# A note on Allen's arc elasticity with arithmetic, geometric and harmonic means.

Introduction

As outlined by Vazquez (2008), discussion and debate on the accuracy and applicability of Allen's arc elasticity has a long history, dating back to the 1930s and continuing into the 1980s, 1990s, and 2000s (e.g., Vazquez 1985; Daellenbach et al. 1991, 1993; Daskin 1992; Phillips 1993, 1994; Vazquez 1995; Pratt 2003; Haque 2005). In a spirited exchange over two approaches, Daellenbach et al. recommend using logarithms to calculate a constant-demand elasticity based on an assumed isoelastic demand curve, while Phillips argues for the arc elasticity based on the geometric mean. According to Daellenbach et al., the isoelastic approach, attributed to Gallego-Diaz (1944-1945) and Holt and Samuelson (1946), provides a strong pedagogical link between economic principles and real world events, a point recognized by Pratt in studies of automotive fuel demand. Not only do Daellenbach et al. find the arc elasticity formula to be pedagogically tedious to calculate and apply, but they also assert that it yields biased results.

Phillips contends that Daellenbach et al. overstate this bias and the suitability of an assumed isoelastic demand curve. He further argues that a geometric mean calculation of the arc elasticity formula retains the positive attributes of Allen's analysis and provides instructors and practitioners with the flexibility of choosing from among at least three approaches: the conventional percentage change method, Dalton's upper and lower elasticity measures, or a standardizing ray technique. In short, the subtle difference between the arc elasticity and isoelasticity approaches exists because the denominator of the logarithmic difference of two prices (quantities) corresponds nicely to the geometric mean. However, such an approximation does not exist in the cases of the arithmetic and harmonic means. Nonetheless, we can compare them by using, for example, the isoelasticity approach as the reference case, as will be done in this analysis.

This note has a twofold purpose. First, it shows that the difference in elasticity values between Allen's geometric mean and the isoelastic approach is negligible for small changes in price and quantity. This suggests that, despite the arguments of Daellenbach et al. and Phillips, instructors and practitioners may view the two distinctly different elasticity approaches as close substitutes for small changes in price and quantity, which make up the vast majority of such changes. Second, this note expands the discussion on elasticity calculations by pointing out that the measures of central tendency used in elasticity formulas need not be limited to the arithmetic and geometric means. In some situations, as seen in rapidly changing markets for commodities and securities, the harmonic mean may be the most accurate measure.

The next section shows that small percentage changes in price lead to negligible differences in elasticity estimates between the isoelastic approach and the geometric mean method. The section following this one extends the analysis to the harmonic mean, followed by the section with illustrations that compare estimates across the four measures of elasticity: isoelastic, arithmetic, geometric, and harmonic. The final section concludes the study.

Allen's Geometric Mean Formula

Phillips (1994) provides an excellent analysis of the evolution of price elasticities since the introduction of Marshall's point elasticity. Motivating the evolution is the broad relevance of elasticity, which has important implications for international trade (e.g., Panagariya et al. 2001; Ketenci and Uz 2011), tax incidence (e.g., Yang and Stitt 1995; McDonald and Yurova 2007), price discrimination (e.g., Villas-Boas 2009), and options pricing (e.g., Greenblatt et al. 2007; Miller and Platen 2010). Within the framework of a linear demand function, Phillips (1994, p. 318) shows that a geometric mean approach to Allen's arc elasticity can be expressed as follows:

[[[eta].sub.G] = ([DELTA]q]/[bar.[q.sub.G]/([[DELTA].sub.p]/ [bar.[p.sub.G]) = [[[DELTA]q]/([q.sub.1][q.sub.2]).sup.0.5]]/[[DELTA]p]/([p.sub.1] [p.sub.2].sup.0.5]], (1)

where [[DELTA].sub.p] is the change in the price (p); [bar.[p.sub.G] is the geometric mean of the price; [[DELTA].sub.q] is the change in the quantity (q); and [bar.[q.sub.G] is the geometric mean of the quantity. Similarly, the elasticity for an assumed isoelastic demand curve (Gallego-Diaz 1944-1945; Holt and Samuelson 1946; Daellenbach et al. 1991) is

