How can I reconcile these two different equations for "Arc Elasticity"?

Question

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} $ ?

Answer 1

There is no paradox.

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.

Answer 2

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."

Read more: http://www.investopedia.com/terms/a/arc-elasticity.asp#ixzz1qPquLt5I

hope it helps