industry, I here take steel as numeraire. The rental price, also known as the factor price, for labor is the real wage:

w_α,L = w_α/p_α,1

The rental or factor price for steel is the cost of a the services of a ton of steel when paid at the end of the year:

w_α,Steel = p_α,1(1 + r)/p_α,1

Likewise, the rental or factor price of corn is:

w_α,Corn = p_α,2(1 + r)/p_α,1

Using these definitions, the condition that, when in use, no extra profits are made and no extra costs are in incurred in producing steel with the Alpha process yields the following equation:

(1/10) w_α,Steel + (1/40) w_α,Corn + w_α,L = 1

Notice that this is a linear equation in three variables. It is illustrated by the blue plane in Figure 1. The factor prices for the Beta process yield another linear equation:

((113/232) w_β,Steel + (275/464) w_α,L = 1

The plane for Beta is shown in red in Figure 1.

At a switch point, both the Alpha and the Beta processes are eligible for adoption by cost-minimizing managers of firms. Accordingly, switch points must lie on the intersection of the two planes described above. The intersection, although difficult to see, is shown in black in the figure.

In discussing rental or factor prices, I have yet to take into account that corn must also be produced. If one substitutes, on the right-hand side in the three equations defining rental prices, the solution of the Alpha system of equations in Section 3, one obtains factor prices as a parametric function of the interest rate. This is the real factor price curve for the Alpha technique and is shown in blue above. The real factor price curve for the Beta technique, in red, is easier to see. (Each real factor price curve lies within the plane of the same color.) For each curve, when it lies on the real factor price frontier is indicated. And the switch points do indeed lie on the intersections of the real factor price curves.

5.0 Conclusion

Does the real factor price frontier in Figure 1 provide a mechanism for analyzing the choice of technique? Is the factor price curve for the cost-minimizing technique always furtherest from the origin?

The wage frontier, where applicable, can be drawn in a two-dimensional diagram for examples with any number produced of produced commodities. If n commodities are produced, Milana's diagram illustrates, roughly, the intersections of hyperplanes of dimension (n - 1). And those intersections will be themselves hyperplanes of dimension (n - 2). Switch points, if any, lie in those intersections. The factor price curves will still be one-dimensional curves, as I understand it, in the appropriate hyperplanes.

Obviously, this cannot be visualized in higher dimensions. Nevertheless, the mathematics still works out. Different valid approaches to finding the cost-minimizing technique in a long-period position, given an exogenous specification of the distribution of income, in some sense, will all yield the same answer. That is the case for the reswitching example presented here.

Some People Who Have Shaped Economics

"The University [of Chicago] is the best investment I ever made in my life." -- John D. Rockefeller

Consider the following people and selected activities:

• Lewis Brown founded the American Enterprise Institute, in 1938.
• Jasper Crane cofounded the Foundation for Economic Education, in 1946.
• Leonard Read cofounded the Foundation for Economic Education, in 1946.
• Harold Luhnow, even before 1947, directed spending for the Volker Fund.
• Sir Antony Fisher funded the Institute for Economic Affairs, around 1956.
• Lord Ralph Harris, first general director of the Institute for Economic Affairs.
• Arthur Seldon, first editorial director of the Institute for Economic Affairs.
• F. A. Harper founded the Institute for Humane Studies, in 1961.
• Charles Koch funded the development of the Virginia school, notably including James Buchanan's work.
• Edwin Feuler, founded the Heritage Foundation, in 1973.
• Edward H. Crane founded the Cato Institute, in 1977.
• Eamonn Butler cofounded the Adam Smith Institute, in 1978.
• Madsen Pirie cofounded the Adam Smith Institute, in 1978.

I've written on the influence of fundings sources on the development of economics before. A developing body of scholarly literature explores the impact of the above list of people. The above list is not complete. For example, John Blundell seems to be an important fellow in the world hinted at above.

I think funding sources have been concentrated on the right. I suppose you can try to make a list not so concentrated on the right. George Soros and the Institute for New Economic Thinking, John Reed of Citicorp and Santa Fe Institute, John Podesta and theCenter for American Progress (CAP) would all be in the list. I do not know where funding for the Economic Policy Institute comes from. It seems to me a distinction exists between investigating ideas and trying to publicize conclusions you already believe.

2019 Nobel Prize Celebrating The Triumph Of Institutionalism?

Elizabeth Warren Echoing A View Institutionalists Understand

This year, the "Nobel prize" in economics went to Abhijit Banerjee, Esther Duflo, and Michael Kremer. They champion empirical economics over theory. Previously, institutionalist economics was described as 'measurement without theory' (Koopmans 1947). Does institutionalist economics parallel the supposed mainstream empirical turn?

Although institutionalists, as far as I know, did not have the resources to create randomized control trials (RCTs), they did collect and analyze statistical data. I think especially of Wesley Clair Mitchell and the National Bureau of Economic Research (NBER).

Institutionalists was not atheoretical, I think. They developed qualitative analytical concepts. I think of C. E. Ayres extension, for example, of the Veblenian dichotomy. Sometime, I intend to read John R. Commons' 1924 book to see how he breaks up a transaction. John Kenneth Galbraith's concept of the technostructure and Alfred Eichner's idea of the megacorp are other examples here. Institutionalists contributed to the development of Industrial Organization. John Maurice Clark had, at least, a verbal description of the business cycle that combined the multiplier and the accelerator.

