Positions On The Philosophy Of Math

What does a mathematical proof show? In what sense, if any, do the objects which mathematicians reason about exist? A program of study on the philosophy of mathematics might consider some of the following views:

• Bourbaki - Structuralism
• Brower - Intuitionism
• Frege - Logicism (?)
• Gödel - Platonism
• Hilbert - Formalism
• Lakatos - Proofs and refutations in the tradition of Popper
• J. S. Mill - Math as empirical generalization
• Poincare - Intuitionism (?)
• Russell - Logicism
• Wittgenstein - Constructivism (?)

And one might look at more recent academic interpretations and commentary, such as Putnam or Kripke's interpretation of Wittgenstein.

Philosophers of math often discuss various interesting bits of mathematics. These include the construction of real numbers as equivalence classes of Cauchy-convergent sequences of rationals, of rational numbers as equivalence classes of ordered pairs of integers, and of integers as functions mapping (subsets of) the natural numbers to the natural numbers. Each construction comes with definitions of <, +, and *. The notion of an isomorphism is important in these constructions.

Those who think mathematics is in need of a foundation have often looked for one in terms of logic and set theory. Different axiom systems have been offered for sets. Russell's theory of types contrasts with the Zermelo-Fraenkel (ZF) system. In these set theories, there is an infinity of orders of infinity. I've always like the proof that the power set of a set, that is, the set of all subsets of a set, cannot be put in a one-to-one relationship with the original set. Thinking about applying that theorem recursively to the set of the natural numbers soon exhausts my imagination. Someday I would like to understand Gödel's proofs that if ZF is consistent, then ZP with the axiom of choice (ZFC) is consistent. And if ZFC is consistent, then Cantor's continuum hypothesis is consistent with ZFC. Paul Cohen went further. He proved, in 1963, the continuum hypothesis is independent of the axioms of ZFC. I guess this relates to model theory. I gather the Löwenheim-Skolem theorem is a surprising result.

Philosophers of mathematics often discuss certain important results from comutability theory and the theory of automata. Among these are Gödel's imcompleteness theorem. (Barkley Rosser, Sr., generalized Gödel’s work, from ω-consistency to consistency.) I gather the unsolvability of Diophantine equations, in general, follows from Gödel's theorem. The existence of uncomputable functions is of interest. Every computer programmer should be aware of the halting problem. I find interesting the Church-Turing thesis, the Chomsky hierarchy, and the question of whether the set of problems that can be solved in polynomial time by a deterministic Turing machine is equivalent to the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.

Oversimplified

"Professor Harcourt reviews a body of doctrine challenging the foundations of standard economics. The challenges, if valid, would confer increased respectability on ways of organizing production and income distribution guided more by politics and oriented less toward profit than the ways most amenable to standard analysis." -- Leland B. Yeager (1976). "Review of Some Cambridge Controversies in the Theory of Capital", American Political Science Review, V. 70, N. 4 (December): 1270-1271

Overlapping Research On Overlapping Generations Models

I've recently read Saverio Fratini's "Reswitching of Techniques in an Intertemporal Equilibrium Model with Overlapping Generations" (Contributions to Political Economy V. 26, N. 1 (2007): 43-59).

Mainstream economists responded to the Cambridge Capital Controversy by claiming that disaggregated models of intertemporal and temporary equilibrium are unaffected. So the question arises whether Cambridge capital-theory “paradoxes” can tell us anything interesting about such models. It is known that multiple equilibria can arise in overlapping generations, single-good, models. So pointing out multiple equilibria is not enough. Fratini addresses this issue. He constructs a reswitching example in which reswitching is necessary for multiple equilibriua to arise at an interest rate in a region where a higher interest rate is associated with more savings.

Here’s the abstract of Fratini’s paper:

"We study an overlapping generation model of Diamond's type. In contrast to Diamond, we do not assume that the technology can be represented by an aggregate neoclassical production function. Rather, we study the case in which (i) the technology consists of a finite number of constant coefficient productive activities, (ii) there are capital goods physically heterogeneous with respect to each other and to the economy's only consumption good and (iii) there are two alternative production methods for obtaining the consumption good. We explore the possible linkage between reswitching of techniques and the multiplicity of stationary state equilibria of the model."

Fratini’s research is quite close to some of my own research. Let me compare and contrast:

• Fratini considers a structure of production in which in each technique, the single consumption good is produced with a different capital good. This approach resembles this example I take from Garegnani, except in Garegnani’s example a continuum of capital goods is available. In Fratini’s model, only two techniques are available.
• Fratini imposes some constraints on his model that I do not consider. For example, roughly, he assumes that in each technique the production of the consumption good is more capital-intensive than the production of the capital good for that technique.
• Fratini is better on the economic intutition. For example, he explains his savings schedule as combining income and well-behaved substitution effects.
• I have described a wider range of overlapping generation models. In particular, I wonder if my needing to include a labor-leisure trade-off in the agents’ utility function (as compared to Section 3 here) to get multiple equilibria associated with the “perverse” switch is a challenge to Fratini’s results. (Keep in mind that I do not impose any restriction on the direction of price Wicksell effects.)
• Fratini briefly analyzes the stability of stationary states, drawing on past results in the economic literature.
• I think I am better on visualization of results. This is probably only because I was willing to do more tedious work within a spreadsheet that is needed to generate Fratini’s Figures 1 and 2 and Table 1.
• Fratini has gone through the labor of writing up his results in a coherent whole and seeing the article through the refereeing and publishing process.

