The Choice Of Technique As A Linear Complementarity Problem

1.0 Introduction

Linear Complementarity Problems (LCPs) have well-known algorithms (as least known by others) to solve them. Of particular interest to me is the Lemke algorithm. I think Christian Bidard or Guido Erreygers was the first to point out that the Lemke algorithm applies to economics in this way. But I do not know they ever specify the details in this post. I often need to step through what others find obvious to understand something.

2.0 The Linear Complementarity Problem (LCP)

Table 1: Parameters and Variables for the LCP

Symbol	Type	Definition
k	Parameter	Problem size, known as the order of the LCP.
M	Parameter	A k x k matrix.
u	Parameter	A k-element column vector.
x	Variable	A k-element non-negative column vector.
z	Variable	A k-element non-negative column vector.

This section specifies the LCP. Let M be a given k x k matrix and u a given k-element column vector (Table 1). Find the k-element column vectors x and z such that:

x - M z = u

x_i ≥ 0, for i = 1, 2, ..., k

z_i ≥ 0, for i = 1, 2, ..., k

x_i z_i = 0, for i = 1, 2, ..., k

The last condition can be specified as the condition that the vector dot product x^T z must be equal to zero. (x^T is the tranpose of x.)

3.0 The Problem Of The Choice of Technique

I now specify inequalities and equalities (for duality conditions) that specify the problem of finding a cost-minizing solution in the analysis of the choice of technique. Table 2 defines the given parameters for this problem. Table 3 defines the vectors to be found. Technology and final demand is taken as given. Given the rate of profits, the level of operation of each process in the technology and the price of each produced commodity is determined by the solution, when a solution exists and is unique.

Table 2: Parameters for a Cost-Minimizing Technique

Symbol	Definition
n	The number of produced commodities.
m	The number of processes, m ≥ n.
A	The n x m input matrix.
a0	The m-element row vector of direct labor coefficients.
B	The n x m output matrix.
y	The n-element column vector of net output, also known as final demand.
r	The rate of profits.

Table 3: Variables for a Cost-Minimizing Technique

Symbol	Definition
q	A m-element non-negative column vector. The level of operation of each process.
p	A n-element non-negative column vector. The price of each produced commodity.

The net output must meet or exceed the specified final demand:

B q - A q ≥ y

A vector is greater than or equal to another if and only if each element of the first is greater than or equal to another. Each process must be operated at a non-negative level.

q_i ≥ 0, for i = 1, 2, ..., m

The above non-negativity conditions complete the specification of the quantity system.

The specification of the price system starts with the following system of inequalities:

(1 + r) A^T p + a₀^T ≥ B^T p

The cost of each process cannot fall below the revenues obtained by that process. Labor is paid out of the surplus at the end of the production period. Each price must be non-negative:

p_i ≥ 0, for i = 1, 2, ..., n

In this specification, a person-year of labor is the numeraire.

I turn to duality conditions. The law of free goods states that any commodity in excess supply has a price of zero. It can be stated as the following equality:

p^T [(B - A) q - y] = 0

The law of non-operated processes asserts that any process in which the cost exceeds revenues is not operated. It is stated like so:

[p^T(B - (1 + r) A) - a₀] q = 0

A (cost-minimizing) solution of the above systems of equalities and inequalities is a long-period position. The prices are prices of production.

4.0 Mapping the Choice of Technique to a LCP

The main inequality in the quantity system can be converted to equality by subtracting excess supplies from the left hand side (LHS). Some manipulation yields the equation in Figure 1. Notice that the two vectors on the LHS in Figure 1 are non-negative. The matrices and the vector on the RHS side are part of the data when finding a cost-minimization system.

Figure 1: Quantity System as an Equality

The same approach can be adopted for the main inequality in the price system. Here I introduce a vector of extra costs for each process. The result is shown in Figure 2.

Figure 2: Price System as an Equality

Two vectors are found in the solution of a LCP. The above suggests that k = n + m is the order of the LCP in which I am interested. Figure 3 specifies the other vector in the solution for the LCP in terms of the vectors to be found in finding a cost-minizing solution.

Figure 3: One Solution Vector in the LCP as Price and Quantity Vectors for Cost-Minimization

Figure 4 specifies the vector taken as a parameter in the LCP. Figure 5 specifies the matrix.

Figure 4: The Given Vector in the LCP for the Cost-Minimization Problem

Figure 5: The Given Matrix in the LCP for the Cost-Minimization Problem

With the mapping shown, the problem of finding a cost-minimizing solution to the problem of the choice of technique is now a LCP. The condition that x^T z must be zero incorporates both the rule of free goods and the law of non-operated processes.

5.0 Conclusion

Solving a LCP provides a solution to a linear program. In this formulation, duality considerations apprently enter. I have seen and even published an article, in 2005, with a LP formulation of the analysis of the choice of technique. Does the LCP formulation yield the same LP?

Above, the LCP is for a model of general joint production. I suppose that I could explicitly formulate the LCP to consider the theory of rent. I do not think that any difficulties arise here, at least as far as setting up the problem. Erreygers (1995) and Kurz and Salvadori (1995) provide an approach.

Can I find some way of illustrating a LCP by partitioning a low-dimensional parameter or solution space for some small problem? Can I actually step through the Lemke algorithm for a small problem?

References

• Erreygers, Guido. 1995. On the uniqueness of cost-minimizing techniques. The Manchester School 63: 145-166.
• Murty, Katta G. 1997. Linear Complementarity Linear and Nonlinear Pogramming
• Lemke, Carlton. 1965. Bimatrix Equilibrium Points and Mathematical Programming. Management Science 11(7): 681-689.

Joan Robinson On The Lack Of A Marginalist Theory Of The Rate Of Profits

John Eatwell conludes a 2019 article with the assertion, "There is no neo-classical theory of the rate of profit." As appropriate for a conclusion, this article demonstrates that this proposition is true.

I recently read Robinson (1972). The first crisis in economic theory in her lifetime, she says, is the failure of economic theory to explain the level of production. Keynes addressed that crisis, albeit mainstream economists these days seem mostly ignorant of his ideas. The second crisis concerns what is produced, of the composition rather than the volume of production. This crisis includes concerns about what governments should spend on, while trying to maintain effective demand. More social spending and less spending of technologies for war would be nice.

But here I want to note these passages:

What is the orthodox theory of profits actually received? Many years ago I set out to write a little book on Marxian economics; when I had written a chapter on Marx's theory of profits, I thought I had to write a chapter on the orthodox theory for comparison, and blest if I could fmd one high or low. Ever since I have been inquiring and probing but I still cannot find out what it is. We have Marshall's theory that the rate of interest is the 'reward of waiting' but 'waiting' only means owning wealth. A man 'may have obtained the de facto possession of property by inheritance or by any other means, moral or immoral, legal or illegal. But if, having the power to consume that property in immediate gratifications, he chooses to put it in such a form as to aflford him deferred gratifications, then any superiority there may be in deferred gratifications over those immediate ones is the reward of his waiting'. In short, a man who refrains from blowing his capital in orgies and feasts can continue to get interest on it. This seems to be perfectly correct, but as a theory of distribution it is only a circular argument...

...Each individual goes on saving or dis-saving till the point where his individual subjective rate of discount is equal to the market rate of interest. There has to be a market rate of interest for him to compare his rate of discount to. But of course the whole thing is quite beside the point once we have accepted the Keynesian view that investment governs saving, not saving investment...

...There is also the problem of the relative levels of different types of earned income. Here we have the famous marginal productivity theory. In perfect competition an employer is supposed to take on such a number of men that the money value of the marginal product to him, taking account of the price of his output and the cost of his plant, is equal to the money wage he has to pay. Then the real wage of each type of labour is supposed to measure its marginal product to society. The salary of a professor of economics measures his contribution to society and the wage of a garbage collector measures his contribution. Of course this is very comforting doctrine for professors of economics but I fear that once more the argument is circular. There is not any measure of marginal products except the wages themselves.

In short, we have not got a theory of distribution. We have nothing to say on the subject which above all others occupies the minds of the people whom economics is supposed to enlighten. -- Joan Robinson

This passage reminded me that Eatwell and Robinson collaborated, in the early 1970s, on a textbook. So Eatwell is drawing on Robinson, as well as Sraffa, in his argument about the lack of an orthodox explanation of how capitalists obtain income. And you can see that Robinson's attempt to find such a theory goes back to 1942. The generalization of the General Theory to the long run was a major part of her research program.

I do not think it makes sense to talk about the rate of profits in the Arrow-Debreu model of intertemporal equilibrium, in John Hicks' Value and Capital model of temporary equilibrium, or in models of dynamic equilibrium paths. In these models, relative prices vary over time.

