Recurrence Of Truncation Without Reswitching: Another Example

Figure 1: Demand for Labor in the Corn Industry

1.0 Introduction

I have previously worked through an example of the recurrence of truncation with neither the reswitching of techniques nor capital-reversing. This post reports the perturbation of several coefficients of production in that example: a_0,1, a_0,2, a_1,2, and a_1,3.

This modification of the example lowers the rate of profits at which the recurrence of truncation occurs, from approximately 87.5 percent to 67.6 percent. That rate of profits may still be taken to be too high to be observed empirically. Also, in practice, you may want the coefficients of production not to vary too much among processes. The ratio of a_0,2 to a_0,1 is now 7.5, much lower than before. Presumably, with many more produced commodities and machines with physical lives of more production cycles, the recurrence of truncation might appear at a lower rate of profits.

2.0 Technology

Two industries exist in the example. One industry produces machines, and the other industry produces corn. Corn is a consumption good, the good for circulating capital, and the numeraire. Machines are fixed capital. Each machine has a physical life of two years. Old machines cannot be transferred between industries. I assume constant returns to scale (CRS) and the free disposal of old machines. Labor is advanced and paid out of the surplus of corn.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs, at a unit level of operation, consist of 4/5 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	4/5	6	43/40	1
Corn	1/16	1/10	13/100	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

3,0 The Choice of Technique

The economic life of a machine is chosen to minimize cost. A system of equations for prices is associated with each technique. This system can be solved. In the solution, the wage is a function of the rate of profit. Each price of a produced commodity is also a function of the rate of profits.

Figure 2 shows the wage curves, for the four techniques in the example. The cost-minimizing technique at each wage or rate of profits is the technique with its wage curve on the outer frontier. The cost-minimizing techniques are indicated on the figure.

Figure 2: Wage Curves In The Example

At any rate, the cost-minimizing techniques, in order of an increasing rate of profits, are Alpha, Gamma, Delta, and Beta. Each pair of techniques at a switch point on the frontier differs in one process. A switch point in which the economic life of a machine differs in both industries would be a fluke case. No fluke switch points exist in this example, without perturbing some coefficients of production.

Table 4 summarizes the results for this numerical example. The bounds on the ranges of the rates of profits are approximate.

Table 4: Cost-Minimizing Techniques

Range	Technique	Truncation
0 ≤ r ≤ 46.3%	Alpha	Machines truncated in both industries.
46.3% ≤ r ≤ 49.6%	Gamma	Machines truncated in machine-production.
49.6% ≤ r ≤ 67.6%	Delta	Machines operated at full physical life in both industries.
67.6% ≤ r ≤ 129.4%	Beta	Machines truncated in corn-production.

At any rate, the machine is truncated in corn-production when both the Alpha and the Beta technique are cost-minimizing. The truncation of the machine in corn-production recurs, being part of the cost-minimizing technique at extremes of low and high rates of profits. This is not, however, an example of the reswitching of techniques.

Around the switch point between Beta and Delta, an increased wage is associated with increased labor inputs in the corn industry, per bushel corn produced in that industry (Figure 1). At each point on the graph, the production system as a whole is in equilibrium, in the sense that no extra profits can be obtained. Both the rate of profits and the price of a new machine vary with the wage. The switch points between Alpha and Gamma and between Beta and Delta appear as horizontal line segments on this graph. Since the economic life of a machine in corn production does not vary at the switch point between Gamma and Delta, that switch point is only a point on the graph. The switch point between Beta and Delta is an example of the reverse-substitution of labor. The example illustrates that the reverse substitution of labor and the recurrence of truncation are independent of the reswitching of techniques and of capital-reversing.

