Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 3] THE MEASUREMENT OF ELASTICITIES 41
This estimate improves as the arc becomes smaller and approaches a point in the limit. Point elasticity of
demand can be found geometrically as shown in Examples 3 and 4.
EXAMPLE 3. We can find the elasticity of the demand curve in Example 1 at point C geometrically as follows. (For easy
reference, Fig. 3-1, with some modifications, is repeated here as Fig. 3-3.) Since we want to measure elasticity at point C,we
have only a single price and a single quantity. Expressing each of the values in the formula for e in terms of distances, we get:
e ¼
DQ
DP P Q
¼ NM
NC NC
ON
¼ NM
ON ¼ 6000
2000 ¼ 3
Note that this value of e is the same as that given by the modified formula in Example 1.
Fig. 3-4
Fig. 3-3
EXAMPLE 4. We can find e at point D for the demand curve of Example 2, as follows. (For easy reference, Fig. 3-2 with
some modifications is repeated as Fig. 3-4.)
We draw a tangent to D y at point D and then proceed as in Example 3. Thus,
e ¼ ML
OM ¼ 4000
2000 ¼ 2
Notice that the price elasticity at D 0 (about 1.78 found in Example 2) differs slightly from the point elasticity of D y at point
D. The difference is due to the curvature of D y and would diminish as C and F move closer to each other.
3.3 POINT ELASTICITY AND TOTAL EXPENDITURES
A straight-line demand curve (extended to both axes) is elastic above its midpoint, has unitary elasticity at
the midpoint, and is inelastic below its midpoint (see Example 5). There are no such generalizations for curvilinear
demand curves (see Problems 3.6 to 3.9). In the special case when a demand curve takes the shape of a
rectangular hyperbola, e ¼ 1 at every point on it (see Problem 3.8).
Regardless of the shape of the demand curve, as the price of a commodity falls, the total expenditures of
consumers on the commodity (P times Q) rise when e . 1, remain unchanged when e ¼ 1, and fall when e , 1
(see Example 5).