John Eatwell, with others, says that economists changed the question. He is not referring to the marginal revolution. Early marginalists tried to explain the rate of profits, in a model of long run equilibrium with a given quantity of capital. All of their approaches were incoherent. The changed question looked only at prices along short run equilibrium paths, where the agents have perfect foresight. Joan Robinson objected that she could not get these models to stand up long enough to knock them down. They failed to be in historical time.

References

• John Eatwell. 2019. 'Cost of Production' and the theory of the rate of profit. Contributions to Political Economy 38(1): 1-11.
• Joan Robinson and John Eatwell. 1973. An introduction to Modern Economics. McGraw-Hill
• Joan Robinson. 1942, 1966. An Essay on Marxian Economics. Palgrave Macmillan.
• Joan Robinson. 1972. The second crisis of economic theory. American Economic Theory 62(1/2): 1-10.

Some Quotations On Rent

1.0 Introduction

This post presents some quotations about rent. I criticize most of these statements. I have great respect for the quoted scholars.

2.0 David Ricardo

Thomas Robert Malthus, David Ricardo, Robert Torrens and Edward West are credited with the first clear and comprehensive analysis of differential land rent and the associated economic relationships. Their pamphlets in February 1815 had a little known precursor at the time in the work of James Anderson. (I modified the Wikipedia entry for these sentences.)

Ricardo explains what the theory is about:

"Rent is that portion of the produce of the earth, which is paid to the landlord for the use of the original and indestructible powers of the soil." -- David Ricardo, Principles

Ricardo clearly distinguishes between rent paid for the services of land and, for example, house rent. The land, if properly cultivated leaves a production process in as good shape as in which it enters. And, by abstraction, the land was not the result of any investment or labor before initially being used for agriculture. Rent on the results of investment that must be periodically renewed is not under consideration.

Here Ricardo explains extensive rent:

"When land of the third quality is taken into cultivation, rent immediately commences on the second, and it is regulated as before, by the difference in their productive powers. At the same time, the rent of the first quality will rise, for that must always be above the rent of the second, by the difference between the produce which they yield with a given quantity of capital and labour.” -- David Ricardo, Principles

Ricardo takes differences in fertility as a matter of nature. He does not recognize that, with heterogeneous inputs, at least, that the order in which lands are taken into cultivation, depends on the distribution between wages and profits. Typically, prices of production differ at different levels of, say, a given wage. And, thus, the order of fertility can vary too.

Furthermore, Ricardo does not recognize that the order of fertility can differ from the order of rentability.

Here Ricardo combines extensive and intensive rent in his analysis:

"It often, and, indeed, commonly happens, that before No. 2, 3, 4, or 5, or the inferior lands are cultivated, capital can be employed more productively on those lands which are already in cultivation. It may perhaps be found, that by doubling the original capital employed on No. 1, though the produce will not be doubled, will not be increased by 100 quarters, it may be increased by eighty-five quarters, and that this quantity exceeds what could be obtained by employing the same capital, on land No. 3.” – David Ricardo, Principles

I have been elaborating on a combination of extensive and intensive rent.

3.0 Piero Sraffa

"No changes in output and (at any rate in Parts I and II) no changes in the proportions in which different means of production are used by an industry are considered…” -- Piero Sraffa (1960)

Part II is on joint production. Part III is the single chapter on the choice of technique, and Sraffa seems to imply that changes in output are allowed in that analysis. Chapter XI, on land, is a six-page chapter in part II. I cannot read section 88 as not considering a change in output.

Here Sraffa points to the need for further analysis (really, the whole book presents exercises for the reader):

"More complex cases can generally be reduced to combinations of the two [extensive and intensive rent] that have been considered. The main type of complication arises from the multiplicity of agricultural products." -- Piero Sraffa (1960)

I am not the first to note that Sraffa is overoptomistic about the simpler cases.

3.0 Alberto Quadrio Curzio

I consider Quadrio Curzio as the scholar who has probed most deeply into the theory of rent in the tradition of post Sraffian price theory. Others include Christian Bidard, Guido Erreygers, Heinz Kurz & Neri Salvadori, and Betram Schefold.

