Paul Krugman Ignorant Of The Cambridge Capital Controversy

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

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

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

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

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

Two Four-Technique Patterns With Markup Pricing

Figure 1: Wage Curves for the Example

1.0 Introduction

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

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

2.0 Technology

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

Table 1: The Technology

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

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

Table 2: Techniques

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

3.0 Prices and the Choice of Technique

I consider prices of production in which:

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

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

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

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

Figure 2: The Price of Iron in the Example

Figure 3: The Price of Steel in the Example

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

4.0 Conclusion

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

Fluke Switch Points in Pure Fixed Capital Systems

I have a working paper at the Centro Sraffa.

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

Algebraic Geometry

An Introduction to Algebraic Geometry

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

Flummery From Robert A. Heinlein

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

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

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

-- Robert A. Heinlein, Starship Troopers

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