[[eta].sub.iso] = (ln [q.sub.2] - ln [q.sub.1])/(ln [p.sub.2] - ln [p.sub.1]). (2)

For infinitesimally small changes in quantity, the numerator of Eq. (2) can be expressed as ln [q.sub.2]-1n [q.sub.1] =d ln q=dq/q. By definition, this is the percentage change in quantity. Similarly, as [p.sub.1] and [p.sub.2] [right arrow] p, the denominator of Eq. (2) measures the percentage change in price. In order to facilitate the comparison between Eqs. (1) and (2), we relate both quantities by expressing [q.sub.2] = [q.sub.1](1 + r), where r is a small percentage. As a result, it follows immediately from Eq. (2) that

ln[q.sub.2] - ln[q.sub.1] = ln(1 + r)[q.sub.1] - ln[q.sub.1] = ln(1 + r). (3)

As well, the percentage change in quantity in Eq. (1) can be written as

[[rq.sub.1]/[[q.sub.1] (1 + r)[q.sub.1]].sup.0.5] = r/(1 + r).sup.0.5]. (4)

To facilitate a comparison between Eqs. (3) and (4), we expand Eq. (3), based on a well-known formula (Kaplan 1973, p. 432), and expand [(1 + r).sup.0.5], the denominator of Eq. (4), around [bar.r] = 0 . As shown in the Appendix, by subtracting the result of expanding r/(1 + r).sup.0.5] from the result of expanding ln(1 + r), we arrive at the difference in the percentage change in quantity. By exactly the same procedure, we obtain the difference in the percentage change in price. In each case, the result is an alternating series of higher order, indicating that small percentage changes in price and quantity lead to negligible differences. This suggests that instructors and practitioners can choose from either approach (geometric mean vs. constant elasticity) and obtain nearly equivalent results. (1)

However, for moderately large changes in price and quantity, (ln [q.sub.2] - In [q.sub.1]), for example, provides only a crude approximation for the percentage change (Fama 1970, p. 393). Consequently, for such changes, which are often used in principles courses for visual clarity, the tree elasticity of an assumed isoelastic demand curve cannot be calculated with precision.

The discussion on elasticity measures can be easily expanded beyond the arithmetic and geometric means to include the harmonic mean. The intention here is not just to investigate analytically the accuracy of each elasticity measure to a true value, but also to introduce a different mean and to compare their locations in the same manner as Phillips (1994), with a simple linear and constant elasticity demand functions.

Allen's Arc Elasticity with the Harmonic Mean

While Phillips presents a strong case for the geometric mean, the harmonic mean--the reciprocal of the arithmetic mean of two reciprocals (in equation form below)--may be more accurate in particular economic situations. Two are presented here. The first one follows from using reciprocals in currency (e.g., 75 yen per U.S. dollar) and commodity markets. For example, the value (price) of a futures contract can be expressed as bushels per \$1000. Consider a commodity market where in the first period a consumer purchases 10 units of a commodity at a price of 30 units per dollar. In the next period, the consumer purchases another 10 units of the commodity but now at a price of 50 units per dollar. From these two transactions, the consumer has spent 10/30 + 10/50 = \$(8/15) for the 20 units. The average price is 37.5 units per dollar [20/\$(8/15)]. The arithmetic mean of 40 units per dollar [i.e., (30 + 50)/2 ] would not correctly represent the transactions. Alternatively, the harmonic mean is more accurate as a measure of central tendency in such situations, as follows:

HM = n / [n. summation over (i=1])] (1/[x.sub.i]). (5) /i=l

Second, assume that the upper-end price [p.sub.U] is uncertain and may diverge considerably from the lower-end price [p.sub.L] (i.e., [p.sub.U]>[p.sub.L], a situation experienced in volatile stock and options markets, as witnessed in the global financial crisis of 2007-09, the flash crash on May 6, 2010 (in which the Dow Jones Industrial Average fell by about 1000 points before rising by nearly 600 points), spikes in stock prices due to acquisition announcements, and in periods of hyperinflation. In these situations, the harmonic mean price may be expressed as follows:

2/(1/[p.sub.U]. + 1/[p.sub.L]) = 2[p.sub.U][p.sub.L]/([p.sub.U] + [p.sub.L]). (6)

While the arithmetic and geometric means are ill-suited--mathematically undefined--to the kind of change in prices suggested by these scenarios, the harmonic mean converges to 2[p.sub.L] . (2) A weak point of the harmonic mean is that it explodes or becomes undefined as the lower price approaches zero (e.g., an unwanted commodity at a point of time). Although the formula for the harmonic mean (HM) appears to be much different compared to the formulas for the arithmetic (AM) and geometric (GM) means, there is a relationship among the three. When n=2, which corresponds nicely to two points on a demand curve, the geometric mean is the geometric mean of the arithmetic and harmonic means, as follows:

[x.sub.1] [x.sub.2] = [([x.sub.1] + [x.sub.2])/2] x [2/(1/[x.sub.1] + 1/[x.sub.2])]. (7)

In addition, a standard theorem states that AM>GM>HM (Young 1992), which serves as a basis for the relationship among the different elasticity measures. For [q.sub.2] > [q.sub.1], it follows that [bar.[q.sub.A]] > [bar.[q.sub.G]] > [bar.[q.sub.H]], where [bar.[q.sub.A]] is the arithmetic mean of the quantity; [bar.[q.sub.G]] is the geometric mean of the quantity; and [bar.[q.sub.H]] is the harmonic mean of the quantity. Because Allen's arc elasticity is based on the same slope, [[DELTA].sub.q] / [[DELTA].sub.p], of a linear demand function, the sizes of the three elasticity measures depend on the magnitudes of the respective means, or, [bar.[q.sub.A]], [bar.[q.sub.G]], [bar.[q.sub.H]], [bar.[p.sub.A]], [bar.[p.sub.G]], and [bar.[p.sub.H]], where [bar.[p.sub.A] is the arithmetic mean of the price; [bar.[q.sub.G]] is the geometric mean of the price; and [bar.[q.sub.H]] is the harmonic mean of the price. Given that [bar.[q.sup.-2.sub.G]] = [bar.[q.sub.A]] x [bar.[q.sub.H]] and that [bar.[q.sup.-2.sub.G]] = [bar.[p.sub.A]] x [bar.[p.sub.H]], it can be shown that the square of [[eta].sub.G] (Allen's geometric mean arc elasticity) is the product of [[eta].sub.A] (Allen's arithmetic mean arc elasticity) and [[eta].sub.H] (Allen's harmonic mean arc elasticity). (3) The result implies that [[eta].sub.G] must always be bounded by [[eta].sub.A] and [[eta].sub.H].

An Illustration

Table 1 provides an illustration of the basic points of this paper. The linear demand function from Phillips (1994, p. 319) q= 12-p is used to calculate Allen's elasticity across the arithmetic, geometric, and harmonic means. Also provided are the isolasticity estimates. To illustrate the means at the upper end of the price range, 9 <_ p < 11, and the corresponding quantity range, 1 [less than or equal to] q [less than or equal to] 3, the following results are obtained:

[bar.p.sub.G] [square root of 99]; [bar.p.sub.A] = (9+ 11)/2 = 10; [bar.p.sub.H] =2/(1/9+ 1/11) =9.9; [bar.[q.sub.G] = [square root of 3]; [bar.qsub.A] = (1 + 3)/2 = 2; [bar.q.sub.H] = 2/(1/1 + 1/3) = 1.5. They lead, in descending order, to the following ranking of arc elasticity measures in absolute value: [[eta].sub.H] = 6.6 > [[eta].sub.G] = 5.745 > [[eta].sub.A] = 5. However, in the corresponding inelastic segment of the demand curve, 1 [less than or equal to] p [less than or equal to] 3), the ranking changes to [[eta].sub.A] > [[eta].sub.G] > [[eta].sub.H]. An examination of Table 1 reveals a number of patterns. First, for small changes in price and quantity, the differences between the isoelastic and geometric mean elasticity measures are small (e.g., 1.409 and 1.414 for 6 [less than or equal to ] p [less than or equal to ] 8), and reducing the price and quantity changes to a single unit would reduce these small differences further. Second, whether in the elastic or inelastic segments of the function, [[eta].sub.G] is always bounded by [[eta].sub.A] and [[eta].sub.H], suggesting that the geometric mean is a prudent choice among the three measures. Third, at 6.600, Allen's harmonic mean elasticity is the greatest among the three measures in the elastic segment of the demand function but, at 0.152, is the smallest among all values on the most inelastic segment, or in the last row.