Institutionalist economics is not a strictly American school of thought. I include Geoffrey M. Hodgson and Gunnar Myrdal as institutionalists. I suppose I should read the Journal of Economic Issues or the Journal of Institutional Economics more frequently. The Association for Evolutionary Economics (AFEE) puts out the JEI.

References

• John R. Commons. 1924. Legal Foundations of Capitalism Macmillan.
• John S. Gambs. 1946. Beyond Supply and Demand: A Reappraisal of Institutional Economics. Columbia University Press.
• Tjalling C. Koopmans. 1947. Measurement without theory. Review of Economics and Statistics 29(3): 161-172.

A Fake Switch Point in an Example With Circulating Capital

Figure 1: A Switch Point and a Fake Switch Point on Wage Curves

1.0 Introduction

In the analysis of the choice of technique, I typically consider examples of technology with a finite number of techniques. For each technique, I find the wage as a function of the rate of profits. The outer envelope of these curves shows the cost-minimizing technique at each rate of profits (or each level of the wage). Points on more than one wage curve are switch points.

This approach is valid when, for example, all techniques produce the same set of commodities, and each commodity is basic, in the sense of Sraffa. That is, all commodities enter directly or indirectly into the production of all commodities.

But another requirement is that prices of all commodities in common between two techniques be identical at a switch point. Points of intersection on wage curves without this property of identical prices are known as fake switch points. I have previously considered fake switch points in (an extension of) an example from Christian Bidard. In this post, I present an example of a fake switch point in an example with single production (or circulating capital) only. It is critical to this example that a non-basic commodity is the numeraire and that the techniques vary in the process used to produce a non-basic commodity.

The necessity to consider prices in the analysis of the choice of technique is, as I understand it, a critical point from Milana. I think he extends this point, though, to examples in which it cannot be used to criticize Sraffians.

2.0 Technology

Table 1 shows the coefficients of production for this example. Coefficients of production specify inputs per unit output. Each process takes a year to complete. Inputs are totally used up in the production of the outputs.

Table 1: Coefficients of Production for The Technology

Input	Corn Industry	Silk Industry
		Alpha	Beta
Labor	1	1	2 Person-Yrs
Corn	1/5	3	(38/15) Bushels
Silk	0	0	0 Square-Yds

The first produced commodity, corn, enters directly into the production of both commodities. It is a basic commodity, in the sense of Sraffa. Silk is a non-basic commodity. It does not enter, either directly or indirectly, into the production of corn.

3.0 Price Equations

I take a square yard of silk as the numeraire. The same rate of profits is assumed to be made in both industries when prices of production prevail. Labor is advanced, and wages are paid out of the net product at the end of the year.

3.1 The Alpha Technique

The following two equations specify prices of production for the Alpha technique:

(1/5) p_{1, α} (1 + r) + w_α = p_{1, α}

3 p_{1, α} (1 + r) + w_α = 1

The variables are:

• r: The rate of profits.
• w_α: The wage, for the Alpha technique.
• p_{1, α}: The price of corn, for the Alpha technique.

The solution, in terms of the rate of profits, is:

w_α(r) = (4 - r)/(19 + 14 r)

p_{1, α}(r) = 5/(19 + 14 r)

3.2 The Beta Technique

The price equations for the Beta technique are:

(1/5) p_{1, β} (1 + r) + w_β = p_{1, β}

(38/15) p_{1, β} (1 + r) + 2 w_β = 1

The solution is:

w_β(r) = 3(4 - r)/[2( 31 + 16 r)]

p_{1, β}(r) = 15/[2( 31 + 16 r)]

4.0 Switch Points

Suppose, at the given rate of profits, the Alpha technique is in use and prices of production for the Alpha technique prevail. Figure 2 shows the cost of producing silk, for each process, at these prices. The advances, at the beginning of the year, for produced inputs are costed up at the going rate of profits. The cost of producing silk with the process in the Alpha technique, under these assumptions, is unity for any feasible rate of profits. Extra costs are not incurred in the Alpha technique. Neither are supernormal profits available.

Figure 2: Cost of Producing Silk at Alpha Prices

But supernormal profits are available for the silk-producing process in the Beta technique if the rate of profits is feasible and exceeds the rate of profits at the switch point. The Beta technique is cost-minimizing here, while the Alpha technique is only cost-minimizing at lower rates of profits. The same conclusion about when each technique is cost-minimizing would be drawn if one started with prices of production for the Beta technique.

The switch point occurs at a rate of profits of 50 percent. The wage is (7/52) square yards per person-years, and the price of corn is (5/26) square yards per bushel at the switch point. Prices of production are the same, at the switch point, whichever technique is used.