Keynes In Historical Time

Alex Tabarrok praises Phelps desire to develop a non-equilibrium, non-hydralic economics. “GVV”, a commenter, points out that Nicholas Kaldor and Joan Robinson have already developed this theme. This led to a question of whether Austrians or Post Keynesians were first to develop this theme. (Peter Boettke also responds to Phelps and comments by pointing to contributions of the Austrian school.)

I think both schools developed a receptiveness to this theme in parallel. Hayek’s student Shackle emphasized the uncertaintly and money in Keynes. Another of Hayek’s students, Ludwig Lachmann, was important, along with Kirzner, in the revival of interest in the U.S. in the 1970s in the Austrian school. And Lachmann lauded Shackle’s interest in disequibrium.

My point in this post is to point out that these themes of disequilibrum, uncertainty, and historical time were in Keynes’ revolution from the start. Consider:

”Actually, however, we have as a rule, only the vaguest idea of any but the most direct consequences of our acts. Sometimes we are not much concerned with their remoter consequences, even though time and chance may make much of them. But sometimes we are intensely concerned with them, more so, occasionally, than with the immediate consequences. Now of all human activities which are affected by this remoter preoccupation, it happens that one of the most important is economic in character, namely, wealth. The whole object of the accumulatin of wealth is to produce results, or potential results, at a comparatively distant, and sometimes at an indefinitely distant, date. Thus the fact that our knowledge of the future is fluctuating, vague and uncertain, renders wealth a peculiarly unsuitable subject for the methods of the [neo]classical economic theory. This theory might work very well in a world in which economic goods were necessarily consumed within a short interval of their being produced. But it requires, I suggest, considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor; and the greater the proportionate part played by such wealth-accumulation the more essential does such amendment become.

By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty; nor is the prospect of a Victory Bond being drawn. Or, again, the expectation of life is only slightly uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any capable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benhamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to be summed.

How do we manage in such circumstances to behave in a manner which saves our faces as rational economic men? We have devised for the purpose a variety of techniques, of which much the most important three are:

1. We assume that the present is a much more serviceable guide to the future than a candid examination of past experience would show it to have been hitherto. In other words we largely ignore the prospect of future changes about the actual character of which we know nothing.

2. We assume that the existing state of opinion as expressed in prices and the character of existing output is based on a correct summing up of future prospects, so that we can accept it as such unless and until something new and relevant comes into the picture.

3. Knowing that our own individual judgement is worthless, we endeavor to fall back on the judgement of the rest of the world, which is perhaps better informed. That is, we endeavor to conform with the behavior of the majority or the average. The psychology of a society of individuals each of whom is endeavoring to copy the others leads to what we may strictly term a conventional judgement.

Now a practical theory of the future based on these three principles has certain marked characteristics. In particular, based on so flimsy a foundation, it is subject to sudden and violent changes. The practice of calmness and immobility, of certainty and security, suddenly breaks down. New fears and hopes will, without warning, take charge of human conduct. The forces of disillusion may suddenly impose a new conventional basis of valuation. All these pretty, polite techniques, made for a well-panelled board room and a nicely regulated market, are liable to collapse. At all times the vague panic fears and equally vague and unreasoned hopes are not really lulled and lie but a little way below the surface.

Perhaps the reader feels that this general philosophical disquisition on the behavior of mankind is somewhat remote from the economic theory under discussion. But I think not. Though this is how we behave in the market-place, the theory we devise in the study of how we behave should not itself submit to market-place idols. I accuse the [neo]classical economic theory of being one of those pretty, polite techniques which tries to deal with the present by abstracting from the fact that we know very little about the future.” -- John Maynard Keynes (1937).

References

• John Maynard Keynes (1937). “The General Theory of Employment”, Quarterly Journal of Economics, V. 51: 209-223.
• Ludwig Lachmann (1976). “From Mises to Shackle: An Essay on Austrian Economics and the Kaleidic Society”,
• G. L. S. Shackle (1988). Business, Time and Thought, New York University Press

Inequality And Voting In U.S.A.

The following figure is from Jeffrey M. Stonecash's "The Income Gap" (Political Science and Politics, July 2006):

Voting Differences Greater When Class Divisions Rise

The upper line is the Gini coefficient, multiplied by 100 to fall on a scale ranging from zero to 100. The higher the Gini coefficient, the moe unequal income is distributed. Notice how it has an upward trend after 1972. The income distribution has been becoming more unequal in the United States for a generation.

The difference is, roughly, the difference between the percentage of the poorest 15% voting Democratic and the percentage of the richest 5% voting Democratic. A higher value here shows that the rich and poor have more different voting patterns.

