Reswitching Pattern In Corn-Tractor Model

Figure 1: Variation in the Cost-Minimizing Technique with Selected Coefficient of Production

This post reports on some work with Steedman's corn-tractor model. I have yet to find an instance of triple-switching. I have found a case of reswitching, though.

Figure 1 above and Table 1 below show how switch points vary with perturbations of the labor needed to produce a bushel of corn in the corn industry with Type I tractors. Only one switch point exists in region 1. Around this switch point a lower rate of profits is associated with the production and use of Type I tractors, an increase in consumption per labor-year, and a decrease in labor inputs in the corn industry per bushel produced. In other words, this switch point conforms to obsolete marginalist intuition.

Table 1: Regions

Region	Description
1	Around switch point, Type I tractors are cost-minimizing at a lower rate of profits, higher wage.
2	Reverse labor deepening. Around switch point, Type I tractors are cost-minimizing at a lower rate of profits, higher wage. Increased wage is associated with more labor employed in corn industry.
3	Reswitching. Type II tractors are cost-minimizing at high and low rates of profits.
4	Type II tractors are cost-minimizing throughout.

A single switch point exists in region 2 as well. This switch point is like that in region 1, except a lower rate of profits around the switch point is associated with an increase in labor inputs in the corn industry. This is a case of the reverse substitution of labor.

Region 3 is an example of reswitching. In region 4, both switch points have disappeared. Figure 2 below is much like Figure 1, but the wage is graphed on the ordinate.

Figure 2: Variation in Wage at Switch Points

I also graphed two of my fluke switch points. Figure 3 is for a switch point on the wage axis. It is at the partition between regions 2 and 3 in the above graphs. Figure 4 is for what I call a reswitching pattern. It is at the partition between regions 3 and 4.

Figure 3: Pattern over Wage Axis

Figure 4: Reswitching Pattern

Consider the switch point at the higher rate of profits in region 3. Around this switch point, a lower rate of profits is associated with adoption of the technique in which Type I tractors are produced and used. Consumption of corn per employed worker is decreased. Suppose, incoherently, that a lower rate of profits is an indication that consumers are more patient and supply more capital. The example clarifies that identifying a type of machine as more capital-using or labor-saving cannot be seen from a physical characterization of machinery alone.

I am still seeking parameters in which triple-switching occurs. To find the above case, I needed to find the roots of a fourth degree polynomial. I am looking for a case in which all roots are real, and three are positive and less than the maximum rate of profits.

It also would be nice to find reswitching in the special case in which the following is true for each type of tractor:

a/b = alpha/beta

Reswitching in this special case contradicts Samuelson's 1962 account of surrogate production functions. He claims that reswitching is impossible here. The difference is that Steedman analyzes depreciation rigorously, not with an approximation of radioactive decay.

Ludwig Von Mises, Male Chauvinist

Ludwig Von Mises expanded his erroneous 1920 essay into a book, Socialism: An Economic and Sociological Analysis. The first edition was published in 1922 and the second edition in 1932.

Since Von Mises is attempting to be more comprehensive, he treats the socialist advocacy of free love in an early chapter. Being au courant, he writes about Freud. He argues that the bourgeois idea of marriage as a contract, binding on both husband and wife, is an improvement on what came before.

But I find it hard to get beyond passages like those below. Here, Von Mises confines women to her supposed role in propagating the human race, as in Margaret Atwood's The Handmaid's Tale:

"The radical wing of Feminism, which holds firmly to this standpoint, overlooks the fact that the expansion of woman's powers and abilities is inhibited not by marriage, not by being bound to man, children, and household, but by the more absorbing form in which the sexual function affects the female body. Pregnancy and the nursing of children claim the best years of a woman's life, the years in which a man may spend his energies in great achievements. One may believe that the unequal distribution of the burden of reproduction is an injustice of nature, or that it is unworthy of woman to be child-bearer and nurse, but to believe this does not alter the fact. It may be that a woman is able to choose between renouncing either the most profound womanly joy, the joy of motherhood, or the more masculine development of her personality in action and endeavor. It may be that she has no such choice. It may be that in suppressing her urge towards motherhood she does herself an injury that reacts through all other functions of her being. But whatever the truth about this, the fact remains that when she becomes a mother, with or without marriage, she is prevented from leading her life as freely and independently as man. Extraordinarily gifted women may achieve fine things in spite of motherhood; but because the functions of sex have the first claim upon woman, genius and the greatest achievements have been denied her." -- Ludwig von Mises, Socialism, pp. 100-101