5.0 A Fake Switch Point

Figure 1, at the top of this post, graphs the wage curves for the two techniques. Consider rates of profits that equate wages:

w_α(r*) = w_β(r*)

The wage curves have two intersections. One is at the switch points, at a rate of profits of 50%. At the maximum rate of profits of 400 percent, the wage is zero. In the Alpha system, the price of corn is (1/15) square yards per bushel, while it is (3/38) square yards per bushel in the Beta system. Since, prices of production vary among techniques at the maximum rate of profits, it is not a switch point. Rather, it is a fake switch point.

6.0 Conclusions

I would like to find another example of a fake switch point in a circulating capital example with a choice of processes for producing a non-basic commodity. I want a fake switch point not at an extreme, with a wage of zero. The example in Stamatis (2001) seems not to work; maybe there is a misprint in the coefficients of production. Both techniques, however, have the structure of Sraffa's "beans".

References

• Carlo Milana. 27 Nov. 2019. Solving the Reswitching Paradox in the Sraffian Theory of Capital
• Georg Stamatis. 2001. Why the comparison and ordering of techniques is impossible. Political Economy 9: 5-44.

The Interest Rate: Prime, Overnight, Or The Rate On T-Bills

As far as I am concerned, cost-push inflation is a manifestation of class conflict between workers and owners. In the late 1970s, Paul Volker and Ronald Reagan took the side of the owners. I am willing to accept that Volker genuinely believed in Milton Friedman's incorrect quantity theory of money. And, since then, workers have been getting a smaller share in increased productivity. Some obituaries of Paul Volker exhibit an understanding of what he did.

But I want to talk about my recollection of how interest rates have been covered in the press from that time. Of course, at any given time, there are a whole range and time structures of interest rates. When Volker drove the interest rate above 20 percent, the focus in news coverage was, as I recall it, on the prime rate, that is, the best interest rate commercial borrowers, such as large corporations, can obtain. My perception is that now, when movements in interest rates are reported on in the press, the emphasis is more likely to be on one of two rates. One is the overnight rate, that banks charge each other overnight. One can hear about the repo market, I guess, in this context. The other much-discussed rate is the rate on short term treasury bills (T bills).

Is my perception accurate? When did this change occur, if so? Is it actually an example of society learning? After all, the Federal Reserve has much more direct control over the latter interest rates and only indirect and tenuous control over the prime rate. Has Volker's demonstration that the quantity theory is wrong been generally taken on board?

The Cambridge Equation, Expanded Reproduction, and Markup Pricing: An Example

1.0 Introduction

I have sometimes set out Marx's model of expanded reproduction, only with prices of production instead of labor values. I assume two goods, a capital good and a consumption good, are produced with constant technology. If one assumes workers spend all their wages and capitalists save a constant proportion of profits, one can derive the Cambridge equation in this model.

The Cambridge equation shows that, along a steady state growth path, the economy-wide rate of profits is determined by the ratio of the rate of growth and the saving rate out of profits. Maybe one should not use causal language here. The Cambridge equation is a necessary, consistency condition for smooth reproduction in a capitalist economy.

This post derives the Cambridge equation with markup pricing, in a highly aggregated model of expanded reproduction. I am curious how far this result generalizes. I am thinking of a model in which, say, n capital goods are produced in Department I and m consumer goods are produced in Department II. At this point, I am not thinking of generalizations in which workers save and therefore own some of the capital stock. Nor am I worrying about fixed capital, depreciation, and technical change.

Table 1: Definition of Variables

Variable	Definition
a01	The person-years of labor hired per unit output (e.g., ton steel) in the first sector.
a02	The person-years of labor hired per unit output (e.g., bushel corn) in the second sector.
a11	The capital goods (measured in tons) used up per unit output in the first (steel-producing) sector.
a12	The capital goods (measured in tons) used up per unit output in the second (corn-producing) sector.
p1	The price of a unit output in the first sector.
p2	The price of a unit output in the second sector.
s1	Relative markup in producing steel.
s2	Relative markup in producing corn.
r̂	The scale factor for the rate of profits.
r	The rate of profits.
σ	The savings rate out of profits.
w	The wage, that is, the price of hiring a person-year.
c	Consumption per worker, in units of bushels per person-year.
X1	The number of units (ton steel) produced in the first sector.
X2	The number of units produced (bushels corn) in the second sector.
g	The rate of growth.

2.0 The Model

Certain quantity equations follow from the assumptions. No produced capital goods remain each year after subtracting those used to reproduce the capital goods used up in throughout the economy and those needed to support the given rate of growth:

0 = X₁ - (1 + g)(a₁₁ X₁ + a₁₂ X₂)

Consumption per person year is the output of the second department:

c = X₂

The model economy is scaled such that one person-year is employed:

a₀₁ X₁ + a₀₂ X₂ = 1

I have the usual price equations, with labor advanced:

p₁ a₁₁ (1 + r̂ s₁) + a₀₁ w = p₁

p₁ a₁₂ (1 + r̂ s₂) + a₀₂ w = p₂

The consumption good is the numeraire:

p₂ = 1

As with Marx in volume 2 of Capital, industries are here grouped into two great departments (Table 1). Means of production (also known as capital goods) are produced in Department I, and means of consumption (or consumer goods) are produced in Department II.

