In order to see this all works for each function, we need to do some algebra. We begin with the initial formula for the elasticity of demand,
Ed = (% change in Qd)/(% change in the good’s own price, P). Let us first substitute the Greek
“delta,” or , for the word “change”. Then our elasticity of demand formula becomes Ed = (% in Qd)/(% in the good’s own price, P), or
Ed = (% in Qd)/(% in P).
In order to apply this formula to estimate an arc elasticity of demand, we need to calculate percentage changes using two different points on our demand function. Call these two points point D and point E. The problem is that for a demand function with a constant downward slope, the elasticity of demand is different at point E than it is at point D. Look at Figure 3.2.
If we try to apply our formula for Ed using the prices and quantities associated with points D and E, we will obtain an elasticity estimate that is neither correct for the elasticity at point D or for the elasticity at point E. It will be close to the elasticity at some midpoint, for the point at triangle shown as point F in Figure 3.2.
Here is how we would have to apply the formula using a finite change in Q and P. First, let QdD be the quantity on the horizontal axis at point D. Let QdE be the quantity on the horizontal axis at point E. Let PD be the price on the vertical axis at point D and PE be the price on the vertical axis at point E, as shown by the dotted lines on Figure 3.2. Then, our equation for the Elasticity of Demand that approximately applies at some mid‐point F would be
Arc elasticities represent average elasticities over a finite change in each of the two variables.
The numbers do not represent the elasticity that exists at a single point on the curve, but an average over a range of points. As a consequence, arc elasticities may not be very accurate.
EdF =% in Qd /% in P, or EdF = Qd /Qd ÷ P/P.
This could be calculated using the formula
EdF = (QdD‐ QdE)/[ (QdD+ QdE)/2] ÷ (PD‐ PE)/[ (PD+ PE)/2].
Note that the 2’s cancel in the division, so this formula can be simplified as EdF = (QdD‐ QdE)/ (QdD+ QdE) ÷ (PD‐ PE)/(PD+ PE).
$0.00
$5.00
$10.00
$15.00
$20.00
$25.00
$30.00
$35.00
$40.00
0 4 8 12 16 20
Price
Quantity
Figure 3.2 Elasticity Varies Along a LInear Demand Function
Elasticity varies from point to point
Demand D
E F
QdD PD
PE
QdE
Technically, EdF is the arc elasticity of demand that approximates the average elasticity that applies between point D and point E, and this formula is the method commonly employed for calculating demand elasticities in a beginning microeconomics. Take a closer look at this equation
EdF = Qd /Qd ÷ P/P.
To divide one fraction by another we invert the bottom fraction and multiply the two fractions. For example (1/2) ÷ 1/8 is the same as writing (1/2) (8/1). So, in this case Qd/Qd ÷
P/P must be equal to (Qd /Qd) (P/P). But this equation can be simplified further by putting the (Qd /P) together with (P/Qd) so that EdF = (Qd /P) ÷ (Qd/P). If we divide by a fraction we invert and multiply so that EdF = (Qd/P)∙(P/Qd). This is the basic definition for the arc elasticity of demand. The formula consists of two components. The first component is (Qd/P). This component is not the slope of the demand function but it is closely related to the slope. Indeed, it is the inverse slope of the linear demand function.
Now recall the demand equation used for the linear demand function in Chapter 2. That demand equation was p = $40 ‐ 2q. If q increases by 1 unit, then p decreases by 2 units. This suggests that for this particular demand function, p/q is ‐2, and then the slope of the demand function is ‐2:1. A more general case is the linear demand function was p = A +bq, where the coefficient b is ordinarily negative (consumers ordinarily buy less when the price increases). In this case, p/q is b and the slope of the demand function is b:1.
However, note that the left‐hand side of the arc elasticity formula is Qd/P, not P/Qd. How do we find Qd/ P if we know P/Qd? That is easy because Qd/P is 1/( P/Qd). For example, if b = P/Qd = ‐2, then Qd/P =1/b = 1/‐2 = ‐ 0.5. So, for a linear demand function with a constant slope, left‐hand of the elasticity of demand formula is 1/slope of the demand function. What about the right‐hand side? That uses the P and Q at each point along the demand function.
The numbers in Table 3.1 for the elasticities of demand (the far right‐hand column labeled Ed) are not averages over a finite change in P and Qd but in fact are point elasticities of demand that are perfectly accurate for each P and Qd combination. Notice that as one moves from left to right along the demand function, the elasticities become less and less negative, until finally we reach 20 units where the demand function intersects the horizontal axis and the demand elasticity becomes zero, since P/Qd = 0/20 =0, 1/b = ‐ 0.5 and ‐ 0.5 0 = 0 where the demand function intersects the horizontal or quantity axis.
At the opposite extreme, where the demand function intersects the vertical or price axis, Qd is zero but P is the positive number $40. A mathematician would say that any number divided by zero is undefined. Mathematicians actually have two possibilities depending on whether you are approaching zero from the negative or positive side. The two possibilities are infinity (∞) and negative infinity (‐∞). For economists we might better say that in instances whereby the quantity demanded, Qd of a good, approaches zero and the demand function is linear but downward sloping, then the elasticity of demand is becoming a very large but negative number, approaching negative infinity (‐∞). That avoids getting into debates with mathematicians as to whether or not it is possible to define a number divided by exactly zero, which happens every time we attempt to calculate an elasticity at Qd = 0.
Table 3.1 Point Elasticities for a Linear Demand Function Qd P b 1/b P/Qd Ed= (1/b)(P/Qd)
0 $40 ‐2 ‐0.5 infinite ‐ infinity
1 $38 ‐2 ‐0.5 38.000 ‐19.000
2 $36 ‐2 ‐0.5 18.000 ‐9.000
3 $34 ‐2 ‐0.5 11.333 ‐5.667
4 $32 ‐2 ‐0.5 8.000 ‐4.000
5 $30 ‐2 ‐0.5 6.000 ‐3.000
6 $28 ‐2 ‐0.5 4.667 ‐2.333
7 $26 ‐2 ‐0.5 3.714 ‐1.857
8 $24 ‐2 ‐0.5 3.000 ‐1.500
9 $22 ‐2 ‐0.5 2.444 ‐1.222
10 $20 ‐2 ‐0.5 2.000 ‐1.000
11 $18 ‐2 ‐0.5 1.636 ‐0.818
12 $16 ‐2 ‐0.5 1.333 ‐0.667
13 $14 ‐2 ‐0.5 1.077 ‐0.538
14 $12 ‐2 ‐0.5 0.857 ‐0.429
15 $10 ‐2 ‐0.5 0.667 ‐0.333
16 $8 ‐2 ‐0.5 0.500 ‐0.250
17 $6 ‐2 ‐0.5 0.353 ‐0.176
18 $4 ‐2 ‐0.5 0.222 ‐0.111
19 $2 ‐2 ‐0.5 0.105 ‐0.053
20 $0 ‐2 ‐0.5 0.000 0.000