Abstract
In this paper, the perception and representation of economic quantity found in the works of Augustin Cournot, Leon Walras, Vilfreto Pareto, and Alfred Marshall will be discussed. An interpretation of the perception and representation of economic quantities and economic variables, specifically relating to the concept of demand, in the fundamental theoretical framework of general equilibrium theory will be provided, particularly from the vantage point of the Debreu conjecture.
Received: October 19, 2010
Revised: November 3, 2010
JEL classification: B13, B23, B3
Mathematics Subject Classification (2010): 62P20, 91B02, 91B42, 91B50
I acknowledge the support for this research through a JSPS Grant in Aid for Scientific Research, no. 22530188.
This paper is the English version of an article written originally in Japanese. The Japanese version was published in both the Mita Journal of Economics 103(1), 2010, and in Epimetheus in Economics, edited by T. Maruyama (Chisen Shokan: Tokyo, 2010). The author is heavily indebted to Professor Philip Culbertson for suggesting many elegant improvements to the manuscript in English.
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Notes
- 1.
In the latter part of the quotation, Debreu does not give an explicit explanation for why the assumption of the perfect divisibility of commodities is acceptable for an economic agent who is transacting a large number of commodities. I would presume that Debreu’s perception in this regard is similar to that of Pareto or Walras, which I will take up in the later part of this paper.
- 2.
Italics are mine.
- 3.
In general, “divisibility” in arithmetic means that a rational number can be expressed by finite digits. Thus, as an analytical concept, divisibility in the sense of economics stands in polar relationship to its sense in mathematics.
- 4.
- 5.
See Cournot [4, p. 36]: En outre, qu’entend-on par la quantité demanďe? Ce n’est sans doute pas celle qui se débite effectivement sur la demande des acheteurs; …
- 6.
Cournot [4, pp. 38–39]; italics added. The English translation is as follows (Cournot [5, pp. 49–50]): 22. We will assume that the function F(p), which expresses the law of demand or of the market, is a continuous function, i.e. a function which does not pass suddenly from one value to another, but which takes in passing all intermediate values. It might be otherwise if the number of consumers were very limited: thus in a certain household the same quantity of firewood will possibly be used whether wood costs 10 francs or 15 francs the stere, and the consumption may suddenly be diminished if the price of the stere rises above the latter figure.
- 7.
Cournot clearly states that he uses the word “the demand” (la demande) and “the sale”(le débit) synonymously. Cournot [4, pp. 38–39]: Le débit ou la demande (car pour nous ces deux mots sont synonymes, et nous ne voyons pas sous quel rappport la théorie aurait à tenir compte d’une demande qui n’est pas suivie de débit), le débit ou la demande, disons-nous, croît en général quand le prix décroît. The English translation is as follows (Cournot [5, p. 46]): The sales or the demand (for to us these two words are synonymous, and we do not see for what reason theory need take account of any demand which does not result in a sale) – the sales or the demand generally, we say, increases when the price decreases.
- 8.
Cournot [6, p. 3]. My English translation is as follows: The proper characteristic of a continuous function is that one can always assign, to each of the variables, values sufficiently close to each other such that the difference of the values taken by the function, on which they depend, falls within any given magnitude.
- 9.
In the quotation from Cournot, he may be interpreted as denying the lower-semicontinuity of individual demand functions. This will be discussed further in Sect. 4, in addition to explaining the concept of semi-continuity.
- 10.
Cournot [4, p. 39]. The English translation is as follows (Cournot [5, p. 50]): If the function F(p) is continuous, it will have the property common to all functions of this nature, and on which so many important applications of mathematical analysis are based: the variations of the demand will be sensibly proportional to the variations in price so long as these last are small fractions of the original price. Moreover, these variations will be of opposite signs, i.e., an increase in price will correspond with a diminution of the demand.
- 11.
Cournot [4, p. 39]. The English translation is as follows (Cournot [5, p. 50]): But the wider the market extends, and the more the combinations of needs, of fortunes, or even of caprices, are varied among consumers, the closer the function F(p) will come to varying with p in a continuous manner. However little may be the variation of p, there will be some consumers so placed that the slight rise or fall of the article will affect their consumptions, and will lead them to deprive themselves in some way or to reduce their manufacturing output, or to substitute something else for the article that has grown dearer, as, for instance, coal for wood or anthracite for soft coal.
