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Allen's arc vs. assumed isoelasticity, pedagogical efficacy vs. artificial accuracy: a comment.

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". Dalton's approach is, of course, not a measure of the elasticity "of the arc" itself. Rather, using the slope of the chord ab, it measures the point elasticities at the extremes of the chord -- at points a and b. Most importantly, while Dalton's article includes a graphical representation of the true demand curve -- given his ex ante assumptions -- the true demand curve is not known, only the two price-quantity combinations are known. Note that this is precisely the dilemma presented to principles students. "Given two price-quantity combinations, and no other information, which combination should be used as the basis for calculating the elasticity over the arc? Point a or point b?"

Schultz |1928~ addressed the issues involved in using real world data to estimate the parameters of the true demand curve. Rather than consider only two, hypothetical price-quantity combinations, he applied the isoelastic functional form to the task of empirically estimating the true demand curve. Recognizing their econometric advantages, Schultz gave a particularly cogent discussion of isoelastic demand curves and their relationship to arc elasticity:

The coefficient of the elasticity of demand may be defined as the ratio of the relative change in the quantity demanded to the corresponding relative change in price, when the relative changes are infinitesimal.

The definition presupposes a knowledge of the demand curve. Consequently, any attempt to derive the coefficient of the elasticity of demand for a commodity without first deriving the equation to the demand curve is apt to lead to difficulties. The coefficient of elasticity relates to a point on the demand curve, and may vary in magnitude from point to point. In giving the coefficient of elasticity of a commodity, one must, therefore, specify the point on the demand curve to which it applies, unless, of course, the coefficient is the same at every point on the curve.

The characteristic of a constant-outlay curve is that any change in price causes a proportionate change in the amount bought . . . percentages are not good measures of proportionate changes . . . .

A much better measure of proportionate changes is to be had by taking the differences between the logarithms of the numbers. The coefficient of elasticity is then simply the ratio of the difference between the logarithms of the quantities to the difference between the logarithms of the prices. When the changes in both quantities and prices are measured logarithmically, there is no difference whatsoever between the point elasticity and the arc elasticity of the constant-outlay curve.

The thing to remember, therefore, is that in order to derive an unequivocal coefficient of elasticity we must first derive the equation of the demand curve, or the law of demand.

These excerpts from Schultz emphasize four points that are particularly relevant to the DKKS article. (1) To the extent the true demand curve exhibits variable elasticity, the definition of elasticity is restricted to infinitesimal price changes. (2) Thus, only with an isoelastic demand curve can a single, "the" elasticity -- one that spans the whole demand curve -- be determined. (3) Rather than assume an isoelastic demand curve, failure to determine the true demand curve is "apt to lead to difficulties." (4) Lastly, but most importantly, before the logarithmic formula can be considered appropriate, it must be determined that the price and quantity do change in a proportionate manner.

The distinction between Dalton's ambiguity and the approach taken by Schultz is significant. Dalton assumed a situation where the true demand curve is unknown and attempted to draw useful conclusions from only two data points. Schultz on the other hand, assumed a data-rich environment and set about constructing the theoretical framework necessary for estimating small segments of the true demand curve. While Dalton addressed a theoretical issue, Schultz, writing eighteen years later, used real-world data to estimate real-world relationships.

In addressing Dalton's arc elasticity ambiguity, Allen |1934~ asked, "Can any meaning be given to the elasticity of demand for the arc |P.sub.1~|P.sub.2~". That is -- except for the special, isoelastic case -- given that points on the demand curve have different elasticities, Allen questioned whether it is meaningful to ascribe a single elasticity measure to an arc of the demand curve? As with any average, the question of whether a single measure can meaningfully represent the range of elasticities over the arc depends on the magnitude of the dispersion of the elasticities along the arc. Acknowledging the limitations of any average, Allen characterized his proposal as an approximation, useful for summarizing the elasticities of small segments of a demand curve.