4.0 Conclusion

The CCC developed a critique of aggregate production functions and aggregate neoclassical theory. But the critique also extends to microeconomics. The recurrence of truncation, like process recurrence (Steedman 2002), re-iterates that the concept of vertically-integrated industries provides a coherent framework for partial analysis. Both process recurrence and this example of the recurrence of truncation challenge claims that, under ideal conditions, the long-period demand for labor from a particular industry is a non-increasing function of the real wage. These demonstrations about industry-specific analyses do not rely on either the reswitching of techniques or capital-reversing.

Mike Woodford Confused: Is Money Like Oil Or Like Sand In The Gears?

The 2026 'Nobel' prize in economics goes to Robert E. Hall and Michael Woodford. I happen to already have an opinion that Woodford's work, and 'New Keynesianism', is misconceived.

Woodford's textbook is widely cited, as I understand it. Robert Solow was Woodford's thesis advisor. He acknowledges Axel Leijonhufvud for introducing him to Wicksell's approach to monetary theory. New Keynesianism is widely drawn on, I gather, by the Federal Reserve and other monetary authorities.

New Keynesianism is named in a misleading way. Keynes, in the General Theory tries to explain persistent unemployment when wages and prices are flexible. The explanation of persistent unemployment with sticky wages or prices or other imperfections is pre-Keynesian. We now know that no tendency can exist for the labor market to clear in any run, even under ideal conditions.

Another issue is confusion about the role of money in neo-Walrasian General Equilibrium Theory. Prices in terms of an arbitrary numeraire are not nominal, that is, money prices. The pure theory is one of barter. Money has no 'essential' role. The Hahn problem is how to modify the theory so that the equilibrium depends, in some essential way, is known as the Hahn problem. You can claim that some commodity or another is stored across time periods, talk about banks, and so on. Rhetorically, money makes the transition to equilibrium smoother. But this way of talking is disconnected with, perhaps contradicted by, the mathematical theory.

Another approach is to model money as a friction. I once explored a cash-in-advance model for a bit. This approach seems confused, too.

I draw heavily on Colin Rogers in this post.

Selected References

• Frank Hahn. 1965. On Some Problems of Proving the existence of an equilibrium in a monetary economy. In Hahn & Brechling (eds.). Theory of Interest Rates
• Frank Hahn and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press
• Colin Rogers. 1989. Money, Interest and Capital: A Study in the Foundations of Monetary Theory . Cambridge University Press.
• Colin Rogers. 2024. Whitehead's fallacy of misplaced concreteness and the unfortunate uselessness of all monetary-macro theory micro-founded on Walrasian-Pareto general equilibrium theory. Cambridge Journal of Economics 48(2): 235-256.
• Colin Rogers. 2026. Understanding the General Theory after 90 Years. Review of Political Economy
• Michael Woodford. 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press.

Anwar Shaikh (1945 - 2026)

Anwar Shaikh has died. Those who knew him or are more familiar with his work will have more to say. I know a few elements of his work:

• His development of the humbug production function (Shaikh 1974). The apparent good empirical fit of aggregate production functions is an accounting identity when functional income distribution is stable.
• His reading of Marx's solution to the transformation problem as the first step in an iterative algorithm converging to prices of production (Shaikh 1977).
• Criticisms of neoricardian economics (Shaikh 1981, 1984).
• Empirical application of the labor theory of value with Leontief input-output matrices (e.g., Shaikh and Tonak 1996). That book goes into depth in distingusihing productive and unproductive labor.

I still have not read his comprehensive book 2016 book emphasizing 'turblent regulation' and 'real competition'. Currently, his archives are available on-line.

Shaik was born in 1945 in Pakistan, before the partition from India. He lived as a child in various cities in various countries. His undergraduate degree was from Princeton, in 1965. He spent a couple of years as an engineer and in Kuwait teaching physics, math, and social science. His doctorate is from Columbia in 1973. He lived in Harlem, and participated in various protests at Columbia. He joined the New School in 1972.