With the exceptions of 2000 and 2004, the electorate seems to increasingly perceive the Democratic party as for the less affluent as income becomes more unevenly distributed. If you want to live like a Republican, vote Democratic. (Stonecash explains the break in 2000 and 2004 on exceptional circumstances - Clinton's impeachment and Iraq. I wonder if it may have something to do with increasing media concentration and dishonesty. An overwhelming proportion of the mass media in the U.S. is owned by a handful of companies.)

Family Resemblances Among Games

Do mainstream economists have a theory of prices? General Equilibrium theory is obviously one candidate for such a theory. But economic theorists have found many problems with it, some internally generated. After examining the failure of General Equilibrium Theory, in a paper in the Cambridge Journal of Economics, Rizvi turned to look at game theory, another candidate:

"It is important to understand the reasons why general equilibrium theory had to be replaced. This has to do with the emergence of arbitrariness results in general equilibrium theory during the 1970s and early 1980s. Because of these results, it became obvious that individual rationality postulates gave no structure to aggregate theorizing: the aggregate realm was undetermined and arbitrary. This vacuum provided an environment in which rational choice game theory, with all of its problems, was able to prosper. Here, too, the conception of rationality was inadequate to the task at hand. Nash play required entirely unbelievable assumptions, and created a plethora of equilibria; the simplest form of subgame perfection was bedeviled by experimental nonconfirmation. These problems gave impetus in turn to the development of boundedly rational and evolutionary approaches to game theory. To the extent that these approaches have had success in giving basis to some notions from rational choice game theory, they do this by, to a large extent, doing away with rationality. Yet since many of these results were obtained in an oversimple setting, it does not seem likely that evolutionary approaches will find widespread economic applications. Another response to the arbitrariness results has been the work on market demand. Explicitly, authors in this tradition do away with individual rationality to the extent possible. What remains, then, of optimization and its relation to best choice? Since best choice has been the basis of normative economics, evaluative concerns are sacrificed in the effort to obtain aggregate-level results of the most straightforward sort. Many of these approaches are to some extent allied to the emergence of experimental economics. But there is controversy concerning the role of experiments in economics. Are they just a source of paradoxes to be resolved in the usual way in theoretical economics, or should experiments form the centerpiece of an inductive approach to economics? If the latter way is found to be attractive, it is doubtful that there is much chance that the view of economic behavior or satisfaction that emerges will be anything like the universal and individually rational agent.

In the course of examining these transformations in economic theory, we can see quite clearly a loss of ambition in terms of generality, and an increasingly tenuous grasp on the concept of rationality. In consequence, there is a loss in the ability to say anything much about normative considerations. To some, the current period of rapid theoretical change and even to some extent pluralism will be attractive; but durable results are hard to find and many of the fundamental concerns of economics have fallen by the wayside." -- S. Abu Turab Rizvi (1999?). "Rationality, Evolution and Games"

Fordism

In keeping with my amateur status, I like the support of references. Other bloggers have different styles.

I think I'll even provide references for an Eric Nilsson post, which I'm almost sure that he is aware of.

The term "Fordism" was used by Antonio Gramsci in his Prison Notebooks. As I understand it, the phrase is also used by those developing the theory of economic regulation and describing social structures of accumulation. A standard reference here, on my list of books to read some time, is Michael Aglietta's A Theory of Capitalist Regulation: The U.S. Experience (Verso, 1979, first published in French in 1976). Lots of work has been done here since 1976.

Koopmans On Friedman's Claimed Methodology

Over on Crooked Timber, "reason" says he likes a James Galbraith quotation I refer to. Galbraith mentions Koopmans. I suppose Galbraith is talking about this:

"Here the 'direct' implications of the postulates, their accuracy in describing directly observed individual behavior, are placed [by Friedman] in a category with which we need to be less concerned.

There are several objections to such a concept of theory construction. In the first place, in order that we shall have a refutable theory at all, the postulates then need to be supplemented by a clear description of the class of implications by which the theory stands or falls. Otherwise, every contradiction between an implication and an observation could be met by reclassifying the implication as a 'direct' one.

This objection is met by Friedman's suggestion that there should in each case be 'rules for using the model,' that is, a specification of the 'class of phenomena the hypothesis is designed to explain.' But a second objection arises out of this answer to the first. To state a set of postulates, and then to exempt a subclass of their implications from verification is a curiously roundabout way of specifying the content of a theory that is regarded as open to empirical refutation. It leaves one without an understanding of the reasons for the exemptions. The impression of ingeniousness that this procedure gives is reinforced by the fact that in each of Professor Friedman's examples he knows more about the phenomenon in question than he lets on in his suggested postulates. He is willing to predict the expert billard player's shots from the hypothesis that the player knows the mathematical formulae of mechanics and computes their application to each situation with lightning speed, even though he (Friedman) knows that most experts at billards do not have these abilities. He is willing to predict the distribution of leaves on a tree from the hypothesis that each leaf seeks a position of maximum exposure to sunlight (given the position of all other leaves), although no one has reported observing a leaf change its location on a tree.

