Sunday, August 30, 2020

Stephen Gordon And Alex Tabarrok Being Stupid On Twitter

Paul Graham jokingly asks, "What phrase signals that the person using it doesn't understand your field?" Stephen Gordon and Alex Tabarrok both respond with "Neoclassical economics".

  • Jamie Morgan (ed.) (2016) What is Neoclassical Economics? Debating the Origins, Meaning, and Significance, New York: Routledge.
  • R. Robert Russell and Maurice Wilkinson (1979). Microeconomics: A Synthesis of Modern and Neoclassical Theory, New York: John Wiley.

Saturday, August 29, 2020

Unpublished Reviews of Sraffa's Book and Related Matters

I have a new working paper.

The article presents previously unpublished material from file D13/12/111 in Sraffa's archives. In particular, it reproduces an English translation of Aurelio Macchioro's review in Annali dell'Istituto Giangiacomo Feltrinelli, a summary by Christopher Bliss of a paper that he read to the Cambridge Political Economy Club, a draft response by James Meade, a rejected paper on Marx by Vittorio Volterra and Moshe Machover, and a paper on the subsistence economy by Gouranga Rao. Correspondence in the Sraffa archives related to these works is also reproduced.

Tuesday, August 25, 2020

2 = +2 = 2/1 = 2.0 = 2.0 + j x 0.0?

"Typically 2 the integer is used for counting, whereas 2 the real number is used for measuring.

But in higher mathematics there's a technical sense in which integers aren't real numbers -- we say instead that they can be identified with real numbers." -- Timothy Gowers

1.0 Introduction

Numbers, in some sense, are only defined in mathematics up to isomorphisms. This post runs quickly through some math to explain what this means.

I begin by assuming knowledge of the natural numbers, {0, 1, 2, ...}, as characterized by the Peano axioms. I also assume an understanding of what it means for two natural numbers to be equal, for one to be greater than another, and for two to be added or multiplied together. Other operations could be built on top of this structure, as needed.

2.0 Integers as Equivalence Classes of Ordered Pairs of Natural Numbers

Consider ordered pairs (a, b) of natural numbers. The point of this may be cryptic for a while. Part of the point is to check that no manipulations of these ordered pairs or definitions rely on anything that is not defined in the natural numbers.

I define equality between two ordered pairs:

(a, b) = (c, d)

if and only if

a + d = b + c

At this point, one should check that, by this definition, an ordered pair is equal to itself (reflexive) and that if an ordered pair is equal to another ordered pair, that ordered pair is also equal to the first (symmetry). One also wants to show that this definition is transitive.

if (a, b) = (c, d) and (c, d) = (e, f) then (a, b) = (e, f)

A relation with these three relations is called an equivalence relation. An equivalence relation breaks a set into non-intersecting equivalence classes. All elements of an equivalence class are equal to one another, and no element in the set outside an equivalence class is equal to any element in that class.

Here are three equivalence classes, by this definition: {(1, 0), (2, 1), (3, 2), ...}, { (0, 0), (1, 1), (2, 2), ...}, and {(0, 1), (1, 2), (2, 3), ...}. Is it becoming clear what is going on here?

Next, I want to define a total order:

(a, b) ≥ (c, d)

if and only if:

b + ca + d

I claim that ≥ is well defined. That is, if

  • (a, b) = (a', b')
  • (c, d) = (c', d')
  • (a, b) ≥ (c, d)

then:

(a', b') ≥ (c', d')

I also want to show that ≥ is reflexive, transitive, and antisymmetric. This relation is antisymmetric if and only if for all ordered pairs of natural numbers,

if (a, b) ≥ (c, d) and (c, d) ≥ (a, b) then (a, b) = (c, d)

Furthermore, the relation is complete. For all pairs of ordered pairs, (a, b) ≥ (c, d) or (c, d) ≥ (a, b). With these properties, ≥ orders equivalence classes, just as well as ordered pairs of natural numbers. And one could use the above to define a relation >.

