As noted at the beginning of the chapter, the price elasticity of demand is equal to the ratio of the percentage change in quantity demanded divided by the percent- age change in price. When calculating the value of E, it is convenient to avoid computing percentage changes by using a simpler formula for computing elastic- ity that can be obtained through the following algebraic operations
E 5 %DQ_____
%DP 5 ____ DQ Q 3 100
_________
DP
___ P 3 100 5 DQ____
DP 3 P __ Q
Thus, price elasticity can be calculated by multiplying the slope of demand (DQyDP) times the ratio of price divided by quantity (PyQ), which avoids making tedious percentage change computations. The computation of E, while involving the rather simple mathematical formula derived here, is complicated somewhat by the fact that elasticity can be measured either (1) over an interval (or arc) along demand or (2) at a specific point on the demand curve. In either case, E still mea- sures the sensitivity of consumers to changes in the price of the commodity.
The choice of whether to measure demand elasticity at a point or over an interval of demand depends on the length of demand over which E is measured.
If the change in price is relatively small, a point measure is generally suitable.
Alternatively, when the price change spans a sizable arc along the demand curve, the interval measurement of elasticity provides a better measure of consumer responsiveness than the point measure. As you will see shortly, point elasticities are more easily computed than interval elasticities. We begin with a discussion of how to calculate elasticity of demand over an interval.
Computation of Elasticity over an Interval
When elasticity is calculated over an interval of a demand curve (either a linear or a curvilinear demand), the elasticity is called an interval (or arc) elasticity. To measure E over an arc or interval of demand, the simplified formula presented
Now try Technical Problem 6.
interval (or arc) elasticity
Price elasticity calculated over an interval of a demand curve:
E 5 ____DDPQ 3 __________Average P Average Q
earlier—slope of demand multiplied by the ratio of P divided by Q—needs to be modified slightly. The modification only requires that the average values of P and Q over the interval be used:
E 5 DQ____
DP 3 __________ Average P Average Q
Recall from our previous discussion of Figure 6.1 that we did not show you how to compute the two values of the interval elasticities given in Figure 6.1. You can now make these computations for the intervals of demand ab and cd using the above formula for interval price elasticities (notice that average values for P and Q are used):
Eab 5 _____ 122 200 3 17 ____ 700 5 22.43 Ecd 5 _____ 122 200 3 8 _____ 1600 5 20.5
Relation When calculating the price elasticity of demand over an interval of demand, use the interval or arc elasticity formula:
E 5 ____ DQ
DP 3 ________ Average P Average Q Now try Technical
Problem 7.
Computation of Elasticity at a Point
As we explained previously, it is appropriate to measure elasticity at a point on a demand curve rather than over an interval when the price change covers only a small interval of demand. Elasticity computed at a point on demand is called point elasticity of demand. Computing the price elasticity at a point on demand is accomplished by multiplying the slope of demand (DQyDP), computed at the point of measure, by the ratio PyQ, computed using the values of P and Q at the point of measure. To show you how this is done, we can compute the point elasticities in Figure 6.1 when Borderline Music Emporium charges $18 and $16 per compact disc at points a and b, respectively. Notice that the value of DQyDP for the linear demand in Figure 6.1 is 2100 (5 12400y224) at every point along D, so the two point elasticities are computed as
Ea 5 2100 3 18 ____ 600 5 23 Eb 5 2100 3 16 ____ 800 5 22
Relation When calculating the price elasticity of demand at a point on demand, multiply the slope of demand (DQyDP ), computed at the point of measure, by the ratio PyQ, computed using the values of P and Q at the point of measure.
point elasticity A measurement of demand elasticity calculated at a point on a demand curve rather than over an interval.
I L L U S T R AT I O N 6 . 2 Texas Calculates Price Elasticity
In addition to its regular license plates, the state of Texas, as do other states, sells personalized or “van- ity” license plates. To raise additional revenue, the state will sell a vehicle owner a license plate saying whatever the owner wants as long as it uses six letters (or numbers), no one else has the same license as the one requested, and it isn’t obscene. For this service, the state charges a higher price than the price for stan- dard licenses.
Many people are willing to pay the higher price rather than display a license of the standard form, such as 387 BRC. For example, an ophthalmologist an- nounces his practice with the license MYOPIA. Others tell their personalities with COZY-1 and ALL MAN.
When Texas decided to increase the price for van- ity plates from $25 to $75, a Houston newspaper re- ported that sales of these plates fell from 150,000 down to 60,000 vanity plates. As it turned out, demand was rather inelastic over this range. As you can calculate us- ing the interval method, the price elasticity was 20.86.
The newspaper reported that vanity plate revenue rose after the price increase ($3.75 million to $4.5 million), which would be expected for a price increase when de- mand is inelastic.