One cannot help but feel uneasy in the face of so much ingenuity. Truth, like peace, is indivisible. It cannot be compartmentalized. Before we can accept the view that obvious discrepancies between behavior postulates and directly observed behavior do not affect the predictive power of specified implications of the postulates, we need to understand the reason why these discrepancies do not matter. This is all the more important in a field such as economics where, as Friedman also emphasizes, the opportunities for verification of the predictions and implications derived from the postulates are scarce and the outcome of such verification often remains somewhat uncertain..." -- Tjalling C. Koopmans (1957). Three Essays on the State of Economic Science, McGraw-Hill: 139-140

Koopmans also discusses methodology, in an attack on American institutionalism, in his "Measurement without Theory" (Review of Economic Statistics, V. 29, N. 3 (August 1947)). I have already pointed out some more recent criticisms of Friedman's methodology.

One Generation Passes And Another Comes, But The World Forever Stays

Léon Walras claimed that the economy would never be in equilibrium:

"Equilibrium in production, like equilibrium in exchange, is an ideal and not a real state. It never happens in the real world that the selling price of any given product is absolutely equal to the cost of the productive services that enter into that product, or that the effective demand and supply of services or products are absolutely equal..." -- Walras (1954: Lesson 18, Section 188).

and again:

"Finally, in order to come still more closely to reality, we must drop the hypothesis of an annual market period and adopt in its place the hypothesis of a continuous market. Thus, we pass from the static to the dynamic state. For this purpose, we shall now suppose that the annual production and consumption, which we had hitherto represented as a constant magnitude for every moment of the year under consideration, change from instant to instant along with the basic data of the problem... Every hour, nay, every minute, portions of these different classes of circulating capital are disappearing and reappearing. Personal capital, capital goods proper and money also disappear and reappear, in a similar manner, but much more slowly. Only landed capital escapes this process of renewal. Such is the continuous market, which is perpetuating tending towards equilibrium without ever actually attaining it, because the market has no other way of approaching equilibrium except by groping, and, before the goal is reached, it has to renew its efforts and start over again, all the basic data of the problem, e.g. the initial quantities possessed, the utilities of goods and services, the technical coefficients, the excess of income over consumption, the working capital requirements, etc., having changed in the meantime. Viewed in this way, the market is like a lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it. But whereas there are days when the surface of a lake is almost smooth, there never is a day when the effective demand for products and services equals their effective supply and when the selling price of products equals the cost of the productive services used in making them. The diversion of productive services from enterprises that are losing money to profitable enterprises takes place in various ways, the most important being through credit operations, but at best these ways are slow. It can happen and frequently does happen in the real world, that under some circumstantces a selling price will remain for long periods of time above the cost of production and continue to rise in spite of increases in output, while under other circumstances, a fall in price, following upon this rise, will suddenly bring the selling price below cost of production and force entrepreneurs to reverse their production policies. For, just as a lake is, at times, stirred to its very depths by a storm, so also the market is sometimes thrown into violent confusion by crises, which are sudden and general disturbances of equilibrium. The more we know of the ideal conditions of equilibrium, the better we shall be able to control or prevent these crises." -- Walras (1954: Lesson 35, Section 322).

For Walras, equilibrium is a property of his theoretical model, never of the economy. His claim is that an equilibrium model can tell us something about actual economies because of tendencies existing in the economy at any time.

Now Walras could have misunderstood and mischaracterized his own theory. Perhaps properties of neoclassical theory prevent it from being applied with the method Walras describes. As I have previously blogged, Sraffians, such as Bharadwaj, Garegnani, Gram, and Petri, have made this case for neo-Walrasian models adopted and developed by economists more recent than Walras. I take the Arrow-Debreu model of intertemporal equilibrium as the canonical example. This is a short-period model, and represents a dramatic change in economic method. In this model, the initial quantites of all capital goods are taken as data. Yet, some of these quantites, that is, those of circulating capital goods, change in any approach to equilibrium. Thus, if an economy is approaching an equilibrium, that equilibrium will not be the Arrow-Debreu equilibrium calculated for the data, as currently existing in the economy.

For the Arrow-Debreu model to be descriptive of existing capitalist economies, such economies must never be out of equilibrium. This is a defect of the theory.

References

• Krishna Bharadwaj (1989). Themes in Value and Distribution: Classical Theory Reappraised, Unwin-Hyman
• Pierangelo Garegnani (1976). “On a Change in the Notion of Equilibrium in Recent Work on Value and Distribution”, reprinted in Keynes’s Economics and the Theory of Value and Distribution (edited by J. L. Eatwell and M. Milgate, 1983), Oxford University Press
• Harvey Gram (1995). “The Role of Perfect Foresight in Krishna Bharadwaj’s Critque of Demand and Supply Equilibrium-Based Theory”, in The Classical Tradition in Economic Thought: Perspectives on the History of Economic Thought: Volume Eleven (edited by I. H. Rima), Edward Elgar
• Murray Milgate (1979). “On the Origin of the Notion of ‘Intertemporal Equilibrium’”, Economica, V. 46, N. 1: 1-10
• Fabio Petri (2004). General Equilibrium, Capital and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, Edward Elgar
• Léon Walras (1954). Elements of Pure Economics or The Theory of Social Wealth (translated by William Jaffé)

By The Way, Paul, Welcome To The Party

Paul Krugman's New York Times column yesterday seems to be describing Hyman Minsky's financial instability hypothesis. The obviousness of the practical relevance of Minsky's hypothesis today is indeed a scary thing.