The next step is to define addition:

(a, b) + (c, d) = (a + c, b + d)

This definition also applies to equivalence classes. The sum of elements from two equivalence classes is in the same equivalence class, whichever elements you start with. Notice that:

(a, b) + (0, 0) = (0, 0) + (a, b)

So one might as well define:

+0 = {(0, 0), (1, 1), (2, 2), ...}

The negative numbers are those equivalence classes less than zero. That is, if (a, b) is in an equivalence class where a > b, then that equivalence class is a negative number. The positive numbers are equivalence classes greater than zero. More on this below.

Addition has some other properties of interest. But I am going to move on to define multiplication:

(a, b) * (c, d) = (ad + bc, ac + bd)

Multiplication is well-defined for equivalence classes. Multiplication also has an identity element:

(a, b)*(0, 1) = (0, 1)*(a, b) = (a, b)

So define +1 as {(0, 1), (1, 2), (2, 3), ...}.

Now, associate every natural number n with an equivalence class:

f( n ) = +n = {{(0, n), (1, 1 + n), (2, 2 + n), ...}

The function f is one-to-one and onto for the set of non-negative integers. It preserves equality:

If n = m, then f( n ) = f( m )

The first equality is a relation in the set of the natural numbers. The second equality was defined above. The function f also preserves order, addition, and multiplication:

  • If nm, then f( n ) ≥ f( m )
  • n + m = f( n ) + f( m )
  • nm = f( n ) * f( m )

The function f is an isomorphism.

Suppose proofs can be given for propositions stated above. Then I have (unoriginally) constructed the integers out of the natural numbers. Negative numbers are not some mystical entities. In this construction, zero is the additive identity. Every integer has an additive inverse, that is, one can negate every integer. Addition is associative and commutative. The integer +1 is the multiplicative identity, and the additive and multiplicative identities are distinct. Multiplication is commutative and associative. Multiplication distributes over addition. Finally, if (a, b) * (c, d) = +0, then (a, b) = (c, d) = +0.

Any set with two binary operations with these properties is an integral domain. I think the set of polynomials with coefficients that are rational numbers, with the usual definitions of polynomial addition and multiplication, is also an integral domain.

So the natural number 2 has been shown to be isomorphic to the positive integer +2. I think this proof of the first equality in the post title is fairly typical of math. It is not particularly difficult, but requires what is called "mathematical maturity". One has to keep track of what can be inferred from definitions, and not let your intuition leap ahead. Even so, your intution will guide you. When writing the above, I kept thinking of (a, b) as "b - a". I would like to say this is the kind of mathematics that Bertrand Russell amused himself with when the Brits put him in prison during World War I. But this would all come after the proof of 1 + 1 = 2 in Principia Mathematica.

3.0 The Field of Quotients of an Integral Domain

Now consider ordered pairs (a, b) of the elements of an integral domain, where b is not +0, the additive identity for the integral domain. Here, I define equality by:

(a, b) = (c, d) if and only if a*d = b*c

I have put aside the structure of the elements of the integral domain, whether they are equivalence classes of ordered pairs of natural numbers or polynomials with rational coefficients or whatever. Addition is defined by:

(a, b) + (c, d) = (a*d + b*c, b*d)

One can show that this definition is well-defined for equivalence classes. The additive identity is the set of ordered pairs {(0, a), a a non-zero element of the integral domain}. Multiplication is defined as:

(a, b)*(c, d) = (a*c, b*d)

The multiplicative identity is the set of ordered pairs equal to (1, 1). Every non-zero ordered pair (a, b) has the multiplicative inverse (b, a).

Define an isomorphism to a subset of the quotient field defined above:

g( a ) = a/1 = {(c, d), where (c, d) = (a, 1)}

Obviously, I have skipped over a lot of steps. But the above is an outline of how to construct the rational numbers and to prove that +2 = 2/1.

4.0 The Reals as Equivalence Classes of Cauchy-Convergent Sequences of Rationals

The set of real numbers is the set of limit points of all convergent sequences of rational numbers. I guess below is Cantor's approach to constructing the real numbers. Dedekind had another approach, and Rudin (1973) presents this construction based on "cuts".

Consider a sequence of rational numbers (a0, a1, a2, ...) where for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| an + 1 - an | < ε

In some sense, the terms in the sequence get closer together. Such a sequence is known as Cauchy-convergent.