In order to measure differences among the three elasticity measures at a given isoelasticity, we present numerical simulations assuming three distinct isoelasticity values: [[eta].sub.iso] = 3, [[eta].sub.iso] = 2, and [[eta].sub.iso] = 0.5 in Table 2, 3, and 4 respectively. (4) For a given analytical price elasticity of [[eta].sub.iso] = 3, dq/dp is evaluated at p= 1 and dp=0.1 via the equation of lnq= -3 ln p, which implies q=1 as p=1. As shown in the fourth column of Table 2, dq/dp = 2.4869 . Using this value, one can calculate three elasticity values per the three mean values of price and quantity. For instance, the arithmetic mean price is 1.05 and the corresponding quantity at p=l.1 is 1/[[(1.1).sup.3] = 0.7513148 . Hence the corresponding mean quantity is 0.8756574, or (1 + 0.7513148)/2. As a result, the arc elasticity can be calculated to be 2.9820, as shown in the fifth column of Table 2 ([[eta].sub.A]). Similarly, one can obtain the geometric and harmonic mean arc elasticity values: [[eta].sub.G] = 3.0091 and [eta].sub.H] = 3.0364. The errors between the isoelasticity [[eta.sub.iso] = 3 and the three other elasticity measures are denoted as err_[[eta].sub.A], err_[[eta].sub.G], and err_[[eta].sub.H]. In this case, the geometric mean arc elasticity (G) has the smallest error, or min err.

As we vary price with other values (e.g., 0.5, 0.25, ...) and set dp=0.1, 20 more runs of elasticity values and errors are reported in Table 2. It shows that the geometric arc mean elasticity values are closest to the isoelasticity ([[eta].sub.iso] = 3) in 15 of the 21 cases. The result is somewhat surprising as isoelasticity and the geometric mean elasticity are considered similar. The simulation shows the arithmetic mean elasticity is closer to 3 at very low prices (i.e., in six cases, in which the geometric mean elasticity significantly increases). It can be explained by the demand function in which a very small price raised to the negative third power can produce a very large quantity. In this case, the geometric mean price/quantity, being less extreme than its arithmetic mean counterpart, cannot fully reflect the original small price/quantity ratio when dq/dp is extremely large (2.6E+ 10 for p=0.015625 as shown in Table 2). There is not, however, a clear-cut formula to determine which elasticity measure gives the best approximation.

Similar results are seen in Table 3 for [[eta].sub.iso] = 2. Except at very low prices (i.e., in five cases) where the arithmetic mean is a more accurate approximation, the geometric mean elasticity values are closer to 2 in 16 cases. On the other hand, as shown in Table 4, by setting isoelasticity at 0.5, we find that the geometric mean elasticity values are closer to 0.5 over the entire rage. This may be explained as follows: unusually small or large prices lead to relatively mild quantities via the inverse of the square root transformation, given the exponent of-0.5. The mild transformation is in agreement with the property of the geometric mean, especially for a small displacement.

We note that the three arc elasticity measures are identical (to the 15th decimal place) in value when used to approximate [[eta].sub.iso] = 1 . To demonstrate, for a given dq/dp--the same for the three elasticity measures--and in the case of the arithmetic mean, [bar.p] and [bar.q] are [p.sub.1]+0.05 and [1/[p.sub.1] + 1/([p.sub.1] + 0.1)]/2, respectively, via the demand function q = 1/p, where [p.sub.1] is the original price. Hence, we can reduce [bar.p]/[bar.q] [top.sub.1]([p.sub.1]+0.1). In the case of the geometric mean, [beta] and are [[[p.sub.1]([p.sub.1] + 0.1)].sup.1/2] and {(1/[p.sub.1])[1/([p.sub.1]+0.1)]}.sup.1/2], respectively. As a result, [bar.p]/[bar.q] is [p.sub.1]([p.sub.1]+0.1). For the harmonic mean, [bar.p] and [bar.q] can be expressed as 2/[(1/p) + 1/(p + 0.1)] and 2/[1/[p.sup.-1] + 1/[(p+0.1).sup.-1], respectively, which reduce to [p.sub.1]([p.sub.1] + 0.1). Therefore, any of the three elasticity measures is equally valid in the case of a rectangular hyperbola demand function.

