# Elasticity and Demand

Edited by Paul Ducham

PRICE ELASTICITY OF DEMAND

As noted above, price elasticity of demand measures the responsiveness or sensitivity of consumers to changes in the price of a good or service. We will begin this section by presenting a formal (mathematical) definition of price elasticity and then show how price elasticity can be used to predict the change in sales when price rises or falls or to predict the percentage reduction in price needed to stimulate sales by a given percentage amount.

Consumer responsiveness to a price change is measured by the price elasticity of demand (E), defined as

E = %ΔQ / %ΔP = Percentage change in quantity demanded / Percentage change in price

Since price and quantity demanded are inversely related by the law of demand, the numerator and denominator always have opposite algebraic signs, and the price elasticity is always negative. The price elasticity is calculated for movements along a given demand curve (or function) as price changes and all other factors affecting quantity demanded are held constant. Suppose a 10 percent price decrease (%ΔP = -10%) causes consumers to increase their purchases by 30 percent (%ΔQ = +30%). The price elasticity is equal to -3 (= +30%/-10%) in this case. In contrast, if the 10 percent decrease in price causes only a 5 percent increase in sales, the price elasticity would equal -0.5 (= +5%/-10%). Clearly, the smaller (absolute) value of E indicates less sensitivity on the part of consumers to a change in price.

When a change in price causes consumers to respond so strongly that the percentage by which they adjust their consumption (in absolute value) exceeds the percentage change in price (in absolute value), demand is said to be elastic over that price interval. In mathematical terms, demand is elastic when |%ΔQ| exceeds |%ΔP|, and thus |E| is greater than 1. When a change in price causes consumers to respond so weakly that the percentage by which they adjust their consumption (in absolute value) is less than the percentage change in price (in absolute value), demand is said to be inelastic over that price interval. In other words, demand is inelastic when the numerator (in absolute value) is smaller than the denominator (in absolute value), and thus |E| is less than 1. In the special instance in which the percentage change in quantity (in absolute value) just equals the percentage change in price (in absolute value), demand is said to be unitary elastic, and |E| is equal to 1. Table 6.1 summarizes this discussion.

Predicting the Percentage Change in Quantity Demanded

Suppose a manager knows the price elasticity of demand for a company’s product is equal to -2.5 over the range of prices currently being considered by the firm’s marketing department. The manager is considering decreasing price by 8 percent and wishes to predict the percentage by which quantity demanded will increase. From the definition of price elasticity, it follows that

-2.5 = %ΔQ / -8%

so, with a bit of algebraic manipulation, %ΔQ = +20% (= -2.5 * -8%). Thus the manager can increase sales by 20 percent by lowering price 8 percent. As we mentioned in the introduction, price elasticity information about industry demand can also help price-taking managers make predictions about industry- or marketlevel changes. For example, suppose an increase in industry supply is expected to cause market price to fall by 8 percent, and the price elasticity of industry demand is equal to -2.5 for the segment of demand over which supply shifts. Using the same algebraic steps just shown, total industry output is predicted to increase by 20 percent in this case.

Predicting the Percentage Change in Price

Suppose a manager of a different firm faces a price elasticity equal to -0.5 over the range of prices the firm would consider charging for its product. This manager wishes to stimulate sales by 15 percent. The manager is willing to lower price to accomplish the increase in sales but needs to know the percentage amount by which price must be lowered to obtain the 15 percent increase in sales. Again using the definition of price elasticity of demand, it follows that

-0.5 = +15% / %ΔP

so, after some algebraic manipulation, %ΔP = -30% (= 15%/-0.5). Thus this manager must lower price by 30 percent in order to increase sales by 15 percent. As we explained in the case of predicting percentage changes in quantity demanded, elasticity of industry demand can also be used to make predictions about changes in market-determined prices. For example, suppose an increase in industry supply is expected to cause market output to rise by 15 percent, and the price elasticity of industry demand is equal to -0.5 for the portion of demand over which supply shifts. Following the algebraic steps shown above, market price is predicted to fall by 30 percent. As you can see, the techniques for predicting percentage changes in quantity demanded and price can be applied to both individual firm demand curves or industry demand curves.