Allen's solution to Dalton's ambiguity simply uses the arithmetic means of the prices and quantities as the basis for calculating the percentage changes. This is, of course, the approach used today in calculating the arc elasticity (also called the "mid-point elasticity"). Again however, it is not a measure of the elasticity "of the arc" ab, but rather it is a measure of the elasticity "across the arc" ab -- at the midpoint of the chord ab:

|Eta~ = (|p.sub.1~ + |p.sub.2~)/(|q.sub.1~ + |q.sub.2~) (|q.sub.2~ - |q.sub.1~)/(|p.sub.2~ - |p.sub.1~) = |Delta~q/|q.sub.arithmetic mean~/|Delta~p/|p.sub.arithmetic mean~

Gallego-Diaz |1944-1945~ appears to have been the first writer to combine the isoelastic demand curve used by Marshall and Schultz with the two price-quantity-combination problem considered by Dalton and Allen. Gallego-Diaz showed there exists a point on the arc that has a point elasticity equal to that of the isoelastic curve that passes through both a and b. Note that Gallego-Diaz does not assume the true demand curve is isoelastic. He simply shows an isoelastic curve exists that coincides with the two price-quantity combinations of points a and b. That is, as shown in Figure 2, one need not know whether the convex to the origin curve DD, the concave to the origin curve dd, or any other function is the true demand curve to assert there is a point in the arc ab that has a coefficient equal to the ratio of the logarithms of the two prices and quantities. Quoting Gallego-Diaz, ". . .the arc elasticity . . . will be worth:

log (|q.sub.2~/|q.sub.1~)/log (|p.sub.2~/|p.sub.1~) which is: log |q.sub.2~ - log |q.sub.1~/log |p.sub.2~ - log |p.sub.1~ . . ." In their development of a graphical approach to measuring arc elasticity, Holt and Samuelsons |1946~ state:

. . . Whenever two price-quantity situations lie along a curve of constant point elasticity, the arc elasticity of demand shall be equal to that constant value, regardless of the size of the step. In that case, the only correct measure of arc elasticity of demand between any two points (|q.sub.1~, |p.sub.1~) and (|q.sub.2~, |p.sub.2~) is given by:

- |Delta~ log q/|Delta~ log p = log |q.sub.2~ - log |q.sub.1~/log |p.sub.2~ - log |p.sub.1~

|emphasis in the original~.

The distinction between Holt and Samuelson's proposal and the point made by Gallego-Diaz is fundamental. Gallego-Diaz asserts -- without knowing the functional form nor the parameters of the true arc -- there exists a single point in the arc that has a point elasticity equal to that calculated by the logarithmic formula. In contrast, the Holt and Samuelson statement that the logarithmic coefficient is the "only correct measure," is fully dependent on some ex ante determination that the true demand curve is, in fact, isoelastic. In one sense, Holt and Samuelson are not estimating the elasticity of the two price-quantity combinations, but are using only two observations to estimate the parameters of an exponential, isoelastic function.

Gabor |1976~ (without mentioning Holt and Samuelson) acknowledges the Gallego-Diaz contribution, ". . . although his |Gallego-Diaz~ original brief note (1944-45) does not appear to have made any lasting impression on his contemporaries". Gabor concludes with a final warning, "However, this expedient |using the logarithmically determined coefficient~ does not in any way validate the assumption that the functions concerned are actually of the form x = |Cy.sup.E~". No one, neither DKKS, Gallego-Diaz, nor Holt and Samuelson present any evidence -- either theoretical or empirical -- to support the assumption that the true, underlying demand curve is, in fact, isoelastic -- of the form x = |Cy.sup.E~.

3. The DKKS Argument

DKKS present their argument in three parts:

(a) Because Allen's arc elasticity formula yields a biased result for large percentage changes, and

(b) As large changes are evident in the real world, then

(c) The isoelastic/logarithmic approach should be used as an alternative to Allen's arc.

The following subsections argue DKKS have constructed a strawman by taking Allen's arc where it was never intended to go. DKKS criticize the ability of Allen's arc to represent large changes across an assumed isoelastic function. This is not what Allen intended to do nor is it currently used in this manner. Rather it was designed to summarize different elasticities along a, nothing-said-to-the-contrary, variable elasticity curve.