Table 2: Value of Outputs by Department and Distribution

Department	Capital	Wages	Profits
I. Capital Goods	a11 X1 p1	a01 X1 w	a11 X1 p1 s2 r̂
II. Consumption Commodities	a12 X2 p1	a02 X2 w	a12 X2 p1 s2 r̂

The overall, economy-wide rate of profits is defined in terms of profits and capital advances, aggregated over both departments:

r = (a₁₁ X₁ p₁ s₂ r̂ + a₁₂ X₂ p₁ s₂ r̂)/(a₁₁ X₁ p₁ + a₁₂ X₂ p₁)

The economy experiences expanded reproduction when it consistently expands each year. In this case, the demand for capital goods from the second department includes the savings of the capitalists receiving profits from that department. Likewise, the demand for consumption goods from the first department excludes the savings of the capitalists in that department. Observing these qualifications, it is easy to mathematically express the condition that the demand for capital goods from the second department match the demand for consumption goods from the first department:

a₀₁ X₁ w + (1 - σ) a₁₁ X₁ p₁ s₂ r̂ = a₁₂ X₂ p₁ + σ a₁₂ X₂ p₁ s₂ r̂

3.0 Some Aspects of The Model Solution

Quantity variables (c, X₁, and X₂) can be found as a function of the rate of growth. Price variables (w, p₁, and p₂) can be found as a function of the scale factor for the rate of profits. These solutions allow one to use the balance equation to find a relation between the scale factor for the rate of profits:

r̂ = (g/σ){1/[s₂ - (1 - g)(s₂ - s₁)a₁₁]}

One can use the above relationship and the solution quantities and prices to find the economy-wide rate of profits:

r = g/σ

Along a path in which the economy steadily expands, the rate of profits must be equal to the quotient of rate of growth and the savings rate out of profits. The rate of profits is dependent on investment and savings decisions, out of the control of the workers. (In a two-class economy in which the workers save at a smaller rate than the capitalists, the Cambridge equation remains valid, with the savings rate in the denominator being that of the capitalists.) It is independent of the technical conditions of the chosen technique, and marginal productivity has nothing to do with it.

4.0 Conclusions

I know that this model can be generalized to hold when any number of consumer goods are produced. I have not yet been able to show the Cambridge equation holds when any number of capital goods are produced.

Elsewhere

• David Graeber's review of Robert Skidelsky's Money and Government: The past and Future of Government.
• A TED talk, by Nick Hanauer, on how complexity economics is replacing "neoliberal" economics. He is especially interested in reciprocity.
• A 2014 interview by Bill Moyers, of Paul Krugman, on Piketty's book.
• Mariana Mazzucato, with a talk on the value of everything. She also has a 2013 TED talk.
• Heinz Kurz on the Cambridge capital controversy.
• Bertram Schefold on the CCC and his recent research.
• John Eatwell on Joan Robinson, including on the "disgrace" of mainstream economists not taking on board the results of the CCC.

Update (6 Dec. 2019): Added link for John Eatwell.

Literature Distinguishing Large Corporations And Finance From Competitive Firms

A considerable body of literature has been published, during the last century, arguing that a movement away from competitive markets must be recognized in trying to describe and understanding contemporary capitalism. The literature I am thinking of emphasizes big business, corporations, and finance. Here are some selections, not all of which I have read:

• Rudolf Hilferding (1910). Finance Capital: A study of the latest phase of capitalist development.
• Adolfe A. Berle and Means (1932). The Modern Corporation and Private Property.
• Bruno Rizzi (1939). The Bureaucratization of the World
• James Burnham (1941). The Managerial Revolution: What is happening in the world.
• Joseph Schumpeter (1950). Capitalism, Socialism, and Democracy
• Josef Steindl (1952). Maturity and Stagnation in American Capitalism.
• Paul A. Baran and Paul M. Sweezy (1966). Monopoly Capital: An essay on the American economic and social order.
• Robin Marris (1967). The Economic Theory of Managerial Capitalism.
• John Kenneth Galbraith (1967). The New Industrial State.
• Michal Kalecki (1971). Selected Essays on the Dynamics of the Capitalist Economy.
• Jonathan Nizan and Shimshon Bichler (2009). Capital as Power: A study of order and creorder.

I find that I have provided a similar list before. If I wanted to include journal articles, I would say something about Paulo Sylos-Labini. There was a nearby literature arguing the convergence of different systems across, say, the first and second worlds. Another related literature develops theories of imperialism, especially in the context of north-south relations.

As I understand it, many of the above writers were influenced by Marx. But I think even those who accepted the labor theory of value for competitive conditions, argued that the developments they were writing about implied that it no longer applies. For instance, Baran and Sweezy replaced surplus value by the (non-quantitative?) concept of the economic surplus. I was surprised to find myself arguing against this conclusion in recent work.

The Rate of Profits is Not the Scale Factor

Figure 1: Rate of Profits Unequal to Scale Factor for Rate of Profits

This post continues the example in the previous post. I modify the prices equations so that the rate of profits in producing corn is (s₁ r̂), and the rate of profits in producing ale is (s₂ r̂). The solution to the price equations are:

p_corn = 16 [16 + (s₁ - s₂) r̂]/[204 + (3 s₁ + 9 s₂) r̂]

p_ale = 32 [10 - (s₁ - 3 s₂) r̂]/[204 + (3 s₁ + 9 s₂) r̂]

w = 4 [51 - (9 s₁ + 5 s₂) r̂ - s₁ s₂ r̂²]/[204 + (3 s₁ + 9 s₂) r̂]

where r̂ is what I have been calling the scale factor for the rate of profits.