As you can see, the concept of elasticity is rather simple. Price elasticity is nothing more than a mathematical measure of how sensitive quantity demanded is to changes in price. We will now apply the concept of price elasticity to a crucial question facing managers. How does a change in the price of the firm’s product affect the total revenue received?

PRICE ELASTICITY

Managers of firms, as well as industry analysts, government policymakers, and academic researchers, are frequently interested in how total revenue changes when there is a movement along the demand curve. Total revenue (TR), which also equals the total expenditure by consumers on the commodity, is simply the price of the commodity times quantity demanded, or

TR = P * Q

As we have emphasized, price and quantity demanded move in opposite directions along a demand curve: If price rises, quantity falls; if price falls, quantity rises. The change in price and the change in quantity have opposite effects on total revenue. The relative strengths of these two effects will determine the overall effect on TR.We will now examine these two effects, called the price effect and the quantity effect, along with the price elasticity of demand to establish the relation between changes in price and total revenue.

Price Elasticity and Changes in Total Revenue

When a manager raises the price of a product, the increase in price, by itself, would increase total revenue if the quantity sold remained constant. Conversely, when a manager lowers price, the decrease in price would decrease total revenue if the quantity sold remained constant. This effect on total revenue of changing price, for a given level of output, is called the price effect. When price changes, the quantity sold does not remain constant; it moves in the opposite direction of price. When quantity increases in response to a decrease in price, the increase in quantity, by itself, would increase total revenue if the price of the product remained constant. Alternatively, when quantity falls after a price increase, the reduction in quantity, by itself, would decrease total revenue if product price remained constant. The effect on total revenue of changing the quantity sold, for a given price level, is called the quantity effect. The price and quantity effects always push total revenue in opposite directions. Total revenue moves in the direction of the stronger of the two effects. If the two effects are equally strong, no change in total revenue can occur.

Suppose a manager increases price, causing quantity to decrease. The price effect, represented below by an upward arrow above P, and the quantity effect, represented by a downward arrow above Q, show how the change in TR is affected by opposing forces:

To determine the direction of movement in TR, information about the relative strengths of the price effect and output effect must be known. The elasticity of demand tells a manager which effect, if either, is dominant.

If demand is elastic, |E| is greater than 1, the percentage change in Q (in absolute value) is greater than the percentage change in P (in absolute value), and the quantity effect dominates the price effect. To better see how the dominance of the quantity effect determines the direction in which TR moves, you can represent the dominance of the quantity effect by drawing the arrow above Q longer than the arrow above P. The direction of the dominant effect—the quantity effect here—tells a manager that TR will fall when price rises and demand is elastic:

If a manager decreases price when demand is elastic, the arrows in this diagram reverse directions. The arrow above Q is still the longer arrow since the quantity effect always dominates the price effect when demand is elastic.

Now consider a price increase when demand is inelastic. When demand is inelastic, |E| is less than 1, the percentage change in Q (in absolute value) is less than the percentage change in P (in absolute value), and the price effect dominates the quantity effect. The dominant price effect can be represented by an upward arrow above P that is longer than the downward arrow above Q. The direction of the dominant effect tells the manager that TR will rise when price rises and demand is inelastic:

When a manager decreases price and demand is inelastic, the arrows in this diagram would reverse directions. Adownward arrow above P would be a long arrow since the price effect always dominates the quantity effect when demand is inelastic.

When demand is unitary elastic, |E| is equal to 1, and neither the price effect nor the quantity effect dominates. The two effects exactly offset each other, so price changes have no effect on total revenue when demand is unitary elastic.

Relation The effect of a change in price on total revenue (TR = P * Q) is determined by the price elasticity of demand. When demand is elastic (inelastic), the quantity (price) effect dominates. Total revenue always moves in the same direction as the variable (P or Q) having the dominant effect. When demand is unitary elastic, neither effect dominates, and changes in price leave total revenue unchanged.