I like this story:

"While at Columbia I wrote to Joan Robinson and brashly asked her if she would be willing to speak at the Economics Department. On receiving an affirmative response, I excitedly rushed to tell the Department Chair. To my astonishment, the department originally declined to invite her but, after some pressure from students, finally acceded. When the time came, she strode into the room, resplendent in a coat given to her by Mao, having just come from conversations with Fidel. She proceeded to say sharply critical things about aggregate production functions and the marginal productivity theory of distribution. The hall was full of students. But a mandatory department meeting just happened to have been scheduled at the same time, and only two faculty members from the Economics Department attended her talk. They spent their time during the discussion period attempting to defend standard theory." -- Anwar Shaikh

Somewhere else, I think, he says he did not realize graduate students were not supposed to invite guest speakers.

Update: Obituary from Vivek Chibber in Jacobin

Very Selective References

• Shaikh, Anwar. 1974. Laws of algebra and laws of production: the humbug production function, Review of Economics and Statistics 61(1).
• Shaikh, Anwar. 1977. Marx's theory of value and the 'transformation problem', in Jesse Schwartz (ed.), The Subtle Anatomy of Capitalism Goodyear Publishing Co.
• Shaikh, Anwar. 1981. The Poverty of algebra: critical notes on Neo-Ricardian economics. In The Value Controversy. New Left Books.
• Shaikh, Anwar. 1984. The transformation from Marx to Sraffa: prelude to a critique of the neo-Ricardians’. In E. Mandel (ed.), Marx, Ricardo, Sraffa. Verso
• Shaikh, Anwar and E. Ahmet Tonak. 1996. Measuring the Wealth of Nations: The Political Economy of National Accounts. Cambridge University Press.
• Shaikh, Anwar. 2001. Anwar M. Shaikh. In: Philip Arestis and Malcolm Sawyer (ed.), A Biographical Dictionary of Dissenting Economists, 2nd edition. Edward Elgar.
• Shaikh, Anwar. 2016. Capitalism: Competition, Conflict, Crises. Oxford University Press.

Finding Rents By Solving A LCP

This post is a continuation of this previous one. I solve a numerical example at a given rate of profits. In this example combining extensive and intensive rent, the order of fertility is opposite the order of rentability.

I find that Andreas Almqvist and others have implemented a pivoting algorithm in Matlab for solving an arbitrary LCP. Mathworks has a description of the interface, [x,z,retcode] = LCPSolve(M,u);. The implementation is here. I provide my main routine as an appendix to this post. A better main program would improve the interface with the user.

I solve my numerical example for a rate of profits of 14 percent. Table 1 presents the solution for four increasing levels of final demand. The order of fertility is Type 3, 2, and 1 lands. This order is the order in which different types of land are taken up for cultivation as final demand expands. Both the order of fertility and the order of rentability vary with the rate of profits.

Table 1: Selected Solutions for the LCP

Cost-Minimizing Technqie	Delta	Kappa	Pi	Omega
Final demand	5 tons 5 bushels	10 tons 10 bushels	30 tons 30 bushels	55 tons 55 bushels
Iterations of Lemke algorithm	6	8	14	10
Operated processes	I and V	I, III, and V	I, II, III, and V	I, II, III, IV, and V
Lands	None scarce. Type 3 partially farmed.	Type 3 pays extensive rent. Type 2 partially farmed.	Types 2 and 3 pay extensive rent. Type 1 partially farmed.	All scarce. Intensive rent paid on Type 3.

The order of rentability is the order of lands, from high rent to low rent per acre, at a given rate of profits and a given level of final demand. When Pi is cost-minimizing in the example, rents are approximately 0.0067 and 0.0128 labor-units per acre on Type 2 and 3 lands, respectively. The order of rentability is Type 3, 2, 1 lands, matching the order of fertility. All rents with expanding final demand are extensive rents up to this point.

Further expansion of final demand leads to the Omega technique being cost-minimizing, as shown in the rightmost column in Table 1. No new type of land is taken up for cultivation, and the order of fertility remains unchanged. Landlords now receive a combination of extensive and intensive rent. Rents are approximately 0.03625, 0.030248, and 0.024781 labor-units per acre on Type 1, 2, and 3 lands respectively. The order of rentability is exactly opposite the order of fertility.

