## The Fisher – Becker Curio

from **Edward Fullbrook**

Yesterday evening **Merijn Knibbe** put up this comment on **Lars Syll**’s post Utility maximization — explaining everything and nothing.

One of the features of the utility model is that it ´explains´ the downward sloping demand curve, a cornerstone of economics. Which means that neoclassical demand theory seems a pretty coherent building with sound foundations.

However.

It does not explain the downward sloping demand curve. It is only consistent with this curve. And in 1962 Gary Becker showed, in an article called ´Irrational behavior and economic theory´, that many models can ´explain´ a downward sloping demand curve when money is limited, including the throw of a dice. Ockhams razorblade requires us to use the simplest model… http://mcadams.posc.mu.edu/econ/Becker,%2520Irrational%2520Behavior.pdf

Becker himself seems not to have grasped the implications of his article, which shredded neoclassical demand theory. Accepting that demand curves very often slope downwards does not mean that one has to accept utility theory.

My paper (http://www.paecon.net/Fullbrook/IntersubjectiveTheoryofValue.pdf) “An Intersubjective Theory of Value”, in *Intersubjectivity in Economics: Agents and Structures*, editor Edward Fullbrook. London and New York: Routledge, 2002, pp. 273-299, includes a subsection on the theoretical anomaly noted by Gary Becker in his 1962 paper. But that anomaly had been noted with somewhat greater depth and sophistication by Irving Fisher in 1920. Neither economist, however, was capable of understanding the profound significance of the anomaly, because to do so requires a bit of abstract algebra, which, unfortunately, is not part of the economist’s standard tool kit. Below is the relevant section form my 2002 paper. The first part of that paper includes a gentle introduction to the mathematical ideas missing from Fisher’s book and Becker’s paper.

13b. The Fisher – Becker CurioGary Becker [1962, 1971] showed that from the existence of budget constraints alone it follows that a

marketdemand curve must slope downward, whether consumer behavior is rational or not. By beginning his analysis at the level of the market, rather than at the level of the individual, Becker shows that there exists a macro budgetary effect which entails “the basic demand relations”. [Becker, 1971, p. 11] In other words, scarcity of funds is a condition for negative sloping market demand curves, and rational consumer behavior is not a necessary condition. In fact, Becker showed that the whole of the subjective side of demand theory is otiose when it comes to the deduction of the general inverse relation between price and quantity demanded at the market level.Becker’s result needed an explanation which would integrate it into a larger theoretical edifice, but none was forthcoming. Without this theoretical grounding, Becker’s “scarcity principle” remained an anomaly and, therefore, survives today only as a curious and obscure footnote to economics, with students continuing to be taught that the downward slope of the market demand curve is due to intrasubjective factors.

But Gary Becker was not the first to point out the budget effect, that is, “the effect of a change in prices on the distribution of opportunities.” [Becker, 1962, p. 6]. With a different emphasis and at a higher level of aggregation, Irving Fisher noticed and expounded on the same macro dimension of

relativeprices forty years earlier. InThe Purchasing Power of Money, Fisher writes:

“if one commodity rises in price (without any change in the quantity of it or of other things bought and sold, and without any change in the volume of circulating medium or in the velocity of circulation), then other commodities mustfallin price. The increased money expended for this commodity will be taken from other purchases.” [1920, p. 178]Note that here Fisher is not considering the absolute level of prices, which he assumes constant, but rather relative prices and how at the macro level there exists an interdependency between them. Fisher realized, in a way that Becker did not, that his discovery posed a major paradox for economics. [1920. p. 180] Furthermore, because Fisher carries the analysis to a higher level of aggregation, he comes much closer than does Becker to discovering the boolean structure of exchange-value. His nearness to this elementary truth becomes apparent when two terms of his foregoing passage are translated into the terminology of the present paper. Substituting “exchange-value per unit” for Fisher’s “price” and “measured exchange-value” for his “money expended” yields the followings statement:

If one commodity rises in exchange-value per unit (without any change in the quantity of it or of other things bought and sold, and without any change in the volume of circulating medium or in the velocity of circulation), then other commodities mustfallin exchange-value per unit. The increased measured exchange-value for this commodity will be taken from [the exchange-value of] other purchases.Thus translated, Fisher’s passage captures “the part of the whole” relation which, as for probability, is one dimension of exchange-value. It captures also the fact that, despite all the concrete individuality that goes into exchange-value’s making, the exchange-value of every unit of every commodity is irreducibly and fundamentally a SOCIAL relation that extends as far as does the economy in which the exchange of the unit takes place.