Table 6.2 summarizes the relation between price changes and revenue changes under the three price elasticity conditions.

Changing Price at Borderline Music Emporium: A Numerical Example

The manager at Borderline Music Emporium faces the demand curve for compact discs shown in Figure 6.1. At the current price of \$18 per compact disc, Borderline can sell 600 CDs each week. The manager can lower price to \$16 per compact disc and increase sales to 800 CDs per week. In Panel A of Figure 6.1, over the interval a to b on demand curve D the price elasticity is equal to -2.43. (You will learn how to make this calculation in Section 6.4.) Since the demand for compact discs is elastic over this range of prices (|-2.43| > 1), the manager knows the quantity effect dominates the price effect. Lowering price from \$18 to \$16 results in an increase in the quantity of CDs sold, so the manager knows that total revenue, which always moves in the direction of the dominant effect, must increase.

To verify that revenue indeed rises when the manager at Borderline lowers the price over an elastic region of demand, you can calculate total revenue at the two prices, \$18 and \$16:

Point a: TR = \$18 * 600 = \$10,800

Point b: TR = \$16 * 800 = \$12,800

Total revenue rises by \$2,000 (= 12,800 - 10,800) when price is reduced over this elastic region of demand. While Borderline earns less revenue on each CD sold, the number of CDs sold each week rises enough to more than offset the downward price effect, causing total revenue to rise.

Now suppose the manager at Borderline is charging just \$9 per compact disc and sells 1,500 CDs per week (see Panel B). The manager can lower price to \$7 per disc and increase sales to 1,700 CDs per week. Over the interval c to d on demand curve D, the elasticity of demand equals -0.50. Over this range of prices for CDs, the demand is inelastic (|-0.50| < 1), and Borderline’s manager knows the price effect dominates the quantity effect. If the manager lowers price from \$9 to \$7, total revenue, which always moves in the direction of the dominant effect, must decrease.

To verify that revenue falls when the manager at Borderline lowers price over an inelastic region of demand, you can calculate total revenue at the two prices, \$9 and \$7:

Point c: TR = \$9 * 1,500 = \$13,500

Point d: TR = \$7 * 1,700 = \$11,900

Total revenue falls by \$1,600 (ΔTR = \$11,900 - \$13,500 = -\$1,600). Total revenue always falls when price is reduced over an inelastic region of demand. Borderline again earns less revenue on each CD sold, but the number of CDs sold each week does not increase enough to offset the downward price effect and total revenue falls.

If the manager decreases (or increases) the price of compact discs over a unitaryelastic region of demand, total revenue does not change. You should verify that demand is unitary elastic over the interval f to g in Panel A of Figure 6.1.

Note in Figure 6.1 that demand is elastic over the \$16 to \$18 price range but inelastic over the \$7 to \$9 price range. In general, the elasticity of demand varies along any particular demand curve, even one that is linear. It is usually incorrect to say a demand curve is either elastic or inelastic. You can say only that a demand curve is elastic or inelastic over a particular price range. For example, it is correct to say that demand curve D in Figure 6.1 is elastic over the \$16 to \$18 price range and inelastic over the \$7 to \$9 price range.

FACTORS AFFECTING PRICE ELASTICITY OF DEMAND

Price elasticity of demand plays such an important role in business decision making that managers should understand not only how to use the concept to obtain information about the demand for the products they sell but also how to recognize the factors that affect price elasticity. We will now discuss the three factors that make the demand for some products more elastic than the demand for other products.

Availability of Substitutes

The availability of substitutes is by far the most important determinant of price elasticity of demand. The better the substitutes for a given good or service, the more elastic the demand for that good or service. When the price of a good rises, consumers will substantially reduce consumption of that good if they perceive that close substitutes are readily available. Naturally, consumers will be less responsive to a price increase if they perceive that only poor substitutes are available.

