# Allen's Arc vs. Assumed Isoelasticity, Pedagogical Efficacy vs. Artificial Accuracy: A Comment

## Article excerpt

1. Introduction

In their recent article in this journal, "Restrictions of Allen's Arc Elasticity of Demand; Time to Consider the Alternative?" Daellenbach, Khandker, Knowles, and Sherony|l991~ (hereafter "DKKS") join a long list of detractors of Allen's |1934~ arc elasticity formula. DKKS demonstrate the bias inherent in the coefficients generated by Allen's arc formula (also known today as the "mid-point" formula) and resurrect Holt and Samuelson's |1946~ proposal as an alternative. Holt and Samuelson proposed that when only two price-quantity combinations are known an isoelastic demand curve be assumed and the logarithmic formula used to calculate a constant demand elasticity (hereafter the "isoelastic/logarithmic approach").

This paper argues DKKS misrepresent the bias of Allen's arc and overstate the suitability of the isoelastic/logarithmic approach to serve as an alternative. This investigation:

(1) Reviews the pertinent elasticity literature.

(2) Analyzes the bias in Allen's arc formula.

(3) Replies to the DKKS concern over large real world changes.

(4) Examines the appropriateness of the isoelastic/logarithmic approach as an alternative to Allen's arc.

This paper concludes:

(1) Allen's arc should remain the basic computational formula for introducing the elasticity concept.

(2) Inclusion of the isoelastic/logarithmic approach should he delayed until advanced classes -- where it can be studied as a special case, useful for empirical studies of small price and quantity changes.

2. The Historical Context

As is often the case, the original writings probably offer the soundest foundation to understanding the controversy. This section reviews the evolution of the arc and isoelasticity approaches to measuring demand elasticity. Irrespective of the terminology used in the original writings, the following conventions are used throughout this paper:

p = price, q = quantity.

|Delta~p = absolute value of a finite change in price = |absolute value of~ |p.sub.1~ - |p.sub.2~.

|Delta~q = absolute value of a finite change in quantity = |absolute value of~ |q.sub.1~ - |q.sub.2~.

dp = infinitesimal change in price; dq = infinitesimal change in quantity.

r = tangent of angle r = p/q; s = tangent of angle s = |Delta~q/|Delta~p/dp.

Figures 1 and 2 are alternative representations of identical price-quantity differences between points a and b.

Marshall |1920~, Schultz |1928~, and Gallego-Diaz |1944-45~, precede Samuelson and Holt |1946~ in considering isoelastic demand curves. In the same time frame, arc elasticity was first introduced by Dalton |1920~ and Allen |1934~.

In his venerated Principles text, Marshall primarily addressed the more general, variable elasticity case. The special, isoelastic case is included in Note III of his Mathematical Appendix -- where he states, "The general equation of demand curves representing at every point an elasticity equal to n is . . . p|q.sup.n~ = C". That is, the constant n is ". . . the proportion in which the amount demanded increases in consequence of a small fall in the price.

Dalton |1920~, concerned with the practical application of the point elasticity definition, objected that Marshall's ". . . elasticity at a point on a curve can tell us nothing of the elasticity corresponding to finite changes in price" |emphasis in the original~. That is, assuming the general case -- where the elasticity at point a does not equal the elasticity at point b -- knowing the elasticity at point a does not reveal the elasticity over some finite arc ab.

Dalton ambiguously proposed using either (|r.sub.1~ x s) or (|r.sub.2~ x s) to calculate the "elasticity across a finite arc". Unfortunately, as Dalton acknowledged, this generates different coefficients, ". . . for any demand curve the elasticity for a given arc is different according to which end of the arc is taken as the base". …

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