3a. Bias in Allen's Arc

Part (a) of the DKKS argument states: the percentage changes in quantity and price are restricted to less than 200 percent by Allen's arc elasticity formula". As an example of this restriction, DKKS construct a somewhat peculiar predicament where the arc elasticity is predetermined to be 0.50, the quantity demanded changes from 3 to 9, and the price change must be calculated in accordance with Allen's arc elasticity formula. Stating that one price was predetermined, the authors assert that no combination of prices "satisfactorily completes" the problem.

There is, of course, a trivial solution to their example -- one price must simply be zero while the other price is any non zero amount. Excluding zero prices, their statement is true -- a 100% quantity change necessarily results in an Allen arc elasticity greater than 0.5.

Particular care must be taken not to misinterpret the DKKS statement that, ". . . the percentage changes in quantity and price are limited to less than 200% by Allen's arc elasticity formula . . .". This statement does not mean "the percentage changes in quantity and price are restricted to 200%." The percentage change itself can be greater than 200%. Allen's method of basing the percentage change on the average amount restricts his representation of the percentage change to 200%.

Consider the limiting case -- according to Allen's formula:

(1) An infinite percentage increase, from zero to any non zero amount -- when calculated as a percentage of the average value:

an infinite % change / half of the infinite % change = 200%.

(2) The same change viewed from the opposite direction, a decrease from any non zero amount to zero is "only" a 100% change -- when calculated as a percentage of the average value:

a 100% change / 50% of the change = 200%.

Allen's formula -- disregarding the direction of change -- represents both an infinite percentage increase and a 100% decrease as a 200% change. There is no restriction on the ability of Allen's formula to accommodate percentage changes greater than 200%. As with any arithmetic mean, the larger measure is understated, the smaller is overstated. However, the understatement of the increase (the larger measure) is no more "important" than the overstatement of the decrease (the smaller measure). In a literal sense, this is of course, not a restriction of ability of Allen's arc formula to represent large (or small) changes -- it is only a bias in its representation.

Regardless, as DKKS note, because Allen's percentage changes in quantity and price are limited to the 200% bound, Allen's arc elasticity measure approaches unity as his percentage changes become large. However, as long as the context in which Allen offered his formula is observed -- an approximation useful for summarizing the elasticities of small segments of a demand curve -- the significance of this bias is problematic. It is only when Allen's arc is misapplied to summarizing the elasticities of large segments of an assumed isoelastic demand curve that its bias is of any concern.

3b. Large Real World Changes

Part (b) of the DKKS argument states: " . . . since large percentage changes are evident in the real world, it is unrealistic only to consider problems with small changes in price or quantity". This assertion fails on four counts: (1) Large percentage changes are most infrequent in the real world. (2) On the rare occasions that large changes do occur they are invariably associated with large changes in the price elasticity itself. (3) Exogenous events of sufficient magnitude to cause large price changes cannot be usefully analyzed with a simple, two dimensional price vs. quantity demanded analysis. (4) In a pedagogical setting, large price-quantity changes are examined solely for visual clarity. Discussion of each of the four points follows:

Note. The notion as to what constitutes a "large" change depends upon the context in which the change is considered. Using infinitesimal calculus, in a point elasticity setting, "large" is simply anything greater than infinitesimal, i.e., any finite change. DKKS clearly use the term in a different context. In their example, quantities changing between 3 and 9 units represent a 200 or 67 percent change, depending upon which quantity is used as the base. Some might view these as huge, not just "large" changes.

(1) While large price changes are statistically possible, compared to the overwhelming number of small market price changes -- in the spirit of Walras' tatonnment -- large price changes are undoubtedly rare events. Certainly large price changes do occur -- especially in commodities markets -- where both supply and demand are relatively inelastic. But when they do occur they are news events -- not typications appropriate to serve as the basis of pedagogical elucidation. To once again call on Marshall's famous dictum, "Natura non facit saltum," markets do not thrash about in violent price and quantity changes.

(2) Demand curves exhibit variable elasticity over large price and quantity changes. It is futile to search for a single coefficient to represent the changing elasticity over a large segment of a demand curve.