Marginalist economics, also known as neoclassical economics, was developed as a generalization of Ricardo’s theory of intensive rent (Campus 1987). It is also an imitation of energetics, a nineteenth century approach in physics emphasizing the conservation of energy (Mirowski 1989). In the example, the owners of the most fertile land, the type of land that is farmed first, ultimately receive the lowest rent per acre. The owners of the least fertile land, the type of land that is farmed last, receive the highest rent per acre. Is this result consistent with the theory of supply and demand? Is it consistent with supposedly intuitive tales some economists tell about prices as signals of relative scarcity?

Appendix: Source Code

% Casts the choice of technique in a model of extensive rent into a LCP and solves it.
%
% Author: Robert L. Vienneau
% Date: 3 October 2026

% 1.0 Define and echo data.
% Define vector of direct labor coefficients.
a0 = [1, 9/10, 3/5, 29/50, 9/20];

% Define input vectors. The first row is inputs of iron; the second of corn.
A = [[9/20, 1/40, 3/2000, 29/500, 2/30]; ...
[2, 1/10, 9/20, 13/100, 13/100]];

n =(size(A))(1);
m = (size(A))(2);

% Define output vector.
B = [[1, 0, 0, 0, 0]; ...
[0, 1, 1, 1, 1]];

% Define land coefficients.
C = [[0, 1, 0, 0, 0]; ...
[0, 0, 49/50, 0, 0]; ...
[0, 0, 0, 2/50, 2]];

k = (size(C))(1);

% Define endowments of land.
t = [100; 100; 100];

% Define final demand (I want to update this often and rerun).
% d = [5; 5]; % Delta. Processes 1 and 5 are operated.
% No land pays rent.
% 6 iterations in outer loop.
% d = [10; 10]; % Kappa. Processes 1, 3, and 5 are operated.
% Type 3 land pays extensive rent.
% 8 iterations in outer loop.
% d = [30; 30]; % Pi. Processes 1, 2, 3, and 5 are operated.
% Types 2 and 3 lands pay rent.
% 14 iterations in outer loop of Lemke algorithm.
d = [55; 55]; % Omega. All processes are operated in the cost-minimizing
% technique All lands pay rent. Type 3 pays extensive rent.
% 10 iterations in outer loop.
% Rents are 0.03625, 0.030248, and 0.024781 labor-units per acre.
% Order of rentability is 1, 2, 3.
% Order of fertility is 3, 2, 1.

% Define rate of profits. (I want to rerun this often and rerun).
r = 0.14;


% Echo data.
printf( "Vector of direct labor coefficients:\n");
a0
printf( "Input matrix:\n");
A
printf( "Output matrix:\n");
B
printf( "Matrix of land coefficients:\n");
C
printf( "Endowments of land:\n");
t
printf( "Final demand:\n");
d
printf( "Rate of profits:\n");
r
printf( "\nNumber of produced commodities:\n");
n
printf( "Number of production processes:\n");
m
printf( "Number of types of land:\n");
k


% 2.0 Define and echo block-structured matrices, from Erreygers (1995).
Atilde = [A; [0, 0, 0, 0, 0]; [0, 0, 0, 0, 0]; [0, 0, 0, 0, 0]];
Btilde = [B; -C];
dtilde = [d; -t];

% Echo data.
printf( "\nNew input matrix:\n");
Atilde
printf( "New output matrix:\n");
Btilde
printf( "New final demand vector:\n");
dtilde


% 3.0 Define and echo matrices for Linear Complementarity Problem (LCP).
% This is my invention, although suggested by Bidard's work.
M = [[zeros(n + k, n + k), Btilde - Atilde]; ...
[(1 + r)*transpose(Atilde) - transpose(Btilde), zeros(m, m)]];
u = [-dtilde; transpose(a0)];

printf( "\nMatrix for LCP:\n");
M
printf( "Vector for LCP:\n" );
u

% 4.0 Solve LCP.
[x,z,retcode] = LCPSolve(M,u);


