Friday, July 23, 2021

Extensive Rent For A Reswitching Example

Figure 1: Wage Curves and Rent
1.0 Introduction

I might as well illustrate an example with extensive rent and reswitching. I find it incredible that the agents in these sorts of models understand the implications of, say, a variation of the distribution of income for their self-interests. Nevertheless, I try to note the consequences of variation in the distribution of income and perturbations of model parameters on prices of production. And I do not worry too much about disequilibria.

2.0 Technology and Requirements for Use

Consider a capitalist economy in which two commodities, iron and corn, are produced. One process is known for producing iron. In the iron industry, workers use inputs of iron and corn to produce an output of iron. The output of the iron industry is one ton with the inputs shown in Table 1. Two processes are known for producing corn. Each corn-producing process operates on a specific type of land. The coefficients of production shown in Table 1 are for an output of one bushel corn. These processes can be thought of as examples of joint production. Their outputs are corn and the same quantity of land used as input, unchanged by the production process. Presumably, some of the labor in these processes is used to maintain the land in a given state. For this post, I assume σt is 17/100.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
IIIIII
Labora0,1 = 1a0,2 = 5191/5770a0,3 = (305/494) e(3/20) - σt
Type 1 Land0c1,2 = 10
Type 2 Land00c2,3 = e(3/20) - σt
Irona1,1 = 9/20a1,2 = 1/40a1,3 = (3/1976) e(3/20) - σt
Corna2,1 = 2a2,2 = 1/10a2,3 = (229/494) e(3/20) - σt

The specification of technology is completed by noting the values of parameters for the quantities available of non-produced means of production. For this numerical example, let there be 100 acres of type 1 land and 100 acres of type 2 land. The iron-producing process and each corn-producing process exhibits constant returns to scale, up to the limits imposed by the endowments of land.

I consider stationary states with a net output consisting solely of corn. A bushel corn is the numeraire. Any one of four techniques can be used to produce corn, depending on the requirements for use. The process for producing iron is part of each technique. Table 2 specifies which types of land are fully or partially farmed in each technique. In the Alpha and Beta techniques, both types of land are cultivated, with one type only partially farmed. In the remaining two techniques, one type of land is left totally farrow. Which techniques are feasible depends on the endowments of the land and on the requirements for use.

Table 2: Techniques
TechniqueType of Land
Type 1Type 2
AlphaFully farmedPartially farmed
BetaPartially farmedFully farmed
GammaPartially farmedFarrow
DeltaFarrowPartially farmed

Suppose requirements for use, that is, net output of corn, exceed 55.112 bushels and fall below 80.90. Delta is not feasible. Beta and Gamma are feasible. With Alpha, corn is in excess supply.

2.0 Prices of Production

I have asserted above that only the Beta and Gamma techniques are feasible, given technology, endowments, and requirements for use. A system of prices of production is associated with each technique. For Beta, type 2 land pays a rent. For Gamma, neither type of land pays a rent.

3.1 Prices for Beta

Suppose managers of firms have adopted the Beta technique. Prices of production satisfy the following system of three equations:

(pβ a1,1 + a2,1)(1 + r) + wβ a0,1 = pβ

(pβ a1,2 + a2,2)(1 + r) + wβ a0,2 = 1

(pβ a1,3 + a2,3)(1 + r) + ρ2 c2, 3 + wβ a0,3 = 1

In these equations, pβ is the price of iron, wβ is the wage, ρ2 is the rent per acre for type 2 land, and r is the given rate of profits. The left-hand side (LHS) of each equation is the cost of operating the corresponding process at a unit level. Costs include the cost of previously produced commodities used as raw material or ancillary inputs, the going rate of profits on these costs, rent, and wages. Since type 1 land is not fully cultivated, it obtains no rent. The right-hand side (RHS) is the revenue obtained from the corresponding process.

For prices of production, costs do not exceed revenue for any operated process. Furthermore, supernormal profits cannot be made in any prices.

3.2 Prices for Gamma

Now suppose instead that the Gamma technique is adopted by managers. Prices of production, in analogous notation, must satisfy the following system of equalities and inequalities:

(pγ a1,1 + a2,1)(1 + r) + wγ a0,1 = pγ

(pγ a1,2 + a2,2)(1 + r) + wγ a0,2 = 1

(pγ a1,3 + a2,3)(1 + r) + wγ a0,3 > 1

3.3 The Choice of Technique

Which system of equations and inequalities prevails for a given rate of profits. The analysis of the choice of technique, in models of extensive rent, can still be based on wage curves. In both the Beta and the Gamma techniques, the first two equations for prices of production are in three variables: the price of iron, the wage, and the rate of profits. Thus, one can solve for the wage as a function of the rate of profits. This is the curve labeled 'Type 1 Land' in the left panel in Figure 1 above.

For the Beta technique, one can solve the last equation for the rent on type 2 land, given the solution from the first two equations. This decomposition of the equations shows that land is a non-basic commodity, in Sraffa's terminology. Hence, a tax on land will not affect the price of iron.

The wage curve for type 2 land can be found from the system of equalities and inequalities for the Delta technique. This wage curve is also shown in Figure 1.

Consider the outer frontier of the wage curves in Figure 1. If requirements for use can satisfied by only cultivating that type of land, then the cost-minimizing technique at a given rate of profits is the corresponding technique. That is, Gamma is cost-minimizing for rates of profits between the switch points.

If the technique for the wage curve on the frontier is not feasible, the corresponding type of land will be fully cultivated. To find the cost-minimizing technique drop down to next wage curve at the given rate of profits. In this example, the cost-minimizing technique corresponds to the wage curve on the inner frontier of the wage curves. So Beta is cost-minimizing at low and high rates of profits. The same rate of profits is made in operating both type 1 and type 2 land, and type 2 land pays a rent.

Whether or not type 2 land is introduced into cultivation alongside partial cultivation of type 1 land depends on the rate of profits. When type 2 land is fully cultivated, less of type 1 land is farmed.

4.0 Conclusion

Type 1 land is partially farmed. Whether or not type 2 land is fully farmed or left farrow depends on distribution. For high and low rates of profits (or low and high wages), type 2 land is fully farmed and owners of type 1 land receive a rent. For intermediate rates of profits (or wages), type 2 land is left farrow, and no land receives a rent.

Employment is greater under Gamma than when the Beta technique is adopted. Thus, around the switch point at the lower wage, an increased wage is associated with each worker benefitting and employment being increased. Owners of type 2 land have a stake in how the social question is being decided among workers and capitalists.

Saturday, July 10, 2021

Some Difficulties In Reading Marx

"Let us take the process of circulation in a form under which it presents itself as a simple and direct exchange of commodities. This is always the case when two owners of commodities buy from each other, and on the settling day the amounts mutually owing are equal and cancel each other. The money in this case is money of account and serves to express the value of the commodities by their prices, but is not, itself, in the shape of hard cash, confronted with them. So far as regards use-values, it is clear that both parties may gain some advantage. Both part with goods that, as use-values, are of no service to them, and receive others that they can make use of. And there may also be a further gain. A, who sells wine and buys corn, possibly produces more wine, with given labour-time, than farmer B could, and B on the other hand, more corn than wine-grower A could. A, therefore, may get, for the same exchange-value, more corn, and B more wine, than each would respectively get without any exchange by producing his own corn and wine. With reference, therefore, to use-value, there is good ground for saying that 'exchange is a transaction by which both sides gain.'" -- Karl Marx, Capital, Chapter 5: Contradictions in the General Formula of Capital.