Some goods for which demand is rather elastic include fruit, corporate jets, and life insurance. Alternatively, goods for which consumers perceive few or no good substitutes have low price elasticities of demand. Wheat, salt, and gasoline tend to have low price elasticities because there are only poor substitutes available—for instance, corn, pepper, and diesel fuel, respectively.

The definition of the market for a good greatly affects the number of substitutes and thus the good’s price elasticity of demand. For example, if all the grocery stores in a city raised the price of milk by 50 cents per gallon, total sales of milk would undoubtedly fall—but probably not by much. If, on the other hand, only the Food King chain of stores raised price by 50 cents, the sales of Food King milk would probably fall substantially. There are many good substitutes for Food King milk, but there are not nearly as many substitutes for milk in general.

Percentage of Consumer’s Budget

The percentage of the consumer’s budget that is spent on the commodity is also important in the determination of price elasticity. All other things equal, we would expect the price elasticity to be directly related to the percentage of consumers’ budgets spent on the good. For example, the demand for refrigerators is probably more price elastic than the demand for toasters, because the expenditure required to purchase a refrigerator would make up a larger percentage of the budget of a “typical” consumer.

The length of the time period used in measuring the price elasticity affects the magnitude of price elasticity. In general, the longer the time period of measurement, the larger (the more elastic) the price elasticity will be (in absolute value). This relation is the result of consumers’ having more time to adjust to the price change.

Consider, again, the way consumers would adjust to an increase in the price of milk. Suppose the dairy farmers’ association is able to convince all producers of milk nationwide to raise their milk prices by 15 percent. During the first week the price increase takes effect, consumers come to the stores with their grocery lists already made up. Shoppers notice the higher price of milk but have already planned their meals for the week. While a few of the shoppers will react immediately to the higher milk prices and reduce the amount of milk they purchase, many shoppers will go ahead and buy the same amount of milk as they purchased the week before. If the dairy association collects sales data and measures the price elasticity of demand for milk after the first week of the price hike, they will be happy to see that the 15 percent increase in the price of milk caused only a modest reduction in milk sales.

Over the coming weeks, however, consumers begin looking for ways to consume less milk. They substitute foods that have similar nutritional composition to milk; consumption of cheese, eggs, and yogurt all increase. Some consumers will even switch to powdered milk for some of their less urgent milk needs—perhaps to feed the cat or to use in cooking. Six months after the price increase, the dairy association again measures the price elasticity of milk. Now the price elasticity of demand is probably much larger in absolute value (more elastic) because it is measured over a six-month time period instead of a one-week time period.

For most goods and services, given a longer time period to adjust, the demand for the commodity exhibits more responsiveness to changes in price—the demand becomes more elastic. Of course, we can treat the effect of time on elasticity within the framework of the effect of available substitutes. The greater the time period available for consumer adjustment, the more substitutes become available and economically feasible. As we stressed earlier, the more available are substitutes, the more elastic is demand.

PRICE ELASTICITY OF DEMAND CALCULATOR

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.

MARGINAL REVENUE, DEMAND, AND PRICE ELASTICITY

The responsiveness of consumers to changes in the price of a good must be considered by managers of PRICE-SETTING FIRM when making pricing and output decisions. The price elasticity of demand gives managers essential information about how total revenue will be affected by a change in price. As it turns out, an equally important concept for pricing and output decisions is marginal revenue. Marginal revenue (MR) is the addition to total revenue attributable to selling one additional unit of output:

MR = ΔTR/ΔQ

Because marginal revenue measures the rate of change in total revenue as quantity changes, MR is the slope of the TR curve. Marginal revenue is related to price elasticity because marginal revenue, like price elasticity, involves changes in total revenue caused by movements along a demand curve.

Marginal Revenue and Demand

As noted, marginal revenue is related to the way changes in price and output affect total revenue along a demand curve. To see the relation between marginal revenue and price, consider the following numerical example. The demand schedule for a product is presented in columns 1 and 2 of Table 6.3. Price times quantity gives the total revenue obtainable at each level of sales, shown in column 3.

