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I'm helping a family friend with an introductory business mathematics course; I have no background in economics, so I've been previewing the applications covered in her textbook, and just read about the concept of elasticity.

The derivation of the formula $E=-\frac{dq}{dp} \cdot \frac{p}{q}$ is straightforward, as are the ideas that when $E<1$, demand reacts weakly to an increase in price, so raising the price will lead to an increase in revenue, and that the opposite will happen when $E>1$. The textbook also mentions* (though only an intuitive explanation is provided) that the price which maximizes revenue is the one that gives unit elasticity (i.e. $E=1$.)

What I'm not quite grasping is what the use of this concept is to begin with. Maximizing revenue would seem to me to just come from $\frac{dR}{dp}=0$, and it's not difficult to see the two are equivalent:

$$R=pq \implies \frac{dR}{dp} = q + p \cdot \frac{dq}{dp}$$

and therefore

$$\frac{dR}{dp} = 0 \iff -p \cdot \frac{dq}{dp} = q \iff E=1$$

and similarly $E<1 \iff \frac{dR}{dp}>0$ and $E>1 \iff \frac{dR}{dp}<0$.

I do understand that $E$ gives us the percent change in demand for each unit percent increase in price, but does that have any practical use other than to maximize revenue? If not, why use $E$ at all rather than just $\frac{dR}{dp}$?

*What the text actually says is that "in ordinary cases the price that maximizes revenue must give unit elasticity." However, I'm unclear on when the derivation I showed above would not work.

A.J.
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  • One thing that's nice is that $E$ is unitless so you can compare between different items whereas just $\frac{dR}{dp}$ has units of $ $/unit$ – Osama Ghani Nov 04 '23 at 23:46
  • @Osama can you provide an example? I'm not sure what you mean by 'compare between different items'. Thanks! – A.J. Nov 05 '23 at 00:00
  • Having a unitless or dimensionless quantity makes it meaningful to talk about the price elasticity of demand of expensive meats vs cigarettes for example. We often say that expensive meats have a higher PED whereas addictive substances like cigarettes have a low PED, but a priori, it doesn't make sense to compare $ $3 $/kg of meat with 2c/cigarette. Is this a fair comparison? What if I did it per pound of meat or per carton of cigarettes? I can manipulate these numbers however I want because there are units associated. But it does make sense to compare the unitless PEDs ($>1$ or $<1$)! – Osama Ghani Nov 05 '23 at 16:19
  • Sorry I do realize that all my PEDs are upside down (I've been doing money/item but it should be item/money) but the idea stands. You can't compare PEDs of items against each other unless you make it unitless/dimensionless. Now, there are other ways of adimensionalizing but this seems to be one that gives us other properties too (such as revenue maximization above!) – Osama Ghani Nov 05 '23 at 16:21
  • @A.J. "so raising the price will lead to an increase in revenue" Usually the definition of the elasticity is: Increasing the price by 1 percent will lead to an increase of the quantity ($q$) by $x$ percent, if $E<1$. That is the main purpose of the elasticity. – callculus42 Nov 05 '23 at 16:48
  • @Osama Thank you for the example, I do understand what you mean now. It does lead to another question, though: of what use is such a comparison? (Please remember - no economics background!) Using your example, if I knew the PED for expensive meats was > 1 while the PED for cigarettes was < 1, what would that information lead to? I do get the logic that addictive substances would tend to have E < 1 while luxury items would tend to have E > 1; but I'm not seeing how comparing them would be useful. – A.J. Nov 07 '23 at 04:04
  • @ callculus42 Thanks for the clarification; I do understand that E itself is comparing the percent changes in price and demand (though I believe increasing the price should lead to a decrease in the quantity demanded.) However, I don't think it is incorrect to say that if E < 1, increasing the price by a small percentage would lead to an increase in revenue, while if E > 1, a similar price increase would lead to a decrease in revenue. – A.J. Nov 07 '23 at 04:17
  • Saying statements like $E > 1$ and $E< 1$ is comparing! That's one application right there. You'll see other uses in things like the Marshall-Lerner condition. As for your second comment for callculus42, one small miscommunication. $E$ (for demand) is the negative of the percent change in quantity divided by the percent change in price so everything works out the right way – Osama Ghani Nov 08 '23 at 03:26

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