Conclusion

Four points conclude the study. First, except for relatively large changes in price (small) and quantity, instructors and practitioners can view the isoelastic and Allen geometric mean calculations as close substitutes in the majority (but not all) of the cases. Second, while the geometric mean may have advantages over the arithmetic mean, researchers and practitioners should be aware of situations in which the harmonic mean elasticity measure may be appropriate, such as in the cases of rapid security or commodity price movements. Third, given that 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 especially when the demand structure exhibits the inverse of a square root relation. Fourth, in the case of a unitary elastic demand curve, the three elasticity measures are identical.

Appendix

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

ln(1 + r) = [[infinity].summation over (i=1] [(-1).sup.n+1](1 + r - 1).sup.n] / n.

This reduces to the following:

ln(1 + r) r[1 - (r/2) + ([r.sup.2]/3) - ([r.sup.3]/4) + ...], for |r| < 1. (1A)

Expanding [(1 + r).sup.-0.5] of Eq. (4) around [bar.r]= 0 readily yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2A)

Subtracting (2A) from (1A) gives rise to the difference in the percent change in quantity, or the numerators of [[eta].sub.G] and [[eta].sub.iso], as follows:

[[DELTA].sub.q] = r(-[r.sup.2]/24 + [r.sup.3]/16 ...). (3A)

By exactly the same procedure, the difference in the percent change in price, or the numerators of [[eta].sub.G] and [[eta].sub.iso], is as follows (where s is a small percent):

[[DELTA].sub.p] s(-[s.sup.2]/24+[s.sup.3]/16 ...). (4A)

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

References

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Vazquez, A. (1985). Geometrical analysis of the arc price elasticity of demand and the change in total revenue. Canadian Journal of Agricultural Economies, 33(3), 387-395.

Vazquez, A. (1995). A note on the arc elasticity of demand. Estudios Eeonomicos, 10(2), 221-228.

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C. W. Yang

Department of Economics, Clarion University, Clarion, PA, USA

C. W. Yang

Department of Economics, National Chung Cheng University, Chia Yi, Taiwan, Republic of China

A. L. Loviseck

Department of Finance, Scton Hall University, South Orange, N J, USA

H. W. Cheng ([mail])

Department of International Business, Ming Chuan University, 250, See. 5, Chung Shan N. Road,

Taipei, Taiwan, Republic of China

e-mail: hwcheng@mail.mcu.cdu.tw

K. Hung

A.R. Sanchez, Jr. School of Business, Texas A&M International University, Laredo, TX, USA

(1) Phillips (1994, Daellenbach 319) notes the similarity in the results between the two approaches for small changes in price, but asserts that an argument for isoelastic demand curves "is referenced here only for comparison purposes." Apart from the debate, Vazquez (1995) formulates a measure of arc elasticity that is in line with the characteristics of point elasticity and Haque (2005) shows the potential of using arithmetic means to approximate harmonic means in the computation of elasticity.

(2) When [p.sub.U] diverges toward infinity, we can divide both numerator and denominator by [p.sub.U] and take the limit. We find that the lim[2[p.sub.U][p.sub.L]/([p.sub.U] + [p.sub.L]] = lim|[2[p.sub.L/(1 +[p.sub.L]/[p.sub.U]] = 2[p.sub.L]. In contrast, both the arithmetic and geometric means are undefined in this situation.