% 5.0 Echo solution.
x
z
retcode


Finding Rent By Solving A Linear Complementarity Problem (LCP)

1.0 Introduction

This post is revisiting this example of extensive and intensive rent. I find writing up examples of rent difficult because they quickly get too complicated. This time, I'm thinking of confining myself to a small range of the rate of profits, where the order of fertility is opposite the order of rentability. This post is the first exposition I know of that explicitly sets out a LCP for rent. Maybe Bidard has something in French. (I should stick to GIFs, not JPEGs as below.)

2.0 Specification of Parameters as Matrices and Vectors

The data for this problem in the analysis of the choice of technique consist of technology, the endowments of lands of three types, and requirements for use. This data is specified by a number of matrices and vectors.

The technology consists of five production processes. Each process takes a year to complete and completely uses its inputs of produced commodities, iron and corn. Corn is an agricultural commodity, grown on a specified type of land. Each type of land comes out of a production process as good as it was when it went into it. Each process exhibits constant returns to scale (CRS), up to the limit imposed by the endowments of land.

Labor inputs are specified by the row vector, a₀, the vector of labor coefficients:

Each labor coefficient is the physical input of labor, measured in, say, person-years, required to operate the corresponding process at unit level. The inputs of iron and corn required to operate each process at unit level are specified by the input matrix A:

Each column of the input matrix is the physical inputs of iron and corn for a process. The columns of the output matrix, B, are the physical outputs of iron and corn for each process, when operated at a unit level:

The matrix of land coefficients, C, is the last matrix for specifying the technology.

Each column of the land coefficients specifies the services of the types of land, in acres, required to operate a process at unit level. Iron is produced without land services, while corn can only be produced on a type of land. The existence of two processes for producing corn introduces the possibility of intensive rent in a model that would otherwise be one of extensive rent.

Another parameter is the available endowments of land, expressed in the column vector t:

Endowments of each type of land are measured in acres.

Requirements for use, d is the final parameter specified in this section:

Requirements for use is also known as final demand. The first element of the above vector is the tons iron in final demand. The second element is the bushels corn in final demand. I am considering the case when both elements are numerically equal. To calculate the order of fertility, I need to consider the choice of technique as final output expands.

3.0 Conditions on Matrices

The above specifications of vectors and matrices is a numerical example of a more general model. Clearly, there can be any number of industrial commodities, types of land, and processes available on each type of land. Here I specify a set of sufficient conditions that the above example satisfies.

Direct labor is needed as an input to operate each process. All of the elements of the vector of labor coefficients are positive. Commodity inputs are assumed to be needed for each process. Thus, each column of A contains some non-zero entries. By assumption, no pure joint production, such as the production of mutton and wool, is possible. Each column of B contains exactly one positive entry. All the rest are zero. Some process produces each (non-land) commodity. Therefore, each row of B contains at least one non-zero entry.

The input and output matrices have a specific structure. Suppose the produced commodities consist of n - 1 industrial commodities and one agricultural commodity, called corn. More specifically, the output matrix has the following structure:

The subscripts represent the size of each submatrix. The upper left submatrix is the identity matrix. The upper right is a matrix of all zeros. The lower left is a row vector of zeros. And the lower right submatrix is a unit row vector. The first n - 1 processes produce the industrial commodities. The remaining processes produce corn.

The input matrix for land is assumed to have a certain structure too. Land is not needed as a direct input to produce the industrial commodities. The elements of the first n - 1 columns of C are all zero. Each process for producing corn requires an input of the services of one type of land. That is, each of the last m - n + 1columns of C contain exactly one non-zero element. Each type of land is used in at least one process for producing corn. Each row of C contains at least one non-zero element.