Some Data, Much Sad

Keynes Against Laissez Faire Before His Revolution

"It is not true that individuals possess a prescriptive 'natural liberty' in their economic activities. There is no 'compact' conferring perpetual rights on those who Have or on those who Acquire. The world is not so governed from above that private and social interest always coincide. It is not so managed here below that in practice they coincide. It is not a correct deduction from the Principles of Economics that enlightened self-interest always operates in the public interest. Nor is it true that self-interest generally is enlightened; more often individuals acting separately to promote their own ends are too ignorant or too weak to attain even these. Experience does not show that individuals, when they make up a social unit, are always less clear-sighted than when they act separately." -- John Maynard Keynes (1926). "The End of Laissez-Faire"

I was inspired to contextualize this Keynes quotation, with the post title, by a reaction on Ezra Klein's blog to Dani Rodrik's dualism. (I like the Keynes quotation that Rodrik chooses.) There are other reactions to Rodrik, including by me in comments.

Update: Atrios alludes to the Lipsey and Lancaster (1956-1957) paper; his title is an allusion to a funny Leijonhufvud paper.

Update II: Does Radek describe Econ 101 as a kind of "noble lie"? Anyways he has some links to others discussion Rodrik's post. He doesn't link to Peter Boettke, though.

Sraffa Correspondence On Demand

"You say, 'I don't see how demand can be said to have no influence on ... prices, unless constant returns ...' I take it that the drama is enacted on Marshall's stage where the claimants for influence are utility and cost of production. Now utility has made little progress (since the 1870ies) towards acquiring a tangible existence and survives in textbooks at the purely subjective level. On the other hand, cost of production has successfully survived Marshall's attempt to reduce it to an equally evanescent nature under the name of 'disutility', and is still kicking in the form of hours of llabour, tons of raw materials, etc. This, rather than the relative slope of the two curves, is why it seems to me that the 'influence of the two things on price is not comparable." -- Piero Sraffa (1971). Letter to Athanasios Asimakopulos, in (1)

"I am sorry to have kept your MS so long - and with so little result. The fact is that your opening sentence is for me an obstacle which I am unable to get over. You write: 'It is a basic proposition of the Sraffa theory that prices are determined exclusively by the physical requirements of production and the social wage-profit division with consumers demand playing a purely passive role.' Never have I said this: certainly not in the two places to which you refer in your note 2. Nothing, in my view, could be more suicidal than to make such a statement. You are asking me to put my head on the block so that the first fool who comes along can cut it off neatly. Whatever you do, please do not represent me as saying such a thing." -- Piero Sraffa (1964). Letter to Arun Bose, in (2)

References

1. Athanasios Asimakopulos (1990). "Keynes and Sraffa: Visions and Perspectives", in Essays on Piero Sraffa: Critical Perspectives on the Revival of Classical Theory (edited by Krishna Bharadwaj and Bertram Schefold), Unwin Hyman.
2. Ajit Sinha (2006). "A Comment on Sen's 'Sraffa, Wittgenstein, and Gramsci'", Journal of Economic Behavior & Organization, V. 61: 504-512.

Samuelson on Economics as Apologetics

"Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the 'ergodic hypothesis' by analogy to the use of this term in statistical mechanics. Remember that the classical economists were fatalists (a synonym for 'believers in equilibrium'!). Harriet Martineau, who made fairy tales out of economics (unlike modern economists who make economics out of fairy tales), believed that if the state redivided income each morning, by night the rich would again be sleeping in their comfortable beds and the poor under the bridges. (I think she thought this a cogent argument against equilitarian taxes.)

Now, Paul Samuelson, aged 20 a hundred years later, was not Harriet Martineau or even David Ricardo; but as an equilibrium theorist he naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theorists hoped not to introduce hystersis phenomena into our model, as the Bible does when it says, 'We pass this way only once' and, in so saying, takes the subject out of the realm of science into the realm of genuine history." -- Paul A. Samuelson, "What Classical and Neo-Classical Monetary Theory Really Was," Canadian Journal of Economics, V 1., N. 1 (1968): 1-15. (Reprinted in Monetary Theory: Selected Readings (edited by Robert W. Clower), Penguin (1969))

Influence Of Tastes On Prices

1.0 Introduction

This post illustrates why the non-substitution theorem includes an assumption of no joint production. I have previously gone a little into the the theory of joint production in an analysis of depreciation. I have also previously illustrated the non-substitution theorem with an example in which the theorem's assumptions are met (part 1, part 2, Kurz and Salvadori on the theorem).

2.0 The Technology

Consider three islands, Alpha, Beta, and Gamma. A competitive capitalist economy exists on each island. These islands are identical in some respects and differ in others. The point is to understand how differences in tastes can be related to other differences, particularly in prices.

