Using an elasticity to evaluate changes in a dependent variable

Question

Often in textbooks and online I see thte definition of an elasticity to be as follows:

1) $\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\partial \log Y}{\partial \log X}$

Thus, when using an elasticity along with a percentage change in X to estimate a percentage change in Y, it would seem to make sense that one would multiply the percentage change in X by the elasticity as below:

2) ${\frac{\Delta Y}{Y}}=\epsilon\cdot(\frac{\Delta X}{X})$

However, in one instance I've seen mention that it is more accurate to estimate the percentage change in Y as follows:

3) ${\frac{\Delta Y}{Y}}=(\frac{X_2}{X_1})^\epsilon-1$

I'm seeking to apply elasticities derived by estimating log-log regression models and would appreciate some guidance as to whether formula #2 or #3 is more appropriate when evaluating impacts on Y resulting from changes in X.

Can someone please explain why formula #3 may be more appropriate than formula #2? And how does one get mathematically from formula #1 to formula #3?

Answer 1

why formula #3 may be more appropriate than formula #2?

Actually, #2 is a linear approximation of #3.

Note that

$\partial \log Z$

can be written/seen as

$\log Z_2 - \log Z_1 = \log\frac{Z_2}{Z_1} (\approx 0)$

where the above difference is infinitesimal, hence the use of $\partial$. In such a case, #1 can be written as

$\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$

Equivalently, the above equality is only true because we are dealing with infinitesimal changes. Which means that if, say, $Y_2 >> Y_1$, the above equality implies an approximation, i.e.

$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)} \approx \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}$

Which leads to #2:

$\frac{\Delta Y}{Y} \approx \epsilon \frac{\Delta X}{X}$

Actually, the smaller $\Delta Y$ and $\Delta X$, the better the approximation.

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how does one get mathematically from formula #1 to formula #3?

$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$

$\Leftrightarrow \log \frac{Y_2}{Y_1} = \epsilon \log \frac{X_2}{X_1} =$ $\log\left[\left(\frac{X_2}{X_1}\right)^\epsilon\right]$

$\Leftrightarrow \frac{Y_2}{Y_1} = \left(\frac{X_2}{X_1}\right)^\epsilon$

$\Leftrightarrow \frac{Y_2}{Y_1} -1 = \frac{Y_2 - Y_1}{Y_1} = \frac{\Delta Y}{Y}= \left(\frac{X_2}{X_1}\right)^\epsilon -1$