- 12.
Walras [18, pp. 57–58]. The English translation is as follows (Walras [19, p. 169]): There is nothing to indicate that the individual demand curves a d, 1 a p, 1 and so on, or the individual demand equations d a = f a, 1(p a ) and so on, are continuous, in other words that an infinitesimally small increase in p a produces an infinitesimally small decrease in d a . On the contrary, these functions are often discontinuous. In the case of oats, for example, surely our first holder of wheat will not reduce his demand gradually as the price rises, but he will do it in some intermittent way every time he decides to keep one horse less in his stable. His individual demand curve will, in reality, take the form of a step curve passing through the point a …. All the other individual demand curves will take the same general form.
- 13.
In his book on calculus [6], Cournot explained a differentiable function by using the word “infinitesimal smallness.” I tend to presume that Cournot [4] avoided the use of these words on purpose. However, in case of Walras, I believe he did not intend to claim the differentiability of the demand function in this expression.
- 14.
Walras [18, p. 58]. The English translation is as follows (Walras [19, p. 169]): And yet the aggregate demand curve A d A p …can, for all practical purposes, be considered as continuous by virtue of the so-called law of large numbers. In fact, whenever a very small increase in price takes place, at least one of the holders of (B), out of a large number of them, will then reach the point of being compelled to keep one horse less, and thus a very small diminution in the total demand for (A) will result.
- 15.
Pareto [15, p. 169]. The English translation is as follows (Pareto [16, p. 122]): 65. Continuous variations and discontinuous variations. The indifference curves and the paths could be discontinuous, and they are in reality. That is, the variations in the quantities occur in a discontinuous fashion. An individual passes from a state in which he has 10 handkerchiefs to a state in which he has 11, and not through intermediate states in which he would have, for example, 10 and 1/100 handkerchiefs, 10 and 2/100 handkerchiefs, etc.
- 16.
Pareto [15, p. 169]; italics are mine. The English translation is as follows (Pareto [16, p. 123]): In order to come closer to reality, we would have to consider finite variations, but there is a technical difficulty in doing so.
Problems concerning quantities which vary by infinitely small degrees are much easier to solve than problem in which the quantities undergo finite variations. Hence, every time it is possible, we must replace the latter by the former; this is done in all the physiconatural sciences. We know that an error is thereby committed; but it can be neglected either when it is small absolutely, or when it is smaller than other inevitable errors which make it useless to seek a precision which eludes us in other ways. This is precisely so a in political economy, for there we consider only average phenomena and those involving large numbers. We speak of the individual, not in order actually to investigate what one individual consumes or produces, but only to consider one of the elements of a collectivity and then add up the consumption and the production of a large number of individuals.
- 17.
Pareto [15, p. 169]. The English translation is as follows (Pareto [16, p. 123]): 66. When we say that an individual consumes one and one-tenth watches, it would be ridiculous to take those words literally. A tenth of a watch is an unknown object for which we have no use. Rather these words simply signify that, for example, one hundred individuals consume 110 watches.
When we say that equilibrium takes place when an individual consumes one and one-tenth watches, we simply mean that equilibrium takes place when 100 individuals consume – some one, others two or more watches and some even none at all – in such a way that all together they consume about 110, and the average is 1.1 for each.
- 18.
Pareto [15, p. 169]. The English translation is as follows (Pareto [16, p. 123]): This manner of expression is not peculiar to political economy; it is found in a great number of sciences.
In insurance one speaks of fractions of living persons, for example, 27 and 37 hundredths of living persons. It is quite obvious there is no such thing as thirty-seven hundredths of a living person!
If we did not agree to replace discontinuous variations by continuous variations, the theory of the lever could not be derived. We say that a lever having equal arms, a balance, for example, is in equilibrium when it is supporting equal weights. But I might take a balance which is sensitive to a centigram, put in one of the trays a milligram more than in the other, and state that, contrary to the theory, it remains in equilibrium.
The balance in which we weigh men’s tastes is such that, for certain goods it is sensitive to the gram, for others only to the hectogram, for others to the kilogram, etc.
The only conclusion that can be drawn is that we must not demand from these balances more precision than they can give.
- 19.
As I cautioned earlier, one must be careful to discern whether he actually meant the “continuity” of the market demand, or its “differentiability.”
- 20.
