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I'm struggling with a math concept dealing with unit elasticity at an undergraduate textbook level. In most problems, we are given a starting price and an ending price as well as corresponding quantities demanded. For example:

When bread was ${$2.00}$ a loaf Bobby wanted ${4}$ loaves, when bread prices raised to ${$2.50}$ Bobby wanted just ${2}$ loaves. What is bobby's price elasticity of demand?

To calculate these we say the percent change in price is ${\% \Delta P = \frac{$2.50-$2.00}{$2.00}=25\%}$. The percent change is measured relative to the starting price. Meanwhile ${\% \Delta Q = \frac{2-4}{4}=-50\%}$. Making elasticity ${-2.0}$.

However, I came across a problem where essentially it asked what is the elasticity for someone who wants to spend exactly ${$10}$ on gasoline. I first tried to think about this intuitively, "If price doubles, she'll be able to get half as much quantity no matter the starting price," which seems like elasticity of ${1}$. However, I set it up algebraically I get the following: $${Q*P=10 \implies Q = 10/P}$$

So I put in some random values of ${P}$, for example ${P_1=$2.00}$ and ${P_2=$2.50}$ for ${\% \Delta P = \frac{$2.50-$2.00}{$2.00}=25\%}$. Meanwhile, For quantity we have ${Q_1 = \frac{10}{2}=5}$ and ${Q_2 = \frac{10}{2.5}=4}$. So that ${\% \Delta Q = \frac{4-5}{5}=-20\%}$.... Which isn't unit elastic.

So I turned to calculus and did the following:

$${Q(P)=\frac{10}{P} \implies \frac{dQ}{Q}*\frac{P}{dP}=\frac{dQ}{dP}*\frac{P}{\frac{10}{P}}=-\frac{10}{P^2}*\frac{P^2}{10}=-1.}$$

Which is unit elasticity. Clearly the discrepancy here is caused by the fact that the derivative has the "starting quantity" = "ending quantity" since it's the same point. But should we be measuring percent change relative to starting price and quantity? Or relative to starting price, but ending quantity? Or does it depend? And why?

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You have to use the midpoint method to resolve this (if I recall correctly).

https://quickonomics.com/how-to-calculate-price-elasticities-using-the-midpoint-formula/

The reason I may not recall correctly is because the second you introduce calculus into the study of economics, you discard these formulae entirely and (try to) forget they ever existed. You seem to already know calculus, which is probably causing you to be even more confused rather than less, and understandably so.

The primary purpose of these non-calculus formulations of elasticity is to teach you the intuition around what elasticity is as a concept - I definitely have strong opinions about how well it accomplishes that. :)

EDIT: Just wanted to add that, if indeed the problem is as you're saying it (come hell or high water, Billy is going to spend \$10 and only \$10 on gas) then your intuition is right - Billy has unit price elasticity of demand for gas, although in a kludgy sort of way. This is supported by your calculus-based analysis.

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  • $\begingroup$ Thank you! The material doesn't mention the midpoint method, at all. I think the complication I was having was the disconnect between my intuition about what unit elasticity means and trying to plug in numbers to prove what I was thinking. In some ways it makes sense that it shouldn't be based on the "starting" or "ending" value since those are relative to our perception of time while it seems elasticity should not depend on such a notion. $\endgroup$ – James Bender Feb 27 at 21:48
  • $\begingroup$ Yeah, the formulas you first encounter are various approximations of the "correct" calculus method. In addition to the linear method and midpoint method, there's also an arc-length method, all of which are clearly flawed to the observant student in a way that can't be addressed in class because at the first and second year level, calculus is verboten. Heck, you'll be lucky if you see an integral in undergraduate at all, unless you're studying at a top-tier school. $\endgroup$ – heh Feb 27 at 22:57
  • $\begingroup$ By the way the reason the calculus method is "correct" is because it produces expressions that you'll see pop out of solutions to more advanced optimization problems. This allows you to talk about things like the optimal Ramsey price in terms of the price elasticities of the products in question, which helps make intuitive sense of some highly abstract concepts. $\endgroup$ – heh Feb 27 at 22:59

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