(3) The result is [(dg/dp).sup.2] ([bar.[p.sup.-2.sub.G]]/ ([bar.[q.sup.-2.sub.G]]) = (dq/dp)([bar.p.sub.A]]([bar.[p.sub.H]]/([bar.[q.sub.A]]([bar.[q.sub.H]]) (dq/dp). However, the fact that [[eta].sup.2.sub.G] = [[eta].sub.A] x [[eta].sub.H] does not imply that the absolute values of the elasticity measures follow the order of [[eta].sub.A] > [[eta].sub.G] > [[eta].sub.H]. That is, although the geometric mean price and quantity values are bounded by the arithmetic and harmonic means, their respective ratios (e.g., [bar].[p.sub.A]//[bar.[q.sub.A]) need not follow this order.

(4) We are greatly indebted to the referee for kindly supplying the simulation results shown in Table 2. The use of constant price elasticity demand functions enables us to compare the accuracy of the approximation. We expand the simulations to include other elasticity values in Tables 3 and 4.

Published online: 25 March 2012

DOI 10.1007/s11293-012-9315-5
```Table 1 Elasticities for a demand curve over selected price and
quantity ranges: q=12-p

Price range   Quantity range   Assumed isoelasticity   [[eta].sub.A]

11-9          1-3              5.475                   5.000
10-8          2-4              3.106                   3.000
9-7           3-5              2.033                   2.000
8-6           4-6              1.409                   1.400
7-5           5-7              1.000                   1.000
614           6-8              0.710                   0.714
5-3           7-9              0.492                   0.500
4-2           8-10             0.322                   0.333
3-1           9-11             0.183                   0.200

Price range   [[eta].sub.G]   [[eta].sub.H]

11-9          5.745           6.600
10-8          3.162           3.333
9-7           2.049           2.100
8-6           1.414           1.429
7-5           1.000           1.000
614           0.707           0.700
5-3           0.488           0.476
4-2           0.316           0.300
3-1           0.174           0.152

Table 2 Elasticity Values (Absolute) for a Constant Price Elasticity
Function of lng=-3lnp
[absolute value
dp    [[eta].sub.iso]   P          of dq/dp]         [[eta].sub.A]

0.1   3                 1          2.4869            2.9820
0.1   3                 0.5        33.704            2.9355
0.1   3                 0.25       406.76            2.7949
0.1   3                 0.125      4,242             2.4754
0.1   3                 0.0625     38,630            2.0078
0.1   3                 0.03125    323,257           1.5817
0.1   3                 0.015625   2.6E+06           1.3060
0.1   3                 0.007813   2.1E+07           1.1554
0.1   3                 0.003906   1.7E+08           1.0780
0.1   3                 0.001953   1.3E+09           1.0390
0.1   3                 0.000977   1.1E+10           1.0195
0.1   3                 2          0.1702            2.9952
0.1   3                 4          0.0112            2.9988
0.1   3                 8          0.0007            2.9997
0.1   3                 16         4.5E-05           2.99992
0.1   3                 32         2.8E-06           2.99998
0.1   3                 64         1.8E-07           2.999995
0.1   3                 128        1.1E-08           2.999999
0.1   3                 256        7.0E-10           2.9999997
0.1   3                 512        4.4E-11           2.99999992
0.1   3                 1024       2.7E-12           2.99999998

dp    [[eta].sub.G]   [[eta].sub.H]   err_[[eta].sub.A]

0.1   3.0091          3.0364          0.0180
0.1   3.0333          3.1344          0.0645
0.1   3.1143          3.4702          0.2051
0.1   3.3556          4.5486          0.5246
0.1   3.9846          7.9079          0.9923
0.1   5.4381          18.697          1.4183
0.1   8.5351          55.778          1.6940
0.1   14.872          191.445         1.8446
0.1   27.638          708.561         1.9220
0.1   53.219          2725.84         1.9610
0.1   104.41          10692.6         1.9805
0.1   3.0024          3.0095          4.8E-03
0.1   3.0006          3.0024          1.2E-03
0.1   3.0002          3.0006          3.1E-04
0.1   3.00004         3.0002          7.8E-05