The definition or meaning of the upper hemi-continuity is reviewed in footnote 24.
- 21.
Note, however, that here the demands as a correspondence satisfy the property of upper hemi-continuity. It simply represents the lack of convex-valuedness of the correspondence at the price vector p A.
- 22.
However, if we take Walras’s explanation [18, p. 57] to indicate changes in quantity demanded of a perfectly divisible commodity resulting from its substitution for an indivisible one, then these figures fail to represent his explanation. The Walras’ explication might seem persuasive at first sight, but if we depict the consumer’s choice in Fig. 6 by the demand function for commodity 2, then, strictly speaking, it seems that his insight may have been misled.
- 23.
In particular, I would like to call attention to the fact that his explanation in the quotation could be understood to mean that an individual demand function as a real-valued function cannot be (lower semi-) continuous although it might be upper semi-continuous.
A real-valued function f : X → ℜ is upper semi-continuous at x ∈ X if the set {z | f(z) < f(x)} is open, and f is upper semi-continuous if it is upper semi-continuous at all x ∈ X.
f : X → ℜ is lower semi-continuous at x ∈ X if the set {z | f(z) > f(x)} is open, and f is lower semi-continuous if it is lower semi-continuous at all x ∈ X.
f is said to be semi-continuous if it is either upper semi-continuous or lower semi-continuous at all x ∈ X.
Note that even if a real-valued function is upper semi-continuous, it need not be upper hemi-continuous when regarded as a correspondence. See a footnote 24 for the concepts of the upper hemi-continuity and the lower hemi-continuity of a correspondence.
- 24.
A correspondence F : X → Y is upper hemi-continuous at x ∈ X if for any open set G ⊃ F(x) in Y, there exists an open set V with x ∈ V such that for every z ∈ V one has F(z) ⊂ G. F is upper hemi-continuous if it is upper hemi-continuous at every x ∈ X.
Fig. 8 
Consumer’s choice
Fig. 9 
Consumer’s demand curve for commodity 1
F is lower hemi-continuous at x ∈ X if for any open ste G in Y with F(x) ∩ G≠∅, there exists an open set V with x ∈ V such that for any z ∈ V one has F(z) ∩ G≠∅. If F is lower hemi-continuous at every x ∈ X, then F is lower hemi-continuous.
- 25.
Since the proofs of the existence of an equilibrium in the general equilibrium model of 1950s and 1960s were carried out in a framework where individual demand correspondences essentially become upper hemi-continuous, the awareness of circumstances under which individual demand correspondences fail to be upper hemi-continuous seems to be fairly limited.
As we typically see in Debreu [7, 4.8, p. 63], an individual demand correspondence may fail to be upper hemi-continuous if the level of wealth of that individual drops to the minimal level so as to sustain the purchase of the least expensive combinations of commodities among all the possible consumptions. Thus, in most cases, existence proofs have been carried out under conditions in which all the economic agents circumvent the situation of the minimum level of their wealth for possible consumptions.
Now, in Fig. 9, the circumstances under p 1 B or p 1 C do not correspond to the “minimum wealth level” among all the possible consumptions. However, the lack of upper hemi-continuity arose from the existence of an indivisible commodity. In the literature these circumstances were clarified by Broome [3], Mas-Colell [14], and Yamazaki [20, 22].
- 26.
- 27.
It may be better to offer a remark on the question whether a demand correspondence, in being a function, implies de facto its being continuous. Within the framework of the Debreu conjecture, all the commodities are perfectly divisible, and in his general equilibrium with a differentiable structure of preference relations, as noted in an earlier footnote, the circumstances under which individual demand correspondence may fail to be upper hemi-continuous are excluded so that the mere fact of being demand functions guarantees the continuity of demand functions.
- 28.
For, as we remarked in an earlier footnote, he used the property of local linearity as a characteristic of a continuous function. What is more, he described this property as a property of differentiable function in his textbook on calculus [6, pp. 9–10].
- 29.
- 30.
- 31.
If one is interested in seeing to what extent such a proposition might be shown to be true, see, e.g., Yamazaki [22, Theorem 14.1 or 14.2, pp. 184–187].
- 32.
See, e.g., Bourbaki [2].
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Yamazaki, A. (2011). On the perception and representation of economic quantity in the history of economic analysis in view of the Debreu conjecture. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 15. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53930-8_5
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