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Chapter 3

Elasticity

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A firm’s market performance depends on several factors, some of which are directly under a manager’s control, while others are not. The individual firm cannot affect market interest rates, the pricing and advertising strategies of rivals, foreign-exchange rates that affect overseas sales or the cost of imports, business cycles, unemployment, or the rate of inflation. On the other hand, managers do control the firm’s organizational structure, pricing, product design and packaging, marketing strategy, research and development expen-ditures, and so on. Changes in any of these factors can have a profound effect on the firm’s bottom line.

In the previous chapter we examined how changes in price and other determi-nants affect the demand for goods and services. In this chapter we will explore these relationships in greater detail by introducing the important concept of elasticity, and how this measure can be used by managers to assess the effects of changes in the firm’s pricing and advertising strategies, business cycle fluctua-tions, changes in the pricing and advertising strategies of rival firms, and other factors that affect the firm’s bottom line. We will begin our discussion with the most important of these elasticity measures—the price elasticity of demand.

PRICE ELASTICITY OF DEMAND

In general, elasticity measures the percent change in the value of a dependent variable given a percent change in the value of an explanatory variable. Sup-pose, for example, the demand for a firm’s product (Qxd) depends on its price (Px), the consumer’s money income (M), the price charge by rival firms (Py), and the firm’s advertising expenditures (A). This relationship may be written

Q xd = f ( Px , M, Py , A).	(3.1)
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56 Chapter 3

On the basis of historical data, it should be possible for a firm to estimate the demand equation for its product, which may be used by a manager to improve the company’s overall performance. Suppose, for example, that the demand for a firm’s product x is given by the equation

Q x = 127 − 50Px .

(3.2)

This linear demand equation is depicted in Figure 3.1. The slope coefficient βx = −50 tells us that the quantity demanded of good x increases (decreases) by 50 thousand units for every $1 decrease (increase) in its price.

The value of the slope may be calculated directly from the information provided in Figure 3.1. As we move along the demand curve from point A to point B, the value of the slope may be calculated as

bx =	ΔQx	=	Q2	− Q1	=	22	−12	= −50.	(3.3)
	ΔPx		P2	− P1		2.10 − 2.30

The value of the slope (βx) measures the change in unit sales resulting from a change in the price of a good or service. How a consumer reacts to a $10 rebate on the purchase of an item selling for $100, however, can be quite different than the same rebate on an item that costs $1,000. A 10 percent rebate may be viewed as a real bargain; 1 percent rebate may be interpreted as a marketing ploy. On the other hand, if a firm offers a 10 percent rebate regardless of price, the consumer’s reaction may be quite different. For the

Figure 3.1

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Elasticity

manager, the question is how will a 10 percent rebate affect the company’s unit sales, revenues, and profits?

Another problem with the slope is that its numerical value depends on how we measure unit sales. Suppose, for example, that unit sales in Eq. (3.3) are measured in units instead of thousands of units. In this case, the value of the slope is β1 = −50,000. Although we are dealing with the same rela-tionship, changing the units of measurement changes the slope’s numerical value. A related problem is comparing consumer sensitivity to similar price changes for different products, such as the demand for automobiles, yards of cloth, gallons of beer, and pounds of beef. To overcome these measurement problems, economists substitute the slope with the price elasticity of demand, which is a dimensionless measure of consumer sensitivity.

The price elasticity of demand measures the percent change in the quan-tity demanded of a good or service given a percent change in its price, that is

%ΔQ

Ex , x = %ΔPx . (3.4)

The first subscript of Ex,x refers the quantity demanded of good x and the second subscript refers to the price of good x. In subsequent sections we will discuss several other elasticity measures that follow this convention.

Solved Exercise

Suppose that the price elasticity of demand for a product is −2. How will the quantity demanded of this product change is price is reduced by 5 percent?

Solution
Substituting these values into Eq. (3.4) we get
−2 =	%ΔQx	.	(3.5)

	−5

Solving, the percent change in the quantity demanded is %∆Qx = 10. That is, a 5 percent reduction in price results in a 10 percent increase in quantity demanded.

Calculating the Price Elasticity of Demand

The price elasticity of demand is the percent change in the quantity demanded given a percent change in price. How we calculate the price elasticity of demand depends on the available information.

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58 Chapter 3

Midpoint Formula

Suppose you are given two price quantity combinations (P1, Q1) and (P2, Q2), such as points A and B in Figure 3.1. Calculating the price elasticity of demand using Eq. (3.4) is not as straightforward as it appears. This is because we normally calculate a percent change by subtracting the starting value from the ending value, and then divide by the starting value. For example, if price increases from P1 to P2, the percent change in price is would be calculated as

	P − P
%ΔPx	=	2	1	100	> 0.	(3.6)
			P1

On the other hand, if the price declines from P2 to P1, the percent change in price is

	P − P
%ΔPx	=	2	1	100	< 0.	(3.7)
			P2

Ignoring the sign change, Eqs. (3.6) and (3.7) are not equal because are the denominators are different. The same applies to calculating the percent change in quantity demanded. The effect on the calculated price elasticity of demand can be significant. For example, the price elasticity of demand when moving from point A to point B in Figure 3.1 is

Ex , x	=	%ΔQx	=	(	22	− 12	)	/ 12	= −9.58.	(3.8)

				(2.10 − 2.30) / 2.30
		%ΔPx

By contrast, the value of the price elasticity of demand when moving from point B to point A is

		%ΔQx		12	− 22	)	/ 22
Ex , x	=		=	(				= −4.77.	(3.9)
				(2.30 − 2.10) / 2.10
		%ΔPx

What is needed is an elasticity measure that does not depend on whether we are moving from point A to point B, or from point B to point A. A con-venient approximation of the price elasticity of demand is the midpoint formula, which is given by the equation

	Q	− Q		P + P
Ex , x	=	2	1		12
		P2	− P1
				Q1 +Q2

	P	+ P
= bx	1	2	,	(3.10)
		+ Q2
Q1

where βx = (Q2 − Q1)/(P2 − P1) = ∆Qx/∆Px is the slope of the demand curve. The midpoint formula allows us to calculate a price elasticity of demand that is invariant with respect to a price increase or decrease. Moving from

point A to point B, the price elasticity of demand using Eq. (3.10) is

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		Elasticity						59
		22 −12	2.30 + 2.10			6.47
Ex , x	=						= −		.	(3.11)
		2.10 − 2.30		12	+ 22			1

It is left as an exercise for the reader to show that this is the same value for the price elasticity of demand when moving from point B to point A. The interpretation of this result is that a one percent increase in price will result in a 6.47 percent decrease in the quantity demanded. Conversely, a one percent decrease in price will result in a 6.47 percent increase in the quantity demanded.

Point Price Elasticity of Demand

As we have seen, the price elasticity of demand has several advantages over the slope as a measure of consumer sensitivity. It is also easy to calculate and interpret. Unfortunately, the midpoint formula suffers from two weaknesses that reduce its usefulness. To begin with, the midpoint formula implicitly assumes that the underlying demand curve is linear. Since demand curves are more appropriately convex, the value Ex,x is sometimes referred to as the arc price elasticity of demand.

Another weakness of the midpoint formula is that it produces a value that depends on the price-quantity combinations selected for its calculation. