Where does the term elasticity (of a function) come from?

Question

Elasticity of a function is a mathematical concept that is widely used in economics. In particular, price elasticity of demand or supply. But generally elasticity in economics is the measurement of how an economic variable responds to a change in another.

Yet I found the term ambiguous and I don't understand why we didn't choose "sensitivity" or "reactivity" instead. To make matters worse, economists often call "Price elasticity of the demand" only, "Elasticity of the demand".

What is the history of the term "elasticity"? Why do we use this term?

Welcome to HSM. Concepts like "sensitivity" and "reactivity" do not capture the economic/statistical notion of regression toward the mean, while elasticity does capture this notion. Elastic means the ability to resume its normal shape after distortion, contraction, or dilatation. — nwr, Jan 17 '19 at 20:46
@nwr: The term elasticity (as used in economics) has nothing to do with regression to the mean or " the ability to resume its normal shape after distortion, contraction, or dilatation". — user103496, Oct 30 '23 at 03:13
Btw: Engineers sometimes use "sensitivity" for the plain derivative, since dy/dx expresses how sensitive y reacts to changes in x. Of course the elasticity is closely related, it's just the derivative in relative terms. — Michael Bächtold, Oct 30 '23 at 08:24
I wonder how closely the stiffness of a function is related to its elasticity? It's not the same thing and is evaluated differently but the two do seem to be linked in some way. — uhoh, Nov 02 '23 at 22:05

user103496 · Answer 1 · 2023-10-30T03:42:49.300

The term elasticity was first used in print by Alfred Marshall (1885, p. 260):

We may want to find a measure of what may be called the elasticity of demand; that is, when a fall of price leads to an increase in the amount demanded, we may want to know the ratio in which the percentage by which the amount demanded has increased stands to the percentage by which the price has fallen.

Marshall then elaborated on elasticity in Principles of Economics (1890, pp. 162 ff):

his willingness to purchase the thing stretches itself out a great deal under the action of a small inducement: the elasticity of his demand, we may say, is great.

I agree with OP that sensitivity would have been a better word.

But it seems that Marshall somehow thought of "stretching" and so used the word elasticity instead. His enormous influence is such that we are now stuck with this word.

Note that economists had been at least somewhat aware of the notion of elasticity, but it was Marshall who highlighted it and made it explicit.

John Maynard Keynes (1925):

I do not think that Marshall did economists any greater service than by the explicit introduction of the idea of "elasticity." ...

it is rather remarkable that the notion was not more clearly disentangled either by Mill or by Jevons. ...

The way in which Marshall introduces Elasticity, without any suggestion that the idea is novel, is remarkable and characteristic. The field of investigation opened up by this instrument of thought is again one where the full fruits have been reaped by Professor Pigou rather than by Marshall himself. ...

Mrs. Marshall tells me that he hit on the notion of elasticity, as he sat on the roof at Palermo shaded by the bath-cover in 1881, and was highly delighted with it.

score 4 · Answer 2 · answered Nov 02 '23 at 17:12

The term elasticity in economics is almost surely linked to the concept of elasticity in physics.

While the introduction of the term elasticity in economics is attributed to Marshall, as far as I know, we haven’t an explicit source about the reason why Marshall chose this term.

But we can make conjectures that are not arbitrary and are based on relevant sources.

It is well known that the mathematization of economics in the second half of the XIX century referred widely to physics, and mathematical economists as Edgeworth, Walras, Pareto, Marshall himself, knew physics, and referred often to physics in their works.

In particular, Pareto wrote his graduation thesis in engineering on Principles of the theory of elastic solid bodies (Engl. transl).

Marshall knew their works, as he explicitly quotes them in his Principles of Economics (1890) .

Moreover, in Appendix C, sec. 2. The method in economics, physics and biology, of the Principles, he explicitly compares the methodology of physics and economics:

It is true that the forces with which economics deals have one advantage for deductive treatment in the fact that their method of combination is, as Mill observed, are that of mechanics rather than of chemistry. That is to say, when we know the action of two economic forces separately—as for instance the influences which an increase in the rate of wages and a diminution in the difficulty of the work in a trade will severally exert on the supply of labour in it—we can predict fairly well their conjoint action, without waiting for specific experience of it .[…] (p. 449)

Marshall himself had a degree in mathematics and physics at the University of Cambridge.

So, the hypothesis that Marshall had in mind the terminology of physics and the elasticity of a body, or the stretching of a wire, is more than reasonable (even if the concept of elasticity in economics and in physics are of course different). $$***$$

In many economics texts it is taken for granted that the term elasticity in economics comes from the terminology of physics.

For an example,

Lionello Rossi, Del concetto di elasticità in economia (Engl. Tr. About the Concept of Elasticity in Economics), 1932

In this article, Rossi seems to take for granted that the concept of elasticity in economics originates from physics, and discusses the actual use in economics of the term elasticity and other terms coming from mechanics as rigidity and anelastic, comparing their use in economic and in physics.

For instance, he writes:

We try […] to clarify the concepts outlined above, and we start making few remarks about the homonymous mechanical concepts[…]

In mechanics a body is defined elastic if, when deformed under the action of a stimulus, has the property to resume its primitive form […]

The concept of rigidity, instead, defines the greater or smaller deformability, and therefore it is not antithetical to the concept of elasticity […]. That is, rigidity denotes a further property of elastic bodies. Anelasticity, instead, it a concept antithetical to that of elasticity[…].

That said about mechanical concepts, what about homonymous concepts of economics?

The only concept that in the common use has a perfect analogy with the concept of mechanics is that of rigidity […]; and the authors seem to agree about it.

Instead, the concept of elasticity, would be, according to Marshall, nothing more than the opposite of the concept of rigidity[…].

Whatever the mechanical concepts are, in economics the Marshallian definition of elasticity as variability and antithetical to rigidity seems obvious, in relations to the denominations that entered the common usage, reserving the name of anelasticity to a different concept, conform to the mechanical one […]. (pp. 20-21, my transl.)

score 1 · Answer 3 · answered Oct 31 '23 at 19:38

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The answer by user103496 gives the history. But they write

But it seems that Marshall somehow thought of "stretching" and so used the word elasticity instead. His enormous influence is such that we are now stuck with this word.

I argue that it is not the case that he just "somehow thought". He envisioned the graph, and realized that a demand schedule that has this measure higher in absolute value, stretches more (compared to a straight line), hence it is "more elastic", hence the measure does measure some notion of elasticity. To wit,

answered Oct 31 '23 at 19:38

Alecos Papadopoulos

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How can you argue what someone thought, without providing some quotes from that person? – Michael Bächtold Nov 01 '23 at 08:04
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@MichaelBächtold Because I argue, I do not assert. – Alecos Papadopoulos Nov 01 '23 at 08:42
"I assert, therefore I am." :-) – uhoh Nov 01 '23 at 18:17
And I'll argue that what you meant to say is that you think that way, not Marshall – Michael Bächtold Nov 01 '23 at 19:55
@MichaelBächtold. Obviously. Interpretation and rationalization has always been part of historiography. – Alecos Papadopoulos Nov 01 '23 at 23:02
It's not historiography if you don't back your claims with historical references. Reminds me of how people like to argue that the reason we write second derivatives as $\frac{d^2y}{dx^2}$ is because "obviously" $\frac{d}{dx}\frac{d}{dx}=\frac{d^2}{dx^2}$. Which is just wrong. – Michael Bächtold Nov 02 '23 at 08:47
As far as I know, we haven’t a sources that say “I, Alfred Marshall, used the term elasticity because…”, but we can do conjectures that are not arbitrary and based on relevant sources. It is well known that the mathematization of economics in the second half of the XIX century was referring mainly to physics, and mathematical economists as Edgeworth, Walras, Pareto knew physics, and surely Marshall knew their works, as he explicitly speaks of them in his Principles. – BakerStreet Nov 02 '23 at 12:42
Even, Pareto wrote a thesis on (Engl. Tr. Principles of the theory of elastic solid bodies).
Marshall knew their works, as he explicitly cites them in his Principles. Marshall himself had a degree in mathematics and physics at the University of Cambridge. So, the hypothesis made by Papadopouls, is not arbitrary, but it relies not only on graphs present in Marshall’s Principles, but on the reasonable hypothesis that Marshall had in mind the elasticity of a body and the stretching of a wire.
– BakerStreet Nov 02 '23 at 12:44
@BakerStreet I could also come up with a reasonable interpretation of "stretching" that has nothing to do with wires: say the price elasticity of demand is 2, then when the price increases by 1% the demand stretches by twice as much (2%). So whose interpretation is closer to Marshalls? – Michael Bächtold Nov 02 '23 at 17:25
@ Michael Bächtold I'm not sure to understand. Your interpretation is the standard definition of elasticity in economics. It has nothing to do with wires, as such. But the name elasticity in economics comes almost surely from physics, and the stretching of a wire can be similar to the graph of demand curves of different elasticity. But the term does't necessarily come from wires. Marshall says 'elasticity' or 'responsiveness', of course elasticity is a suggestion coming from physics, he never said that it was the same concept as in physics. Physics terminology was widesperad among ecomists. – BakerStreet Nov 02 '23 at 17:43
@BakerStreet Rereading the answer of user103496, I see that the interpretation in my comment is precisely what Marshall wrote. It is "physical" and easy to understand. So I don't see why the wire interpretation in this answer is even necessary. (It would in particular require some probably non trivial calculations, to show, that an infinitely long wire that is hanging between (1,0) and (0,infinity), would hang lower if it is more elastic. Can you provide it?) – Michael Bächtold Nov 02 '23 at 17:49
@Michael Bächtold The wire is only a possible suggestion, just an inspiration, when someone sees a graph of demand curves. I mentioned it because of the answer of Papadopoulos, but it is not important, the relevant point is that economists were accustomed to the terminology of physics, it is a fact, and it is not surprising that Marshall used this terminology in a different context, outside physics, and with a different meaning. It make no sense to look for specific analogies between elasticity in physics and in economics. It is only a matter of terminology. – BakerStreet Nov 02 '23 at 18:14
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@BakerStreet you started your comments saying that we don't have a source where Marshall explains why he used the term elasticity, but we have: they're quoted in the first answer. So I don't see the point of this and your answer, where you try to come up with completely different ideas of what Marshal might have thought, without backing them up with quotes from him. – Michael Bächtold Nov 02 '23 at 21:57
And it's not in some vague or obscure sense that Marshall "thought of stretching". He explains it very well in the short quote in user103496's answer. Instead of "the demand stretches itself" he might also have said it increases or expands. – Michael Bächtold Nov 02 '23 at 22:05
In my answer I am pointing out that economics was intertwined with physics, and that the terminology of physics was known among economists, this is the important aspect of the matter, and it is almost sure that it is the reason why the term was introduced in economics. I ansewered the question in the title: where does the term elasticity comes from? Not who introduced the term. My answer is not a comment on the answer of user103496, which sais who but doesn't says why. – BakerStreet Nov 02 '23 at 22:29
The interesting part of the question from the point of view of the history of economic thought is about the relationship between physics and economics, a very important relationship in the history of mathematization of economics. The origin of the modern mathematization of economics was there, the economics theory which attempted, at least in part, to resort to classical mechanics, for instance introducing and formalizing the concept of equilibrium, that came from physics. – BakerStreet Nov 02 '23 at 22:33
It is no coincidence that Marshall used the term elasticity, but it came from the cultural context. – BakerStreet Nov 02 '23 at 22:43

Where does the term elasticity (of a function) come from?

3 Answers3