Quadrio Curzio draws a conclusion:

"Rent greatly complicates the relations between wages and profits" – Alberto Quadrio Curzio (1980: 238)

This is a conclusion that I want to avoid. I want to say something more than that it is complicated. I think I may have achieved this by demonstrating, for example:

• That the orders of fertility and rentability can each exhibit a kind of reswitching independent of each other.
• The the orders of fertility and of rentability may be completely opposite of one another.
• That the order of fertility may depend on techniques in which intensive rent is obtained, even though the cost-minimizing technique at the given output only pays extensive rent.

Here Quadrio Curzio and Pellizzari say something more:

"Even when the intensive rent disappears, the effects of intensive cultivation persist in the different productive processes applied to the different lands, which affect the extensive differential rents" – Alberto Quadrio Curzio & Fausta Pellizzari (2010: 46).

I think my third bulleted point above explains this observation a bit more.

References

• Quadrio Curzio, Alberto. 1980. Rent, income distribution, and orders of efficiency and rentability, in Pasinetti, L. L. (ed.) Essays on the Theory of Joint Production.
• Quadrio Curzio, Alberto and Fausta Pellizzari. 2010. Rent, Resources, Technologies.
• Ricardo, David. 1951. On the Principles of Political Economy and Taxation.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities.

John Stuart Mill Further Demonstrates The Marx Follows From Ricardo

I want to here consider further evidence that Ricardo's views lead to something like Marx's theory of surplus value. My argument is that John Stuart Mill read Ricardo in a way that supports this thesis. And that he correctly had Ricardo's value as an absolute value. In particular, he had Ricardo's labor value to be much like Marx’s. (I do not want to argue that distinctions exist between Marx and Ricardo's theories of value.)

I previously noticed that Mill's had, in his Principles of Political Economy, an account of the source of profits as what Marx described as the exploitation of labor. Here I turn to his earlier work, "On Profits, And Interest," in his 1844 Essays on some unsettled questions of political economy. This is in volume IV of The Collected Works of John Stuart Mill.

Mill says that Ricardo had a notion of labor values distinct from exchangeable value. According to Terry Peach, this was not Mill's later position. As far as Mill's own theories, Joseph Schumpeter, for one, has him halfway between classical political economy and neoclassical economics.

Anyways, Mill explains that Ricardo saw that the rate of profits could only rise as wages fall:

"Profits, then (meaning not gross profits, but the rate of profit), depend (not upon the price of labour, tools, and materials - but) upon the ratio between the price of labour, tools, and materials, and the produce of them: upon the proportionate share of the produce of industry which it is necessary to offer, in order to purchase that industry and the means of setting it in motion." -- J. S. Mill, p. 262

"And thus we arrive at Mr. Ricardo's principle, that profits depend upon wages; rising as wages fall, and falling as wages rise." -- J. S. Mill, p. 262

And then Mill explains what Ricardo meant by value:

"The rate of profits depends not upon absolute or real wages, but upon the value of wages.

If, however, by value, Mr. Ricardo had meant exchangeable value, his proposition would still have been remote from the truth. Profits depend no more upon the exchangeable value of the labourer's remuneration, than upon its quantity. The truth is, that by the exchangeable value is meant the quantity of commodities which the labourer can purchase with his wages; so that when we say the exchangeable value of wages, we say their quantity, under another name.

Mr. Ricardo, however, did not use the word value in the sense of exchangeable value.

Occasionally, in his writings, he could not avoid using the word as other people use it, to denote value in exchange. But he more frequently employed it in a sense peculiar to himself, to denote cost of production; in other words, the quantity of labour required to produce the article; that being his criterion of cost of production. Thus, if a hat could be made with ten days' labour in France and with five days' labour in England, he said that the value of a hat was double in France of what it was in England. If a quarter of corn could be produced a century ago with half as much labour as is necessary at present, Mr. Ricardo said that the value of a quarter of corn had doubled." -- J. S. Mill, p. 263

Mill goes on to reject Ricardo's claims, without modification. He has something like his version of the transformation problem. Mill argues that the rate of profits falls as the cost of production of wages rises, where Mill now includes profits on dated labor inputs in cost of production. He has something like Sraffa's more rigorous distinction between basic and non-basic commodities

Mill argues that the trend in the rate of profits varies with decreasing returns in agriculture and with improvements in production. The rate of profits declines if the former dominates. Marx wanted to avoid this explanation.

Elsewhere

• Sraffa's Reswitching Silliness: Uncomprehending ignorance.
• The New School, in Manhattan, is one of the few places in the United States that teaches political economy in a rigorous way. They have problems.
• Maybe I should subscribe to Catalyst if Post Keynesians are going to publish there.
• The Place of Sraffa in the History of Economic Thought, a session at the annual ASSA conference.

