Fixed Capital And Extensive Rent

This post is a problem statement.

Models of the production of commodities with circulating capital alone have certain nice properties. I refer to models in which commodities are produced by means of commodities, with a certain circular structure in production. Direct labor inputs are assumed to be necessary to operate each process in the technology.

The choice of technique can be analyzed in models with circulating capital alone by constructing the outer envelope of the wage curves for each technique. Each wage curve slopes down. The wage, for a technique, is lower the higher the rate of profits. The cost-minimizing technique at a given rate of profits is unique, except at switch points. The "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang 2019). The wage and prices of production are unique functions of the rate of profits. If a technique exists with a defined wage and prices of production at a given rate of profits, then a cost-minimizing technique exists. A market algorithm (Bidard 1990) converges, without going into a cycle.

None of these properties necessarily hold in models with joint production. For example, Bidard & Klimovsky (2004) define fake switch points as intersections on the outer wage frontier at which the cost-minimizing technique does not vary.

But they do in a model of pure fixed capital, with the exception that wage curves can slope up when not on the frontier. These models are non-interlocked systems in which old machines cannot be transferred among sectors. Each process produces exactly one finished good, such as a consumption good; a good used as circulating capital; or a new, possibly long-lived machine. Old machines are intermediate goods. Old machines cannot be consumer goods. Every finished good has exactly one primary process for producing it, in which an intermediate good does not enter as an input. In each sector, the secondary processes completely use up the old machine produced by the primary process in that sector, with no other intermediate goods as inputs. They produce the same finished good, possibly jointly with an intermediate good. Joint utilization of machines does not exist in any process. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete.

Most of the properties of circulating capital also hold in models of extensive rent. Extensive rent occurs when multiple types of land must be cultivated to satisfy requirements for use. Only one process is operated on each type of land. All but one of the types of cultivated land are fully farmed, except in fluke cases, to the extent of their endowment. With only one price prevailing for corn and only one rate of profits being obtained in the system of prices of production, different amounts of rent per acre must be paid on the different types of land that are fully farmed. The type of land that is only partially farmed is not scarce and does not pay a rent. The cost-minimizing technique can be found from wage curves, but it does not correspond to the technique on the outer frontier. Requirements for use determine which techniques are feasible, and only a feasible technique can be cost-minimizing.

My claim is that a model combining fixed capital and extensive rent lacks more properties characteristic of circulating capital than either does alone. For example, the wage frontier, defined by the wage curves for cost-minizing techniques at each rate of profits, can slope upward. A numerical example can demonstrate this. So I am curious if an elaboration of the model specified in this post works for this purpose.

Tables 1 and 2 specify the technology for a simple model producing multiple commodities and combining fixed capital and extensive rent. A numerical example results by setting each coefficient of production not equal to zero or unity to some positive value.

Table 1: Inputs for Processes Comprising the Technology

Input	Industry
	Machine	Corn
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	c1,1 = 0	c1,2	c1,3	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4	c2,5
Corn	a1,1	a1,2	a1,3	a1,4	a1,5
New Machines	a2,1 = 0	a2,2 = 1	a2,3 = 0	a2,4 = 1	a2,5 = 0
Type 1 Old Machines	a3,1 = 0	a3,2 = 0	a3,3 = 1	a3,4 = 0	a3,5 = 0
Type 2 Old Machines	a4,1 = 0	a4,2 = 0	a4,3 = 0	a4,4 = 0	a4,5 = 1

Table 2: Outputs for Processes Comprising the Technology

Output	Industry
	Machine	Corn
	I	II	III	IV	V
Corn	b1,1 = 0	b1,2	b1,3	b1,4	b1,5
New Machines	b2,1 = 1	b2,2 = 0	b2,3 = 0	b2,4 = 0	b2,5 = 0
Type 1 Old Machines	b3,1 = 0	b3,2 = 1	b3,3 = 0	b3,4 = 0	b3,5 = 0
Type 2 Old Machines	b4,1 = 0	b4,2 = 0	b4,3 = 0	b4,4 = 1	b4,5 = 0

In the tables, each column specfies the inputs and outputs of corn, new machines, and old machines of each type needed to operate that process at a unit level. Inputs of labor and each of the two types of land are also specified. I assume constant returns to scale, up to the limits imposed by endowments of land. Each process requires the same time, a year, to complete. The first process uses inputs of labor and corn to manufacture new machines. The second and third processes produce corn on the first type of land. The remaining two processes also produce corn, but on the other type of land.