Here, Von Mises suggests that women are not capable of the intellectual contemplation possible for men:

"It is clear that sex is less important in the life of man than of woman. Satisfaction brings him relaxation and mental peace. But for the woman the burden of motherhood begins here. Her destiny is completely circumscribed by sex; in man's life it is but an incident. However fervently and whole-heartedly he loves, however much he takes upon himself for the woman's sake, he remains above the sexual. Even woman are finally contemptuous of the man who is utterly engrossed by sex. But woman must exhaust herself as lover and as mother in the service of the sexual instinct. Man may often find it difficult, in the face of all the worries of his profession, to preserve his inner freedom and so to develop his individuality, but it will not be his sexual life which distracts him most. For woman, however, sex is the greatest obstacle.

Thus the meaning of the feminist question is essentially woman's struggle for personality. But the matter affects men not less than women, for only in co-operation can the sexes reach the highest degree of individual culture. The man who is always being dragged by woman into the lower spheres of psychic bondage cannot develop freely in the long run. To preserve the freedom of inner life for the woman, this is the real problem of women; it is part of the cultural problem of humanity." -- L. von Mises, op. cite., pp. 102-103

Here, Von Mises says that it does not matter that women cannot vote or hold elected office, as well as a lot of other vicious nonsense:

"...And now man and woman are equal before the law. The small differences that still exist in private law are of no practical significance. Whether, for example, the law obliges the wife to obey her husband is not particularly important; as long as marriage survives one party will have to follow the other and whether husband or wife is stronger is certainly not a matter which paragraphs of the legal code can decide. Nor is it any longer of great significance that the political rights of women are restricted, that women are denied the vote and the right to hold public office. For by granting the vote to women the proportional political strength of the political parties is not on the whole much altered; the women of those parties which must suffer from the changes to be expected (not in any case important ones) ought in their own interests to become opponents of women's sufferage rather than supporters. The right to public office is denied women less by the legal limitations of their rights than by the peculiarities of their sexual character. Without underestimating the value of the feminists' fight to extend woman's civil rights, one can safely risk the assertion that neither women nor the community are deeply injured by the slights to women's legal position which still remain in the legislation of civilized states.

The misconception to which the principle of equality before the law is exposed in the field of general social relationships is to be found in the special field of relations between those sexes. Just as the pseudo-democratic movement endeavours by decree to efface natural and socially conditioned inequalities, just as it wants to make the strong equal to the weak, the talented to the untalented, and the healthy to the sick, so the radical wing of the woman's movement seeks to make women the equal of men. [Footnote: To examine how far the radical demands of feminism were created by men and women whose sexual character was not normally developed would go beyond the limits set to these expositions.] Though they cannot go so far as to shift half the burden of motherhood on to men, still they would like to abolish marriage and family life so that women may have at least all that liberty which seems compatible with childbearing. Unencumbered by husband and children, woman is to move freely, act freely, and live for herself and the development of her personality.

But the difference between sexual character and sexual destiny can no more be decreed away than other inequalities of mankind. It is not marriage which keeps woman inwardly unfree, but the fact that her sexual character demands surrender to a man and that her love for husband and children consumes her best energies..." -- L. von Mises, op. cite., pp. 104-105

Von Mises' praise for Mussolini's fascism appears to be more well-known that his stated indifference to women not being allowed to vote.

Goal: Perturb Special Case Of Steedman's Corn-Tractor Model

1.0 Introduction

I would like to illustrate triple switching, in the corn-tractor model, with one of my one-dimensional diagrams. I have a triple-switching example, from Bertram Schefold, but the wage-rate of profit frontier is not visually striking in it. Such an example would not be worthy of a research paper. But perhaps I could modify a section of my recent working paper to submit somewhere. Besides, posing a new problem might motivate me to update my computing technology.

2.0 Technology

The corn-tractor model is a fixed capital model, an adaption of the Samuelson-Gargenani model. Labor and tractors are used to produce new tractors. Labor and tractors are also used to produce corn. Corn is the consumption good and the numeraire. Table 1 shows the coefficients of production for a particular type of tractor.

Table 1: Inputs for The Technology

INPUTS	Industry
	Tractor	Corn
Labor	b person-years	beta person-years
Tractors (of any age)	a tractors	alpha tractors
Corn	0	0
OUTPUTS	1 new tractor	1 bushel corn

Tractors last n years in the tractor industry and v years in the corn industry. Although not apparent in the table, this is an example of joint production. Every process for producing a new tractor also produces tractors one year older than the tractors used as inputs, except for the process using (n - 1)-year old tractors as an input. Similarly, every process for producing corn also produces tractors one year older, except for the process using (v - 1)-year old tractors. Tractors operate with constant efficiency for their physical life, albeit with different efficiencies in the two industry. As assumed in pure fixed capital models, old tractors cannot be transferred between industries.