0.1   3.00001         3.00004         2.0E-05
0.1   3.000002        3.000010        4.9E-06
0.1   3.0000006       3.000002        1.2E-06
0.1   3.0000002       3.0000006       3.1E-07
0.1   3.00000004      3.0000002       7.6E-08
0.1   3.00000001      3.00000004      1.9E-08

dp    err_[[eta].sub.G]   err_[[eta].sub.H]   min_err

0.1   0.0091              0.0365              G
0.1   0.0333              0.1344              G
0.1   0.1143              0.4702              G
0.1   0.3556              1.5486              G
0.1   0.9846              4.9079              G
0.1   2.4381              15.697              A
0.1   5.5351              52.778              A
0.1   11.873              188.45              A
0.1   24.638              705.56              A
0.1   50.219              2722.8              A
0.1   101.41              10690               A
0.1   2.4E-03             9.5E-03             G
0.1   6.1E-04             2.4E-03             G
0.1   1.5E-04             6.2E-04             G
0.1   3.9E-05             1.6E-04             G
0.1   9.7E-06             3.9E-05             G
0.1   2.4E-06             9.8E-06             G
0.1   6.1E-07             2.4E-06             G
0.1   1.5E-07             6.1E-07             G
0.1   3.8E-08             1.5E-07             G
0.1   9.5E-09             3.8E-08             G

Table 3 Elasticity values (Absolute) for a constant price
elasticity function of lnq=-21np

[absolute value
dp    [[eta].sub.iso]   p          of dq/dp]         [[eta].sub.A]

0.1   2                 1          1.7355            1.9955
0.1   2                 0.5        12.222            1.9836
0.1   2                 0.25       78.367            1.9459
0.1   2                 0.125      442.47            1.8491
0.1   2                 0.0625     2,181             1.6701
0.1   2                 0.03125    9,660             1.4506
0.1   2                 0.015625   40,212            1.2654
0.1   2                 0.007813   162,980           1.1442
0.1   2                 0.003906   6.5E+05           1.0751
0.1   2                 0.001953   2.6E+06           1.0383
0.1   2                 0.000977   1.1E+07           1.0193
0.1   2                 2          0.2324            1.9988
0.1   2                 4          0.0301            1.9997
0.1   2                 8          0.0038            1.9999
0.1   2                 16         4.8E-04           1.99998
0.1   2                 32         6.1E-05           1.999995
0.1   2                 64         7.6E-06           1.999999
0.1   2                 128        9.5E-07           1.9999997
0.1   2                 256        1.2E-07           1.99999992
0.1   2                 512        1.5E-08           1.99999998
0.1   2                 1024       1.9E-09           1.999999995

dp    [[eta].sub.G]   [[eta].sub.H]   err_[[eta].sub.A]

0.1   2.0022          2.0091          0.0045
0.1   2.0083          2.0333          0.0164
0.1   2.0284          2.1143          0.0541
0.1   2.0870          2.3556          0.1509
0.1   2.2326          2.9846          0.3299
0.1   2.5373          4.4381          0.5494
0.1   3.0879          7.5351          0.7346
0.1   3.9840          13.872          0.8558
0.1   5.3514          26.638          0.9249
0.1   7.3634          52.219          0.9617
0.1   10.267          103.41          0.9807
0.1   2.0006          2.0024          1.2E-03
0.1   2.0002          2.0006          3.0E-04
0.1   2.00004         2.0002          7.7E-05
0.1   2.00001         2.00004         1.9E-05
0.1   2.000002        2.00001         4.9E-06
0.1   2.000001        2.000002        1.2E-06
0.1   2.0000002       2.0000006       3.0E-07
0.1   2.00000004      2.0000002       7.6E-08
0.1   2.00000001      2.00000004      1.9E-08
0.1   2.000000002     2.00000001      4.8E-09

dp    err_[[eta].sub.G]   err_[[eta].sub.H]   min_err

0.1   0.0022              0.0091              G
0.1   0.0083              0.0333              G
0.1   0.0284              0.1143              G
0.1   0.0870              0.3556              G
0.1   0.2326              0.9846              G
0.1   0.5373              2.4381              G
0.1   1.0879              5.5351              A
0.1   1.9840              11.872              A