Neither DKKS nor Holt and Samuelson offered an economic explanation of why an isoelastic demand function should be assumed over small, much less large, changes. Contrarily, Marshall, |1920, p. 87~ succinctly states:

The elasticity of demand is great for high prices, and great, or at least considerable, for medium prices; but it declines as the price falls; and gradually fades away if the fall goes so far that satiety level is reached."

Marshall clearly considered isoelasticity to be a special case -- relegating it to a note in his Mathematical Appendix. Likewise, in the International Encyclopedia of the Social Sciences entry for "Demand and Supply" Boulding |1968, p. 98~ states:

There is no reason to suppose in fact, however, that these |supply or demand~ functions are more likely to be logarithmic than linear in absolute terms, and for many purposes the absolute concepts are preferable. A logarithmically linear demand curve with constant relative elasticity . . . would not intersect either axis . . . implying that the price would have to be infinite before cutting off purchases altogether and that, at a zero price and infinite quantity would be taken. This clearly is absurd.

Econometric studies are typically restricted to examining small changes in price for the very reason that such restrictions enable the researcher to take advantage of isoelastic functional forms. As Culyer |1985, p. 46~ notes in his principles text:

Although constant elasticity demand curves arc often used in empirical work, the restriction on the elasticity in this way is not as severe as it may seem. It is usually the case that available data permit estimation of the elasticity only for a relatively small portion of the demand curve -- that for which variable price data actually exist. It may therefore be quite possible to obtain a good estimate of the elasticity on the assumption that it is constant in this region. The danger lies in extrapolating the constant elasticity outside this region; the further outside it goes, the more arbitrary the restriction will become. |Emphasis in the original.~

(3) Large price changes are invariably associated with shifts in otherwise fixed demand curves. The 1973-74 Arab oil embargo resulted in the most dramatic price changes seen in recent times. The two year time period in which petroleum prices quadrupled was undoubtedly accompanied by a myriad of functional shifts -- not merely movements along fixed supply and demand relationships -- but shifts in the curves themselves.

While no one would seriously suggest the application of a naive, two dimensional price vs. quantity demanded analysis to such a dramatic event, the exercise may none-the-less be informative. From 1973 to 1974, by restricting output from 19.8 to 15.5 million barrels per day, Arab oil suppliers were able to raise their price of crude oil from $3.07 to $12.38 per barrel |Clark, 1990, pp. 233-237~. Applying these data to the formulae given above yields a coefficient = 0.18 for the isoelastic/logarithmic approach and 0.20 for Allen's arc. Even for this atypical case (worst case scenario -- very inelastic and unrealistically assuming the relationship is stable and isoelastic) Allen's arc exceeds the logarithmic coefficient by only 15%. It is doubtful that student understanding is significantly compromised by extreme cases generating inconsistencies of this magnitude.

(4) Any large deviation of the coefficient generated by Allen's arc elasticity approach from an assumed isoelastic function is the result of choosing -- in a pedagogical setting -- price-quantity combination differences that would be unreasonably large in the real world. Parenthetical caveats should suffice to quell the misgivings of instructors concerned with the large change bias of Allen's formula.

Referring again to his Principles text, Marshall perceived the demand schedule as ". . . how much he would be willing to purchase at each of the prices . . ." |1928, p. 81~. For Marshall, the demand curve represents an instantaneous "snapshot" of alternative, price-quantity combinations. It is not a menu from which one may choose how real world price changes would actually cause the quantity demanded to change over time. Marshall continues:

. . . a list of demand prices represents the changes in the price . . . other things being equal; yet other things are seldom equal in fact over periods of time sufficiently long for the collection of full and trustworthy statistics. There are always occurring disturbing causes whose effects are commingled with, and cannot be easily separated from, the effects of that particular cause which we desire to isolate. This difficulty is aggravated by the fact that in economics the full effects of a cause seldom come at once, but often spread themselves out after it has ceased to exist. |Emphasis in the original.~

While the assumption of an instantaneous time period can be relaxed to some extent, two dimensional long run functions are simplistic artifacts of the "instantaneous snapshots" moving over time. Large, long run movements capture the effect that all the other causal variables have on quantity demanded and attribute them solely to changes in price. In this context, attempts to "more accurately" measure the own price elasticity of demand are not justified.