Corn is the numeraire and the only consumption good. The total acres of each type of land are also part of the data. Requirements for use are specified by the quantity of corn in the net output of the economy.

Table 3 lists the techniques of production available. In Alpha, Beta, Gamma, and Delta, land is not scarce. Only one type of land is farmed, and the quantity farmed does not exceed its endowment. Beta differs from Alpha in that a machine of the first type is used for its full physical life. Likewise, Delta differs from Gamma in the same way for a machine of the second type. Both types of land are farmed in the remaining eight techniques, and ownership of one type of land obtains a rent. The techniques vary in which land receives a rent and in the economic lifetime of a machine.

Table 3: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, IV	Fallow	Partially farmed
Delta	I, IV, V	Fallow	Partially farmed
Epsilon	I, II, IV	Fully farmed	Partially farmed
Zeta	I, II, III, IV	Fully farmed	Partially farmed
Eta	I, II, IV, V	Fully farmed	Partially farmed
Theta	I, II, III, IV, V	Fully farmed	Partially farmed
Iota	I, II, IV	Parially farmed	Fully farmed
Kappa	I, II, III, IV	Parially farmed	Fully farmed
Lambda	I,II, IV, V	Parially farmed	Fully farmed
Mu	I, II, III, IV, V	Parially farmed	Fully farmed

Which techniques are feasible for a given level of net output? What are the quantity flows? What are the price systems? Which techniques are cost-minimizing? Can I specify numerical values that illustrate interesting phenomena? These questions might be worth answering.

This combination of fixed capital and extensive rent should also demonstrate that approaches to the analysis of the choice of technique customized for each apply to their combination. If the price of some machines or rent on a farmed type of land is negative at a given rate of profits, that technique cannot be cost-minimizing at that rate, for example.

Three Cases In Which The Labor Theory of Value Is A Simple Theory Of Price

1.0 Introduction

This post should be mostly uncontroversial. The labor theory of value (LTV) explains prices in certain special cases.

2.0 An Economy of Self-Employed Artisans

First, consider a pre-capitalist economy with no employer and employees. Commodities are manufactured for the market by self-employed artisans. They might have titles like carpenters, weavers, or blacksmiths. In this case, market prices tend to be (proportional to) labor values.

Adam Smith described this case. Karl Marx can be read as presenting this case in chapter 1 of volume 1 of Capital. In that chapter, he has independent producers. He introduces concepts in a certain order. Can you find anything in that chapter about workers being paid wages?

3.0 When the Rate of Profits is Zero

I now turn my attention to a sort of capitalist economy. Employers hire workers for wages. The workers produce commodities for the employers to obtain the revenues from selling them on the markets.

But here the capitalists do not have the ability to generally obtain a profits. Market prices tend towards labor values.

Joseph Schumpeter wrote about this case in The Theory of Economic Development. He has profits being temporary, as innovating entrepreneurs destroy the smooth reproduction of a capitalist economy, until it tends to settle down again.

4.0 When Capital-Intensity Does Not Vary Among Industries

I now consider a capitalist economy in which the capitalists are generally able to obtain a rate of profits. Suppose capital-intensity does not vary among industries. The ratio of the total value of the capital goods used up in production to the labor directly used in producing each commodity is the same, whatever commodity is being produced. You might as well use labor values when adding up capital goods.

Then market prices tend towards labor values.

Karl Marx wrote about this case in much of the first volume of Capital. Obviously, he does not write about eigenvalues and Leontief input-output matrices.

5.0 Conclusion

Modern reformulations of classical political economy, of course, are not confined to the above special cases. I find of particular interest the mathematical dual of the last case. David Ricardo and Karl Marx discuss this dual case - not with that jargon. Ricardo's last essay, unpublished in his lifetime, was about that. Marx discussed this dual case towards the start of the third volume of Capital.