0.1   3.3514              24.638              A
0.1   5.3634              50.219              A
0.1   8.2669              101.41              A
0.1   6.0E-04             2.4E-03             G
0.1   1.5E-04             6.1E-04             G
0.1   3.9E-05             1.5E-04             G
0.1   9.7E-06             3.9E-05             G
0.1   2.4E-06             9.7E-06             G
0.1   6.1E-07             2.4E-06             G
0.1   1.5E-07             6.1E-07             G
0.1   3.8E-08             1.5E-07             G
0.1   9.5E-09             3.8E-08             G
0.1   2.4E-09             9.5E-09             G

Table 4 Elasticity values (Absolute) for a constant price elasticity
function of lng=-0.51np

[absolute value
dp    [[eta].sub.iso]   p          of dq/dp]         [[eta].sub.A]

0.1   0.5               1          0.4654            0.5003
0.1   0.5               0.5        1.2322            0.5010
0.1   0.5               0.25       3.0969            0.5035
0.1   0.5               0.125      7.2024            0.5106
0.1   0.5               0.0625     15.193            0.5275
0.1   0.5               0.03125    28.966            0.5592
0.1   0.5               0.015625   50.591            0.6069
0.1   0.5               0.007813   82.682            0.6658
0.1   0.5               0.003906   1.3E+02           0.7279
0.1   0.5               0.001953   2.0E+02           0.7864
0.1   0.5               0.000977   2.9E+02           0.8370
0.1   0.5               2          0.1704            0.5001
0.1   0.5               4          0.0614            0.50002
0.1   0.5               8          0.0219            0.500005
0.1   0.5               16         7.8E-03           0.500001
0.1   0.5               32         2.8E-03           0.5000003
0.1   0.5               64         9.8E-04           0.50000008
0.1   0.5               128        3.5E-04           0.50000002
0.1   0.5               256        1.2E-04           0.500000005
0.1   0.5               512        4.3E-05           0.500000001
0.1   0.5               1024       1.5E-05           0.5000000003

dp    [[eta].sub.G]   [[eta].sub.H]   err_[[eta].sub.A]

0.1   0.4999          0.4994          0.0003
0.1   0.4995          0.4979          0.0010
0.1   0.4982          0.4930          0.0035
0.1   0.4946          0.4792          0.0106
0.1   0.4861          0.4479          0.0275
0.1   0.4695          0.3941          0.0592
0.1   0.4433          0.3238          0.1069
0.1   0.4088          0.2510          0.1658
0.1   0.3688          0.1869          0.2279
0.1   0.3268          0.1358          0.2864
0.1   0.2855          0.0974          0.3370
0.1   0.49996         0.4999          7.4E-05
0.1   0.49999         0.49996         1.9E-05
0.1   0.499998        0.499990        4.8E-06
0.1   0.4999994       0.499998        1.2E-06
0.1   0.4999998       0.4999994       3.0E-07
0.1   0.49999996      0.4999998       7.6E-08
0.1   0.499999991     0.49999996      1.9E-09
0.1   0.499999998     0.499999991     4.8E-09
0.1   0.4999999994    0.499999998     1.2E-09
0.1   0.4999999998    0.4999999994    3.0E-10

dp    err_[[eta].sub.G]   err_[[eta].sub.H]   min_err

0.1   0.0001              0.0006              G
0.1   0.0005              0.0021              G
0.1   0.0018              0.0070              G
0.1   0.0054              0.0208              G
0.1   0.0139              0.0521              G
0.1   0.0305              0.1059              G
0.1   0.0567              0.1762              G
0.1   0.0912              0.2490              G
0.1   0.1312              0.3131              G
0.1   0.1732              0.3642              G
0.1   0.2145              0.4026              G
0.1   3.7E-05             1.5E-04             G
0.1   9.5E-06             3.8E-05             G
0.1   2.4E-06             9.6E-06             G
0.1   6.1E-07             2.4E-06             G
0.1   1.5E-07             6.1E-07             G
0.1   3.8E-08             1.5E-07             G
0.1   9.5E-09             3.8E-08             G
0.1   2.4E-09             9.5E-09             G
0.1   6.0E-10             2.4E-09             G
0.1   1.5E-10             6.0E-10             G
```
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