# How can I reconcile these two different equations for “Arc Elasticity”?

Well, I've encountered a problem which seemed me like a wrong answered one so, I Google'd for the formulas of both "Arc Elasticity" and "Arc Elasticity of Demand" So far, I've found myself in some kind of paradox that is caused by some different educational theories about Economics.

Here, I'm giving the numbers of the problem which was asked for the "Arc Elasticity": $$P_0=100, \; \; \; Q_0=25 \\ P_1=300, \; \; \; Q_1=15 \\ \large{\bf{E_{arc}^{d}}}= \; ?$$

And, here're some formulas about the "Arc Elasticity":

$$Formula \; 1:$$ $$\large{E_{arc}^{d}= \; \frac{\frac{Q_1-Q_0}{Q_1+Q_0}} {\frac{P_1-P_0}{P_1+P_0}}} \\\$$ $$Formula \; 2:$$ $$\large{E_{arc}^{d}= \; \frac{\frac{Q_1-Q_0}{\frac{Q_1+Q_0}{2}}} {\frac{P_1-P_0}{\frac{P_1+P_0}{2}}}} \\\$$

So, which formula I've to apply to find the $E_{arc}^{d}$ ?

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## migrated from economics.stackexchange.comMay 1 '12 at 19:55

This question came from our site for professional and academic economists and analysts.

Thanks for the migration, Ilmari. – Kerim Atasoy May 2 '12 at 12:59

Did you try putting in the numbers, into those two formulae?

What did you find, when you did?

Put in the numbers, and for each formula, just calculate the two numerators (the small numerator on the large numerator, and the small numerator on the large denominator) and the two denominators (the small denominator on the large numerator, and the small denominator on the large denominator) on each. Write these two sets of fractions down next to each other, and compare them. What do you notice?

Can you understand why you got that result?

Spoiler (mouse over the box below to see the spoiler text, but only once you've followed the above advice and worked out what's going on):

Formula 1 is exactly equivalent to formula 2, and has just had the algebra simplified. Formula 2 comes from the geometric interpretation of the arc elasticity. Formula 1 comes about by taking Formula 2, and dividing top and bottom by 2.

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Yes sir, both of them just worked fine. Thank you... :) I guess, I didn't understand the problem well enough to solve. Seems awkward, doesn't it... :) THANK YOU again... :) – Kerim Atasoy Mar 27 '12 at 18:37
But, what about the second formula? Why they show us something like that? For just to calculate and find the equation easier...? I didn't get it actually... – Kerim Atasoy Mar 27 '12 at 18:41
By the way, for now, I'm not able to "submit" the "queries" those are related to acceptance of the current answers and comments but will remember that later... :) – Kerim Atasoy Mar 27 '12 at 18:47

Both formulas are the same (mathematically speaking) It's the second formula that makes sense and is easy to interpret.

Definition of Arc Elasticity in comparison with simple elasticiy:

Price elasticity of demand is the percentage change in quantity demanded for a unit change in price. Arc elasticity computes the percentage change between two points in relation to the average of the two prices and the average of the two quantities, rather than the change from one point to the next. This provides the average elasticity for the arc of the curve between the two points. Hence, the term "arc elasticity."

hope it helps

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Yes, thank you very much... So, should I use the notation for "Arc Elasticity" formula only with $\; \large{E_{arc}} \;$ here, not with $\; \large{E_{arc}^{d}} \;$ , right...? By the way, I got $\; \large{\frac{-1}{2}}={-0.5} \;$ as answer. So, does the current approach fit well enough for this problem now...? Thank you for your all help... – Kerim Atasoy Mar 28 '12 at 13:44
Finally, I understand it now: According to one of my textbooks, both $\; \large{E_{arc}^{d}} \;$ and $\; \large{E^{d}} \;$ should get understood different from each other. Well, it's just one of the books and it inculudes its own specific language about some subjects I think... So, it might be a good idea to search more about these subjects. Here, some other convenient links to share: en.wikipedia.org/wiki/Arc_elasticity en.wikipedia.org/wiki/Price_elasticity_of_demand en.wikipedia.org/wiki/Elasticity_%28economics%29 Thank you everyone. :) – Kerim Atasoy Mar 30 '12 at 22:11
Well.. as you might have already deduce it out, economists were wondering with part in the elasticity formula should be on the numerator side: x1 (initial) or x2(the later) , where x would be a var. Obviously you get different results of an elasticity if you put differenet x-es. So they thought: why not just put the average :D The interpretation in your case: (-0.5) is: if x changes by 10%, then y will change by -0.5*10%, where the arc elasticity is: e=/frac{/frac{/delta x}{(x_1+x_2)/2}}{/frac{/delta y}{(y_1+y_2)/2}} – moldovean Apr 3 '12 at 20:41
I still need to study more about these subjects, they really make me confused recently... :) – Kerim Atasoy Apr 3 '12 at 21:36