I want to show that, only in exceptional cases can the the markups s₁ and s₂ be rescaled such that the scale factor is equal to the economy-wide rate of profits, whatever the distribution of income. For concreteness, assume that the rate of profits is four times as high in the corn industry, as compared to the ale industry. That is, introduce a new parameter α such that:

s₁ = α

s₂ = (1/4) α

Suppose the wage is unity and workers receive the entire standard commodity. Then both the scale factor for the rate of profits and the rate of profits are identically zero percent.

Next, consider the other extreme case, where the wage is zero. The maximum rate of profits is 300 percent. For the scale factor to also be three, the numerator for the wage, in the third equation above must be zero, when a rate of profits of 300 percent and the above scale factors are substituted in. One thereby obtains a quadratic equation:

3 α² + 41 α - 68 = 0

The positive solution is:

α = (1/6)( -41 + 2,497^1/2)

For any scale factor for the rate of profits, one can find the wage with these markups. For any wage, one can find the economy-wide rate of profits:

r = 3 (1 - w)

The simplicity of the above equation results from taking the standard commodity as the numeraire. The graph at the top of this post shows the difference between the rate of profits and the scale factor, as a function of the proportion of the standard commodity paid out in wages. The rate of profits and the scale factor are equal only at the two extremes. I guess that in a model with more commodities, the difference does not come out looking like a quadratic function.

This counterexample demonstrates that, in general, one cannot rescale markups such that the scale factor for the rate of profits is the economy-wide rate of profits, however distribution stands. Part of Marx's point in Capital is that observers who focus on surface phenomena will not perceive underlying value relations.

"The relations connecting the labour of one individual with that of the rest appear, not as direct social relations between individuals at work, but as what they really are, material relations between persons and social relations between things."

I guess I have shown that the existence of persistent ratios between the rate of profits in various industries, as with a corporate sector and a sector of small firms and proprietorships, provides another layer of confusion in economic analysis. Here is a source of another illusion created by competition.

Aside: I have stumbled across the International Symposium on Marxian Theory (ISMT). They have a series of books out, with a focus on reacting to the complete works of Marx and Engels (MEGA²). MEGA² is published under the auspices of the Internationale Marx-Engels-Stiftung (IMES) in Amsterdam and of other groups in other countries. Apparently, volume 1 of Capital varied among editions, and Marx had several drafts. Furthermore, MEGA² apparently has editions of volumes 2 and 3 that show exactly how Engels edited them. These facts re-enforce the point of this post, not that I want to read all of these variants.

An Example Of The Labor Theory Of Value

Figure 1: Variation of Prices of Production with Wages and Markups

1.0 Introduction

This post documents an example in my working paper, The Labor Theory of Value and Sraffa's Standard Commodity with Markup Pricing.

2.0 Technology

Consider a simple economy in which corn and ale are each produced from inputs of labor, corn, and ale. Inputs for unit outputs are shown in the columns in Table 1. Obviously, the units of measure should not be taken serious. Inputs are totally used up in the production of outputs. I abstract from the existence of fixed capital, land, and joint production.

Table 1: The Technology

Input	Industry
	Corn	Ale
Labor	1 Person Year	1 Person-Year
Corn	(1/8) Bushels	(3/8) Bushels
Ale	(1/16) Pints	(1/16) Pints

The standard net product consists of (9/16) bushels corn and (3/16) pints ale. The Perron-Frobenius root of the Leontief input-output matrix is 1/4. (The other eigenvalue is (-1/16). The maximum rate of profits is 300 per cent. Labor values are (64/51) person years per bushel corn and (80/51) person-years per pint ale.

3.0 Price Equations

Equations for prices of production are:

[(1/8) p_corn + (1/16) p_ale]( 1 + r̂) + w = p_corn

[(3/8) p_corn + (1/16) p_ale]( 1 + s₂ r̂) + w = p_ale

(9/16) p_corn + (3/16) p_ale = 1

I have taken the standard commodity as the numeraire. This allows one to freely move back and forth, when evaluating aggregates, from labor values to monetary units.

The rate of profits in producing corn is 100 r̂ percent, while it is s₂ r̂ percent in producing ale. I am assuming there are persistent barriers to entry or some reason why the rate of profits persistently varies between industries. Some economists talk about dual markets. I can also point to John Kenneth Galbraith's The New Industrial State for a contrast of corporations in the planning system and more traditional firms. Anyways, the solution of these equations is:

p_corn = 16 [16 + (1 - s₂) r̂]/[204 + (3 + 9 s₂) r̂]

p_ale = 32 [10 - (1 - 3 s₂) r̂]/[204 + (3 + 9 s₂) r̂]

w = 4 [51 - (9 + 5 s₂) r̂ - s₂ r̂²]/[204 + (3 + 9 s₂) r̂]

These equations show that prices of production vary from labor values when the rate of profits is positive. Furthermore, these are not straight lines, although the curvature is not visually impressive in the figure at the top of this post.