Each technique is associated with a solving subsystem (Quadrio Curzio & Pellizzari 2010), as defined by an n-element row vector of labor coefficients and a nxn square matrix of input coefficients:

A solving subsystem resembles the vector of direct labor coefficients and the input-output matrix for a model with circulating capital alone. The first (n ― 1) labor coefficients and columns in the solving subsystems are from the industrial processes specified by the technology. The last labor coefficient and last column are from a corn-producing proces. So far, I have only considered solving subsystems for extensive rent.

Suppose all produced commodities are basic commodities and that the technique represented by each of the matrices in solving subsystems is productive. A commodity is basic if it enters, directly or indirectly, into the production of all commodities. For a matrix representing a productive technology, there exists some level of operation of these processes such that a surplus product remains after the replacement of the commodity inputs.

What about solving subsystems for intensive rent? Consider two processes that produce corn on the same type of land. These would be the last two processes in the numeric example above. Sraffa (1960) shows how a linear combination of these two processes can eliminate the input of the land services. Let this linear process be the last column in a matrix where the first n - 1 columns for the input matrix are the processes for producing the industrial commodities. The corresponding vector of labor coefficients and this matrix constitute the solving subsystem for a technique that pays intensive rent.

Consider the special case in which all the elements of the solving subsystems for techniques that pay intensive rent are non-negative. Furthermore, as with the solving subsystems for extensive rent, all produced commodities are basic commodities, and all of these matrices are productive. I speculate that under these assumptions, a cost-minimizing technique, possibly combining intensive and extensive rent, exists at a given rate of profits, if a feasible technique exists. And I claim such a solution is unique, except at switch points and for a fluke case when a type of land is just fully farmed. The numeric example demonstrates that this special case is non-empty.

4.0 Block Structured Matrices

Following Erreygers (1995) and Kurz & Salvadori (1995), the choice of technique can be formulated in terms of certain block structured matrices. The new matrix of input coefficients is:

The new matrix of output coefficients is:

The new matrix of final demand is:

The new row vector of prices is:

The vector p is the n-element vector of prices of produced commodities, iron and corn in the example. The vector ρ is the k-element vector of rents for the different types of land.

5.0 The Choice of Technique

The problem of the choice of technique is to find a m-element column vector q of levels of operation of the processes and a (n + k)-element row vector of prices that satisfy a certain set of inequalities and equalities.

The quantity of each commodity must meet or exceed the requirements for use:

The above inequality includes the condition that the use of land services cannot exceed those available from endowments. Each process must be operated at a non-negative level:

The two inequalities complete the quantity system.

Two inequalities also specify the price system. The cost of no process falls below revenues. In other words, no extra profits are possible:

In this formulation, a unit of labor is the numeraire. All prices are non-negative:

Finally, the cost-minimizing technique must satisfy two duality conditions. The law of free goods holds. Any commodity in excess supply, including a type of land, has a price of zero:

The law of non-operated processes holds. Any process in which costs exceed revenues is not operated:

Given the rate of profits, a solution to this combination of equalities and inequalites is a cost-minimizing technique.

6.0 Mapping Block-Structured Matrices to a Linear Complementary Problem

I now consider a mapping of these block-structured matrices and vectors to a LCP. The top of one column vector is the excess supply of commodities and land:

The bottom of this column vector is the excess of costs over revenues for each process:

A vector is defined that contains both levels of operation of each process and the prices of each commodity:

A vector is defined for the right-hand side of an equation in the LCP:

The LCP can be specified in terms of these vectors and matrices.

7.0 The Choice of a Technique as a Linear Complementary Problem

The LCP is to find column vectors x and z such that:

All the elements of x and z are non-negative:

Elements of x are positive only when elements of z are zero and vice-versa:

The solution of the LCP specifies:

• Which processes are operated
• The levels of operation of these processes
• The prices of produced commodities
• The rents of land, including rents of zero for non-scarce land.

