What is the proper way to read and/or express "price elasticity of demand"?

Question

The natural language phrase such as square root of x for a function $Sqrt(x)$ precedes the input with the word of. When we talk of a bivariate function, such as $x/y$ aka $Ratio(x,y)$, we say simply x divided by y or, possibly ratio of x to y.

So, the phrase, price elasticity of demand would imply, not a bivariate function, but a function $PriceElasticity(demand)$.

Moreover, when we speak of causation in mathematical models, we usually, likewise, speak in terms of the future being a function of the past. Therefore, the thing named by the function is the effect; the response of the model. The input to the function is the stimulus.

So this leads one to interpret "price elasticity of demand" as a change in demand resulting in a change in something called "price elasticity".

Lest people think I'm being just a mite too persnickety about all this natural language stuff, there does seem to be a major gap between the economics profession and even the college educated public affected by economic policy, resulting in what might charitably be called "an impedance mismatch".

After looking into this a bit, and discovering that the economists' "elasticity" has little to do with the long history of the word in mathematical physics, I'm wondering if something more along the lines of the bivariate function ratio might better serve both the public and the economics profession.

Or am I simply displaying ignorance of the concepts behind "elasticity" as used in economics?

Answer 1

Demand is a function

In economics, "demand" is an unfortunate shorthand for "demand function". Less frequently and even more unfortunately it is also used for quantity demanded at the current price. Many textbooks make the effort to use "demand schedule" instead of "demand".

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Functions and operators

The elasticity of function $f$ is another function $\epsilon_f$ describing some aspect of the function $f$.

The linguistic structure is similar to derivative of function $f$.

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Multivariate functions

Why not just have "elasticity of the demand function" then, why put the word "price" in there? General demand functions have multiple variables, such as income or prices of other goods. Elasticity can also be calculated according to these variables, i.e. income elasticity and cross-price elasticity exists. Thus adding the word "price" is necessary.

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Language

One could indeed phrase it as "elasticity of function $D_1$ with respect to $p_1$" instead of "price elasticity of $D_1$". This would make it sound more like partial derivative of $f$ w.r.t. $x_i$ or the bivariate ratio. But the second phrase is a bit shorter, well defined, and understood by most people who use it.

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Disclaimer: I am not a native English speaker.

Answer 2

What is the proper way to read and/or express “price elasticity of demand”?

Price elasticity of demand is for simple demand function defined as following:

$$EL_D = \frac{\partial Q(p)}{\partial p} \frac{p}{Q(p)}$$

so the price elasticity of demand is a product of a slope of a demand function evaluated at some specific price and a ratio of that price to the demand function evaluated at the said price. The elasticity express by how much quantity demanded changes (in % terms) when price changes by $x\%$.

Also,

So this leads one to interpret "price elasticity of demand" as a change in demand resulting in a change in something called "price elasticity".

This is purely subjective, I think "price elasticity of demand" leads one to interpret the sentence as response/stretching of demand as a result of change in price which is exactly what it describes.

Lest people think I'm being just a mite too persnickety about all this natural language stuff, there does seem to be a major gap between the economics profession and even the college educated public affected by economic policy, resulting in what might charitably be called "an impedance mismatch"

I don't think this makes much sense:

1. I don't see any evidence for some widespread confusion about the term, I teach economics to undergraduates and I rarely see students having problem grasping the concept. I never seen any data or had experience of some general confusion about the concept of elasticity in particular.

2. Even if we would grant the premise that the name price elasticity of demand is somehow causing confusion among laymen, this confusion can be in present day cleared instantly by just googling the term. I can imagine that avoiding "an impedance mismatch" could have some utility in and prior to 20th century, but not much now.

3. I can't possibly see how "bivariate function ratio" would be more natural or less confusing to layman. It is not even accurate since elasticity can be function of multiple variables in principle. Following, EMFE by Hammond et al pp 407. the general formula for elasticity is given by:

$$EL_i z = \frac{x_i}{f(\mathbf{x})}\frac{\partial f(\mathbf{x})}{\partial x_i}$$

where $\mathbf{x}$ can be arbitrary large vector. So calling it "bivariate function ratio" is not even accurate. If you want to be accurate an elasticity is "a product of a slope of a function evaluated at some specific input with a ratio of input to function evaluated at the said input holding all other inputs constant." However, not only is that mouthful I doubt for regular person that would be more clear description.

4. In virtually all fields of science there are words/jargon that is not completely natural. When a layman person hears words like "rational number" or "irrational number" they often make association with rationality in a sense of 'being smart' rather than the fact that rational refers to an ability of the number to be expressed as a ratio/fraction of two integers. So would it make sense to now go and relabel these ratio and non-ratio numbers, just because it would have more 'natural'/easier interpretation for laymen?

   I think not. Not only would that yield very little benefit in present day where the correct definition can be easily obtained, it would make it harder to relate any mathematical text written before this change to new texts. In the end people would have to learn both definitions just for the sake of continuity. Equivalently, in economics people would still have to learn what elasticity of demand means even if we would suddenly decide to switch to new terminology, because of the continuity in economics research and people working on new ideas often have to cite old work, sometimes even decades old. This would just create more trouble than it is worth. It is no accident that in all branches of science terminology is rarely 'updated', with many branches of science even using Latin terms long after using Latin-based terminology makes any sense.