Two sequences (a0, a1, a2, ...) and (b0, b1, b2, ...) are equal if and only if for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| an - bn | < ε

This definition will yield equivalence classes. Addition of sequences is defined termwise. Should multiplication also be defined termwise? What about order?

Anyway, for any rational number r, the equivalence class of the sequence (r, r, r, ...) is isomorphic in the reals to that rational number.

5.0 Complex Numbers as a Field Extension

I have already stepped through a construction of the field of complex numbers.

6.0 Observations and Higher Mathematics

So that is what numbers are, in some sense. In each of these constructions, I have attempted to preserve some properties of the more primitive domain. I do not know how much freedom one would want to permit me for the properties of, say, equality addition, or multiplication.

Does "+", the symbol for addition have the same meaning when adding natural numbers, integers, rationals, real numbers, and complex numbers?

"This discussion prompts us to ask: do the arithmetical operations have the same meaning in each of these calculi? For instance, in the domain of integers, is subtraction the same operation as in the domain of natural numbers? Furthermore, what do we mean here by 'the same'? If this implies that the operations must satisfy the same conditions, then the questions must be answered negatively. For in the domain of natural numbers the expression a - b is admissible only if a > b, while in the domain of integers this restriction is removed; obviously this is an important distinction. Consequently, there is not, strictly speaking, one subtraction but as many different operations with this name as there are domains of numbers. We should not be deceived regarding this situation by the fact that we use the same signs +, -, :, etc., at the various levels. If we put the statements of these concepts side by side, it becomes clear how far the analogy between them goes and where it stops." -- Friedrich Waismann (1951: 61).

I am not at all sure, however, that Waismann would answer the question in the post title in the negative:

"In the construction above, the integers were first constructed and then rational numbers. Is there an innate necessity for this sequential order? Couldn't we first introduce the rational numbers without signs and then the distinction between positive and negative numbers? Certainly! We would not thereby obtain another system of rational numbers; rather the system so constructed would prove to be isomorphic to the one considered above, since every relation of one system could be mapped on a similarly constructed relation in the other, and conversely." -- Friedrich Waismann (1951: 65).

Waismann famously drew on Wittgenstein in developing his views, though Wittgenstein might have come to disavow them.

Some advanced mathematics raises further questions about how symbols can have meaning. Gödel's first incompleteness theorem seperates the notion of provable from a set of axioms from the question of truth. According to his second incompleteness theorem, consistency, if true, cannot be proven within a sufficiently interesting system of axioms and derivation rules. The Löwenheim-Skolem theorem shows that axioms are not enough to fix the meaning of mathematical objects. I was explaining to a colleaque that a power set is always "bigger" than the original set. And I raised the question of whether the set of all subsets of the natural numbers can be put in a one-to-one correspondence with the reals. (I can understand what the quesion is, at least, given a naive acceptance of the construction of the reals.) I told my colleague that it was not clear what the answer was or even what it would mean for the question to have an answer. Apparently, Gödel thought that mathematicians might one day come to agree on an answer.

I think how mathematical terms are used in these systems affects the meaning of mathematical terminology in ordinary life. Some strict separation of these uses cannot be drawn, as far as I am aware. I conclude by noting that some classics (for example, Putnam 1981 and Kripke 1982) of Anglo-American analytical philosophy argue difficulties in understanding how mathematics means extends to (much of?) the remainder of language.

References
  • John B. Fraleigh. 2003. A First Course in Abstract Algebra, 7th edition.
  • Kurt Gödel. 1947. What is Cantor's continuum problem? American Mathematical Monthly 54: 515-525. Reprinted in Kurt Gödel: Collected Works, Volume II.
  • Saul A. Kripke. 1982. Wittgenstein on Rules and Private Language Harvard University Press
  • Hilary Putnam. 1981. Reason, Truth and History Cambridge University Press.
  • Walter Rudin. 1976. Principles of Mathematical Analysis, 3rd edition
  • Friedrich Waisman. 1951. Inroduction to Mathematical Thinking: The Formulation of Concepts in Modern Mathematics.

Saturday, August 22, 2020

Elsewhere

These seem to be resources for providing the student with an overview of schools of thought, fields, and the history of economics.

Saturday, August 15, 2020

Maybe I Should Order One Of These Books

Tuesday, August 11, 2020

2 + 2 = 5

"One must be able to say at all times - instead of points, straight lines, and planes -- tables, chairs, and beer mugs." -- David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

Consider the Fibonacci sequence: 1, 2, 3, 5, 8, 13, ... The first two terms in this sequence are 1 and 2. After this, each term is the arithmetic sum of the previous two terms. Let A be the set of elements in this series. Let s be the function mapping an element in A onto another element of A, where s(n) is the term in the sequence following n. I want to read s as "successor". (I do not claim originality for the following use of the Fibonacci series.)

Notice that the set A and the successor function satisfy the following properties:

  1. 1 is in the set A.
  2. For any n in the set A, its successor s(n) is in the set A
  3. There does not exist an elment n of A such that its successor s(n) = 1.
  4. For all n and m in A, if s(n) = s(m), then n = m.
  5. Suppose a set B contains 1. And further suppose that for any n in B, the successor s(n) is also in B. Then the set A is a subset of B.

The above five properties are the Peano Axioms. The last property is known as the principle of induction.

Some, such as the formalist David Hilbert, would say that mathematical objects are defined by axioms. And the above axioms define the set of natural numbers. Taking the succesor of a number is defined as adding one. And you can define taking the succesor of the successor of a number as adding 2. So, by these definitions, one has:

1 + 1 = s(1) = 2
2 + 2 = s(s(2)) = 5

(By the way, Hilbert was what the kids these days call "woke":

"David Hilbert sought to bring [Emmy Noether] into the mathematics department at the University of G├Âttingen in 1915, but other faculty objected. 'What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?' one professor complained. Hilbert was indignant. 'I do not see that the sex of the candidate is an argument against her admission,' he retorted. 'We are a university, not a bath house.' (See here.))

Bertrand Russell objected to formalism. He thought that whatever mathematicians meant by numbers, these meaning should allow us to assert that most people have one nose, two eyes, and five fingers on each hand. I find his Introduction to Mathematical Philosophy the most approachable statement of his logicist position. Russell found a way to define numbers based on set theory. He required the universe to contain a countable infinity of things, in some sense. His theory of types was a weakness of his approach. Frank Ramsey comes into the story here, with a theory I do not understand. Anyways, various ways can be found to define number in Zermelo-Fraenkel set theory, which I gather avoid the problems with the theory of types.

But this story is incredible. Did people not know how to count or what they were doing until the twentieth century? Maybe a logical foundation for mathematics is not needed. Rather, one might try to provide an anthropological description of what people do when they are counting, adding, multiplying, manipulating infinite series, and so on. Does this capture the "must" in following a rule, though? When a judge consults a law book, he does not treat it as a work in anthropolgy. And who is in doubt on how to proceed when they come to a sum they have never calculated before? Here I am gesturing towards the work of Ludwig Wittgenstein.

I am near my intellectual limits, but I know I have hardly exhausted positions that have been taken over the last century on the philosophy of mathematics. By the way, if I take Jean Piaget seriously, the most advanced mathematics some tried to teach me is crucial to understanding how children think and important for anybody that wants to attempt to design curricula for mathematics.

This post should have a bibliography.

Saturday, August 08, 2020

"So two and two now make five?"

... If you speak of a revolution in mathematics, you are likely to be treated to the ironic response, "Oh, so two and two now make five?" We can answer: "Why not? Suppose we ask for the delivery of two articles each weighing two pounds; they are delivered in a box weighing one pound; then in this package two pounds and two pounds will make five pounds!"

"But you get five pounds by adding three weights, 2 and 2 and 1.""

"True, our operation '2 and 2 make 5' is not an addition in the usual sense. But we can define the operation to make the result hold true. We can imagine a packaging such that the operation might be, for instance:

  • 2 and 2 make 5; 2 and 3 make 6 (if the box weighs one pound in each case),
  • 3 and 3 make 8; 3 and 4 make 9 (if the box weighs two pounds), etc.

"The symbol 'etc.' here stands for a rule of the game, which would define how much a and b would 'make' for all pairs of integers a and b. We would thus define an operation and we could study its properties."

This answer would undoubtably seem frivolous to our our questioner. It would seem to him to be a game; the study we could make of this game would not seem to be a part of mathematics; besides, we would not have invented anything, for we would merely be thinking of the sum of three terms (of which one was understood), while he was thinking of a real addition of two terms.

Lucienne Felix (1960) The Modern Aspect of Mathematics, Basic Books: 3-4.

I am not sure what I think of Nicolas Bourbaki, a man who never existed. But this is the sort of mathematics some need to understand if they are going to help keep the Internet running or explore the solar system.

Thursday, August 06, 2020

Nicholas Georgescu-Roegen On Mathematical Methods In Economic Science

I have long been convinced that many mainstream economists, however they performed under hazing, do not understand mathematics.

"T. C. Koopmans, perhaps the greatest defender of the use of the mathematical tool in economics, countered the criticism of the exaggeration of mathematical symbolism by claiming that the critics have not come forward with specific complaints. The occasion was a symposium held in 1954 around a protest by David Novick. But, by an irony of fate, some twenty years later one of the most incriminating corpora delicti of empty mathematization got into print with the direct help of none other than Koopmans. R. J. Aumann had already published in Econometrica an article dealing with the problem of a market in which there are as many traders as the real numbers, that is, as many as all the points on a continuous line. In 1972, Koopmans presented to the National Academy of Sciences a paper by Donald Brown and Abraham Robinson for publication in its official periodical. The authors assumed that there are more traders even than the elements of the continuum. Now, since the authors of both these papers and Koopmans are well versed in mathematics, they must have known the result proved long ago by George Cantor, namely, that even an infinite space can accomodate at most a denumerable infinity of three-dimensional objects (as the traders must necessarily be)." -- Nicholas Georgescu-Roegen (1979) Methods in Economic Science. Journal of Economic Issues, XIII (2): 317-328.

I do not intend to get a Twitter account. Yesterday(?) some economist illustrated what they learn by tweeting a screenshot of a couple of pages from Mas-Colell, Whinston, and Green..

Tuesday, August 04, 2020

Gautam Mathur Introduces Edward Nell To Piero Sraffa

I have been exploring Sraffa's correspondence after the publication of his book. Here is a letter dated June 18, 1962, from Mathur to Sraffa (D3/12/111: 298):

Dear Mr. Sraffa,

In Nuffield there is a senior research student Edmund Nell who is attached to the faculty of Literae Humaniores[?] and is researching into the significance of concepts in economics. He is highly interested in the type of analysis you have proposed, and has been trying to work his way through it. In Oxford there is no other person whom I have met or heard of who has made a more detailed study of your book.

He would very much like to meet you, to discuss some of his problems, and would be writing to you direct regarding it. In the meantime, he has asked me to introduce him to you. This letter is meant to perform that function, though imperfectly.

I shall be in Cambridge on 25th, 26th, 27th, and would like to see you sometime. I shall ring you up on arrival. I am leaving England on the 30th June.

With regards,

Yours sincerely

Gautam Mathur

Nell's 14 July 1962 letter (D3/12/111: 299-302) is the longest I have found in the archives so far:

Dear Mr. Sraffa,

I believe Gautam Mathur wrote to you about me recently. I had rather hoped that I might be able to see you in Cambridge before you left for the summer, but I'm afraid I just left it too late. Anyway, I don't know if I really have anything to say that would be of much interest to you (and perhaps I'm a little afraid I'm on the wrong track completely.) I was greatly impressed and excited by Production of Commodities, when it was brought to my attention last year by Luigi Pasinetti. Subsequently I have tried to use it in my own work, which is an attempt to criticise (what I take to be) the neo-classical theory of general equilibrium in production and exchange, which I think to be nonsense. (Roughly speaking, I cannot see either what utility functions and production functions are supposed to be predicated of, or how in general preference orderings and production possibilities could be defined independently of each other. Nor do I see what the neo-classical theorists mean when they talk of a (homogeneous) commodity. Indeed, it seems to me that the implications of the notion of an artefact are quite inconsistent with the neo-classical doctrine of "substitution" on both sides of the market.)

Perhaps I should give some background information. I came to Oxford from Princeton, where I studied Politics, as a Rhodes Scholar. I read PPE at Magdalen, received a First, and was elected to Nuffield, where I have been for the past three years. I am now working on a D.Phil. thesis which might be call something like, "Agent and Object in the Theory of Production and Exchange".

Let me sketch briefly the problem that worries me, what I take to be the solution of it, and the way I propose to support this solution.

The general question that troubles me is this: I do not see at all clearly what is meant when one uses a mathematical formula to represent social behavior. Are the actual, historical acts of particular persons supposed to be represented? Or is what is represented supposed to be the relations of various roles in institutions? Many further questions present themselves: How are the value-ranges of the variables defined? Are all the concepts that regulate role-behavior such as to "have number"? Is the relation of a particular act to the general concept of the action of which it is an instance the same as the relation of a value to the variable of which it is a value. And so on. These rather general worries take a much more particular and concrete form, however, when one comes to consider the neo-classical theory of general equilibrium. For example, one wants to know how the value-range of the variable standing for the quantity of some good demanded by an "individual" can be defined independently: a) of the quantity of any and all goods he supplies - whatever does he use the good for? And if what he uses it for is not fixed, then on what grounds does he "prefer" the good to others? To want something is to want it for something. b) of the quantities of other goods that he consumes. Goods are the goods they are because they have certain technological properties, which imply that they must be used in certain (relatively) fixed proportions with other goods; light bulbs require sockets, lampshades lamps.

Even more worrying, however, is the idea that the set of variable demands for each commodity by each individual can be defined independently of the set of variable supplies of each commodity by each firm. For instance it would seem that to make plausible the idea of substituting one good for another in demand, one must require that each good be carefully and minutely distinguished from each other good. But the more sharply we define a good, the more we specify its technological qualities, and therefore the more we determine both the kinds of things and the relative proportions of things that go into its make-up; hence the more plausible we make the idea of substitution in demand, the less plausible we make this same idea in supply. More generally, however, to say that a certain amount of any artefact exists, has been produced, is to imply that someone has been the producer of it; hence has acted in a role, has certain abilities, and has used up something in order to produce the artefact. Any production implies some consumption, and the qualities of the thing produced tell us what kinds of things have been consumed.

A further worry arises over the notion of "price". Even very elementary reflection suggests that exchange must take place between many holders of many different goods, and there seems no reason to assume at the outset that the price of a good can be adequately represented by a scalar variable1.

Finally, even supposing that these problems about the definitions of the value-ranges of the variables could be overcome, there remains the question of just what it is that has been represented: Have we represented here the behavior of actual persons, so that conclusions drawn from the theory could be used for prediction? Or have we represented the structure of institutions, so that conclusions drawn from the theory would serve to make explicit the way people ought to behave - and what the results of their doing of their duty must be? Even if the mathematical representation is valid, there remains the questions of exactly what has been represented, and what the representation can be used for.

These considerations may be summed up in four problems:

a) to discover and classify all the kinds of statements there are about agents acting upon, with, by, through, etc. to produce objects; that is, to analyze the agent-verb-object relation.

b) to see what differences there are in the way various forms of agent-verb-object are demonstrated; and also to see if any of these differ in the way they are demonstrated from the way propositions of the form φa and aRb are demonstrated. That is, what difference does the notion of "agency" make?

c) to see if statements of the agent-verb-object form are formally similar to (hence representable by) mathematical statements; and to see what are the conditions for this to be possible.

d) to see what limitations there are upon the use of a formal representation of statements of the agent-verb-object kind; in particular to see if these can be used for prediction. (My argument tentatively is that prediction and mathematical calculation are incompatible for purely logical reasons.)

So far I have just tried to give a very rough sketch of the problems that bother me and the general position I take on them. I don’t want to prolong this letter, so I won’t try to summarize my work on these four questions. Instead, I should like to mention briefly how I have drawn on Production of Commodities in my work, and indicate a problem that has arisen out of this.