A large amount of theory exists for this problem in mathematical programming.

8.0 Conclusion

The above LCP differs from my previous exposition of a LCP for the choice of technique in that:

• It supports a theory of rent and is in terms of block matrices that include land.
• It does not allow for a positive steady-state rate of growth.

I know of solutions for my numeric example. The LCP, as I see it, allows one to find bounds on final demand and the rate of profits for which a solution is optimal. I do not know that I need iterate through an algorithm yet to solve thje LCP.

A Bibliography Of Articles On Rent, Natural Resources, and the Space Economy

I have various results in examples with rent, and unsolved problems exist in the literature. Vienneau (2022) shows an example of the reswitching of the order of fertility and of the order of rentability in non-overlapping regions of the parameter space. D’Agata (1983) shows that, with intensive rent, the choice of technique can be non-unique away from switch points. Some wage curves along the frontier can slope up, and a cost-minimizing technique may not exist because of a cycle in a market algorithm. I have a well-behaved example with extensive and intensive rent. So some theorem might demarcate where intensive rent does not cause these issues. In my example, the order of fertility and of rentability are exactly opposite for some range of the rate of profits. In another reswitching example, intensive and extensive rent arise with a technology in which multiple agricultural commodities can be grown on each type of land. I would like to see a simple example of reswitching with exhaustible resources, if that makes sense

The following bibliography is not complete, but the later works are probably sufficient to locate additional earlier works. I probably should have more references for ecological economics.

• Baranzini, M. L., Rotondi C., and Scazzieri R. (eds.) 2015. Resources, Production and Structural Dynamics, Cambridge: Cambridge University Press.
• Barnes, Trevor and Eric Sheppard. 1984. Technical choice and reswitching in space economies. Regional Science and Urban Economics 14(3): 345-362
• Bidard, Christian. 2010. The dynamics of intensive cultivation. Cambridge Journal of Economics 34 (6): 1097-1104.
• Bidard, Christian. 2014. The Ricardian rent theory: an overview. Centro Sraffa Working Papers 8.
• Bidard, Christian. 2018. Ricardo and Ricardians on the order of cultivation. Journal of the History of Economic Thought 40(3): 389-399.
• Bidard, C and G. Erreygers. 2001a. The corn–guano model. Metroeconomica 52(3): 243–253
• Bidard, C and G. Erreygers. 2001b. Further reflections on the corn–guano model. Metroeconomica 52(3): 254–268
• Bidard, C and G. Erreygers. 2020. Exhaustible resources and classical theory. Oeconomia 10(3).
• D'Agata, Antonio. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica 35(1-2): 147-158.
• Erreygers, Guido. 1995. On the uniqueness of cost-minimizing techniques. Manchester School 63(2): 145-166.
• Erreygers, Guido. 2026. Land, rent and reproduction. On Walras’s criticism of the classical theory of rent. Revue d'histoire de la pnesée économique 1(21)
• Fratini, Saverio M. 2016. Rent as a share of product and Sraffa's price equations. Cambridge Journal of Economics 40(2): 599-613.
• Gehrke, Christian. 2013. Rent, as a share of the produce, not governed by proportions. In: Levrero, E. S., A. Palumbo, and A. Stirati (eds.) Sraffa and the Reconstruction of Economic Theory: Vol. Three
• Gibson, Bill and Darryl McLeod. 1983. Non-produced means of production in Sraffa’s system: basics, non-basics and quasi-basics. Cambridge Journal of Economics 7(2): 141-150
• Huang, Biao. 2018. An exhaustible resources model in a dynamic input–output framework: a possible reconciliation between Ricardo and Hotelling. Journal of Economic Structures 7(1).
• Ianni, Guido. 2026. Class struggle and the international division of surplus: the distributive surface in the open economy. Metroeconomica
• Kurz, H. D. and N. Salvadori. 2011. Exhaustible resources: rents, profits, royalties and prices. In: Caspari, V (ed) The evolution of economic theory: essays in honour of Bertram Schefold. Routledge, London.
• Kurz, H. D. and N. Salvadori. 2015. The classical approach to exhaustible resources: Parrinello and the others. In: Kurz, H. D. and N. Salvadori N. Revisiting classical economics: studies in long-period analysis. Routledge, London
• Metcalfe, J. S. and I. Steedman. 1972. Reswitching and primary input use. Economic Journal 82(325): 140-157.
• Montani, Guido. 1975. Scarce natural resources and income distribution. Metroeconomica 27: 68-101.
• Montet, C. 1979. Reswitching and primary input use: a comment. Economic Journal 89(355): 642-647.
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Marx: Why Have Any Science At All?

