A Structure In Parameter Space

Table 1: A Common Structure Example

1.0 Introduction

This post presents partitions of (a part of) parameter space for two examples of models of prices of production with a choice of technique. The examples have a different structure and are parametrized differently. Yet, I want to argue, the partitions are the same, at some level of abstraction.

2.0 Thing 1

The first example is an instance of the Samuelson-Garegnani model. Table 1 presents the coefficients of production for this example. Each coefficient specifies the units of input needed to produce a unit output of the commodity for the given industry. Corn is the numeraire, and is not an input into any industry. It is non-basic in Sraffa's terminology. Three processes are available for producing corn, each distinguished by the capital good used in that process. In a technique, the process that produces that capital good and the given corn-producing process are operated. The techniques are labeled Alpha, Beta, and Gamma, depending on the corn-producing process. The example is fully specified by assigning values to a_2,2 and a_3,3.

Table 1: The Coefficients of Production for First Example

Input	Industry
	Iron	Copper	Uranium	Corn
				Alpha	Beta	Gamma
Labor	1	17328/8281	1	1	361/91	3.63505
Iron	1/2	0	0	3	0	0
Copper	0	a2,2	0	0	1	0
Uranium	0	0	a3,3	0	0	1.95561
Corn	0	0	0	0	0	0

Given the technique (with the parameters assigned numberical values) and, say, a feasible rate of profits, one can solve for the real wage and prices of production. At each feasible rate of profits, the choice of technique is determined by cost. It is the cost-minimizing one. In general, the choice of technique varies with the rate of profits. One could also take the wage as given, and find the rate of profits as a function of the wage.

To go on, I draw upon my publications. I call a switch point in which two wage curves are tangent at that point a reswitching pattern (of switch points). A three-technique pattern (of switch points) is a switch point in which three wage curves intersect. Both of these patterns exist in this example.

The left panel in Figure 1 above partitions a portion of the parameter space for this model. I label each region by the cost-minimizing techniques along the wage frontier, in the order of an increasing rate of profits. The partitions are labeled by the corresponding pattern of switch points. This case illustrates how a perturbation of coefficients of production can lead to the emergence of the reswitching of techniques.

3.0 Thing 2

In the second example, two commodities are produced, and they both are basic commodities in all three techniques. Table 2 presents the coefficients of production for this example. I have introduced technological change in this model The coefficients of production for producing corn with the Beta and Gamma processes decrease exponentially. They decrease at different rates for the two processes, but all coeficients for a process descrease at the same rate.

Table 2: The Coefficients of Production for Second Example

Input	Industry
	Iron	Corn
		Alpha	Beta	Gamma
Labor	1	5191/5770	(305/494) e(3/20) - σt	(19/20) e(3/10) - φt
Iron	9/20	1/40	(3/1976) e(3/20) - σt	(1/40) e(3/10) - φt
Corn	2	1/10	(229/494) e(3/20) - σt	(1/10) e(3/10) - φt

Given the coefficients at a particular time, one can find prices of production and the cost-minimizing technique. Here, too, the analysis of the choice of technique can be performed by constructing the wage frontier. The model is parametrized by (σ t) and (φ t). The right panel in Figure 1 shows a portion of this parameter space for this model.

4.0 Conclusion

I want to say that the two panels in Figure 1 depict the same structure. The labeling of techniques is different, and one panel requires the other be rotated and stretched, as is typical of topological structures. In both cases, the loci for two reswitching patterns intersects. At that intersection, one wage curve on the wage frontier is tangent to two other wage curves, each at a separate switch point. Each locus for the reswitching patterns terminates at a point of tangency for a locus of three-technique patterns. And a three-technique pattern forms an arc connecting those two points of tangency.

Other common structures can be found in the parameter spaces of models of prices of production with the choice of technique.l

John Stuart Mill Illustrates Charles Mills' Racial Contract

Here is John Stuart Mill stating a principle that sounds noble, and then immediately making a strange caveat.

"The object of this Essay is to assert one very simple principle, as entitled to govern absolutely the dealings of society with the individual in the way of compulsion and control, whether the means used be physical force in the form of legal penalties, or the moral coercion of public opinion. That principle is, that the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. That the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. His own good, either physical or moral, is not a sufficient warrant. He cannot rightfully be compelled to do or forbear because it will be better for him to do so, because it will make him happier, because, in the opinions of others, to do so would be wise, or even right. These are good reasons for remonstrating with him, or reasoning with him, or persuading him or entreating him, but not for compelling him, or visiting him with any evil, in case he do other wise. To justify that, the conduct from which it is desired to deter him must be calculated to produce evil to some one else. The only part of the conduct of any one, for which he is amenable to society, is that which concerns others. In the part which merely concerns himself, his independence is, of right, absolute. Over himself, over his own body and mind, the individual is sovereign.