Anyways, here is a question. Suppose labor coefficients happen to be a left-hand eigenvector of the Leontief input-output matrix, a very special case. When prices of production are defined with equal rates of profits across all industries, prices of production are labor values in this special case. (The specification of the numeraire does not matter.) Does this property still hold under the sort of markup pricing which I am assuming?

Update (8 Nov 2019): A supporter in email points out a special case. Let s₁ = 5/13 and let s₂ = 5/29. Then prices of production are labor values. The scale factor for the rate of profits is: r̂ = 3 (1 - w). That is, the scale factor is the rate of profits. Presumably, with these relative markups, relative prices are relative labor values, whatever the numeraire.

Keen's Debunking Economics Most Popular Among Popular Critiques

Table 1: Selected Critiques

Author	Book	Number Ratings	Mean Rating
Moshe Adler	Economics for the Rest of Us	21	4
Rod Hill & Tony Myatt	The Economics Anti-Textbook	13	4
Steve Keen	Debunking Economics, 1st edition	25	3 to 4
	Debunking Economics, 2nd edition	56	4 to 5
Paul Ormerod	The Death of Economics	10	3 to 4
John Quiggin	Economics in Two Lessons	2	4
John Weeks	Economics of the 1%	13	4 to 5

Steve Keen seems to be the most popular of those writing internal critiques of economics directed towards the common reader. I selected the above books and looked at rankings on Amazon's United States website. You can spend lots of time reading the comments.

I am not sure about how to characterize this genre. I am more focused on theory than offering political programs. Would Robert Reich's Saving Capitalism be excluded? But what about memoirs, such as John Perkins' Confessions of an Economic Hit Man, Stiglitz' Globalism and its Discontents, Thaler's Misbehaving, or Kahneman's Thinking Fast and Slow? These books seem to have much more ratings than the ones I list in the above table.

Why is Keen's book more popular than the other ones in the table? Keen often overstates his case. One reviewer said he confuses necessary conditions with sufficient ones. I'm covered here; I suggested to him, before publication of the first edition, that well-behaved aggregate excess demand curves might exist in special, numeric, cases even if all consumers did not have identical and homothetic preferences. But those who know of Alan Kirman's work, with others, in the 1970s know Keen has a point. You cannot find any other condition than Gorman form that is sufficient to have well-behaved aggregate excess demand curves. And this is true of many other of Keen's points. I had not realized before reading Keen that the standard textbook presentation of perfect competition assumes managers of firms are systematically mistaken in their understanding of the demand curves they face.

Anyways, neoclassical economics is mostly wrong or useless for internal, logical reasons.

The Labor Theory of Value and Sraffa's Standard Commodity with Markup Pricing

I have uploaded a working paper with the post title.

Abstract: This article demonstrates relationships that are transparent in Sraffa's standard system hold even when relative rates of profit vary persistently among industries. Even with such variations, total constant capital, total variable capital, total surplus value, and the rate of profits are unaltered by evaluation at labor values and at prices of production in Sraffa’s standard system. These results buttress those who see in the standard commodity a solution for Marx’s so-called transformation problem.

Actually Existing Socialism In A Capitalist Setting?

Elements of a post capitalist society are and have been developing in actually existing capitalism. This post points out a couple of examples.

The Green Bay Packers is a community-owned (non-profit) football team in the National Football League (NFL). One can find some arguing that they are socialist. And some are concerned to refute this claim.

Decades ago, some universities in the United States set up research and development organizations that then became independent, not-for-profit companies. For example, here is the web site for SRC, formerly Syracuse Research Corporation. This means, apparently, that they re-invest what they make. IRS Publication 557 explains how to apply for status as a 501(c) organization.

A quick Google search gets me to the National Center for Employee Ownership. They explain how a Employee Stock Ownership Plan (ESOP) works.

The cooperative movement is of interest in this context. I gather the Mondragon Corporation, in Spain, is the most well-known example. But I want to turn to producer cooperatives in dairy. The Lowville Producers Dairy Cooperative is one near me. Apparently, the National Milk Producers Federation is a federation of such cooperatives. The United States Department of Agriculture (USDA) provides background. I see that they confirm what I know anecdotally, that not all dairy farmers are members of a coop.

I guess some theory is needed to make sense of any claim that, say, producer coops are an example of socialism or to obtain a general understanding of such organizations. I have only read Hodgson (1998) and Jossa (2005) in the list of references below. From Hodgson, as I recall, I learned that an issue with cooperatives is start-up finance. It may be that producer cooperatives are more efficient than capitalist firms and still be smaller than one would hope. Jossa (2005) argues that cooperatives are consistent with Marx's vision. He draws on Vanek's distinction between worker-managed firms (WMFs) and labor-managed firms (LMFs). In WMFs, workers provide the finance, while in a LMF, the firm borrows. Anyways, here is some literature to explore.