The sheer volume of his work makes Karl Marx difficult to read. Here I concentrate on the three volumes of Capital. In the Progress Publishers edition, they consist of 867 pages, 551 pages, and 948 pages, respectively. Is there a one-semester class in which students are expected to read all of that? If I were a scholar, I suppose I would be required to learn German. I suppose one ought to also look at the secondary and tertiary literature, maybe from one's home country. (The fact that I can assume such literature exists, wherever you are coming from, attests to Marx's importance.)

Another difficulty is in understanding why Marx chose his order of exposition. The introduction to the Grundrisse is an important text on method here, although I gather Marx came to think of the order in the main text of the Grundrisse as backwards. As I understand it, Marx works from higher levels of abstraction to lower levels, with the concrete being overdetermined, in some sense. These levels are supposed to be real abstractions. Reality is not generated out of thought, as in Hegel. One might respond to the first objection here by saying that the distinction between fixed and circulating capital is a volume 2 issue and can be ignored in volume 1. The overall arc is to consider production in volume 1, circulation combined rather mechanically with production in volume 2, and then the unity of production and circulation in volume 3. The decomposition of surplus value into profits, interest on monetary loans, rent, wages for non-productive workers (such as clerks hired by banks or lawyers) cannot be fully explained until volume 3. And Marx never even gets to taxes and the state. But, according to Lenin, I do not understand Capital since I have never read Hegel's Logic, as one might expect of one who has read some of Bertrand Russell.

Capital is an immanent critique, and it is not always easy to be sure of Marx's attitude to what he is writing about. I think that Marx does not expect prices of production to be proportional to labor values, for example. He says as much in a footnote at the end of chapter 5 in volume 1. It does not help the reader that he does not explain this as fully as he ever will until towards the start of volume 3, more than a thousand pages later. The distinction between labor values and exchange values seems to be of no matter in much of the middle of volume 2, where he explains how the wear and tear of long-lived machinery, the value of ancillaries such as fuel to keep machinery running and to light a factory, and raw materials that actually appear changed in the product all contribute to a commodity's value.

I think a major difficulty some have is that a Marx is developing a systems view or structural approach, in some sense. I find many bourgeois commentators seem to object to the labor theory of value based on the feelings of those engaged in a single transaction or on a supposed analysis of a single market. This standpoint seems besides the point to me. When, in volume 1, Marx talks about the capitalist, he is treating the capitalist as "capital personified", and so on.

I think a major point of Marx is why so many have necessary illusions about capitalism. So many perceive human social powers as relationships inherent in objects, such as money capital goods, or land, or inherent in institutions such as markets. This confusion and fetishism by apologists, at a superstructural level, has a role in keeping capitalism going. Marx tries to explain such illusions created by markets.

An issue I have is with readings of Marx as presenting a scientific, positivistic theory of prices and distribution. Even if not fully in the spirit of Marx, I find empirical work with input-output tables of interest, though.

Saturday, July 03, 2021

Structural Dynamics With Extensive Rent

Figure 1: Variation in Switch Points with Time
1.0 Introduction

This post continues my effort to understand how fluke cases can partition parameter spaces in models of prices of production with extensive rent. Some background for this post is here, here, and here.

2.0 Technology

The technology is described by the coefficients of production in Table 1. I assume that requirements for use are such that they cannot be satisfied by cultivating only two types of land. After fully cultivating two types, the third type must also be introduced into cultivation, at least partially. See this post for a slightly longer description of the technology.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
IIIIIIIV
Labora0,1 = 1a0,2 = 1/2a0,3 = 3a0,4(t) = 2.0743 e-0.03648 t
Type 1 Land0b1,2 = 100
Type 2 Land00b2,3 = 10
Type 3 Land000b3,4 = 1
Irona1,1 = 0a1,2 = 1/2a1,3 = 1/8a1,4(t) = 0.3551 e-0.06337 t
Corna2,1 = 1/2a2,2 = 0a2,3 = 0a2,4(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. Only techniques in which all three types of land must be cultivated are listed.

Table 2: Techniques
TechniqueType of Land
Type 1Type 2Type 3
AlphaFully farmedFully farmedPartially farmed
BetaPartially farmedFully farmedFully farmed

3.0 Prices

I consider the system of prices of production. Profits, rent and wages are paid out of the surplus at the end of the year. Each of the four processes contributes an equation to the system of price equations. A bushel corn is the numeraire, and rent must be zero on at least one type of land.

4.0 Variations of the Choice of Technique

In Figures 1 and 2, thin vertical lines partition time into numbered regions. In each numbered region, the variation of the cost-minimizing technique with distribution is qualitatively invariant. The partitions are labeled with a type of patterns of switch points. Region 8 is a narrow range of time between a pattern for the r-order of fertility, which occurs first, and a pattern over the axis for the rate of profits for the order of rentability.

Figure 2: Variation in Switch Points with Time (Start Enlarged)

The maximum rate of profit, the rate of profits at switch points, and the rate of profits for which the rent per acre for the two types of land that pay rent are equal are plotted as functions of time. Heavy solid lines, aside from the bound on the infeasible region, are for switch points on the inner frontier of the wage curves. These solid lines bound areas in which the cost-minimizing technique, Alpha, Beta, or Gamma, is as shown. Dashed lines are for switch points on the outer frontier. For each technique, the dashed lines bound areas in which the order of fertility does not change. Dotted lines are for the rate of profits at which rents are equal. The dashed lines bound areas, for each technique, in which the order of rentability is invariant. Table 3 summarizes how, within each numbered region, the choice of technique, the order of fertility, and the order of rentability vary with the rate of profits.