I like to point out that Marx had sense. Engels lent him the necessary pence.

The Unpaid Working Day As The Source Of Surplus Value

As demonstrated by Ian Steedman in the 1970s, the working day can be broken out into time paid to the worker and unpaid time. Steedman is echoing results established previously by Michio Morishima.

This decomposition takes the technique in use as given. A complication arises for the general case of joint production. The production of wool and mutton, both from sheep, is often cited as an example of joint production. The proper treatment of fixed capital, such as lathes and other machinery, is another example of joint production.

When calculating the part of the day that is required to produce the commodities that the workers buy out of their wages, the technique is not given. In the circulating capital case, the processes being operated, are scaled down. But in the case of general joint production, a different set of processes might be operated. The labor required to produce a net product of a given commodity basket, contrary to Marx, is not a sum of the labor values of the commodities in the basket. An explanation is provided in the closing chapters of Morishima's and Steedman's respective books. The labor required to produce a commodity basket is the solution of a linear program (LP) in which the processes are operated to produce the basket such that labor input is a minimum.

The decomposition of the working day is not dependent on Marx's theory of value. The unpaid part of the working day is the source of surplus, part of which is paid out as the revenue of the capitalist.

Illusions Of The Market

This decomposition of the working day is not necessarily apparent. The employer of a worker at Walmart sees they are being paid so much an hour. All of labor appears to be paid. Marx is quite aware of this appearance:

"Hence, we may understand the decisive importance of the transformation of value and price of labour-power into the form of wages, or into the value and price of labour itself. This phenomenal form, which makes the actual relation invisible, and, indeed, shows the direct opposite of that relation, forms the basis of all the juridical notions of both labourer and capitalist, of all the mystifications of the capitalistic mode of production, of all its illusions as to liberty, of all the apologetic shifts of the vulgar economists." -- Karl Marx, Capital, Vol. 1, chap 19.

These necessary illusions are not just for wages. A financial analyst does not calculate the rate of return for a firm based on how much a company expends on payroll. Rather the entire capital investment is examined, with conventional accounting for depreciation.

Here is another apposite paragraph from Marx:

"Vulgar economy actually does no more than interpret, systematise and defend in doctrinaire fashion the conceptions of the agents of bourgeois production who are entrapped in bourgeois production relations. It should not astonish us, then, that vulgar economy feels particularly at home in the estranged outward appearances of economic relations in which these prima facie absurd and perfect contradictions appear and that these relations seem the more self-evident the more their internal relationships are concealed from it, although they are understandable to the popular mind. But all science would be superfluous if the outward appearance and the essence of things directly coincided. Thus, vulgar economy has not the slightest suspicion that the trinity which it takes as its point of departure, namely, land - rent, capital - interest, labour - wages or the price of labour, are prima facie three impossible combinations." -- Karl Marx, Capital, Vol 3, chap 48: The Trinity Formula

It is no refutation of Marx to note that things look different in appearance.

Concluding Observations

I try to read Marx under the influence of the later Wittgenstein. Some sort of deeper essence does not exist that appears in empirical reality. Marx tries to be more systematic than Wittgenstein, but both are gathering observations for a purpose. Only a change in our form of life will remove the need for these reminders.