# Why does price elasticity of demand formula work?

Hi there,

I understand how to use the price elasticity of demand formula (mid-point method), shown in the picture above.

However, I'm not too sure why exactly it works.

Why are dividing the change in quantity and price by the averages of the quantity and price change? Is that a way of calculating the percentage change?

And why is elasticity a fraction and not a ratio? Would it be better to write: as price changes by 5 percent, quantity decreases by 10 percent, and write that as 5:10?

• I have seen this or some other version of the "mid-point" or "average" formula for price elasticity. Such formulae are not correct. Price elasticity is about what happens at a point--it makes little sense to try to compute it using a formula such as the above. I suspect such formulae were invented so that high-school students would have something to "do" (without actually understanding what they're doing). (Also, the two instances of "/2" in the numerator and denominator are superfluous--suggesting whoever wrote the above formula probably isn't perfectly sure what she's doing.)
– user18
Feb 13 '20 at 3:17
• Does this answer your question? Elasticity of demand equals -1 but income decreases! Feb 13 '20 at 3:26
• @KennyLJ "Also, the two instances of "/2" in the numerator and denominator are superfluous--suggesting whoever wrote the above formula probably isn't perfectly sure what she's doing." I disagree with this. Even though it is superfluous, the /2 serves a didactic purpose. Feb 13 '20 at 3:28
• @Giskard: I disagree. In practice, most students will simply blindly apply the above "formula" without understanding why it works at all. (Indeed, the formula doesn't actually "work".) They won't stop to think, ah the "/2" is because we're thinking about the midpoint. The "/2" just makes the formula look more complicated and the students more bewildered. Students are further encouraged to hold the usual harmful notion that theirs is just to memorize and apply formulae, rather than reason why.
– user18
Feb 13 '20 at 4:01
• @KennyLJ "Students are further encouraged to hold the usual harmful notion" I mean, you cannot force people to learn if they do not want to. Arc elasticity is a real thing though, some textbooks do mention it. It has a Wikipedia page as well, with the /2 included. Luckily it is Wikipedia, so you can edit it! Feb 13 '20 at 11:08

Why are dividing the change in quantity and price by the averages of the quantity and price change? Is that a way of calculating the percentage change?

Yes, elasticity is a measure of proportional response. The math you've shown is the "midpoint" method. It's basically a linear approximation of elasticity, and you'll abandon it once calculus starts showing up in your learning. I would suggest not spending a huge amount of time trying to understand it, but only for this reason. As long as you understand that elasticity of demand says something about how consumers respond to price changes at a certain price level, that's the key message.

And why is elasticity a fraction and not a ratio?

It's really neither - it's a quotient of quotients, which yields a pure (that is, unitless) number. You can check this for yourself if you explicitly write out the units for the things you're looking at. This is called "dimensional analysis" and can be extremely useful for interpreting complicated expressions. The way you do it is to replace each quantity with the thing it "counts" in [square brackets]:

$$\frac{2(Q_2 - Q_1)}{Q_1 + Q_2} \longrightarrow \frac{[purenumber]([apples]-[apples])}{[apples]+[apples]} \longrightarrow \frac{[purenumber][apples]}{[apples]} \longrightarrow [purenumber]$$

Would it be better to write: as price changes by 5 percent, quantity decreases by 10 percent, and write that as 5:10?

In general I would caution against using ratio notation except when playing with statistics (and even then, I personally find them tricky to work with). Here, you may risk confusing a ratio of absolute changes (not what you want to express) for a ratio of percentage changes (this is what elasticity actually is).

• Thanks a lot, very helpful. When using calculus to find elasticity, you'd be using limits, so would you be calculating one point? Feb 13 '20 at 18:27
• Yes, the elasticity would be defined in terms of derivatives, and would potentially have a unique value at every point (for well-behaved demand functions).
– heh
Feb 13 '20 at 19:45

The formula gives an approximation for the (point) price elasticity $$E(P)$$ called the arc elasticity. If you are given a differentiable demand function $$Q(P)$$, then the (point) price elasticity of demand at price $$P$$ is defined as $$E(P)=Q'(P)\frac{P}{Q(P)}$$. If the demand function is not given, you cannot use this exact definition. But if you are given two prices $$P_1$$, $$P_2$$ and the two corresponding quantities $$Q_1$$, $$Q_2$$, then you can approximate the point elasticity by approximating $$Q'(P)$$ by $$\frac{Q_2-Q_1}{P_2-P_1}$$ (remember the definition of the derivative as the limit of this expression if the price difference goes to zero), at least if the two prices are reasonably close.

Now what do you substitute for the expression $$\frac{P}{Q(P)}$$? You could take $$\frac{P_1}{Q_1}$$, thereby approximating $$E(P_1)$$, or you could take $$\frac{P_2}{Q_2}$$, thereby approximating $$E(P_2)$$. In both cases your approximation deviates from the corresponding point elasticity, in one case overestimating it and in the other case underestimating it. (I'm assuming here that the sign of the second derivative of the demand function is constant between the two prices, which makes sense if those are close together.) To decrease your approximation error, i.e. the deviation from the real elasticity, you can choose a compromise and instead use the midpoint between the two prices and the midpoint between your two quantities instead, as is done in your formula. This means that you implicitly assume that the demand function is approximately linear between the two prices, which, again, is reasonable if the two prices are nearby. So this arc elasticity formula gives a good approximation for the point elasticity $$E(\frac{P_1+P_2}{2})$$ at the midpoint between your two prices, or, sloppily speaking, "at the given price range".