3c. The Isoelastic/Logarithmic Alternative

Part (c) of the DKKS argument proposes the isoelastic/logarithmic approach be adopted as an alternative to Allen's arc. Utilizing the arguments constructed thus far, for the following reasons, this solution is not warranted:

(1) DKKS state, "We believe this definition |the logarithmic formula~ has desirable properties and merits consideration. Despite its "desirable properties," neither DKKS, Gallego-Diaz, nor Holt and Samuelson give any economic justification for assuming the demand curve is, in fact, isoelastic. Either an economic rationale appropriate to large changes must be given, or the analysis restricted to approximations of small changes (in which case Allen's arc suffices).

(2) In the absence of any economic justification, should the true demand curve be concave to the origin, a convex to the origin isoelastic function may introduce more error than Dalton and Allen's simple linear approximation. At the extremely large changes that concern DKKS, concavity of the demand curve is not an unlikely event:

(a) As previously noted, Boulding |1968~ suggests that at prohibitively high prices the true demand curve is likely to be elastic, at quantities approaching satiety the true demand curve is likely to be inelastic, such that there is some segment over which the demand curve must be concave to the origin.

(b) Likewise, from the vantage point of a small number of imperfect competitors, a concave to the origin, kinked demand curve is incompatible with a constant elasticity.

(3) Even if the isoelastic functional form is assumed, knowledge of only two price quantity combinations is insufficient to empirically estimate the specific parameters of large segments of an unknown demand curve.

(4) The notion that the isoelastic approach generates coefficients that are more accurate than Allen's Arc formula is entirely artificial. It is fully dependent on the ex ante assumption that the true demand curve is, in fact, isoelastic.

(5) The assumption of an isoelastic demand curve is not a cure-all. Large price changes are likely to be associated with changes in the elasticity of the demand function. Likewise, large price changes are likely to require time periods over which the demand curve itself would be expected to shift.

(6) In the pedagogical setting, student understanding of the assumptions prerequisite to employing an isoelastic demand curve is a separate, more advanced task than understanding the underlying concept of elasticity itself. Likewise, student determination of the specific parameters of the true demand curve is a separate, more advanced task than understanding the elasticity concept.

(7) Finally, leaving the price and quantity measures in non-logarithmic form facilitates student understanding and direct measurement of total revenue and total revenue changes. The student can calculate the total revenue change and then approximate the elasticity -- with sufficient accuracy for pedagogical purposes.

4. Conclusion-Allen's Arc Remains the Pedagogical Choice

Why, after sixty years of continuous criticism, is Allen's arc so firmly entrenched in the microprinciples curriculum? Because, in large part, the issues facing today's students parallel those faced by Allen in 1934. Allen's arc formula offers an effective tool which -- given the amount of data assumed to be available -- is intuitively appealing and offers an appropriate trade-off between ease of use and accuracy. As shown in Figures 1 and 2, without any knowledge of the true demand curve, the student:

(1) Is simply given two price-quantity combinations |p.sub.1~|q.sub.1~ and |p.sub.2~|q.sub.2~,

(2) Uses the price and quantity changes across the chord ab to calculate the different point elasticities, first at one point and then at the other point.

(3) Observing the difference in the coefficients, uses some version of Allen's arc formula to calculate an elasticity at the mid-point of the chord ab.

Klein |1962~ distinguishes between pedagogical and working models. He maintains pedagogical concepts are appropriate for illustrating basic, underlying tendencies, ". . . not suitable for practical decisions in public or business policy". Klein asserts that working models, in contrast -- doing more than illustrate fine points in theory -- must be larger, more complex and more precise. In this spirit, Allen's approach remains the pedagogical choice for the following reasons:

(1) Allen's arc elasticity formula is currently used only as an introductory, pedagogical device -- useful for engaging the beginning principles student in actually calculating an elasticity coefficient. Its use is virtually unknown in working models.