Table 3: Description of Partitions with Rate of Profits Given
RegionRange of rTechniqueOrder of FertilityOrder of Rentability
10 < r < r1,2Alpha1 ,2, 3ρ1 > ρ2 > 0. ρ3 = 0
r1,2 < r < rmax, 3ρ2 > ρ1 > 0. ρ3 = 0
20 < r < r1,2Alpha1 ,2, 3ρ1 > ρ2 > 0. ρ3 = 0
r1,2 < r < r*ρ2 > ρ1 > 0. ρ3 = 0
r* < r < rmax, 32, 1, 3
30 < r < r1,2Alpha1 ,2, 3ρ1 > ρ2 > 0. ρ3 = 0
r1,2 < r < r*ρ2 > ρ1 > 0. ρ3 = 0
r* < r < r**2, 1, 3
r** < r < rmax, 1Beta2, 3, 1ρ2 > ρ3 > 0. ρ1 = 0
40 < r < r*Gamma1 ,3, 2ρ1 > ρ3 > 0. ρ2 = 0
r* < r < r1,2Alpha1 ,2, 3ρ1 > ρ2 > 0. ρ3 = 0
r1,2 < r < r**ρ2 > ρ1 > 0. ρ3 = 0
r** < r < r***2, 1, 3
r*** < r < rmax, 1Beta2, 3, 1ρ2 > ρ3 > 0. ρ1 = 0
50 < r < r1,3Gamma1 ,3, 2ρ1 > ρ3 > 0. ρ2 = 0
r1,3 < r < r*ρ3 > ρ1 > 0. ρ2 = 0
r* < r < r**3, 1, 2
r** < r < r2,3Beta3 ,2, 1ρ3 > ρ2 > 0. ρ1 = 0
r2,3 < r < r***ρ2 > ρ3 > 0. ρ1 = 0
r*** < r < rmax, 12, 3, 1
60 < r < r1,3Gamma1 ,3, 2ρ1 > ρ3 > 0. ρ2 = 0
r1,3 < r < r*ρ3 > ρ1 > 0. ρ2 = 0
r* < r < r**3, 1, 2
r** < r < r2,3Beta3 ,2, 1ρ3 > ρ2 > 0. ρ1 = 0
r2,3 < r < rmax,1ρ2 > ρ3 > 0. ρ1 = 0
70 < r < r*Gamma1 ,3, 2ρ3 > ρ1 > 0. ρ2 = 0
r* < r < r**3, 1, 2
r** < r < r2,3Beta3 ,2, 1ρ3 > ρ2 > 0. ρ1 = 0
r2,3 < r < rmax,1ρ2 > ρ3 > 0. ρ1 = 0
80 < r < r*Gamma3, 1, 2ρ3 > ρ1 > 0. ρ2 = 0
r* < r < r2,3Beta3 ,2, 1ρ3 > ρ2 > 0. ρ1 = 0
r2,3 < r < rmax,1ρ2 > ρ3 > 0. ρ1 = 0
90 < r < r*Gamma3, 1, 2ρ3 > ρ1 > 0. ρ2 = 0
r* < r < rmax,1Beta3 ,2, 1ρ3 > ρ2 > 0. ρ1 = 0

Some of the fluke cases that partition the parameter space in this example arise in models with circulating capital alone and no scarce unproduced means of production. Some are specific to models with land. At the time for the partition between regions 2 and 3, a fluke switch point between the Alpha and Beta techniques occurs on the inner wage frontier with a wage of zero. This fluke is associated with the emergence of a range of low rates of profits in which the Beta technique is cost-minimizing. The partition between regions 3 and 4 is characterized by a fluke switch point on the inner wage frontier with a rate of profits of zero. This fluke is associated with the emergence of a range of high rate of profits in which the Gamma technique is cost-minimizing.

The three-technique pattern of switch points, defining the partition between regions 4 and 5, is probably the most visually noticeable fluke case depicted in Figure 8. For the coefficients of production at this instant in time, the wage curves for the Alpha, Beta, and Gamma techniques intersect at a single switch point. This switch point is simultaneously on the inner and the outer frontiers of the wage curves. For the rate of profits at which this switch point is defined, all three types of lands are equally fertile, and none of them pay any rent. None of the types of land need be fully cultivated prior to some other type being taken into cultivation.

Fluke cases I christen patterns for the r-order of fertility are specific to models of rent. A fluke switch point on the outer frontier also lies on the wage axis at the time which partitions regions 7 and 8. At this time, the range of the rate of profits in which the order of fertility is type 1, type 3, and type 2 land vanishes over the wage axis. The partition between regions 1 and 2 qualitatively resembles the partition between regions 5 and 6. A switch point on the outer frontier of wage curves occurs at a rate of profits of 100 percent, which is also the maximum rate of profits of the Beta technique. This fluke case is associated with a disappearance of a range of the rate of profits at which the order of fertility is type 2, type3, and type 1 lands.

A pattern for the w-order of fertility occurs at a time of approximately 6.0619. The switch point between type 1 and type 3 lands of the outer occurs at the same wage as the maximum wage for type 2 land. This switch point is associated with the disappearance of a range of wages at which the order of fertility is type 3, type 1, and type 2 lands, given the wage. Since the rate of profits is taken as given throughout this post, this example of a pattern for the w-order of fertility is merely an aside.

Patterns in the order of rentability are also specific to models with land. Consider the partition between regions 6 and 7. This is a pattern over the wage axis for the order of rentability. At a rate of profits of zero, the rent on an acre of type 1 and type 3 land is identical. This fluke case is associated with the disappearance of a range of the rate of profits in which the order of rentability is type 1, type 3, type 2 lands. The partition between regions 8 and 9 is a pattern over the axis for the rate of profits for the order of rentability.

5.0 Conclusions

This post continues to extend my research program. It illustrates that a divergence between the order of fertility and the order of rentability is not a fluke case.

Saturday, June 26, 2021

Background Needed For Debates On The Labor Theory Of Value Versus Marginalism

A Propertarian Getting The Better Of The Wrong Side Of The Argument

I have been looking at some debates on YouTube. Not only is there a culture of twitch streamers debating. There are some, such as Noah Cortez "Booksmarts", critiquing debates. By the way, here is a video providing an overview of (Resnick and) Wolff's academic work, seemingly inspired by the prominence Wolff is gaining with his YouTube presence.

I suppose that this post is aspirational, and some of these points can be argued over. I guess, this post is related. I'd like for the debaters to:

  • Recognize that such a debate topic is orthogonal to a debate on socialism versus capitalism. Thus, Barone's economic calculation problem is not on-topic, without an argument.
  • Know the distinction between use value and exchange value, as in Aristotle (arguably), Adam Smith, Ricardo, and Marx. Use value is about physical properties of commodities, not preferences or utility functions.
  • Know something of the theory of revealed preference, ordinal utility, and cardinal utility.
  • Distinguish between market prices and natural prices, also known as prices of production. Prices of production are defined when outputs are at the level of what Adam Smith calls effectual demand.
  • Distinguish between the short run and long run in, for example, Marshall. Some, maybe, should know that Mises' evenly rotating economy is compatible with an analysis of prices of production. Walras' general equilibrium is a theory of the long run.
  • Recognize prices of production are consistent with utility maximization, as in marginalism.
  • Distinguish prices of production from embodied labor values, also known as employment multipliers. Maybe debaters should be able to contrast labor values with labor commanded.
  • Recognize supply and demand are not functions of price in Smith, Ricardo, and Marx, but are such in marginalism. Well-defined functions for supply and demand were abandoned, more or less, in marginalist long run theory after the CCC.
  • Understand that variations of organic composition of capital suggests, as explained by Marx in his account of the transformation problem, that prices of production cannot be expected to be proportional to labor values.
  • Known that Marx claimed, in Volume 3, that the rate of profits used in calculating prices of production comes from the system of labor values. Max also claimed that total prices and total profits were equal to total labor values and total surplus values, or so I think. I call these some of Marx's invariants.
  • Have some familiarity with national income accounting and Leontief matrices.
  • Have some awareness of empirical results on the labor theory of value as a theory of price.
  • Have some awareness of empirical results on trends in the OCC, the rate of exploitation, and the rate of profits.
  • Have some view on what empirical results test marginalism. Does behavioral economics show that it is both falsifiable and false?

By the way, I do not have a high opinion of Böhm-Bawerk's Karl Marx and the Close of his System.

Saturday, June 19, 2021

On My Research Program

"I know not how I may seem to the world, but as to myself I seem to have been only like a boy playing on the sea-shore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." -- Isaac Newton (apocryphal?)

For the past couple of years, I have been pursuing a research program that I seem to have stumbled upon. I am looking for fluke switch points, where these fluke switch points partition certain parameter spaces. The analysis of the choice of technique is supposed to be qualitatively invariant, in some sense, in each region formed by these partitions, but varies among regions.