But the newspaper quoted the assistant director of the Texas Division of Motor Vehicles as saying, “Since the demand droppeda the state didn’t make money from the higher fees, so the price for next year’s per- sonalized plates will be $40.” If the objective of the state is to make money from these licenses and if the numbers in the article are correct, this is the wrong thing to do. It’s hard to see how the state lost money by increasing the price from $25 to $75—the revenue increased and the cost of producing plates must have
decreased because fewer were produced. So the move from $25 to $75 was the right move.
Moreover, let’s suppose that the price elasticity be- tween $75 and $40 is approximately equal to the value calculated for the movement from $25 to $75 (20.86).
We can use this estimate to calculate what happens to revenue if the state drops the price to $40. We must first find what the new quantity demanded will be at $40.
Using the arc elasticity formula and the price elasticity of 20.86,
E 5 ____ DQ
DP 3 __________ Average P Average Q
5 60,000 2 Q__________ 75 2 40 3 (75 1 40)/2 ______________
(60,000 1 Q)/2 520.86 where Q is the new quantity demanded. Solving this equation for Q, the estimated sales are 102,000 (rounded) at a price of $40. With this quantity de- manded and price, total revenue would be $4,080,000, representing a decrease of $420,000 from the revenue at $75 a plate. If the state’s objective is to raise revenue by selling vanity plates, it should increase rather than decrease price.
This Illustration actually makes two points. First, even decision makers in organizations that are not run for profit, such as government agencies, should be able to use economic analysis. Second, managers whose firms are in business to make a profit should make an effort to know (or at least have a good approximation for) the elasticity of demand for the products they sell. Only with this information will they know what price to charge.
aIt was, of course, quantity demanded that decreased, not demand.
Source: Adapted from Barbara Boughton, “A License for Vanity,” Houston Post, October 19, 1986, pp. 1G, 10G.
Point elasticity when demand is linear Consider a general linear demand function of three variables—price (P), income (M), and the price of a related good (PR)
Q 5 a 1 bP 1 cM 1 dPR
Suppose income and the price of the related good take on specific values of __M and
__P R, respectively. Recall from Chapter 2 when values of the demand determinants (M and PR in this case) are held constant, they become part of the constant term in the direct demand function:
Q 5 a9 1 bP
where a9 5 a 1 c __M 1 d __P R. The slope parameter b, of course, measures the rate of change in quantity demanded per unit change in price: b 5 DQyDP. Thus price elasticity at a point on a linear demand curve can be calculated as
E 5 b P __
Q
where P and Q are the values of price and quantity at the point of measure. For example, let’s compute the elasticity of demand for Borderline Music at a price of $9 per CD (see point c in Panel B of Figure 6.1). You can verify for yourself that the equation for the direct demand function is Q 5 2,400 2 100P, so b 5 2100 and
E 5 2100 9 _____ 1,500 5 2 3 __ 5 5 20.6
Even though multiplying b by the ratio PyQ is rather simple, there happens to be an even easier formula for computing point price elasticities of demand. This alternative point elasticity formula is
E 5 P ______
P 2 A
where P is the price at the point on demand where elasticity is to be measured, and A is the price-intercept of demand.1 Note that, for the linear demand equation Q 5 a9 1 bP, the price intercept A is 2a9yb. In Figure 6.1, let us apply this alterna- tive formula to calculate again the elasticity at point c (P 5 $9). In this case, the price-intercept A is $24, so the elasticity is
E 5 ______ 9 2 24 9 5 20.6
which is exactly equal to the value obtained previously by multiplying the slope of demand by the ratio PyQ. We must stress that, because the two formulas b P __
Q and P
______
P 2 A are mathematically equivalent, they always yield identical values for point price elasticities.
1This alternative formula for computing price elasticity is derived in the mathematical appendix for this chapter.
Relation For linear demand functions Q 5 a9 1 bP, the price elasticity of demand can be computed using either of two equivalent formulas:
E 5 b P __
Q 5 ______ P P 2 A
where P and Q are the values of price and quantity demanded at the point of measure on demand, and A (5 2a9yb) is the price-intercept of demand.
Point elasticity when demand is curvilinear When demand is curvilinear, the formula E 5 ____ DQ
DP 3 P __ Q can be used for computing point elasticity simply by sub- stituting the slope of the curved demand at the point of measure for the value of DQyDP in the formula. This can be accomplished by measuring the slope of the tangent line at the point of measure. Figure 6.2 illustrates this procedure.
In Figure 6.2, let us measure elasticity at a price of $100 on demand curve D.
We first construct the tangent line T at point R. By the “rise over run” method, the slope of T equals 24y3 (5 2140y105). Of course, because P is on the vertical axis and Q is on the horizontal axis, the slope of tangent line T gives DPyDQ not DQyDP. This is easily fixed by taking the inverse of the slope of tangent line T to get DQ/DP 5 23y4. At point R price elasticity is calculated using 23y4 for the slope of demand and using $100 and 30 for P and Q, respectively
ER 5 ____ DQ
DP 3 P __ Q 5 2 3 __ 4 3 100 ____ 30 5 22.5
140
100 90
40
0
Price (dollars)
30 105
Quantity R
T S
T'
Q D
ER = –2.5
ES = –0.8 F I G U R E 6.2
Calculating Point Elasticity for Curvilinear Demand
Now try Technical Problems 8–9.
