# Different methods to calculate price elasticity

I was testing different methods to calculate price elasticities in simple theoretical scenarios and I encountered a seemingly discrepancy between two very popular methodologies.

Methodology 1: use the elasticity definition of % volume change for 1% price change, i.e. $$\epsilon=\frac{\%\Delta V}{\%\Delta P}$$, with $$V$$ and $$P$$ the Volume and the Price respectively. For example if $$(V,P)_1=(3,5)$$ and $$(V,P)_2=(2,7)$$, then $$\%\Delta V=-33\%$$ and $$\%\Delta P = 40\%$$ so the elasticity is $$\epsilon=-0.83$$.

Methodology 2: elasticity can be found as the slope of the linear regression of $$\ln(V)$$ vs. $$\ln(P)$$, since in the eq. $$\ln(V)=\epsilon\ln(P)+\beta$$ we have $$\epsilon=\frac{\delta V}{V}\frac{P}{\delta P}$$. However, a simple linear regression with the data shown in Methodology 1 leads to an elasticity of $$\epsilon=-1.21$$.

Can anyone explain to me why these two methods return different results?

Hint: Suppose you use Methodology 1 but take the percentages of the values in $$(V,P)_2$$ instead of $$(V,P)_1$$. Then $$\%\Delta V= 50\%$$ and $$\%\Delta P = -28\%$$ giving $$\epsilon=-1.79\%$$. What does that tell you about Methodology 1?
• As I understand it true elasticity at a point on a demand curve is defined as $\epsilon=\frac{dV}{dP}\frac{P}{V}$. Thus elasticity is a property of the curve at a point and does not depend on the size or direction of a particular movement along the curve. The point of my example is to illustrate that the result of Methodology 1 does depend on the direction of movement so can only yield an approximation to true elasticity. Having said that I see that a number of websites define elasticity in terms of percentage changes. Apr 26, 2023 at 10:45
For a differentiable function $$V(P)$$ the elasticity, more precisely the point elasticity, at price $$P$$ is defined as $$\frac{dV}{dP}\frac{P}{V}$$. This corresponds to the local rate of change of the corresponding percentage. Introductory textbooks approximate this value by substituting a $$1\%$$ change in price for the true (infinitesimal) percentage change, which is small enough for all practical purposes and avoids the use of calculus.