I have tried to define a taxonomy for such fluke switch points. Using an example, I have explored structural dynamics. Analyzing fluke switch points in models of fixed capital lets me see maybe more deeply into the incoherence of Austrian and marginalist approaches. Lately, I have been considering a parameter space of relative markups. Even more recently, I have been considering models of extensive rent.

Mathematically, these post-Sraffian models I have been exploring are open. Given parameters characterizing the technology and relative markups among industries, the distribution of income can vary with one degree of freedom. But if the wage, for example, and the size and composition of the net product is given, the rate of profits and the price of each commodity is determined. This is a matter of mathematics, close to accounting. Is my approach compatible with Ajit Sinha's reading of Sraffa's work as an approach akin to geometrical reasoning?

I do not know that Tony Lawson would accept that these models are ontologically open in his sense. Similarly, Nicholas Georgescu-Roegen made a distinction between what he called arithmorphic and dialetic reasoning. In my approach, I emphasize how quantitative perturbations of parameters leads to qualitative change in admittedly static models. I limit myself to discovering structures at the level of mesoeconomics, not visible at the level of individual transactions or an individual (non-vertically integrated) industry.

I rarely comment on whether or not I am taking an empirical economy as given. My approach certainly relates to Leontief input-output matrices, which can be constructed or approximated from National Income and Product Accounts (NIPAs). The econometrician would probably also want prices indices for individual industries. At a given point of time, one might say a process is dominant in each industry. But some firms might be still operating old processes, and others might be introducing new processes with which they hope to make super normal profits for a time. This observation provides some justification for considering the choice of technique. I often postulate continuous declines in coefficients of production, following Pasinetti's lead, or variations in relative markups. Is this a matter of counter-factual reasoning that Sraffa would reject?

I am aware that my models are not set in historical time. This is a point of contention for some Post Keynesian, such as Lars Syll. Do at least some of my partitions have implications for the dynamics of how or whether market prices approach prices of production? This is a question that I will continue not to address. I found intriguing this talk by Ian Wright, with accompanying handout.

Research in flukes may lead to more acceptance of possibility of reswitching, capital reversing, reverse substitution of labor, recurrence of processes. I wonder if somebody that understands something about algebraic geometry could summarize my approach more shortly, but even more abstractly. I hope and wish that I can read sometime somebody extending this research. Some sort of structures definitely seem to exist in these parameter spaces.

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.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
IIIIIIIV
Labora0,1 = 1a0,2 = 1/2a0,3 = 3a0,4(t) = 2.0743 e-0.03648 t
Type 1 Land0b1,2 = 100
Type 2 Land00b2,3 = 10
Type 3 Land000b3,4 = 1
Irona1,1 = 0a1,2 = 1/2a1,3 = 1/8a1,4(t) = 0.3551 e-0.06337 t
Corna2,1 = 1/2a2,2 = 0a2,3 = 0a2,4(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.

Table 2: Techniques
TechniqueType of Land
Type 1Type 2Type 3
AlphaFully farmedFully farmedPartially farmed
BetaPartially farmedFully farmedFully farmed
GammaFully farmedPartially farmedFully farmed
DeltaFully farmedPartially farmedFallow
EpsilonFully farmedFallowPartially farmed
ZetaPartially farmedFully farmedFallow
EtaFallowFully farmedPartially farmed
ThetaPartially farmedFallowFully farmed
IotaFallowPartially farmedFully farmed
KappaaPartially farmedFallowFallow
LambdaFallowPartially farmedFallow
MuFallowFallowPartially 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.

Monday, June 14, 2021

Engels to Sombart in 1895

This transcription is taken from here.

Dear Sir:

Replying to your note of the 14th of last month, may I thank you for your kindness in sending me your work on Marx; I had already read it with great interest in the issue of the Archiv which Dr. H. Braun was good enough to send me, and was pleased for once to find such understanding of Capital at a German University. Naturally I can't altogether agree with the wording in which you render Marx’s exposition. Especially the definitions of the concept of value which you give on pages 576 and 577 seem to me to be rather all-embracing: I would first limit them historically by explicitly restricting them to the economic phase in which alone value has up to now been known, and could only have been known, namely, the forms of society in which commodity exchange, or commodity production, exists; in primitive communism value was unknown. And secondly it seems to me that the concept could also be defined in a narrower sense. But this would lead too far, in the main you are quite right.

Then, however, on page 586, you appeal directly to me, and the jovial manner with which you hold a pistol to my head made me laugh. But you need not worry, I shall "not assure you of the contrary." The logical sequence by which Marx deduces the general and equal rate of profit from the different values of s/C = s/(c + v) produced in various capitalist enterprises is completely foreign to the mind of the individual capitalist. Inasmuch as it has a historical parallel, that is to say, as far as it exists in reality outside our heads, it manifests itself for instance in the fact that certain parts of the surplus value produced by capitalist A over and above the rate of profit, or above his share of the total surplus value, are transferred to the pocket of capitalist B whose output of surplus value remains as a rule below the customary dividend. But this process takes place objectively, in the things, unconsciously, and we can only now estimate how much work was required in order to achieve a proper understanding of these matters. If the conscious co-operation of the individual capitalists had been necessary to establish the average rate of profit, if the individual capitalist had known that he produces surplus value and how much of it, and that frequently he has to hand over part of his surplus value, then the relationship between surplus value and profit would have been fairly obvious from the outset and would presumably have already been described by Adam Smith, if not Petty.

According to Marx's views all history up to now, in the case of big events, has come about unconsciously, that is, the events and their further consequences have not been intended; the ordinary actors in history have either wanted to achieve something different, or else what they achieved has led to quite different unforeseeable consequences. Applied to the economic sphere: the individual capitalists, each on his own, chase after the biggest profit. Bourgeois economy discovers that this race in which every one chases after the bigger profit results in the general and equal rate of profit, the approximately equal ratio of profit for each one. Neither the capitalists nor the bourgeois economists, however, realise that the goal of this race is the uniform proportional distribution of the total surplus value calculated on the total capital.

But how has the equalisation been brought about in reality? This is a very interesting point, about which Marx himself does not say much. But his way of viewing things is not a doctrine but a method. It does not provide ready-made dogmas, but criteria for further research and the method for this research. Here therefore a certain amount of work has to be carried out, since Marx did not elaborate it himself in his first draft. First of all we have here the statements on pages 153-156, III, I, which are also important for your rendering of the concept of value and which prove that the concept has or had more reality than you ascribe to it. When commodity exchange began, when products gradually turned into commodities, they were exchanged approximately according to their value. It was the amount of labour expended on two objects which provided the only standard for their quantitative comparison. Thus value had a direct and real existence at that time. We know that this direct realisation of value in exchange ceased and that now it no longer happens. And I believe that it won’t be particularly difficult for you to trace the intermediate links, at least in general outline, that lead from directly real value to the value of the capitalist mode of production, which is so thoroughly hidden that our economists can calmly deny its existence. A genuinely historical exposition of these processes, which does indeed require thorough research but in return promises amply rewarding results, would be a very valuable supplement to Capital.

Finally, I must also thank you for the high opinion which you have formed of me if you consider that I could have made something better of volume III. I cannot share your opinion, and believe I have done my duty by presenting Marx in Marx's words, even at the risk of requiring the reader to do a bit more thinking for himself. ...