Marginal revenue, shown in column 4, indicates the change in total revenue from an additional unit of sales. Note that marginal revenue equals price only for the first unit sold. For the first unit sold, total revenue is the demand price for 1 unit. The first unit sold adds \$4—the price of the first unit—to total revenue, and the marginal revenue of the first unit sold equals \$4; that is, MR = P for the first unit. If 2 units are sold, the second unit should contribute \$3.50 (the price of the second unit) to total revenue. But total revenue for 2 units is only \$7, indicating that the second unit adds only \$3 (= \$7 - \$4) to total revenue. Thus the marginal revenue of the second unit is not equal to price, as it was for the first unit. Indeed, examining columns 2 and 4 in Table 6.3 indicates that MR < P for all but the first unit sold.

Marginal revenue is less than price (MR < P) for all but the first unit sold because price must be lowered in order to sell more units. Not only is price lowered on the marginal (additional) unit sold, but price is also lowered for all the inframarginal units sold. The inframarginal units are those units that could have been sold at a higher price had the firm not lowered price to sell the marginal unit. Marginal revenue for any output level can be expressed as

MR = Price - Revenue lost by lowering price on the inframarginal units

The second unit of output sells for \$3.50. By itself, the second unit contributes \$3.50 to total revenue. But marginal revenue is not equal to \$3.50 for the second unit because in order to sell the second unit, price on the first unit is lowered from \$4 to \$3.50. In other words, the first unit is an inframarginal unit, and the \$0.50 lost on the first unit must be subtracted from the price. The net effect on total revenue of selling the second unit is \$3 (= \$3.50 - \$0.50), the same value as shown in column 4 of Table 6.3.

If the firm is currently selling 2 units and wishes to sell 3 units, it must lower price from \$3.50 to \$3.10. The third unit increases total revenue by its price, \$3.10. In order to sell the third unit, the firm must lower price on the 2 units that could have been sold for \$3.50 if only 2 units were offered for sale. The revenue lost on the 2 inframarginal units is \$0.80 (= \$0.40 * 2). Thus the marginal revenue of the third unit is \$2.30 (= \$3.10 - \$0.80), and marginal revenue is less than the price of the third unit.

It is now easy to see why P = MR for the first unit sold. For the first unit sold, price is not lowered on any inframarginal units. Since price must fall in order to sell additional units, marginal revenue must be less than price at every other level of sales (output).

As shown in column 4, marginal revenue declines for each additional unit sold. Notice that it is positive for each of the first 5 units sold. However, marginal revenue is 0 for the sixth unit sold, and it becomes negative thereafter. That is, the seventh unit sold actually causes total revenue to decline. Marginal revenue is positive when the effect of lowering price on the inframarginal units is less than the revenue contributed by the added sales at the lower price. Marginal revenue is negative when the effect of lowering price on the inframarginal units is greater than the revenue contributed by the added sales at the lower price.

Relation Marginal revenue must be less than price for all units sold after the first, because the price must be lowered in order to sell more units. When marginal revenue is positive, total revenue increases when quantity increases. When marginal revenue is negative, total revenue decreases when quantity increases. Marginal revenue is 0 when total revenue is maximized.

Figure 6.4 shows graphically the relations among demand, marginal revenue, and total revenue for the demand schedule in Table 6.3. As noted, MR is below price (in Panel A) at every level of output except the first. When total revenue (in Panel B) begins to decrease, marginal revenue becomes negative. Demand and marginal revenue are both negatively sloped.

Sometimes the interval over which marginal revenue is measured is greater than one unit of output. After all, managers don’t necessarily increase output by just one unit at a time. Suppose in Table 6.3 that we want to compute marginal revenue when output increases from 2 units to 5 units. Over the interval, the change in total revenue is \$5 (= \$12 - \$7), and the change in output is 3 units. Marginal revenue is \$1.67 (=ΔTR/ΔQ = \$5/3) per unit change in output; that is, each of the 3 units contributes (on average) \$1.67 to total revenue. As a general rule, whenever the interval over which marginal revenue is being measured is more than a single unit, divide ΔTR by ΔQ to obtain the marginal revenue for each of the units of output in the interval.

