# Using an elasticity to evaluate changes in a dependent variable

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?

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.

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

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

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

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

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

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

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

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

• Any question @Anon ? – keepAlive Oct 17 '19 at 11:36