All three islands have the same Constant Returns to Scale technology available. They also exhibit the same rate of profits, and have fully adapted production to their conditions. The technology consists of processes to produce rye and wheat, where workers use inputs of rye and wheat to produce rye and wheat available at the end of the time period associated with each process. This time is a year. That is, each production process requires a year to complete. Each process fully uses up their inputs in producing their outputs.

The processes here exhibit joint production. A process is an example of joint production when its output consists of more than one good. The production of wool and mutton is a well-known example. With joint production, there is room in the economy for processes producing the same set of outputs in different proportions, as in this example. Table 1 shows processes, some subset of which are chosen by the firms at the ruling prices in this example. I think a better example might fully specify a larger technology from which to chose processes.

Table 1: The Technique of Production

Inputs Hired At Start Of Year	Predominately Rye Process	Predominately Wheat Process
Labor	1 Person-Year	1 Person-Year
Rye	1/8 Bushel	3/8 Bushel
Wheat	1/16 Bushel	1/16 Bushel
Outputs	1 Bushel Rye & 1/2 Bushel Wheat	1/2 Bushel Rye & 1 Bushel Wheat

Notice that the net output of the predominately rye process, for an unit level of operation, consists of 7/8 bushel rye and 7/16 bushel wheat. The net output of the predominately wheat process consists of 1/8 bushel rye and 15/16 bushel wheat. Linear combinations, in which each process is operated at some non-negative level, can produce only some ratios of rye and wheat. Hence, these processes cannot meet all possible requirements for use, where such requirements include both consumption needs and requirements for growth. If no process exhibits joint production, any ratio of outputs can be be met by some line combination of processes. This contrast in joint production leads to requirements for use being able to influence which goods are commodities, that is, positively priced.

3.0 Quantity Flows

The employed labor force grows at a rate of 100% on each island. Each island differs, however, in the mix of outputs that they produce.

The population on Alpha wants to eat only rye. They do not and will not consume wheat. Table 2 shows the quantity flows per employed laborer on Alpha. Notice that the commodity inputs purchased at the start of the year total 1/8 bushel rye and 1/16 bushel wheat. Since the rate of growth is 100%, 1/4 bushel rye and 1/8 bushel wheat will be needed for inputs into production in the following year. This leaves 3/4 bushels rye available for consumption at the end of the year per employed worker. There is also an excess output of 3/8 bushels wheat per worker, freely disposed of each year.

Table 2: Quantity Flows On Alpha Island Per Worker

Inputs	Rye Process
Labor	1 Person-Year
Rye	1/8 Bushels Rye
Wheat	1/16 Bushels Wheat
Outputs	1 Bushel Rye & 1/2 Bushel Wheat

GROSS OUTPUTS PER WORKER: (1 Bushel Rye, 1/2 Bushel Wheat)

CAPITAL PER WORKER: (1/8 Bushel Rye, 1/16 Bushel Wheat)

CONSUMPTION PER WORKER: 3/4 Bushel Rye

A linear combination of the two processes that exactly satisfies requirements for use arising for 100% growth on the Alpha island operates the predominately wheat-producing process at a negative level. This makes no economic sense. No other possibility arises other than that shown in Table 2.

The Beta population eats only wheat. Table 3 shows the quantity flows on Beta. Here the same sort of calculations reveal that Beta has 3/4 bushel wheat available for consumption at the end of the year, per employed worker.

Table 3: Quantity Flows On Beta Island Per Worker

Inputs	Rye Process	Wheat Process
Labor	1/4 Person-Year	3/4 Person-Year
Rye	1/32 Bushels Rye	9/32 Bushels Rye
Wheat	1/64 Bushels Wheat	3/64 Bushels Wheat
Outputs	1/4 Bushel Rye & 1/8 Bushel Wheat	3/8 Bushel Rye & 3/4 Bushel Wheat

GROSS OUTPUTS PER WORKER: (5/8 Bushel Rye, 7/8 Bushel Wheat)

CAPITAL PER WORKER: (5/16 Bushel Rye, 1/16 Bushel Wheat)

CONSUMPTION PER WORKER: 3/4 Bushel Wheat

Gamma's quantity flows, shown in Table 4, are one possible intermediate case. Gamma has 9/14 bushel rye and 3/7 bushel wheat available for consumption at the end of the year per employed worker.

Table 4: Quantity Flows On Gamma Island Per Worker

Inputs	Rye Process	Wheat Process
Labor	25/28 Person-Year	3/28 Person-Year
Rye	25/224 Bushels Rye	9/224 Bushels Rye
Wheat	25/448 Bushels Wheat	3/448 Bushels Wheat
Outputs	25/28 Bushel Rye & 25/56 Bushel Wheat	3/56 Bushel Rye & 3/28 Bushel Wheat

GROSS OUTPUTS PER WORKER: (53/56 Bushel Rye, 31/56 Bushel Wheat)

CAPITAL PER WORKER: (17/56 Bushel Rye, 1/8 Bushel Wheat)

CONSUMPTION PER WORKER: (9/14 Bushel Rye, 3/7 Bushel Wheat)

4.0 Price System

Since these economies have adapted to their requirements for use, stationary prices prevail. Assume a rate of profits of 100%, identical across all three islands. Also assume the wage is paid at the end of the year.

