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 percentage change in price. When calculating the value of E, it is convenient to avoid computing percentage changes by using a simpler formula for computing elasticity that can be obtained through the following algebraic operations:
Thus price elasticity can be calculated by multiplying the slope of demand (ΔQ/ΔP) times the ratio of price divided by quantity (P/Q), 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 measures 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 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 = ΔQ/ΔP * 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 = +200/-2 * 17/700 = -2.43
Ecd = +200/-2 * 8/1600 = -0.5
Relation When calculating the price elasticity of demand over an interval of demand, use the interval or arc elasticity formula:
E = ΔQ/ΔP * Average P/Average Q
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 (ΔQ/ΔP), computed at the point of measure, by the ratio P/Q, 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 ΔQ/ΔP for the linear demand in Figure 6.1 is 100 ( 2400/24) at every point along D, so the two point elasticities are computed as
Ea = -100 * 18/600 = -3
Eb = -100 * 16/800 = -2
Relation When calculating the price elasticity of demand at a point on demand, multiply the slope of demand (Q/P ), computed at the point of measure, by the ratio P/Q, computed using the values of P and Q at the point of measure.
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 = a + bP + cM + dPR
Point elasticity when demand is curvilinear
When demand is curvilinear, the formula E = ΔQ/ΔP * P/Q can be used for computing point elasticity simply by substituting the slope of the curved demand at the point of measure for the value of ΔQ/ΔP 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 -4/3 (=-140/105). Of course, because P is on the vertical axis and Q is on the horizontal axis, the slope of tangent line T gives ΔP/ΔQ not ΔQ/ΔP. This is easily fixed by taking the inverse of the slope of tangent line T to get ΔQ/ΔP = -3/4. At point R price elasticity is calculated using -3/4 for the slope of demand and using $100 and 30 for P and Q, respectively:
ER= ΔQ/ΔP * P/Q = -3/4 * 100/30 = -2.5
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 absolute rate at which quantity demanded changes as price changes (ΔQ/ΔP) remains constant, the proportional rate of change in Q as P changes (%ΔQ/%ΔP) varies along a linear demand curve. To see why, we can examine the basic formula for elasticity, E = ΔQ/ΔP * P/Q. Moving along a linear demand does not cause the term ΔQ/ΔP to change, but elasticity does vary because the ratio P/Q changes. Moving down demand, by reducing price and selling more output, causes the term P/Q 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 P/Q and |E| to increase. Thus P and |E| vary directly along a linear demand curve.
For movements along a curved demand, both the slope and the ratio P/Q vary continuously along demand. For this reason, elasticity generally varies along curvilinear 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 demand takes the form Q = aPb, the elasticity is constant along the demand curve and equal to b. 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.
Figure 6.3 shows a constant elasticity of demand function, Q = aPb, with the values of a and b equal to 100,000 and -1.5, respectively. Notice that price elasticity equals -1.5 at both points U and V where prices are $20 and $40, respectively:
Clearly, you never need to compute the price elasticity of demand for this kind of demand curve since 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 curvilinear demand, there is no general rule about the relation between price and elasticity, except for the special case of Q = aPb, which has a constant price elasticity (equal to b) for all prices.
I L L U S T R AT I O N 6 . 1
Texas Calculates Price Elasticity
In addition to its regular license plates, the state of Texas, as do other states, sells personalized or “vanity” 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 standard 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 announces his practice with the license MYOPIA. Others tell their personalities with COZY-1 and ALL MAN. Arabid Star Trek fan has BM ME UP.
In 1986, Texas increased the price for such plates from $25 to $75. The Houston Post (October 19, 1986) reported that before the price increase about 150,000 cars in Texas had personalized licenses. After the increase in price, only 60,000 people ordered the vanity plates. As it turned out, demand was rather inelastic over this range. As you can calculate, the price elasticity is -0.86. Thus revenue rose after the price increase, from $3,750,000 to $4,500,000.
But the Houston Post article 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 personalized 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 since fewer were produced. So the move from $25 to $75 was the right move.
Moreover, let’s suppose that the price elasticity between $75 and $40 is approximately equal to the value calculated for the movement from $25 to $75 (-0.86). 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 -0.86,
E = ΔQ/ΔP * Average P/Average Q
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 demanded 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.