## Thursday, June 17, 2021

### Fluke Cases for the Order of Fertility

 Figure 1: Wage Curves for Fluke Case for r-Order of Fertility
1.0 Introduction

This post illustrates two fluke cases that can arise in a model with land and extensive rent. I call these a pattern of switch points for the r-order of fertility and a pattern of switch points for the w- order of fertility. I have previously described a fluke case in the order of rentability, which can be either over the wage axis or over the axis for the rate of profits. These fluke cases can arise in an analysis in which a parameter space is partitioned by fluke cases such that in each of the resulting regions the analysis of the choice of technique does not qualitatively vary, in some sense.

2.0 Technology

The technology is described by the coefficients of production in Table 1. Let there be T1 = 100 acres of type 1 land, T2 = 80 acres of type 2 land, and T3 = 40 acres of type 3 land. See this post for a slightly longer description of the technology.

 Input Iron Industry Corn Industry I II III IV Labor a0,1 = 1 a0,2 = 1/2 a0,3 = 3 a0,4(t) = 2.0743 e-0.03648 t Type 1 Land 0 b1,2 = 1 0 0 Type 2 Land 0 0 b2,3 = 1 0 Type 3 Land 0 0 0 b3,4 = 1 Iron a1,1 = 0 a1,2 = 1/2 a1,3 = 1/8 a1,4(t) = 0.3551 e-0.06337 t Corn a2,1 = 1/2 a2,2 = 0 a2,3 = 0 a2,4(t) = 0.3343 e-0.2906 t

Table 2 lists the techniques for this example. Feasiblity of a technique is determined by requirements for use.

 Technique Type of Land Type 1 Type 2 Type 3 Alpha Fully farmed Fully farmed Partially farmed Beta Partially farmed Fully farmed Fully farmed Gamma Fully farmed Partially farmed Fully farmed Delta Fully farmed Partially farmed Fallow Epsilon Fully farmed Fallow Partially farmed Zeta Partially farmed Fully farmed Fallow Eta Fallow Fully farmed Partially farmed Theta Partially farmed Fallow Fully farmed Iota Fallow Partially farmed Fully farmed Kappaa Partially farmed Fallow Fallow Lambda Fallow Partially farmed Fallow Mu Fallow Fallow Partially farmed

3.0 Fluke Case for the r-Order of Fertility Over the Axis for the Rate of Profits

I start with the assumption that the rate of profits is taken as given. I go through a numerical algorithm to find a particular value of the parameter t. So suppose rent is not paid on a given type of land and that profits, rent, and wages are paid out of the surplus product at the end of the yearly cycle of production. With a bushel corn as numeraire, you can figure out the wage and the price of iron as a function of the rate of profits. And you can get the wage curves shown in Figure 1, at the top of this post.

Now suppose requirements for use are such that they can only be satisfied with one type of land totally farmed and a second type partially farmed. Given the rate of profits, one might consider a vertical line in Figure 1. If the rate of profits is less than r*, land of Type 1 will be farmed fully first, and land of Type 2 will only be farmed to the extent that are mandated by requirements for use. That is, the Delta technique will be adopted. On the other hand, for a rate of profit between r* and 100 percent, the Zeta technique will be adopted. Figure 2 shows rents in this case. Only the lands fully cultivated pay a rent.

 Figure 2: Rent for Fluke Case for r-Order of Fertility

But suppose requirements for use are greater. They can only be satisfied by fully cultivating two types of land and partially cultivating the remaining tye. Then, given the rate of profits, the Alpha technique will be adopted. The rate of profits can only range from zero to r*. The order of fertility, from most fertile lands to least fertile, is Type 1, Type 2, and Type 3.

Figure 3 shows rents in this subcase. For a small rate of profits, the order of fertility matches the order of rentability. Not so much for a higher, feasible rate of profits.

 Figure 3: Rent for Fluke Case for r-Order of Fertility (Cont'd)

Suppose 'slow time' was to increase, with a consequent reduction in the coefficients of production on Type 3 land, other than for the acres of land needed to produce a bushel corn. The wage curve for Type 3 land in Figure 1 would move outward. A range of high rates of profits would appear in which the Alpha technique is cost-minimizing, but in which the order of fertility is Type 2, Type 1, Type 3 lands. The value of time approximately equal to 0.05171 is an edge case just before, at a range of high rate of profits, an order of fertility appears that matches the order of rentability.

