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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?

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2 Answers 2

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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?

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  • $\begingroup$ Thanks for you answer. I don't think I get your point though, the definition of elasticity used in Methodology 1 prescribes a "before" and "after" the price change which should be compared against in a specific order $\endgroup$
    – neutrino
    Apr 25, 2023 at 19:40
  • $\begingroup$ 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. $\endgroup$ Apr 26, 2023 at 10:45
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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.

If you are just given data points with relatively large percentage differences in prices, as in your numerical example, then you cannot calculate or approximate this point elasticity. Instead, one usually calculates the arc elasticity. The method is slightly different from your Methodology 1, however.

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