If I knew more, I think I might not agree with Sombart's take on Marx.

If somebody started going on about a theory of value, without context, you might expect them to talk about what people do or should want, about what things are or should be worth in some moral sense. Maybe such a discussion should draw on the philosophical branches of aesthtics or ethics. Or one might expect to see an exposition of some substantial sociological theory explaining the tastes of people, perhaps drawing on history and where they are in society. Or maybe you might expect a formal characterization of utility functions and revealed preferences.

None of the above, though, have anything to do with Marx's law of value. I think he is clear that he does not expect people in a capitalist economy to be conscious of the labor value embodied in commodities. He says such in the section on commodity fetishism:

A commodity is therefore a mysterious thing, simply because in it the social character of men's labour appears to them as an objective character stamped upon the product of that labour; because the relation of the producers to the sum total of their own labour is presented to them as a social relation, existing not between themselves, but between the products of their labour... the value relation between the products of labour which stamps them as commodities, have absolutely no connection with their physical properties and with the material relations arising therefrom. There it is a definite social relation between men, that assumes, in their eyes, the fantastic form of a relation between things...

...Hence, when we bring the products of our labour into relation with each other as values, it is not because we see in these articles the material receptacles of homogeneous human labour. Quite the contrary: whenever, by an exchange, we equate as values our different products, by that very act, we also equate, as human labour, the different kinds of labour expended upon them. We are not aware of this, nevertheless we do it. -- K. Marx, Capital, vol. 1.

The law of value is about a process in a capitalist economy that takes place behind people's backs. You might find somebody telling you that when they go shopping, they do not make decisions on the basis of the relative time it takes to make the commodities which they are choosing between. Nor do businessmen make investment decisions on the basis of the labor embodied in produced commodities, including the labor embodied in the means of production needed to manufacture these commodities. If somebody points out these facts to you, they are agreeing with Marx, not refuting him.

I read Engels as re-iterating these points when he says that the law of value, more or less, is "completely foreign to the mind of the individual capitalist". I also see Engels here echoing what has become known as the historical transformation problem.

Saturday, June 12, 2021

Flukes In A Modification Of An Example With Land

Figure 1: A Partition of a Slice of the Parameter Space
1.0 Introduction

This post presents another example from Woods (1990). It is a modification of this example.

That previous example demonstrates that the order of fertility - the order in which lands of various types and that support different processes of production are taken into cultivation - varies with distribution. If the wage were different, the order of fertility could be different. Furthermore, the order of rentability - the order of lands from high rent to low rent - also varies with distribution. The order of fertility is not necessarily the order of rentability.

This post demonstrates another issue in the order of fertility. It depends, in this sort of open model, if one takes the wage or the rate of profits as given. This variation introduces complications in comparing the order of rentability with the order of fertility.

Some might think that those who succeed under capitalism might do so because of some physical characteristics. Those who are well off might wonder if that is because they "are necessarily better than those who obey, and if strength of body or of mind, wisdom or virtue are always found in particular individuals, in proportion to their power or wealth: a question fit perhaps to be discussed by slaves in the hearing of their masters, but highly unbecoming to reasonable and free men in search of the truth" (Rousseau). Those who believe so are confusing social relationships between people with as relationships between things. And they just do not know price theory.

You might tell me that most mainstream economists are deluded by illusions created by competition. I would reply, "Yes, sure."

2.0 Technology

As previously, this is an example (Table 1) of a capitalist economy in which two commodities, iron and corn, are produced. One process is known for producing iron. In the iron industry, workers use inputs of corn to produce an output of iron. Three processes are known for producing corn. Each corn-producing process operates on a specific type of land. These processes can be thought of as examples of joint production. Their outputs are corn and the same quantity of land used as input, unchanged by the production process. Presumably, some of the labor in these processes is used to maintin the land in a given state.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
IIIIIIIV
Labora0,1 = 1a0,2 = 1/2a0,3 = 3a0,4 = 2
Type 1 Land0b1,2 = 100
Type 2 Land00b2,3 = 10
Type 3 Land000b3,4
Irona1,1 = 0a1,2 = 1/2a1,3 = 1/8a1,4 = 1/3
Corna2,1 = 1/2a2,2 = 0a2,3 = 0a2,4 = 1/4

I complete the specification of technology with unchanged parameters for the quantities available of non-produced means of production. For this numerical example, 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. Each process exhibits constant returns to scale, up to the limit imposed by the given quantities of the types of land.

I continue to consider stationary states with a net output consisting solely of corn. A bushel corn is the numeraire. In each possible technique, iron is produced by operating process I. Table 2 specifies which types of land are fully or partially farmed in each technique.

Table 2: Techniques
TechniqueType of Land
Type 1Type 2Type 3
AlphaFully farmedFully farmedPartially farmed
BetaPartially farmedFully farmedFully farmed
GammaFully farmedPartially farmedFully farmed
DeltaFully farmedPartially farmedFallow
EpsilonFully farmedFallowPartially farmed
ZetaPartially farmedFully farmedFallow
EtaFallowFully farmedPartially farmed
ThetaPartially farmedFallowFully farmed
IotaFallowPartially farmedFully farmed
KappaaPartially farmedFallowFallow
LambdaFallowPartially farmedFallow
MuFallowFallowPartially farmed

3.0 Prices of Production

One can construct wage curves, for each of the Kappa, Lambda, and Mu techniques. In these cases, since final output is such that no type of land is fully cultivated, all rents are zero. Figure 2 depict the wage curves, with each curve labeled by the type of land that is partially farmed under the corresponding technique.

Figure 2: Wage Curves

3.1 One Type of Land Fully Cultivated, One Partially Farmed

These wage curves can be used to analyze rents when requirements for use are high enough such that they cannot be satisfied by only cultivating one type of land. Accordingly, suppose they can only be satisfied when farmers introduce a second type of land into cultivation, after fully farming the most fertile land. This assumption implies parameters lie within Region 3 in the analysis elaborated in Section 4.

Suppose the wage is taken as given. For a low wage, between zero and approximately 0.261 bushels per person-year, Type 2 land is the most fertile. Its wage curve lies on the outer envelope in Figure 2. So the Zeta technique will be adopted. On the other hand, consider a higher wage up to 3/10 bushels per person-year. The Delta technique, in which Type 1 land is fully cultivated and Type 2 land is partially cultivated, is cost-minimizing.

The same techniques are cost-minimizing, under these assumptions about requirements for use, if the rate of profits is taken as given. For a low rate of profits, below approximately 62 percent, the Delta technique is cost minimizing. For a higher rate of profits, up to 100 percent, farmers will adopt the Zeta technique.

Figure 3 graphs rent against the wage. When the Zeta technique is operated, Type 2 land is fully cultivated and is the only type of land that pays a rent. The analogous conclusion holds for Type 1 land. The rate of profits increases to the left in the figure.

Figure 3: Rent When Two Types of Land are Taken In Cultivation

The assumption, in this example, that only one land is fully cultivated illustrates the dependence of the order of fertility on distribution. It does not allow, however, one to contrast the importance of whether the wage or the rate of profits is taken as given. Nor does it illustrate the difference between the order of fertility and the order of rentability.