Rent With Multiple Agricultural Commodities

1.0 Introduction

My next problem might be to explore how to apply an analysis of the orders of efficiency and rentability to a model of rent with multiple agricultural commodities. I would like the possibility of both extensive and intensive rent. This post outlines the structure of a simople numeric example. I have not written down the price systems for each technique. I need to do that to be sure Table 2 is correct.

I should prioritize submitting an article with my most recent model of rent. I should also work through examples in the problems for the approriate chapter in Kurz and Salvadori (1995). As I understand it, nobody has investigated the orders of efficiency and rentability in a model like this. I think Kurz and Salvadori have an existence proof in a special case of joint production, different from the special case I would develop.

Table 1 shows the structure of the technology I am thinking of investigating. Each agricultural commodity can be produced by two processes. The processes differ in which of the two types of land they are operated on, as well as in other coefficients of production.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	0	c1,2	0	c1,4	0
Type 2 Land	0	0	c2,3	0	c2,5
Iron	a1,1	a1,2	a1,3	a1,4	a1,5
Wheat	a2,1	a2,2	a2,3	a2,4	a2,5
Rye	a3,1	a3,2	a3,3	a3,4	a3,5
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

Table 3 lists the techniques of production.

• Alpha, Epsilon, and Zeta have the same solving subsystem. Epsilon and Zeta pay extensive rent on type 2 land.
• Delta, Eta, and Theta have the same solving subsystem. Eta and Theta pay extensive rent on type 1 land.
• Iota pays intensive rent on type 1 land. The solving subsystem has joint production.
• Kappa pays intensive rent on type 1 land. The solving subsystem has joint production.
• Lambda pays intensive rent on type 2 land. The solving subsystem has joint production.
• Mu pays intensive rent on type 2 land. The solving subsystem has joint production.
• Nu pays intensive rent on both types of land. The solving subsystem has joint production.

I do not seem to have a technique that pays both extensive and intensive rent. If type 1 land were fully farmed under Epsilon, would the price system not be overdetermined?

Table 2: Technique

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

Anyways, this post presents some thoughts about future research I might explore.

John Stuart Mill Explains Profits As The Result Of The Exploitation Of Labor

I find this passage based on guidance from Samuel Hollander:

"The cause of profit is, that labour produces more than is required for its support. The reason why agricultural capital yields a profit, is because human beings can grow more food, than is necessary to feed them while it is being grown, including the time occupied in constructing the tools, and making all other needful preparations: from which it is a consequence, that ff a capitalist undertakes to feed the labourers on condition of receiving the produce, he has some of it remaining for himself after replacing his advances. To vary the form of the theorem: the reason why capital yields a profit, is because food, clothing, materials, and tools, last longer than the time which was required to produce them; so that if a capitalist supplies a party of labourers with these things, on condition of receiving all they produce, they will, in addition to reproducing their own necessaries and instruments, have a portion of their time remaining, to work for the capitalist. We thus see that profit arises, not from the incident of exchange, but from the productive power of labour; and the general profit of the country is always what the productive power of labour makes it, whether any exchange takes place or not. If there were no division of employments, there would be no buying or selling, but there would still be profit. If the labourers of the country collectively produce twenty per cent more than their wages, profits will be twenty per cent, whatever prices may or may not be. The accidents of price may for a time make one set of producers get more than the twenty per cent, and another less, the one commodity being rated above its natural value in relation to other commodities, and the other below, until prices have again adjusted themselves; but there will always be just twenty per cent divided among them all." -- John Stuart Mill, Principles of Political Economy, Book II, Chapter XV, Of Profits, Section 5.

I find the above close to Marx. At the start of the chapter, Mill says that profits are the sum of interest as a payment for abstinence, "indemnity for risk", and "remuneration for the labour and skill required for superintendence". Apparently, Mill regarded this disaggregation as consistent with describring profit as the result of laborers working for more time than needed to reproduce their own necessaries and instruments. For purposes of this post, I do not go into what is wrong with the account of interest as the sum of these components.

I do not draw a connection to Mill's avowal of socialism. But then, I do not read Marx's account of exploitation as an ethical argument for socialism either.

The above quotation is evidence for those who want to argue that Marx has a certain continuity with Ricardo's theory. I do not mean to assert that differences do not exist, as well.