4.0 Fluke Case for the w-Order of Fertility Over the Wage Axis

Now suppose instead that the wage is taken as given outside the system of equations for prices of production. I take a time ofapproximately 1.2411 for defining the coefficients of production. I get the wage curves in Figure 4. For a low given wage, the Zeta technique is cost-minimizing. For a high feasible given wage, the Delta technique will be operated. Figure 5 graphs rents. In this subcase, it does not matter if the wage or the rate of profits is taken as given. The story is analogous.

 Figure 4: Wage Curves for Fluke Case for w-Order of Fertility

 Figure 5: Rents for Fluke Case for w-Order of Fertility

But suppose requirements for use are such that three types of land must be cultivated, with one only partially cultivated. Then the Alpha technique is cost-minimizing, whatever the wage as long as it is feasible. Figure 6 graphs rents in this case. The order of fertility matches the order of rentability for low wages, but not for high feasible wages. If time were to increase, however, a range for high wages would appear in which the order of fertility matches the order of rentability. Figure 4 illustrates a fluke switch point.

 Figure 6: Rents for Fluke Case for w-Order of Fertility (Cont'd)

5.0 Conclusion

So there are two new fluke switch points, where these flukes arise in models of extensive rent. I keep on thinking I am discovering theoretical possiblities, possibly through sheer bloody-mindedness, that nobody has noted before.

Anonymous said...

If I do not understand it incorrectly what is argued in your posts on land as a particular case of fixed capital is that contrary to simple production models there is a difference if one takes as the independent variable r or w. Do I grasp it correctly?

Blissex said...

«I keep on thinking I am discovering theoretical possiblities, possibly through sheer bloody-mindedness, that nobody has noted before»

I surely agree with the "bloody-mindedness" :-) and I think it is good, but I also think that "nobody has noted before" is because nobody has been looking for them before, because of ideological reasons: if the optimization landscape topology is "bumpy" and "kinky" (for example because capital is not "putty" or "leets") and has local maxima then the distribution of income is an accident of history, and that violates "internal consistency" with the most important of JB Clark's "three parables", the one that (abset government "interference" with the "free markets") income distribution is solely and directly justified by productivity.

All your "theoretical possibilities" show that plusvalue distribution is an accident of circumstnaces, so they they are not "internally consistent", therefore: they are not particularly notable from the point of view of someone accepting the notion that the optimization topology is "bumpy" and "kinky", and they are "invalid" or "heresy" for those aiming for "internal consistency".

Robert Vienneau said...

Yes, in models with rent, it matters whether or not one takes r or w as given. In figuring out the cost-minimizing technique, one cannot always look exclusively at the outer envelope of wage curves, since for a large enough economy, the outer curve may be for a technique that has already used all of the land.

Blissex, I sometimes think that you seem to focused on marginalism as a justification for an ideology about income distribution. But when I try to explain my hobby to people in real life, I go on about that and about purges in economics departments and about how what is in most of the textbooks was shown to be without foundation even before I was born. But I think I have something with some sort of structures in parameter spaces, where I do not bother worrying about that setting.

Blissex said...

«you seem to focused on marginalism as a justification for an ideology about income distribution.»

Actually that and even more so the other way round too :-)

«But I think I have something with some sort of structures in parameter spaces, where I do not bother worrying about that setting.»

Ahhhhhh.... I misunderstood then, because the impression I got is that you are coming up with arguments about reswitching, the Cambridges Capital Controversy, LTV, to demolish repeatedly neoclassical approaches. Because that things like reswitching happen because optimization landscapes are "bumpy" is otherwise totally obvious and does not merit finding so many examples of. The LTV and "fertility" topics are more interesting, because the relate to a really big historical debate, that about plusvalue, where it comes from, how to make grow, how it is distributed, which is the central axis of political economy studies.

My guess now that you are generating examples not as counterexamples to neoclassical theory, but in an attempt to see if there is some structure of those examples to find some general rules that characterize that shape of the optimization landscape that give rise to "anomalies" in at least some class of model. That may be a very ambitious project, unless the models are quite simple.
Without thinking too much about my guess is that it is almost as hard (optimistically) as finding general rules for stability regions in chaos theory models (which may be a hint).

But when you talk about the LTV and "fertility" you are also addressing, and I am not criticizing you for it, very different if related topics. and I mentions neoclassical concepts only as paragons to those I think are more fruitful, because for better or worse neoclassicism is something we all know to some extent.