It is, perhaps, hardly necessary to say that this doctrine is meant to apply only to human beings in the maturity of their faculties. We are not speaking of children, or of young persons below the age which the law may fix as that of manhood or womanhood. Those who are still in a state to require being taken care of by others, must be protected against their own actions as well as against external injury. For the same reason, we may leave out of consideration those backward states of society in which the race itself may be considered as in its nonage. The early difficulties in the way of spontaneous progress are so great, that there is seldom any choice of means for overcoming them; and a ruler full of the spirit of improvement is warranted in the use of any expedients that will attain an end, perhaps otherwise unattainable. Despotism is a legitimate mode of government in dealing with barbarians, provided the end be their improvement, and the means justified by actually effecting that end. Liberty, as a principle, has no application to any state of things anterior to the time when mankind have become capable of being improved by free and equal discussion. Until then, there is nothing for them but implicit obedience to an Akbar or a Charlemagne, if they are so fortunate as to find one. But as soon as mankind have attained the capacity of being guided to their own improvement by conviction or persuasion (a period long since reached in all nations with whom we need here concern ourselves), compulsion, either in the direct form or in that of pains and penalties for non-compliance, is no longer admissible as a means to their own good, and justifiable only for the security of others."

-- J. S. Mill, On Liberty

The second paragraph in that quotation above is no abstract theoretical observation. As I understand it, Mill, following his father, had a day job in the East India Company, eventually becoming Chief Examiner of Correspondence. I gather that that was a fairly prominent position.

Mill, in On Liberty is not writing about a social contract, unlike Hobbes, Locke, and Rousseau, for example. But this book is a classic of liberal political philosophy that, when looked at from a subaltern position, has a dark racial underside. A reader of Charles Mills might be sensitized to see this.

Reference

• John Stuart Mill. 1859. On Liberty
• Charles Mills. 1997. The Racial Contract. Cornell University Press

Summary of Some Conclusions From My Research Program

This blog, over years, presents a welter of fluke cases. I created many of the numerical examples to illustrate the reswitching of techniques, capital reversing, or some such so-called 'perversity'. Fluke cases can be combined. For example, a fluke switch point at a rate of profits of zero can also be a fluke switch point at which three wage curves intersect. Or two switch points on the wage frontier can both be fluke switch points at which four wage curves, not necessarily the same, intersect. Numerical examples remain to be developed for some possibilities.

Selected Fluke Cases

Pattern of switch points over the wage axis

Pattern of switch points for the reverse substitution of labor

Pattern of switch points over the axis for the rate of profits

Reswitching pattern of switch points

Three-techniques pattern of switch points

Four-technique pattern of switch points

Pattern of switch points for the w-order of fertility

Pattern of switch points for the r-order of fertility

Pattern over the wage axis for the order of rentability

Pattern over the axis for the rate of profits for the order of rentability

Pattern for the requirements for use

The analysis and construction of fluke cases yields insights into the analysis of the choice of technique in the system of prices of production. The reswitching of techniques, capital-reversing, process recurrence, the reverse substitution of labor, the extension of the lifetime of a machine at a lower wage, and the divergence between the order of fertility and the order of rentability are not fluke cases. These possibilities can be contrasted with genuine fluke cases, in which the perturbation of parameters destroys characteristics specifying such cases. These fluke cases partition parameter spaces into regions where these so-called 'perverse' phenomena arise.

These phenomena are also more or less independent of one another. Reswitching does not occur without process recurrence, but process recurrence can arise without reswitching. The association of a smaller rate of profits around a switch point with the truncation of the lifetime of a machine may or may not be accompanied by capital-reversing. Capital reversing can arise with or without a reverse substitution of labor and vice versa. Variations in the order of fertility need not accompany variations in the order of rentability. Nor need variations in the order of rentability be accompanied by variations in the order of fertility. The divergence between the order of fertility and the order of rentability can arise in an example of the reswitching of techniques, but reswitching is not necessary for such divergences. These specific examples do not exhaust the possible combinations of Sraffa effects.

The demonstration and visualization of these results is presented in an open and disaggregated model of prices of production. The functional distribution of income between wages, rents, and profits is not specified. The approach illustrated in these blog posts, in some sense, provides an even more open model. Prices of production are consistent with the smooth reproduction of a capitalist economy. In specifying prices of production, technology, relative rates of profits among industries, and requirements for use are frozen. Fluke cases are found, on the other hand, by perturbing parameters that specify these givens for prices of production.