But really their interest was not in a theory of prices. Ricardo and Marx both investigated labor values in their explanations of profits, of the ability of capitalists to obtain returns on their investments.

Prohibitions On Tactics In Negotiating Your Wage

Pro-capitalists often depict wages as being determined by free negotiations between employers and employees. They ignore conventions and norms surrounding such negotiations. And laws regulate and prohibit various tactics, especially some that require collective action in solidarity with other workers.

In explaining these restrictions, I concentrate on vignettes in the United States of America. A theory of dual labor markets applies here. In the formal sector, you can expect benefits, paid vacations, defined limitations on working hours, and weekends off. In the other sector, not so much. I have seen talk about the precarity. It is clear that we need more labor unions in the USA. The United Auto Workers has been promising lately.

The rise of labor unions in the USA occurred against massive reactionary violence on the part of the minions of oligarchs, including the state. For example, President McKinley sent the Army to Idaho in 1899 to round up striking workers and others - they did not care who - and put them in the 'bullpen'. This was basically a concentration camp. The robber barons could also call on their own private police force. I refer especially to the Pinkerton Detective Agency.

I skip ahead to the National Labor Relations Act of 1935, also known as the Wagner act. This law legalized labor unions, collective bargaining, and strikes. It prohibited company unions. And it established the National Labor Relations Board (NLRB).

The Taft-Hartley Act was passed in 1947, over Truman's veto. It permitted states to pass so-called 'Right to Work' laws. It prohibited sympathy strikes and general strikes. Union leaders would like to sign contracts in which all jobs in certain grades or categories are union jobs. Employees hired for those jobs can be non-union, but they must pay union fees. A non-union worker might as well join the union to have their voice heard. The misnamed 'right to work' laws prohibit such free contracts. A sympathy strike is called by a union in solidarity with other workers. For example, pilots might go on strike when flight attendants are striking. You could imagine the workers in a city all going on strike in support of a political issue. But this is prohibited in the USA.

The above presents an overview of some elements of history in the USA in the setting in which workers 'freely' negotiate their wage. I focus on the prospect of labor organizing. A more comprehensive history would include the perception of those running the Federal Reserve that they should raise interest rates when wages rise and the general hostility to labor, particularly in the Republican party, since Reagan was president. Strangely enough, Reagan was the only president of a labor union (the Screen Actors Guild) ever to be elected president.

Ludwig Von Mises, Crackpot Conspiracy Theorist?

I get this from Sam Tanenhaus' new biography of William F. Buckley, Jr. The index entry for Von Mises is in error.

The John Birch Society was an extreme reactionary organization in the USA. Bob Welch was its leader and founded it near the start of 1959. Welch had a book, The Politician, later known as the Blue Book, that explained what he was about. According to Welch, George Marshall, Dwight Eisenhower, Earl Warren, the Dulles brother, and on and on were all conscious, dedicated Soviet agents.

Buckley had a magazine, National Review. And he was known for trying to show that American conservatives had views worth taking seriously. They are not all crackpots. So he tried to convey an impression that Bob Welch was not welcome in his movement.

The John Birch Society had a monthly magazine, American Opinion. Von Mises was on its board and had articles published in it. I have read quite a bit of Von Mises and about him too, mostly hagiographies. I never met this historical bit before.

Skilled Labor, Labor Values, Prices Of Production

1.0 Introduction

Suppose not all labor in a capitalist economy is 'unskilled'. Some jobs require specific skills that somehow command a premium. How can heterogeneous labor be handled in modern treatments of classical and Marxian political economy?

My inclination is just to assume that relative wages for different labor categories are stable. I read Ricardo and others as doing the same:

"I must not be supposed to be inattentive to the different qualities of labour, and the difficulty of comparing an hour's or a day's labour, in one employment, with the same duration of labour in another. The estimation4 in which different qualities of labour are held, comes soon to be adjusted in the market with sufficient precision for all practical purposes, and depends much on the comparative skill of the labourer, and intensity of the labour performed. The scale, when once formed, is liable to little variation." -- David Ricardo, Principles

But in this post, I want to consider a different approach. I assume that skilled labor is produced by prior training. This post is only a start.