References

• Geoffrey M. Hodgson (1998). Economics and Utopia: Why the Learning Economy is not the End of History. Routledge.
• Bruno Jossa (2019). The Political Economy of Cooperatives and Socialism, Routledge.
• Bruno Jossa (2005). Marx, Marxism and the cooperative movement. Cambridge Journal of Economics 29: 3-18.
• Jaroslav Vanek (1970). The General Theory of Labor-Managed Market Economies. NCOL.
• Jaroslav Vanek (1971). The Participatory Economy: An Evolutionary Hypothesis and a Strategy for Development. Cornell University Press.
• Jaroslav Vanek (1977). The Labor-Managed Economy: Essays. Cornell University Press.

Structural Economic Dynamics and Fake Switch Points

Figure 1: A Pattern Diagram with Joint Production

1.0 Introduction

This post completes an example. I analyzed bits of this example here and here. This post may make no sense if you have not read a long series of previous posts or, maybe, the papers highlighted here and here. I am interested in how and if my approach to analyzing and visualizing variations in the choice of technique with technical progress extends to joint production. The example suggests fake switch points do not pose an insurmountable obstacle for such an extension.

2.0 Technology

I repeat the specification of technology.

I postulate an economy in which two commodities, corn and linen, can be produced from inputs of corn, linen, and labor. Managers of firms know of three processes (Tables 1 and 2) to produce corn and linen. Each process produces net outputs of corn and linen as a joint product. Inputs and outputs are specified in physical units (say, bushels and square meters) per unit level of operation of the given process. Inputs are acquired at the start of the year, and outputs are available for sale at the end of the year.

Table 1: Inputs for The Technology

Input	Process
	(a)	(b)	(c)
Labor	eσ0,1(1 - t)	eσ0,2(1 - t)	eσ0,3(1 - t)
Corn	20	20	30
Linen	20	20	30

Table 2: Outputs for The Technology

Output	Process
	(a)	(b)	(c)
Corn	21	23	36
Linen	27	25	34

I assume that requirements for use are such that two processes must be operated to satisfy those requirements. I need to investigate the implications of this assumption further. Apparently, for this example, it implies that the economy is not on a golden rule steady state growth path, with the rate of profits equal to the rate of growth. Anyway, with this assumption, three techniques - Alpha, Beta, and Gamma - can be operated. Table 3 specifies which processes are operated for each technique.

Table 3: Techniques

Techniques	Processes
Alpha	a, b
Beta	a, c
Gamma	b, c

The technology, as I have defined it, is parameterized. I consider the following specification for the rate of decrease in labor coefficients.

σ_0,1 = 2

σ_0,2 = σ_0,3 = 5/2

Bidard & Klimovsky's example arises when t is unity.

3.0 Prices and the Choice of Technique

A system of two price equations arises, for each technique. I assume the labor coefficient is treated as a constant over the period of production - say, a year. With linen as numeraire, these equations for the Alpha technique are:

(20 p₁ + 20)(1 + r) + [e^{σ_0,1(1 - t)}] w = 21 p₁ + 27

(20 p₁ + 20)(1 + r) + [e^{σ_0,2(1 - t)}] w = 23 p₁ + 25

One can these equations for two variables in terms of, say, the rate of profits. For each technique, its wage curve shows the wage as a function of the rate of profits. One cannot generally base the choice of technique, under joint production, on figuring out which technique contributes to the outer frontier at a given rate of profits.

Instead, one can calculate profits and losses, with the given rate of profits and a technique's price system for the processes not in the technique. This exercise only makes sense when the rate of profits, the wage, and prices are non-negative for the starting technique. The technique is cost-minimizing only if no extra profits can be made with processes outside the technique.

I deliberately frame this as a combinatorial argument. Bidard likes what he calls a market algorithm, where, when one identifies a process earning extra profits, one introduces the process into the technique. In the case of joint production, it is not clear which process should be dropped. Furthermore, examples exist in which a cost-minimizing technique exists but cannot be reached from certain starting points with the market algorithm.

4.0 Patterns

I have constructed the figure at the top of the post to illustrate how the choice of technique varies with technical progress in this example. The dashed lines highlight features of the example that do not bear on the choice of technique. The light vertical solid lines divide time into numbered regions. Table 3 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.

Table 3: Regions

Regions	Techniques
1	Gamma, No Production, Alpha
2	Gamma, No Production, Alpha
3	Gamma, Alpha & Gamma, Alpha
4	Alpha & Gamma, Alpha
5	Beta, Alpha & Gamma, Alpha

I could say a lot more about the example. I will note that in region 1, the wage increases with the rate of profits, for the Alpha technique, in the interval for the rate of profits where both wages and the price of corn are positive. In region 2, the wage decreases with the rate of profits, for the Alpha technique. The division between regions 2 and 3 is associated with that interval for the rate of profits for Alpha transitioning to have a non-empty intersection with the similar interval for the Gamma technique. for

5.0 Conclusion

This post has illustrated that one type of my types of pattern diagrams can apply to joint production. This type illustrates how the relationship between the choice of technique and distribution varies with technical progress. It can be constructed even in cases, such as joint production, where the choice of technique cannot necessarily be based on wage-rate of profits curves and their outer frontier.