Linear demand equations are frequently employed for purposes of empirical demand estimation and demand forecasting. The relation between a linear demand equation and its marginal revenue function is no different from that set forth in the preceding relation. The case of a linear demand is special because the relation between demand and marginal revenue has some additional properties that do not hold for nonlinear demand curves.

When demand is linear, marginal revenue is linear and lies halfway between demand and the vertical (price) axis. This implies that marginal revenue must be twice as steep as demand, and demand and marginal revenue share the same intercept on the vertical axis. We can explain these additional properties and

Relation When inverse demand is linear, P = A + BQ (A > 0, B < 0), marginal revenue is also linear, intersects the vertical (price) axis at the same point demand does, and is twice as steep as the inverse demand function. The equation of the linear marginal revenue curve is MR = A + 2BQ.

Figure 6.5 shows the linear inverse demand curve P = 6 - 0.05Q. (Remember that B is negative because P and Q are inversely related.) The associated marginal revenue curve is also linear, intersects the price axis at \$6, and is twice as steep as the demand curve. Because it is twice as steep, marginal revenue intersects the quantity axis at 60 units, which is half the output level for which demand intersects the quantity axis. The equation for marginal revenue has the same vertical intercept but twice the slope: MR = 6 - 0.10Q.

Marginal Revenue and Price Elasticity

Using Figure 6.5, we now examine the relation of price elasticity to demand and marginal revenue. Recall that if total revenue increases when price falls and quantity rises, demand is elastic; if total revenue decreases when price falls and quantity rises, demand is inelastic. When marginal revenue is positive in Panel A, from a quantity of 0 to 60, total revenue increases as price declines in Panel B; thus demand is elastic over this range. Conversely, when marginal revenue is negative, at any quantity greater than 60, total revenue declines when price falls; thus demand must be inelastic over this range. Finally, if marginal revenue is 0, at a quantity of 60, total revenue does not change when quantity changes, so the price elasticity of demand is unitary at 60.

Except for marginal revenue being linear and twice as steep as demand, all the preceding relations hold for nonlinear demands. Thus the following relation (also summarized in Table 6.4) holds for all demand curves:

Relation When MR is positive (negative), total revenue increases (decreases) as quantity increases, and demand is elastic (inelastic). When MR is 0, the price elasticity of demand is unitary.

The relation among marginal revenue, price elasticity of demand, and price at any quantity can be expressed still more precisely. The relation between marginal revenue, price, and price elasticity, for linear or curvilinear demands, is

MR = P(1 + 1/E)

where E is the price elasticity of demand and P is product price. When demand is elastic (|E| > 1), |1/E| is less than 1, 1 + (1/E) is positive, and marginal revenue is positive. When demand is inelastic (|E| < 1), |1/E| is greater than 1, 1 + (1/E) is negative, and marginal revenue is negative. In the case of unitary price elasticity (E=-1), 1 + (1/E) is 0, and marginal revenue is 0.

To illustrate the relation between MR, P, and E numerically, we calculate marginal revenue at 40 units of output for the demand curve shown in Panel A of Figure 6.5. At 40 units of output, the point elasticity of demand is equal to -2 [= P/(P - A) = 4/(4 - 6)]. Using the formula presented above, MR is equal to 2 [= 4(1 - 1/2)]. This is the same value for marginal revenue that is obtained by substituting Q = 40 into the equation for marginal revenue: MR = 6 - 0.1(40) = 2.

Relation For any demand curve, when demand is elastic (|E| > 1), marginal revenue is positive. When demand is inelastic (|E| < 1), marginal revenue is negative. When demand is unitary elastic (|E| = 1), marginal revenue is 0. For all demand and marginal revenue curves:

MR = P(1 + 1/E)

where E is the price elasticity of demand.