2.0 Technology

I consider the simplest model of the production of commodities for making my point. Table 1 specifies the technology for this example. Each column shows the bushels corn, person-years of skilled labor, and the person-years of unskilled labor paid by the capitalist operating that process to produce one unit of output. I assume constant returns to scale.

Table 1: Technology

Input	Sector
	Training	Corn
Corn	a1,1 Bushels	a1,2 Bushels
Skilled Labor	c1,1 Person-Years	c1,2 Person-Years
Unkilled Labor	a0,1 Person-Years	a0,2 Person-Years
OUTPUT	One Person-Year	One Bushel

I assume the skilled labor produced by the training sector only has skills for the next year. After working for one year, a skilled worker needs to be retrained. In this way, the model resembles a model with only circulating capital and no fixed capital. The inputs to the training sector do not include the workers who emerge as skilled workers. Rather, those workers are customers, paying the capitalist, who obtains returns from operating the process in the training seector.

All coefficients of production are assumed to be positive. I assume the Hawkins-Simon conditions:

a_{1, 2} < 1

c_{1, 1} < 1

a_{1, 1} c_{1, 2} < (1 - a_{1, 2})(1 - c_{1, 1})

These asumptions ensure that the economy can produce a surplus product.

3.0 Labor Values

I first want to calculate labor values. Labor values are defined in terms of unskilled labor. Introduce the following two variables:

• v₁ is the (unskilled) labor value of corn, in unskilled person-years per bushel.
• v₂ is the (unskilled) labor value of skilled labor, in unskilled person-years per skilled person-year.

The labor value of a bushel corn is the sum of the labor values of the inputs needed to produce it.

v₁ = a_{0, 2} + a_{1, 2} v₁ + c_{1, 2} v₂

The specification of a process for training skilled labor allows for a parallel definition of the labor value of skilled labor:

v₂ = a_{0, 1} + a_{1, 1} v₁ + c_{1, 1} v₂

The above is a system of two equations in two unknowns. The following abbreviation is useful in setting out the solution:

denom = (1 - a_{1, 2})(1 - c_{1, 1}) - a_{1, 1} c_{1, 2}

The solution is:

v₁ = [a_{0, 1} c_{1, 2} + a_{0, 2}(1 - c_{1, 1})]/denom

v₂ = [a_{0, 1}(1 - a_{1, 2}) + a_{0, 2} a_{1, 1}]/denom

The Hawkins-Simon conditiones guarantee that labor values are positive. Despite the existence of heterogeneous labor, the labor value of corn, in terms of one type of labor is well-defined.

4.0 Prices of Production

I skip exploring various ratios and other aspects of the system of labor values. I introduce the following variables for prices of production:

• p₁ is price of corn, in numeraire-uints per bushel.
• w₁ is the wage of unskilled labor, in numeraire units per person-year.
• w₂ is the wage of skilled labor, in numeraire units per person-year.
• r is the rate of profits, a pure number.

The system of equations for prices of production includes an equation for corn:

(p₁ a_1,2)(1 + r) + c_1,2 w₂ + a_0,2 w₁ = p₁

In this equation, wages of skilled and unskilled labor are paid out of the surplus, at the end of the year. The following equation applies to the training sector:

(p₁ a_1,1)(1 + r) + c_1,1 w₂ + a_0,1 w₁ = (w₂ - w₁)/(1 + r)

The Right Hand Side is the present value of the skill premium obtained by a trained worker.

Specifying the numeraire removes one degree of freedom for the system of prices of production. One degree of freedom remains.If the wage of unskilled labor is taken as given, the system is closed.

5.0 Conclusion

Obviously, this analysis can go in many directions. What would it mean for the organic composition of of capital not to vary between the training sector and the corn-producing sector? How does Marx's account of exploitation work here? Can skilled workers exploit unskilled worker, as in Ian Steedman's book? How can I draw on the theory of fixed capital to account for skills lasting for multiple periods? Can I introduce risk and distinguish between the rate of profits and the interest rate used for time-discounting? How does the the theory of rent apply to inate talents? (Bootlickers will insist this is the dominant case.)