Very shortly, I have tried to classify social actions according to the implications respectively about the agent and about the object(s) of a statement containing as verb the concept of the action. It seems that on the basis of such a study of verbs the concept of the action. It seems that on the basis of such a study of verbs one can make some general classification of institutions, and for each class, one can state the general form of the institutions in it, showing the logical relations of the elements of these institutions. The classes of institutions that stand out as particularly interesting I call "productions" and "performances". These involve, respectively, verbs of making and breaking, and verbs appropriate to what Austin has called "performative utterances". Now the connection with Production of Commodities comes in this way: a set of "productions" will do very nicely as an interpretation of the model there, but it seems more difficult to fit in performances (commands, orders, directives, advice, undertakings, making contracts, judging, passing sentence, etc.) since such roles do not produce any definite quantifiable product which can be shown both as output and as input. Yet a society is inconceivable without these institutions. How are these to be treated? They differ from non-basics in the fact that they are essential.

A second problem arises in that the "plus" sign is used to represent the relations of various commodities to each other in each industry. This seems harmless enough, but a verbal description would contain such prepositions as "on", "with", "by", "in", "through", "to", etc., each of which expresses quite a different relation. So long as neither substitution nor growth is contemplated, and the only point of the mathematical representation is to compute the ratios of exchange required for the possibility of reproduction, no harm seems done. I wonder about switches in methods of production, though.

I'm afraid this letter is very disorganized, and probably not at all clear. On the other hand, my purpose in writing it has been not so much to say something clearly, as to convey a general approach - to communicate an attitude rather than to make a statement. In short, I am writing in the hope that something I've said here may arouse your interest in one or more of my various muddles, and that we might correspond further.

Yours sincerely,

Edward J. Nell

1 That is, a variable whose value-range is a set of cardinal or ordinal numbers. To assume this is to assume away the possibility that a price of a good might change so as to be different but neither greater nor less. But this is exactly what would happen necessarily to at least the price of one good, if one carried out "substitution" of one good for another in an industry, in the system of equations in Section II of Production of Commodities - i.e assuming total proportions constant - hence that somewhere else the reverse substitution takes places. (But it is also the case if just one good is just increased in total amount.)

Sraffa wrote his response on 18 August 1962 (D3/12/111: 303):

Dear Mr. Nell

Thank you for your letter of 14 July which has reached me with delay as I have been moving about.

I greatly sympathise with your critical attitude to the neoclassical theory of equilibrium in production and exchange. On the other hand I have only partly understood what you say on particular points. This is probably due to your assumption that I am familiar with the language and technique of philosophy, which I am not.

I should very much like to discuss some of your criticisms, if we could meet sometime. My movements will be rather unpredictable next year, which will be a sabbatical for me. I am however likely to be in Cambridge during the latter part of September and in October, and if at that time you can get in touch by writing at Trinity perhaps we can arrange to a meeting.

Yours sincerely

P. S.

I have not included strikethroughs, handwritten emendations, and such-like above.

Saturday, August 01, 2020

Jonathan Nitzan On The Factual And Logical Invalidity Of Neoclassical Economics

Neoclassical Political Economy

You may have seen the above overly polite video.

I have not done this in a while, even before the pandemic. I used to, when I visited an academic bookstore or a college library, skim through textbooks, concentrating on introductory or intermediate microeconomics. Economics is in an extraordinary state, where the textbooks are full of nonsense that has been known to be logically invalid for half a century.

Here is an example. David D. Friedman (Milton's son) has made the 1990 or second edition of his Price Theory: An Intermediate Text available online. And it contains this manure:

This conclusion is useful for seeing how various changes affect the distribution of income. Suppose the number of carpenters suddenly increases, due to the immigration of thousands of new carpenters from Mexico. Both before and after the change, carpenters receive their marginal revenue product. Both before and after, they receive a wage equal to the marginal value of the last hour of leisure they give up.

But the wage after the migration is lower than the wage before. Since the supply of carpenters is higher than before, the equilibrium wage is lower. At that lower wage more carpenters are hired and their marginal product is therefore lower. With lower wages, the existing carpenters work fewer hours (assuming a normally shaped supply curve for their labor) and, when they are working fewer hours, have more leisure and value the marginal hour of leisure less. Some carpenters--those with particularly good alternative occupations--find that, at the lower wage, they are better off doing something else. The marginal cost to the worker of working an additional hour falls, either because the marginal hour is worked by one of the old carpenters who is now working fewer hours or because the marginal carpenter is now one of the new immigrants. -- David D. Friedman Chapter 14

Is Friedman a liar or a fool?