Whatever practical conclusions can be drawn from this widening of the horizon remain on a high level of abstraction. Characteristics of the conflict over the functional distribution of income between wages and profits can depend on struggle within the class of capitalists. Landlords, in as much as they their interests are reflected in the existence and size of the rent of specific types of land, are also affected by the conflicts between workers and capitalists and within the class of capitalists. Variations in technology, in causes of persistent differences in the rate of profits among industries, and in the requirements for can change these characteristics of these conflicts.

Mark Twain On The Wages Of Whiteness

In Mark Twain's novels, Huckleberry Finn is just a kid in what you might think is the most despised group in society. His mother ran away, and his father, who rarely is home to look after him, is the town drunk. Huck does not go to school, dresses in rags, and often sleeps outside in some barrel down by the waterfront. But Huck is quite conscious that some hard-working adults are looked down on worse than him by respectable people.

'That's all right. Now, where you going to sleep?'

'In Ben Roger's hayloft. He lets me, and so does his pap's nigger man, Uncle Jake. I tote water for Uncle Jake whenever he wants me to, and any time I ask him he gives me a little to eat if he can spare it. That's a mightly good nigger, Tom. He likes me, becuz I don't ever act if I was above him. Sometime I've set right down and eat with him. But you needn't tell that. A body's got to do things when he's awful hungry he wouldn't want to do as a steady thing.'

The Adventures of Tom Sawyer

Remember that Jim is a grown man. Huck in a canoe gets separated from Jim on the raft, which has a wigwam in the center, in a fog in the night. When Huck gets back, Jim is asleep. Huck tells Jim that he was never seperated; Jim was dreaming.

'En when I wake up en fine you back agin, all safe en soun', de tears come, en I could a got down on my knees en kiss yo' foot, I's so thankful. En all you wuz thinking 'bout wuz how you could make a fool uv old Jim wid a lie. Dat truck dah is trash; en trash is what people is dat puts dirt on de head er dey fren's en makes em ashamed.'

Then he got up slow and walked to the wigwam, and went in there without saying anything but that. But that was enough. It made me feel so mean I could almost kissed his foot to get him to take it back.

It was fifteen minutes before I could work myself up to go and humble myself to a nigger; but I done it, and I warn't even sorry for it afterwards, neither. I didn't do him no more mean tricks, and I wouldn't done that one if I'd a knowed it would make him feel that way.

The Adventures of Huckleberry Finn

In quickly googling to check that my title was apropos, I stumbled upon this Adolph Reed essay about W. E. B. Du Bois.

The Four Circuits Of Capital

The Four Circuits of Capital

Marx describes three circuits of capital in the opening chapters of Volume 2 of Capital. But when I draw a diagram, as above, a fourth circuit seems to be missing. So I have added the circuit of advanced capital.

The circuit of advanced capital begins with commodities, consisting of means of production and labor power, in the hands of or under the direction of capitalists. They have purchased these commodities with monetary advances. The capitalists, at this point, care about the use values of these commodities, and they have purchased them to be in specific proportions.

The sphere of circulation is left behind in the next step in the circuit. Workers apply labor to the means of production, under the formal and real subsumption of capital, to produce another set of commodities. These commodities consist of both means of production and means of consumption, depending on the department in which a capitalist operates. But the capitalists do not care about use values at this point. The commodities are, for him, values.

In the next step, the capitalists sell produced commodities for money. That is, values are realized. Money need not be cash or coins. Some capitalists might sell commodities to other capitalists for bills of exchange. Realized value includes surplus value, added by workers but not paid out in wages.

Although not shown in my diagram above, capitalists may use some of the money they have obtained to purchase necessaries and luxuries. But some money is used to complete the circuit, by purchasing means of production and labor power. Usually, one would expect this completion to allow the circuit to continue on an expanded scale.

The time of turnover of capital is the sum of the time of production (including, for example, time for aging processes in which labor is barely expended) and the time of circulation. In some sense, the time of circulation does not enter the circuit of commodity capital. The industrial capitalist can realize the value of produced commodities by selling to merchants, not directly to consumers, and thereby continue the circuit. But the time of the circulation does enter into the three remaining circuits, that of the circuit of money capital, the circuit of advanced capital, and the circuit of productive capital.

The circuits of capital emphasize that, for Marx, capital is a process for extracting surplus value. Capital is neither money; physical commodities, whether means of production or labor power; nor produced commodities. It is all of these in a social process, where one embodiment of capital follows another.

(I have been reading Resnick and Wolff. They draw on Louis Althusser and emphasize a over-determined, process-oriented reading of Marx. I do not see how one gets much in the way of quantative results from this reading, but their qualitative results can be of interest.)

• Resnick, Stephen A. and Richard D. Wolff. 2006. New Departures in Marxian Theory. Routledge.