As it turns out, the alternative formula E 5 P/(P 2 A) for computing point elas- ticity on linear demands can also be used for computing point elasticities on cur- vilinear demands. To do so, the price-intercept of the tangent line T serves as the value of A in the formula. As an example, we can recalculate elasticity at point R in Figure 6.2 using the formula E 5 Py(P 2 A). The price-intercept of tangent line T is $140
ER 5 P ______ P 2 A 5 _________ 100 2 140 100 5 22.5 As expected, 22.5 is the same value for ER obtained earlier.
Since the formula E 5 Py(P 2 A) doesn’t require the slope of demand or the value of Q, it can be used to compute E in situations like point S in Figure 6.2 where the available information is insufficient to be able to multiply slope by the PyQ ratio. Just substitute the price-intercept of T9 (5 $90) into the formula E 5 Py(P 2 A) to get the elasticity at point S
ES 5 P ______ P 2 A 5 _______ 40 2 90 40 5 20.8
Relation For curvilinear demand functions, the price elasticity at a point can be computed using either of two equivalent formulas:
E 5 ___ DQ DP 3 __ P
Q 5 ______ P P 2 A
where DQyDP is the slope of the curved demand at the point of measure (which is the inverse of the slope of the tangent line at the point of measure), P and Q are the values of price and quantity demanded at the point of measure, and A is the price-intercept of the tangent line extended to cross the price-axis.
We have now established that both formulas for computing point elasticities will give the same value for the price elasticity of demand whether demand is linear or curvilinear. Nonetheless, students frequently ask which formula is the
“best” one. Because the two formulas give identical values for E, neither one is better or more accurate than the other. We should remind you, however, that you may not always have the required information to compute E both ways, so you should make sure you know both methods. (Recall the situation in Figure 6.2 at point S.) Of course, when it is possible to do so, we recommend computing the elasticity using both formulas to make sure your price elasticity calculation is correct!
Elasticity (Generally) Varies along a Demand Curve
In general, different intervals or points along the same demand curve have differing elasticities of demand, even when the demand curve is linear. When demand is linear, the slope of the demand curve is constant. Even though the
Now try Technical Problem 10.
absolute rate at which quantity demanded changes as price changes (DQyDP) remains constant, the proportional rate of change in Q as P changes (%DQy%DP) varies along a linear demand curve. To see why, we can examine the basic formula for elasticity, E 5 DQ____
DP 3 P __ Q . Moving along a linear demand does not cause the term DQyDP to change, but elasticity does vary because the ratio PyQ changes. Moving down demand, by reducing price and selling more output, causes the term PyQ to decrease which reduces the absolute value of E. And, of course, moving up a linear demand, by increasing price and selling less output, causes PyQ and |E| to increase. Thus, P and |E| vary directly along a linear de- mand curve.
For movements along a curved demand, both the slope and the ratio PyQ vary continuously along demand. For this reason, elasticity generally varies along cur- vilinear demands, but there is no general rule about the relation between price and elasticity as there is for linear demand.
As it turns out, there is an exception to the general rule that elasticity varies along curvilinear demands. A special kind of curvilinear demand function exists for which the demand elasticity is constant for all points on demand. When de- mand takes the form Q 5 aPb, the elasticity is constant along the demand curve and equal to b.2 Consequently, no calculation of elasticity is required, and the price elasticity is simply the value of the exponent on price, b. The absolute value of b can be greater than, less than, or equal to 1, so that this form of demand can be elastic, inelastic, or unitary elastic at all points on the demand curve. As we will show you in the next chapter, this kind of demand function can be useful in statis- tical demand estimation and forecasting.
Figure 6.3 shows a constant elasticity of demand function, Q 5 aPb, with the values of a and b equal to 100,000 and 21.5, respectively. Notice that price elasticity equals 21.5 at both points U and V where prices are $20 and $40, respectively
EU 5 P ______ P 2 A 5 __________ 20 2 33.33 20 5 21.5 EV 5 P ______ P 2 A 5 __________ 40 2 66.67 40 5 21.5
Clearly, you never need to compute the price elasticity of demand for this kind of demand curve because E is the value of the exponent on price (b).
Relation In general, the price elasticity of demand varies along a demand curve. For linear demand curves, price and |E| vary directly: The higher (lower) the price, the more (less) elastic is demand. For a cur- vilinear demand, there is no general rule about the relation between price and elasticity, except for the special case of Q 5 aP b, which has a constant price elasticity (equal to b) for all prices.
Now try Technical Problem 11.
2See the appendix at the end of this chapter for a mathematical proof of this result.