4.1 Prices on Alpha

Recall that there is excess production of wheat on Alpha. "If there is excess production of [wheat], [wheat] becomes a free good" (J. Von Neumann, "A Model of General Economic Equilibrium," Review of Economic Studies, 1945-1946: 1-9). Asuming the wage is paid at the end of the year, the price system given by Equation 1 will be satisfied:

(1)

where w is the wage and r is the rate of profits. I have implicitly assumed in the above equation that the price of a bushel rye is $1. The wage can be ound in terms of the rate of profits:

(2)

Since the rate of profits is 100%, the wage on Alpha is 3/4 bushel rye per person-year.

4.2 Prices on Beta and Gamma

The price system given by Equations 3 and 4 will be satisfied on Beta and Gamma:

(3)

(4)

where p is the price of a bushel wheat. The wage can be found in terms of the rate of profits:

(5)

The price of wheat, in terms of the rate of profits, is given by Equation 6:

(6)

Given a rate of profits of 100%, the wage on Beta and Gamma is 3/2 bushel rye per person-year, and the price of a bushel wheat is 2 bushels rye.

5.0 Conclusions

Under the conditions satisfied by this example, in which the economies on different islands are fully adapted to tastes, the prices shown in Table 5 prevail. Differences in tastes between Beta and Gamma are associated with unchanged prices, even in this context. Different tastes on Alpha, however, are associated with a difference in which goods have positive prices and a consequent difference in the wage.

Table 5: Summary of Prices

	Alpha	Beta & Gamma
Wheat (Bushel)	0 Bushel Rye	2 Bushel Rye
Labor (Person-Year)	3/4 Bushel Rye	3/2 Bushel Rye

Note that the quantity flows specified above show the wage entirely consumed and profits entirely invested.

Note that if only goods with a positive price were shown in the techniques chosen on the respective islands, the input-output matrices would be square in all cases (1x1 on Alpha and 2x2 on the other two islands). I think that this property can arise in some cases where wages are not entirely consumed and profits not entirely invested. As I understand it, however, it is a theorem that the input-output matrices are square under this golden-rule condition.

If wages were the same across all three islands, then the rate of profits would vary between Alpha, on the one hand, and Beta and Gamma, on the other. Since the rate of growth is equal to the rate of profits, the rate of growth for a fully adjusted economy would be determined endogeneously. The different choices of the workers on how to consume their wages would result in a difference in the rate of growth between Alpha and the Beta and Gamma islands.

Even though differences in tastes can be associated with differences in prices, it is not clear that this example illustrates a model consistent with the neoclassical (scarcity) theory of value:

"..in a production context...it makes no sense to talk of 'endowments' of given physical quantities if these physical quantities, to be carried over from one period to another, are the unknowns to be determined. It makes no sense to talk of 'scarce' resources, if these resources can be produced in whatever quantities may be needed by the economic system...

When all inputs are themselves produced, a change in the composition of demand simply means that more of some inputs and less of other inputs will have to be produced, while the optimum technique remains the same. In other words, the process of adaptation to any given change in the composition of final demand is, in a production context, radically different from the one considered by traditional theory. Whereas, with given and fixed inputs (the traditional case), the only way to adapt is through a change of technique which may allow the substitution of some inputs for others, in a production context in which all inputs are themselves produced the obvious way to adapt is to produce the inputs which are needed and to cut down production of those which are no longer needed. There is no question of changing the technique. Input substitution, in a production context, has no role to play...

Another route which has been pursued to minimize the importance of the new results...consists in attributing the irrelevance of substitution to the 'very special' case of no joint production and constant coefficients [ = constant returns to scale -RLV ]. But the inconsistency of this contention is here brought into sharp relief by the very analysis of the previous pages...

As already pointed out...the joint production and nonconstant coefficients case is more complicated than, but not basically different from, the case concerning single products and constant coefficients. The complication arises from the fact that a change of the composition of demand may entail a change of the optimum technique and of the price structure. However, this does not enable us to say anything about the direction in which the input proportions will change.

...It is precisely the unambiguous direction in which relative prices and input proportions are related to each other that justifies talking of 'substitution.' But there is nothing of the sort in a production context. No general relation exists between the changes in the price structure and changes in the input proportions. More specifically, no monotonic inverse relation exists, in general, between the variation of any price, relative to another price, and the variation of the proportions among the two inputs to which these two prices refer. When this is so, to talk of 'substitution' among these inputs no longer makes any sense." -- Luigi L. Pasinetti (1977). Lectures on the Theory of Production, Columbia University Press: 186-188

Steven Horwitz, Confused?

"A Mengerian understanding of the market process rejects the claim that an economy can be fruitfully understood through the use of simultaneous equations and equilibrium constructs… The Austrian approach rejects equilibrium theory as a description of actual economic events (although some Austrians would retain it as the never-achieved endpoint of economic activity) in favor of other theoretical and metaphorical devices." -- Steven Horwitz (2000: 8)

I don't know why Horwitz identifies "simultaneous equations" with equilibrium.