If you want to apply this approach empirically, you must answer some of these questions. The approach can be extended to more sectors and more types of labor. I have seen some input-output tables broken down to have two types of labor.

John Stuart Mill, Avowed Socialist

I have been reading some secondary literature on John Stuart Mill. He was explicitly against meritocracy, although that term was not available until Young's satirical novel. Here is Mill:

"If some Nero or Domitian were to require a hundred persons to run a race for their lives, on condition that the fifty or twenty who came in hindmost should be put to death, it would not be any diminution of the injustice that the strongest or nimblest would, except through some untoward accident, be certain to escape. The misery and the crime would be that any were put to death at all. So in the economy of society; if there be any who suffer physical privation or moral degradation, whose bodily necessities are either not satisfied or satisfied in a manner which only brutish creatures can be content with, this, though not necessarily the crime of society, is pro tanto a failure of the social arrangements. And to assert as a mitigation of the evil that those who thus suffer are the weaker members of the community, morally or physically, is to add insult to misfortune. Is weakness a justification of suffering? Is it not, on the contrary, an irresistible claim upon every human being for protection against suffering?" – J. S. Mill

The above is not necessarily an argument for socialism. It is consistent with Mill's investigation of what a society organized around private property might be.

In his autobiography, Mill explicitly says that he became a socialist:

"Our [Mill and Harriet Taylor] ideal of ultimate improvement went far beyond Democracy, and would class us decidedly under the general designation of Socialists. While we repudiated with the greatest energy that tyranny of society over the individual which most Socialistic systems are supposed to involve. we yet looked forward to a time when society will no longer be divided into the idle and the industrious; when the rule that they who do not work shall not eat, will be applied not to paupers only, but impartially to all; when the division of the produce of labour, instead of depending, as in so great a degree it now does, on the accident of birth, will be made by concert, on an acknowledged principle of justice; and when it will no longer either be. or be thought to be. impossible for human beings to exert themselves strenuously in procuring benefits which are not to be exclusively their own, but to be shared with the society they belong to." -- J. S. Mill

Mill is the canonical example of a classical liberal. His short book On Liberty is still read. How can he also be a socialist?

Reference

• Helen McCabe. 2021. John Stuart Mill, Socialist. McGill-Queen's University Press.
• Matthew McManus. 2024. The Political Theory of Liberal Socialism. Routledge.
• Michael Young. 1958. The Rise Of The Meritocracy: 1870-2033.. Pelican.

Rent As Joint Production

I here think about Sraffa's price equations for rent. I try to formulate them explicitly as the system of price equations in the chapter on joint production. I suppose this is a case of me setting out something that many take as obvious.

The data for Sraffa's price equations include the specification of a number of processes producing commodities. Suppose the l process is operated by a capitalist farmer. And they pay rent on the uth type of land. The price equation is:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l)(1 + r) + rho_u c_u,l + w a_0,l = p_l

Above, rho_u is rent per acre, and c_u,l are the acres of land per unit output for this process. I hope the remaining symbols are obvious.

Let f_u be the purchase price of an acre of land of the uth type. The above equation can be rewritten as:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l + f_u c_u,l)(1 + r) + w a_0,l = p_l + f_u c_u,l

where:

rho_u = f_u r

The above equality is equivalent to an infinite sum:

f_u = rho_u/(1 + r) + rho_u/(1 + r)² + rho_u/(1 + r)³ + ...

In an explicit formulation of a model of rent as one of joint production, no elements of the input and output matrices, as seen in the price system, are negative. The price of an acre of land is the present value of the stream of rent payments expected to be received on that land in the future.

The above works for intensive rent. One of the lands has a price of zero in the case of extensive rent. How is that expressed in the price equations? I suppose only commodities, that is, goods with positive prices, enter into the equations. Air, being free, does not appear in the price equation for some chemical process that requires nitrogen in some way, for example. But then the solution to a problem of the choice of technique is taken as given. I have a difficulty with many of the chapters in the part on joint production in Sraffa's book.