3.2 Two Types of Land Fully Cultivated, One Partially Farmed

Accordingly, suppose requirements for use are such that more than two types of land must be cultivated. Which type of land is partially farmed can be determined by looking at the inner envelope of the wage curves in Figure 2. Given the wage, the order of fertility, from most fertile to least fertile, is Type 2, Type 1, and Type 3 lands. The switch point occurs at a wage exceeding the maximum wage when Type 3 lands pay no rent.

Figure 4 graphs rent versus the wage in this case. For low wages, the order of rentability, from Type 2 to Type 1 to Type 3 lands, matches the order of rentability. For high wages, the order of rentability is Type 1, Type 2, Type 3. The order of rentability no longer matches the order of fertability.

Figure 4: Rent When Three Types of Land are Taken In Cultivation

By constrast, suppose the rate of profits is taken as given. For low rates of profits, the order of fertility is Type 1, Type 2, Type 3. For high rates of profits, it is Type 2, Type 1, Type 3. Figure 5 graphs the rent per acre, in this case, against the rate of profits. I also indicate the location of the switch point for wage curves on this graph. The order of rentability matches the order of fertility for low and high rates of profits, but not for intermediate rates of profits.

Figure 5: Another View of Rent When Three Types of Land are Taken In Cultivation

4.0 Partition of Part of the Parameter Space

I now do my usual thing of partitioning some selection of the parameter space. See Figure 1 at the top of this post. In each numbered region, the analysis of the choice of technique does not change, in some sense. Table 3 provides a brief characterization of each region, with perturbations in the given wage.

Table 3: Description of Partitions with Wage Given
RegionWageRentsTechniqueSummary
10 < w < w*ρ1 = ρ2 = ρ3 = 0LambdaType 2 partially farmed. Types 1 and 3 fallow.
w* < w < wmax, 1ρ1 = ρ2 = ρ3 = 0KappaType 1 partially farmed. Types 2 and 3 fallow.
20 < w < w*ρ1 = ρ2 = ρ3 = 0LambdaType 2 partially farmed. Types 1 and 3 fallow.
w* < w < wmax, 2ρ1 > 0. ρ2 = ρ3 = 0DeltaType 1 fully farmed. Type 2 partially farmed. Type 3 fallow.
30 < w < w*ρ2 > 0. ρ1 = ρ3 = 0ZetaType 2 fully farmed. Type 1 partially farmed. Type 3 fallow.
w* < w < wmax, 2ρ1 > 0. ρ2 = ρ3 = 0DeltaType 1 fully farmed. Type 2 partially farmed. Type 3 fallow.
40 < w < w1ρ2 > ρ1 > 0. ρ3 = 0AlphaOrder of rentability matches order of fertility, given wage.
w1 < w < wmax, 3ρ1 > ρ2 > 0. ρ3 = 0AlphaOrder of rentability differs from order of fertility, given wage.

Section 3 emphasizes that how the choice of technique is analyzed depends on whether or not the wage or the rate of profits is taken as given. Table 4 describes the regions in Figure 1, assuming the rate of profits is given. The interesting aspect of these tables comes from contrasting the description of Region 4 in Tables 3 and 4.

Table 4: Description of Partitions with Rate of Profits Given
RegionRate of ProfitsRentsTechniqueSummary
10 < r < r*ρ1 = ρ2 = ρ3 = 0KappaType 1 partially farmed. Types 2 and 3 fallow.
r* < r < rmax, 2ρ1 = ρ2 = ρ3 = 0LambdaType 2 partially farmed. Types 1 and 3 fallow.
20 < r < r*ρ1 > 0. ρ2 = ρ3 = 0DeltaType 1 fully farmed. Type 2 partially farmed. Type 3 fallow.
r* < r < rmax, 2ρ1 = ρ2 = ρ3 = 0LambdaType 2 partially farmed. Types 1 and 3 fallow.
30 < r < r*ρ1 > 0. ρ2 = ρ3 = 0DeltaType 1 fully farmed. Type 2 partially farmed. Type 3 fallow.
r* < r < rmax, 1ρ2 > 0. ρ1 = ρ3 = 0ZetaType 2 fully farmed. Type 1 partially farmed. Type 3 fallow.
40 < r < r1ρ1 > ρ2 > 0. ρ3 = 0AlphaOrder of rentability matches order of fertility, given rate of profits.
r1 < r < r*ρ2 > ρ1 > 0. ρ3 = 0AlphaOrder of rentability differs from order of fertility, given rate of profits.
r* < r < rmax, 3ρ2 > ρ1 > 0. ρ3 = 0AlphaOrder of rentability matches order of fertility, given rate of profits.

5.0 Conclusion

This post is not about what is wrong with aggregating capital. Nor is it about aggregate production functions. It is an exploration of an open model, and this model is therefore not a marginalist general equilibrium model. It synthesizes some ideas in Sraffa (1925) and Sraffa (1960). I think especially of Sraffa's comments on Wicksteed's distinction between descriptive and functional curves, between spurious and genuine margins.

References
  • Piero Sraffa. 1925. Relazioni fra costo e quantita prodotta. Annali di Economia 2. Trans. by A. Roncaglia and J. Eatwell.
  • Piero Sraffa. 1960 The Production of Commodities by Means of Commodities Cambridge University Press.
  • J. E. Woods. 1990. The Production of Commodities: An Introduction to Sraffa. Atlantic Highlands: Humanities Press.

Wednesday, June 09, 2021

Does The Existence Of Wine Refute The Labor Theory Of Value?

Victor Magarino Debating "Ubersoy" On The Labor Theory Of Value

The above is one in a series (Richard Wolff versus "Destiny", Slavoj Zizek versus Jordan Peterson) in which the pro-capitalist/anti-socialist side is represented by somebody seemingly almost completely ignorant of the topic they are pretending to discuss. Magarino needs a better interlocutor. I can find some related debates on YouTube with a more level playing field.

That said, I think Magarino needs a better answer to the question in this post title. I do not think that this question is answered by pointing out that some work is needed to maintain the vats while wine is aging. J. R. McCulloch was supposedly extremely confused on this question when it was first(?) posed.

Wine illustrates two challenges to the LTV: the use of natural resources in limited supply and the effects of time. Land poses some difficulties for the definition of prices of production, but these difficulties do not lead to the restoration of the marginalist theory of supply and demand. The latter challenge is addressed in the so-called transformation problem.

First, suppose that some high quality wine is made from varietal grapes grown on only certain lands of varying fertility, and these lands can be only used for these grapes. Oenophiles want a distinct terroir for this wine. If the amount of this wine that is in an economy's overall social product is taken as given, one can solve for prices of production with no problem. The rent of those lands fully under cultivation drops out of the price system. Labor values can also be calculated for the given production system, using the marginal land. Which land is marginal, however, is found endogenously with variations in the level and composition of final demand.

Now suppose lands of various qualities are not so rigidly specialized. Several production processes can be operated, in parallel, on different plots of land of a given quality, and some of these processes might produce different commodities. This is a problem of intensive rent. As I understand it some of the problems of joint production arise here. No system of prices of production might exist for a rate of profits of zero, even though a solution might exist for a range of postive rates of profits. The analysis of the choice of technique might yield a non-square Leontief matrix, and the system of prices of production need not be unique. The level and composition of final demand might enter into the determination of prices of production in a way that contrasts with how final demand does not matter for single production. I do not see any necessity, however, to address requirements for use by introducing unobservable utility functions.