If fake switch points are not shown, this type of pattern diagram does not depend on the specification of the numeraire. If the ordinate in Figure 1 were the wage, instead of the rate of profits, it would be upside down, in some sense. A different numeraire would rescale the wage. When corn is numeraire, only one fake switch point exists. It, too, would be a horizontal line segment. But fake switch points are fake precisely because they do not impact the choice of technique. They can be left off the diagram.

The example also illustrates new types of patterns for dividing adjacent regions. Under joint production, a technique can be associated with non-negative prices and a wage for an interval of the rate of profits that does not include a rate of profits of zero. Both the Alpha and the Beta technique exhibit this possibility in the example. And we can divide regions based on when the range of rate of profits in which such a technique becomes cost-minimizing comes to include zero or begins to interact with the range in which another technique is cost-minimizing

This example also illustrates that the cost-minimizing technique may not be unique in a range of rates of profits. I think this non-uniqueness is qualitatively different than how non-uniqueness can arise in models with only circulating capital. In circulating capital models, non-uniqueness is associated with two techniques having identical wage curves. Not so here.

I do not intend to write this example up any more extensively. I have no so-called paradoxical behavior here, such as reswitching, reverse capital-deepening, or the reverse substitution of labor. I may go on to explore where techniques are described by rectangular matrices, with more produced commodities than processes, and there is a dependence on the requirements for use.

References

• Bidard, Christian and Edith Klimovsky (2004). Switches and fake switches in methods of production. Cambridge Journal of Economics. 28 (1): 89-97.

Elsewhere

• Here is a post from a blog devoted to cybercommunism. The blogger is glowing about Paul Cockshoot's work on refuting Hayek's supposed refutation of the possibility of a post-capitalist society.
• William Milberg writes about how it is becoming more common to use the word "capitalism", a word mainstream economists had mostly stopped using.
• Herbert Giants and Rakesh Khurana write about the corrupting effects of neoclassical economics on what is taught in business school and then practiced by corporate elites.
• Osita Nwanevu writes, in The New Republic, about the enthusiasts that showed up at last weekend's Third MMT Conference.
• Lisa Schweitzer studies urban environments. In a blog post, she expresses irritation at Paul Romer's arrogance, admittedly filtered through a glowing New York Times article.
• A long time ago, Connie Bruck profiled George Soros in the New Yorker. Soros consciously thinks of himself as building on Karl Popper's The Open Society and its Enemies.

Variation in Standard Commodity with Relative Markups

I am not sure about the economic logic in this post. Maybe somebody like D'Agata or Zambelli could do something with this. These ideas were suggested to me by email with a sometime commentator.

I start out with notation for Sraffa's price system, modified in an unusual way to allow for persistent variations in the rate of profits among industries:

• a₀ is a row vector of labor coefficients in each of n industries.
• A is a Leontief input-output matrix, where a_{i, j} is the quantity of the ith commodity needed as input to produce one unit of the jth commodity.
• S is a diagonal matrix, where all off-diagonal elements are zero. s_{j, j} is the markup on non-labor costs in the jth industry.
• p is a row vector of prices.
• w is the wage.
• r is the scale factor for the rate of profits.

The coefficients of production, as expressed in the labor coefficients and the Leontief matrix are given parameters. Relative markups are also taken as given. Prices, the wage, and the scale factor for the rate of profits are the unknowns to be determined. My problem is to find a numeraire such that the wage and the scale factor for the rate of profits trade off in a straight-line relationship, at least when labor is advanced and wages are paid out of the net product:

r = R (1 - w)

I assume all elements of A are non-negative and that all elements of a₀ and all diagonal elements of S are positive. The economy is assumed to be viable, that is, as capable of producing a surplus product. For simplicity, assume that the Leontief matrix is indecomposable. More generally, I need A S to be a Sraffa matrix.

For my purposes here, I formulate price equations as so:

p A S (1 + r) + a₀ w = p

Consider the case when wages are zero and the scale factor for the rate of profits is at its maximum R:

p A S (1 + R) = p

Or:

p A S = (1/(1 + R)) p

I observe that prices are a left-hand eigenvector of the matrix A S, with (1/(1 + R)) the corresponding eigenvalue. To ensure that prices are positive, of the n eigenvalues, choose the maximum. The maximum eigenvalue is also known as the Perron-Frobenius root of A S.

Let y^* be a right-hand eigenvector of A S corresponding to its Perron-Frobenius root. Let q^* be gross output such that the net output is y^*:

y^* = q^* - A q^*

These quantities flow define the standard system here, when scaled so as employ a unit quantity of labor:

a₀ q^* = 1

The net output of the standard system is the desired numeraire:

p y^* = 1

With this definition of the standard system, the ratio of physical gross outputs to circulating capital inputs varies among commodities. This result contrasts with Sraffa's standard system. I suppose I could restore this property by choosing q^*, not y^*, to be an eigenvector. Either way, the ratio of net outputs to circulating capital inputs varies among industries. Either way, the relative ratios of commodities in the standard industry depends on relative markups.

Do Marx's invariants hold with the above definition of the standard system? I expect not. Nevertheless, does this mathematics provide some insight into classical or Marxist political economy?