(2) Mastery of the elasticity concept is prerequisite to determining the true, underlying demand relationship -- as would be required in implementing the isoelastic/logarithmic approach over large changes.

(3) When its use is limited to small changes -- where the use of linear approximations is well established -- Allen's approach requires the true demand curve be neither presumed nor predetermined.

(4) Consequently, the isoelastic/logarithmic approach should be considered only in advanced courses where its "desirable properties" can be taken-up as a special case useful for empirical analysis of small price and quantity changes.

If, according to Mr. Churchill, "Democracy is the worst form of government, except for all the others," then, at least in a pedagogical setting, "Allen's arc must be the worst form of elasticity, except for all the others."

5. Further Research Needed

Further research is needed to evaluate alternative approaches to teaching the elasticity concept. Questions that might be addressed include:

(1) Does mathematical manipulation of the formulae truly increase student mastery of the elasticity concept? We tend to characterize the elasticity of oil during the Arab oil embargo as, "So inelastic that a 25% reduction in quantity enabled a quadrupling of price." Rather than quote the generally accepted short run elasticity of approximately 0.20, would it not suffice to simply state the demand for oil is inelastic in the short run? This level of generality appears to be sufficient to differentiate between goods that are complements and substitutes and between normal and inferior goods.

(2) Rather than use percentages to "gloss over" the "units problem," following Allen's original proposal, might there be pedagogical advantages to his second, more direct approach -- finding a suitable |Lambda~ with which to standardize the slope of the demand curve |Allen, 1934, p. 226~?

(3) Rather than get bogged-down in the computational minutiae of the various measurement formulae, would it not be more useful to consider the most likely shapes of ceteris paribus supply and demand curves over their entire range -- between their maximum price-zero quantity demanded and their zero price-maximum quantity demanded?

(4) Finally, do we really need to calculate a single, summary elasticity coefficient "along the arc or across the chord?" Students appear to have little difficulty with the "mark-up vs. mark-down" problem -- e.g., that a 50|cents~ mark-up from 50|cents~ to $1.00 is a 100% change, while the same 50|cents~ marked down from $1.00 to the original 50|cents~ is only a 50% change. They recognize the importance of which point is used as the base -- the need to distinguish the direction of the change, up or down." Might not the notion of a "directional elasticity" be more effective than the arc/mid-point formula. After all, an appreciation of the "which base to use" dilemma is prerequisite to understanding why the arc/mid-point formula is needed in the first place.


Allen, R. G. D., "The Concept of Arc Elasticity of Demand," Review of Economic Studies, Vol. 1 (3), July 1934, pp. 226-29.

Boulding, Kenneth E., "Demand and Supply," The International Encyclopedia of the Social Sciences, New York: Macmillan, 1968, pp. 96-104.

Clark, John G., The Political Economy of World Energy, Hemel Hempstead, Hertfordshire, Great Britain: Harvester Wheatsheaf, 1990.

Culyer, A. J., Economics, Oxford: Basil Blackwell, 1985.

Daellenbach, L. A., A. W. Khandker, G. J. Knowles, and K. R. Sherony, American Economist, Vol. 35 (2), Spring 1991, pp. 56-61.

Dalton, Hugh, Some Aspects of the Inequality of Incomes in Modern Communities, New York: E. P. Dutton, 1920.

Gabor, Andre, "A Further Note on Arc Elasticity," Bulletin of Economic Research, Vol. 28 (2), 1976, p. 127.

Gallego-Diaz, J., "A Note on the Arc Elasticity of Demand" Review of Economic Studies, Vol. 12 (2), July 1944-45, pp. 114-15.

Holt, C. C. and P. A. Samuelson, "The Graphic Depiction of Elasticity of Demand," The Journal of Political Economy, Vol. 54 (4), Aug. 1946, pp. 354-57.

Klein, Lawrence, An Introduction to Econometrics, Englewood Cliffs: Prentice-Hall, 1962.

Marshall, Alfred, Principles of Economics, 8th edition, London: Macmillan, 1920.

Schultz, Henry, Statistical Laws of Demand and Supply, Chicago: University of Chicago Press, 1928.