I am not sure what these difficulties with intensive rent have to say about the LTV. As I recall, Ian Steedman's examples with negative labor values and positive prices arose in models of joint production.

But put these issues with land aside, and consider the issue of time. (I do not take this discussion to be about the time preferences of consumers.) First, the mere fact that some production process takes a long time to complete does not pose any difficulty for the LTV, in and of itself. Ricardo is quite clear on this in Section III, Chapter 1 of the third edition of his Principles. One way of understanding the LTV is as a theory of relative prices. Relative prices of production vary from relative labor values only if their capital-intensities (also known as the Organic Composition of Capital) vary from one another, in some sense. I gather that the point of the wine example is to suggest that here is a commodity that has a much higher OCC than the average. I agree that, in a simple analysis, this suggests that the LTV might not apply here. (In a full analysis, ranking commodities by the capital-intensities of their production methods poses some difficulties.) Once again, though, this analysis is fully consistent with a revived classical political economy. A much debated question is whether or not an analysis of prices of production receives support or needs to be supplemented by considering labor values.

One difficulty that I have with objections to the LTV based on this example of wine is that it is about an individual price of an individual commodity. The LTV is about the generality of produced commodities in a capitalist economy. Magarino tends to emphasize empirical results found from Leontief input-output matrices for empirical economies across specific countries and specific times. Whatever one might think of the level of aggregation, the methodologies of these investigations, the question of concrete versus abstract labor, and so on, this is the right setting for investigation. One might say it is a question of mesoeconomics. Based on my understanding of the transformation problem and of prices of production, I find surprising how well the LTV is supported empirically as a theory of relative prices.

Update (12 June 2021): I quickly read Chapter 13, production time, last night in volume 2 of Marx's Capital. It discusses how production time can be longer than the working period. He specifically mentions wine as an example of a commodity whose production process might require time for the forces of nature to act without labor being expended. Whatever interest this fairly concrete discussion may have, it does not address the question in the post title.

Tuesday, June 01, 2021

Flukes In An Example With Land

Figure 1: A Partition of a Slice of the Parameter Space
1.0 Introduction

This post tells a story in which owners of a certain type of land find the amount of their land needed to produce net output declines. Wages stay constant, and the rent for some landlords increases. This example is generalized from Woods (1990).

2.0 Technology

This an example (Table 1) of a capitalist economy in which two commodities, iron and corn, are produced. One process is known for producing iron. In the iron industry, workers use inputs of corn to produce an output of iron. Three processes are known for producing corn. Each corn-producing process operates on a specific type of land. These processes can be thought of as examples of joint production. Their outputs are corn and the same quantity of land used as input, unchanged by the production process. Presumably, some of the labor in these processes is used to maintin the land in a given state.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
IIIIIIIV
Labora0,1 = 1a0,2 = 1/2a0,3 = 3a0,4 = 1
Type 1 Land0b1,2 = 100
Type 2 Land00b2,3 = 10
Type 3 Land000b3,4
Irona1,1 = 0a1,2 = 1/2a1,3 = 1/8a1,4 = 1/10
Corna2,1 = 1/2a2,2 = 0a2,3 = 0a2,4 = 0

The specification of technology is completed by noting the values of parameters for the quantities available of non-produced means of production. For this numerical example, 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. Each process exhibits constant returns to scale, up to the limit imposed by the given quantities of the types of land.

I consider stationary states with a net output consisting solely of corn. A bushel corn is the numeraire. Table 2 lists the possible techniques, along with the range of person-years employment consistent with each technique. The process for producing iron is part of each technique. Table 2 specifies which types of land are fully or partially farmed in each technique.

Table 2: Techniques
TechniqueType of LandLabor Force
Type 1Type 2Type 3MinMaximum
AlphaFully farmedFully farmedPartially farmed350350 + (44/b3,4)
BetaPartially farmedFully farmedFully farmed250 + (44/b3,4)350 + (44/b3,4)
GammaFully farmedPartially farmedFully farmed100 + (44/b3,4)350 + (44/b3,4)
DeltaFully farmedPartially farmedFallow100350
EpsilonFully farmedFallowPartially farmed100100 + (44/b3,4)
ZetaPartially farmedFully farmedFallow250350
EtaFallowFully farmedPartially farmed250250 + (44/b3,4)
ThetaPartially farmedFallowFully farmed44/b3,4100 + (44/b3,4)
IotaFallowPartially farmedFully farmed44/b3,4250 + (44/b3,4)
KappaaPartially farmedFallowFallow0100
LambdaFallowPartially farmedFallow0250
MuFallowFallowPartially farmed044/b3,4

I might as well step through arithmetic for one of these techniques to illustrate how to find the range of employment compatible with it. Consider Lambda, in which type 2 land is partially farmed. When this type of land is totally farmed, process III is operated at the level of T2/b2,3. The quotient of 80 acres and 1 acre per bushels is 80 bushels, the maximum gross quantity of corn that can be produced with this technique. The amount of iron required as input into process III at this level is a2,3 q3, or the product of 80 bushels and 1/8 tons per bushel, that is 10 tons. Producing 10 tons iron requires an input of a2,1 q1 = (1/2) 10 = 5 bushels, leaving a net output of 75 bushels for the economy as a whole. The maximum level of employment with the Lambda technique is a0,1 q1 + a0,3 q3, that is, 1 x 10 + 3 x 80 or 250 person-years. In a stationary state, the level of consumption is 75/250 = 3/10 bushels per person-year.

3.0 Prices of Production

Each of the four processes yields an equation for the system of prices of production. These equations are:

(1/2)(1 + r) + w = p
(1/2) p (1 + r) + ρ1 + (1/2) w = 1
(1/8) p (1 + r) + ρ2 + 3 w = 1
(1/10) p (1 + r) + b3, 4 ρ3 + w = 1

The variables are:

  • p: The price of iron (bushels per ton)
  • w: The wage (bushels per person-year).
  • r: The rate of profits.
  • ρj, j = 1, 2, 3: Rent for type j land (bushels per acre).

All must be non-negative. The left-hand sides shows the cost of operating the process, including the rate of profits on capital goods. Profits, rent, and wages are paid out at the end of the year. The right-hand sides of these equations show the revenues obtained by operating each process at a unit level. For prices of production, the revenues obtained, for operated processes, just cover costs. Costs exceed revenues for non-operated process.

In this model of extensive rent, a final equation must be satisfied by the rents:

ρ1 ρ2 ρ3 = 0

This equation states that the rent on at least one type of land is zero.

4.0 The Order of Fertility and the Order of Rentability

Land of a type that is left uncultivated, either partially of fully, is in excess supply and pays no rent. Three possibilities for the price system arise. Suppose type 1 land has a rent of zero (ρ1 = 0). Then the first two equations form a system in three variables: the price of iron, the wage, and the rate of profits. They have one degree of freedom in their solution. The wage, as a function of the rate of profits, is the curve labeled ‘Type 1’ in Figure 2. By the same logic, the curve labeled ‘Type 2’ arises from the first and third equation. The first and fourth equation yield the curve labeled as ‘Type 3’.