What Fisher could not do was explain how the micro and macro levels of “causation” were linked. His insight, like Mendel’s discoveries concerning heredity, lacked a contemporaneous means of explanation. Abstract algebra was little known, and without this set of tools

the boolean structurewhich links exchange-value’s two levels of “causation” could not be identified.4

I recall the following paragraph from your paper which captures a different aspect of the same dependency:

“The exchange-value of the total quantity of money exchanged equals that of the goods for which it exchanges, or, in an economy where all markets clear, the exchange-value of the aggregate endowment of goods (E9). This means that it is the aggregate endowment that acts as the primary standard of exchange-value (E8), with the exchangevalue of each of the individual token-money units being determined as equal parts of the aggregate endowment’s exchange-value (E10).”

What follows from this is that expansion in the aggregate endowment by the factor N must produce change in the exchange value of the total quality of money by the factor 1/N. In plain language, assuming that the amount of money in circulation and velocity of circulation remain constant, the purchasing power of money increases when the volume of (cleared) goods and services increases.

But a certain distributive anomaly can be detected in the above relationship. If the provision of goods and services increases the purchasing power of money (and therefore also the value of financial assets denominated in the same currency) then it is not only the producer that benefits from production. All holders of money and financial assets gain wealth (in the sense of increased purchasing power) without contributing anything whatsoever to the process of production.

Does that mean that the producer of value is entitled to some kind of compensation?

I have examined this question in my paper in RWER issue 70.

No, what you claim follows from the quoted paragraph does not follow. You have confused the terms “aggregate endowment” with “the exchange-value of the aggregate endowment”, as confusing “the bag of groceries” (a non-numerical quantity) with the “cost of the bag of groceries” (a numerical quantity). Your reasoning also assumes that “exchange-value” or “market-value” has the algebraic structure which the paper is arguing that it does not have. The two paragraphs that follow the one that you have quoted, and where Γ stands for the aggregate endowment, partly explain the situation.

“This inverse relation between the quantity of money and the exchange-value of its units corresponds to the negative non-additive law derived earlier with respect to the exchange-value of commodities (E3). But here, with the exchange-value of money, is a very special or pure case of this law, pure in the sense that it is characterized by a fully specified and invariant function–the rarest of all phenomena in the social sciences. Other things remaining unchanged, a percentage change x in the quantity of money exchanged decreases the exchange-value of a unit of money by 1/1+x. In other words, for a given Γ, a money unit’s elasticity of exchange-value with respect to quantity of money exchanged is unitary, that is, a constant.

This section has used the empirical phenomenon of inflation to reveal the structure of the economic process by which token money comes to have the property of exchange-value. If, as with most quantitative orders, the determination of magnitude proceeded from the part to the whole, then an increase in the quantity of money exchanged would not only increase the exchange-value of MV, but would also leave the exchange-value of the individual tokens unchanged. In such a world, monetary inflation would be impossible. Instead, other things being equal, increasing MV affects the exchange-value of individual tokens in the same way as increasing the number of elementary events in an equiprobable probability space affects each of those events’s probability.

Summary of Structural Properties of Exchange-Value Discovered”

Thank you Edward. I have indeed made a mistake, but rather in the following statement:

’What follows from this is that expansion in the aggregate endowment by the factor N must produce change in the exchange value of the total quality of money by the factor 1/N’

Which should read instead

’What follows from this is, other things remaining unchanged, that expansion in the aggregate endowment by the factor N must produce change in the exchange value of the unit of money for which the aggregate endowment is exchanged by the factor 1/N’

I have also omitted a step in the logical progression of my argument so I will try again.

Your Premises:

1: “The exchange-value of the total quantity of money exchanged equals (…) the exchange-value of the aggregate endowment of goods” and

2: “Other things remaining unchanged, a percentage change x in the quantity of money exchanged decreases the exchange-value of a unit of money by 1/1+x.”

The same logic must apply to the aggregate endowment:

3: Other things remaining unchanged, a percentage change x in the aggregate endowment exchanged decreases the exchange-value of individual goods constituting the aggregate endowment by 1/1+x, in order to satisfy Premise 1.

That the exchange-value of individual goods has changed by 1/1+x implies that the exchange-value of a unit of money exchanged for those goods has changed by x% (more goods can be acquired for the same number of units of money). And that what I say follows from your argument.

In other words, the purchasing power of money (assuming MV remains constant) increases with expansion in the aggregate endowment, so that whoever holds a given number of units of money is able to acquire more real value with the same units, what is a kind of enrichment effect built into the system of token representation of value.

I hope I am not implying too much here.

Demand curves can slope down or up (Giffen good). A principal use of utility theory is that it allows the construction of measures of average price or quantity changes for consumers. (I dislike the term ‘index’.)

Claude Hillinger (2008). Measuring Real Value and Inflation. Economics: The Open-Access, Open-Assessment E-Journal, 2 (2008-20): 1—26. http://dx.doi.org/10.5018/economics-ejournal.ja.2008-20