A Note on Allen’s Arc Elasticity with Arithmetic, Geometric and Harmonic Means

Abstract

Discussion and debate on the application of Allen’s arc elasticity has continued into the 21st century. This note demonstrates three points. First, perceived differences between Allen’s geometric mean elasticity and a constant demand elasticity based on an assumed isoelastic demand curve are negligible for small changes in price and quantity, which comprise the vast majority of such changes. Second, in some cases of rapid security or commodity price movements, the harmonic mean may provide the most accurate elasticity estimates across measures of central tendency. Third, because the arithmetic and harmonic means serve as bounds for the geometric mean, an elasticity based on the geometric mean may be considered a prudent choice among these three on this basis alone.

Appendix

To expand Eq. (3), first note that:

$$ \ln \left( {1 + r} \right) = \sum\limits_{{i = 1}}^{\infty } {{{\left( { - 1} \right)}^{{n + 1}}}} {{{{{\left( {1 + r - 1} \right)}^n}}} \left/ {n} \right.}. $$

This reduces to the following:

$$ \ln \left( {1 + r} \right) = r\left[ {1 - \left( {{{r} \left/ {2} \right.}} \right) + \left( {{{{{r^2}}} \left/ {3} \right.}} \right) - \left( {{{{{r^3}}} \left/ {4} \right.}} \right) + ...} \right],\,\,{\text{for}}\,\,\left| r \right| < 1. $$

(1A)

Expanding ${\left( {1 + r} \right)^{{ - 0.5}}}$ of Eq. (4) around $\overline{r} = 0$ readily yields:

$$ \matrix{{*{20}{c}} {{{r} \left/ {{{{\left( {1 + r} \right)}^{{0.5}}}}} \right.} = r\left[ {1 - 0.5{{\left( {1 + \overline{r}} \right)}^{{ - 1.5}}}r + {{{0.75{{\left( {1 + \overline{r}} \right)}^{{ - 2.5}}}{r^2}}} \left/ {{2!}} \right.} - 1.875{{{{{\left( {1 + \overline{r}} \right)}^{{ - 3.5}}}{r^3}}} \left/ {{3!}} \right.} + ...} \right]} \\ { = r\left[ {1 - {{r} \left/ {2} \right.} + 0.375{r^2} - 0.3125{r^3} + ...} \right]} \\ } $$

(2A)

Subtracting (2A) from (lA) gives rise to the difference in the percent change in quantity, or the numerators of ${\eta_G}$and ${\eta_{{iso}}}$ _, as follows:

$$ {\Delta_q} = r\left( {{{{ - {r^2}}} \left/ {{24}} \right.} + {{{{r^3}}} \left/ {{16}} \right.}...} \right). $$

(3A)

By exactly the same procedure, the difference in the percent change in price, or the numerators of ${\eta_G}$ and ${\eta_{{iso}}}$ _, is as follows (where s is a small percent):

$$ {\Delta_p} = s\left( {{{{ - {s^2}}} \left/ {{24}} \right.} + {{{{s^3}}} \left/ {{16}} \right.}...} \right). $$

(4A)

Eqs. (3A) and (4A) represent alternating series for small r and s values which are of negligible magnitudes.