Figure 2: Wage Curves Unaltered by Pertubations in Coefficients of Production for Land

Suppose that final demand is low enough that it can be satisfied by only cultivating one type of land, and that land is not even fully cultivated. Then all rents are zero, and the outer envelope of the wage curves shows which technique is cost-minimizing. In this simple model, only type 3 land will be cultivated to produce corn. That is, the most fertile or efficient land is type 3, whatever the distribution of income. If final demand is high enough that cultivating type 3 land, along with producing iron, cannot yield a sustainable economy, a second type of land must be cultivated. Which less efficient process will be adopted depends on distribution. Even so, type 3 land will then pay a rent. Consider an even higher level of final demand, and suppose fully cultivating two types of land is not enough to satisfy the demand. Then the inner envelope of the wage curves shows the last type of land to be cultivated, and that land pays no rent. This analysis allows one to order lands, given, say, the wage, by their order of fertility.

The above analysis allows one to identify which lands will pay rent and which will pay no rent. Given the wage or the rate of profits, the remaining prices of production can be found for the marginal no-rent land. Positive rents can be found then from the price equations for the corresponding processes.

In the example, suppose two lands are fully cultivated, and one type of land is only partially cultivated. Figure 3 plots rents versus the wage for one value of the parameter b3, 4. For any given wage, one can sort land from the one with the highest rent, to a lower rent, to the type of land that pays no rent, given these assumptions. This is the order of rentability.

Figure 3: Rents With a High Coefficient of Production for Type 3 Land

Figure 4 plots rents as a function of the wage for a slightly lower b3, 4. This perturbation leaves unchanged the location of wage curves in Figure 2. It does result in increased rents on type 3 lands though.

Figure 4: A Fluke Coefficient of Production for Type 3 Land

Figure 5 shows rents at an even lower b3, 4. I consider Figure 4 to be fluke case. Notice at the highest feasible wage in Figure 4, the rent per acre for type 1 and type 3 lands are equal. This fluke is associated with the disappearance of the range of the highes wages in which the order of rentability differs from the order of fertility.

Figure 5: Rents With a Low Coefficient of Production for Type 3 Land

5.0 A Partition of Part of the Parameter Space

Figure 1, at the top of this post, depicts a partition of a projection of the parameter space suggested by the above exploration. Employment is plotted against b3,4 the coefficient of projection for the land cultivated in the last process listed in Table 1. Each partition is a fluke case, in some sense. In each numbered region, the variation of the analysis of the choice of technique with distribution does not change qualitatively, while it does vary among regions. Table 3 specifies, for each region, which technique is cost-minimizing at a given wage and the range of employment. I consider the order of types of land by rent to be part of the analysis of the choice of technique. Table 3 specifies this order as well, along with its variation with the wage. A wage at which this order of rentability changes that is not a switch point in Figure 2 appears in regions 4, 6, and 7.

Table 3: Description of Partitions
RegionWageTechniqueSummary
10 < w < wmaxMuType 3 land partially farmed. ρ1 = ρ2 = ρ3 = 0
20 < w < w*IotaType 3 land fully farmed, type 2 land partially farmed. ρ1 = ρ2 = 0, ρ3 > 0
w* < w < wmaxThetaType 3 land fully farmed, type 1 partially farmed. ρ1 = ρ2 = 0, ρ3 > 0
30 < w < w*IotaType 3 land fully farmed, type 2 land partially farmed. ρ1 = ρ2 = 0, ρ3 > 0
w* < w < wmaxGammaGiven wage, order of rentability is order of fertility. ρ2 = 0, ρ3 > ρ1 > 0
40 < w < w*IotaType 3 land fully farmed, type 2 land partially farmed. ρ1 = ρ2 = 0, ρ3 > 0
w* < w < w1GammaGiven wage, order of rentability is order of fertility. ρ2 = 0, ρ3 > ρ1 > 0
w1 < w < wmaxGammaGiven wage, order of rentability is not order of fertility. ρ2 = 0, ρ1 > ρ3 > 0
50 < w < w*BetaGiven wage, order of rentability is order of fertility. ρ1 = 0, ρ3 > ρ2 > 0
w* < w < wmaxGammaGiven wage, order of rentability is order of fertility. ρ2 = 0, ρ3 > ρ1 > 0
60 < w < w1BetaGiven wage, order of rentability is not order of fertility. ρ1 = 0, ρ2 > ρ3 > 0
w1 < w < w*BetaGiven wage, order of rentability is order of fertility. ρ1 = 0, ρ3 > ρ2 > 0
w* < w < wmaxGammaGiven wage, order of rentability is order of fertility. ρ2 = 0, ρ3 > ρ1 > 0
70 < w < w1BetaGiven wage, order of rentability is not order of fertility. ρ1 = 0, ρ2 > ρ3 > 0
w1 < w < w*BetaGiven wage, order of rentability is order of fertility. ρ1 = 0, ρ3 > ρ2 > 0
w* < w < w2GammaGiven wage, order of rentability is order of fertility. ρ2 = 0, ρ3 > ρ1 > 0
w2 < w < wmaxGammaGiven wage, order of rentability is not order of fertility. ρ2 = 0, ρ1 > ρ3 > 0

To illustrate the orders of fertility and rentability, consider a line segment running upwards through regions 1, 2, 4, and 7. Along such a line, employment increases. For a positive wage less than w*, the cost-minimizing technique is Mu, Iota, and Beta, depending on how much employment has expanded with the net final demand for corn. Thus, for a low wage, land is introduced into production, in order of declining fertility, as type 3, type 2, and type 1. In region 7, the order of lands from high rent to low, if the wage is between w1 and w*, is type 3, type 2, and type 1. If the wage is less than w1, the order from high rent to low, is type 2, type 3, and type 1. Whether or not the order of rentability matches the order of fertility can vary with distribution.

On the other hand, if the wage exceeds w*, the sequence of techniques associated with increasing final demand is Mu, Theta, and Gamma. That is, land, in order of decreasing fertility, is type 3, type 1, and type 2. The maximum wage decreases as less fertile land is taken under cultivation. The order of fertility itself depends on distribution. In this range of wages, region 7 also illustrates the dependence of the order of rentability on distribution.

Figures 3, 4, and 5 illustrate the partition around regions 7 and 6. A decrease of the coefficient b3,4 is associated with the loss of the possibility of a mismatch of the order of rentability and the order of fertility at high wages. A further decrease leads into region 5, where a mismatch in these orders is no longer possible at low rents. An even further decrease is associated with the least fertile lands being taken out of cultivation and the possibility of a larger economy.

Region 3 has an interesting property. The number of types of land that is farmed varies with distribution. For a low wage, workers in the corn industry are employed on types 2 and 3 land; type 1 land is left fallow and, along with type 2 land, pays no rent. For a high enough wage, types 1 and 3 land are fully farmed, and type 2 land is partially farmed and pays no rent. A landlord whose income is generated solely from ownership of type 1 land is better off if wages are high. Incidentally, here is a case in which the quantity demanded of a type of land is greater only when its price is higher.

6.0 Conclusion

This post illustrates that complications are introduced into my analysis of fluke switch points by considering models of land. In particular, I have located fluke parameters in which pertubations around these parameters result in the order of rentability varying from the order of fertility. This qualitative change is independent of switch points on intersecting wage curves. Obviously, this is one relatively simple example. I have a long ways to go in thinking about land.

References
  • J. E. Woods. 1990. The Production of Commodities: An Introduction to Sraffa. Atlantic Highlands: Humanities Press.