In Daellenbach, Khandker, Knowles, and Sherony |1991~ we recognize that Allen's definition of elasticity has weathered numerous challenges. Bounds on Allen's arc elasticity formula are identified and associated with Vaughn's |1988~ notion of bias. A graphic analysis illustrates that for relatively small percentage changes and for elasticities close to unity, the bias is small. A log based definition is recommended to supplement the conventional Allen formula and at least five reasons for the recommendation are provided.

In his comment on our paper, Phillips |1993~ selectively reviews the elasticity literature, considers the bias in Allen's formula, and provides observations on several of our reasons for recommending use of the logarithmic approach. He concludes that Allen's should remain the computational formula for introducing elasticity and the inclusion of the log form should be delayed until advanced classes.

Phillips oversimplifies and misrepresents our position. He contends that our argument for use of the log approach relies on the single, sufficient condition that the Allen formula is biased for large percentage changes. Contrary to his characterization, our recommendation is not predicated solely on large price changes. Our recommendation is based on evidence that there are problems associated with the Allen index. The evidence is taken from the long list of detractors, and includes our analysis of the 200% limiting case and its link to the bias measure described by Vaughn.

Phillips' references to and interpretation of what constitutes a "small" change, a "large" change, and a "huge" change are entwined with problems relating to price changes at the extremes of satiation and zero quantity demanded. The issue is further clouded when he switches reference from individual demand to market demand without adequate explanation or differentiation. We contend that price changes that are large and not in the vicinity of the extremes are real world events. Elasticity is a sufficiently robust concept that can be used to evaluate these situations. Moreover, if we follow Phillips' foray into the infinitesimal and restrict ourselves to his interpretation of point elasticity as defined by Marshall, there would be very little of interest that could be said about demand responsiveness to discrete price changes in a real world context.

We agree wholeheartedly with Phillips' call for further research to evaluate alternative approaches to teaching elasticity at the principles level. Although we share his concern for pedagogy, we believe his research questions are too narrowly focused. The questions that should be considered here include: What constitutes good pedagogy in introductory economics course? Does good pedagogy establish solid links between the principles we teach and the real world?

We believe elasticity and other concepts are best understood when students can work with applications and when their applications are unencumbered by computation minutiae. As a result, we are not convinced by Phillips' arguments that Allen's approach remains the pedagogical choice. The Allen formula is tedious to calculate and therefore apply. It lacks conceptual simplicity. The formula is rarely, if ever, encountered elsewhere.

Finally, we find Phillips' quibbles with the natural log formula lacking and stand by our recommendation to instructors to experiment with the log based definition. The logarithmic approach is suitable for pedagogical purposes. It is less cumbersome to calculate. Most important, as a tool for analysis it enjoys practical application. We believe Allen's formula withstood log-elasticity challenges in part because calculating log based elasticities required an inhibiting log table or slide rule. Ubiquitous calculators and spreadsheets enable the log based definition to be viable today, even at the principles level.


Daellenbach, L. A., A. W. Khandker, G. J. Knowles, and K. R. Sherony. "Restrictions of Allen's Arc Elasticity of Demand: Time to Consider the Alternative?", American Economist, Vol. 35(2), Spring 1991, pp. 56-61.

Phillips, W. A. "Allen's Arc vs. Assumed Isoelasticity: Pedagogical Efficacy vs. Artificial Accuracy," American Economist, This Issue.

Vaughn, M. B. "The Arc Elasticity of Demand: A Note and Comment." Journal of Economic Education, Vol. 19 (3), Summer 1988, pp. 254-258.

L.A. Daellenbach is Professor of Economics. A.W. Khandker, G.J. Knowles, and K.R. Sherony are Associate Professors of Economics at University of Wisconsin-LaCrosse.
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Title Annotation:includes authors' reply; response to L.A. Daellenbach, A.W. Khandker, G.J. Knowles and K.R. Sherony, American Economist, vol. 35, no. 2, p. 56, Spring 1991
Author:Phillips, Wm. A.
Publication:American Economist
Date:Mar 22, 1993
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