Consider this applet. I think one can characterize the underlying mathematics as a system of (countably infinite) simultaneous equations. Yet, one can hardly say that the interest in this mathematics lies in an equilibrium point, at least above a certain value of a parameter.

And that mathematics has economic applications. A Ricardian model can yield a logistic equation (Bhaduri). So can a cobweb cycle with an affine supply function and a quadratic demand function (Goodwin).

Advocates of the Austrian school should strive to write so one cannot read them as not being able to do math, instead of as simply choosing not to do math.

References

• Bhaduri, Amit (1993). Unconventional Economic Essays: Selected Papers of Amit Bhaduri, Oxford University Press
• Horwitz, Steven (2000). Microfoundations and Macroeconomics: An Austrian Perspective, Routledge
• Goodwin, Richard M. (1990). Chaotic Economic Dynamics, Oxford University Press

Peter Klein, Mistaken?

I get this quote second-hand, from Peter Lewin (1999):

"[N]o firm can become so large that it is both the unique producer and user of an intermediate product; for then no market based transfer prices will be available, and the firm will be unable to calculate divisional profit and loss and therefore unable to allocate resources correctly between divisions." - Peter Klein (1996: 15). "Economic Calculation and the Limits of Organization", Review of Austrian Economics; V. 9, N. 2: 3-28.

I have already explained that Sraffa's price theory can be read as instructions to accountants. Klein is being too categorical; correct accounting rules can exist in some cases where he says they cannot. (I am sympathetic to the idea that accounting rules, in practice, contain a large conventional element. Sraffa's instructions to accountants need more information than is usually available, and cases may exist - joint production when the so-called Golden Rule does not hold - where the instructions cannot, even in principle, be applied.)

Ten Principles of Feminist Economics

I've previously referenced expositions of principles of institutional economics and principles of heterodox economics. Geoff Schneider and Jean Shackelford have put forth some principles of feminist economics, also. Since in keeping with an antireductionist view the first principle is that there is no such thing as a definitive list of principles, I will not reproduce the list here.

Milton Friedman's Elegant Tombstones

"Professor Friedman's demonstration [in Capitalism and Freedom] that the capitalist market economy can coordinate economic activities without coercion rests on an elementary conceptual error. His argument runs as follows. He shows first that in a simple market model, where each individual or household controls resources enabling it to produce goods and services either directly for itself or for exchange, there will be production for exchange because of the increased product made possible by specialization. But 'since the household always has the alternative of producing directly for itself, it need not enter into any exchange unless it benefits from it. Hence no exchange will take place unless both parties do benefit from it. Cooperation is thereby achieved without coercion'...So far, so good. It is indeed clear that in this simple exchange model, assuming rational maximizing behavior by all hands, every exchange will benefit both parties, and that no act of coercion is involved in the decision to produce for exchange or in any act of exchange.

Professor Friedman then moves on to our actual complex economy, or rather to his own curious model of it:

'As in [the] simple model, so in the complex enterprise and money-exchange economy, cooperation is strictly individual and voluntary provided: (a) that enterprises are private, so that the ultimate contracting parties are individuals and (b) that individuals are effectively free to enter or not to enter into any particular exchange so that every exchange is strictly voluntary...'

...Proviso (b) is 'that individuals are effectively free to enter or not to enter into any particular exchange', and it is held that with this proviso 'every exchange is strictly voluntary'. A moment's thought will show that this is not so. The proviso that is required to make every transaction strictly volunatry is not freedom not to enter into any particular exchange, but freedom not to enter into any exchange at all. This, and only this, was the proviso that proved the simple model to be voluntary and noncoercive; and nothing less than this would prove the complex model to voluntary and noncoercive. But Professor Friedman is clearly claiming that freedom not to enter into any particular exchange is enough: 'The consumer is protected from coercion by the seller because of the presence of other sellers with whom he can deal...The employee is protected from coercion by the employer because of other employers for whom he can work...'

One almost despairs of logic, and of the use of models. It is easy to see what Professor Friedman has done, but it is less easy to excuse it. He has moved from the simple economy of exchange between independent producers, to the capitalist economy, without mentioning the most important thing that distinguishes them. He mentions money instead of barter, and 'enterprises which are intermediaries between individuals in their capacities as suppliers of services and as purchasers of goods'...as if money and merchants were what distinguished a capitalist economy from an economy of independent producers. What distinguishes the capitalist economy from the simple exchange economy is the separation of labor and capital, that is, the existence of a labor force without its own sufficient capital and therefore without a choice as to whether to put its labor in the market or not. Professor Friedman would agree that where there is no choice there is coercion. His attempted demonstration that capitalism coordinates without coercion therefore fails...

...The logical humanist liberal will regret that ... and the fallacies make Capitalism and Freedom not a defense but an elegant tombstone of liberalism." -- C. B. Macpherson, "Elegant Tombstones", Canadian Journal of Political Science, March 1968.

Heterodox Economics Conference

Apparently, the Association for Heterodox Economics Conference just finished in Bristol. You and I can download drafts of papers presented.