With these assumptions, no choice of the economic life of a machine arises. The tractor will be used for its full physical life in each industry. Only three coefficients need to be specified for each type of tractor: a, beta, and (alpha b). The last is a matter of scaling, of selecting units of measure for labor or tractors, I guess. Without loss of generality, one can set alpha to unity throughout.

3.0 A Special Case

To find a triple-switching example, it is apparently sufficient to set n = v = 2. Tractors last for two years in both the tractor and the corn industry. Eventually, I want to consider two types of tractors. The choice of technique is a matter of choosing the type of tractor to produce and use. (I always find it mysterious how Steedman and his co-authors find their examples. One might think this model was thoroughly analyzed decades ago and did not have anything new to tell us.)

The remainder of this post specifies the solution, in a stationary state, for this special case.

4.0 Quantity Flows

Consider a stationary state in which employment is one person-year, across the four operated processes. Let q₁ be the number of new tractors produced in each process in the tractor industry. Let q₂ be the bushels corn produced in each process in the corn industry. These quantities are as follows:

q₁ = alpha/{2 [2 beta + (alpha b - a beta)]}

q₂ = (2 - a)/{2 [2 beta + (alpha b - a beta)]}

One can check this solution. Total employment, L, is:

L = 2 b q₁ + 2 beta q₂ = 1

The number of new tractors produced is (2 q₁), and the number of new tractors used in production processes is the sum of (a q₁) and (alpha q₂). The new tractors used as inputs replace, at the end of the year, the one-year old tractors used in each industry. So this is a stationary state with employment of one person-year.

The gross output of corn is also the net output, since corn is not used as an input in production. That is, consumption per person-year in a stationary state, c, is:

c = 2 q₂ = (2 - a)/[2 beta + (alpha b - a beta)]

5.0 Prices of Production

The system of equations for prices of production are set out in terms of five price variables:

• p₀: The price of a new tractor.
• p₁: The price of a one-year old tractor used in the tractor industry.
• p₂: The price of a one-year old tractor used in the corn industry.
• w: The wage, in units of bushels per person-year, paid to the workers at the end of the year.
• r: The rate of profits, assumed to be the same in each of the four production processes.

Sraffa shows how to eliminate the prices of old tractors from the system. This analysis derives the price of an annuity. The following variable is convenient in setting out the solution of the price equations:

denom(r) = [(alpha b - a beta) r²

+ [beta + 2 (alpha b - a beta)] r

+ 2 beta + (alpha b - a beta)

The wage, as a function of the rate of profits, is:

w = [-a r² + (1 - 2 a) r + (2 - a)]/denom(r)

I call the above function the wage curve. The price of a new tractor, also as a function of the rate of profits, is:

p₀ = b (r + 2)/denom(r)

The price of a one-year old tractor is:

p₁ = p₂ = b (r + 1)/denom(r)

I think that, in this model, as long as the life of a tractor is the same in producing tractors and corn, the price of an old tractor of a given age does not vary between the two industries.

The wage curve is also the tradeoff for consumption per worker and the steady state rate of growth. Accordingly, the wage at a rate of profits of zero is the same as consumption per worker, c, found as a result of the solution of the quantity equations.

6.0 Conclusion

I guess I should create a spreadsheet for this special case, but with a choice of two types of tractors. My problem is to find a set of six coefficients of production, three for each type of tractor, such that the two wage curves intersect at three points, with positive rates of profits but below the maximum. Finding such is probably tedious. Then, I would like to consider perturbations of the coefficients, maybe exponential decreases with time in labor inputs. And finally, I would like a diagram of, say the wage, for switch points and the maximum, graphed against time.

Update 9/9/2024: Added calculation and comment about price of old tractors.

Elsewhere

• Louis-Philippe Rochon on Paul Davidson.
• James Galbraith, Thomas Palley, and Matias Venengo on Paul Davidson.
• The Economist on Donald Harris. Response in letters to the editor (Both behind paywall).
• A blog post from Matias Vernengo commenting on the above.
• The Washington Post on policy advice from Donald Harris for Jamaica (Behind paywall).
• Maybe I want to read Vanessa Wills' Marx's Ethical Vision, which may be in tension with views I have set out.
• Gordon Katic on the insidious elitist upshot of behavioral economics.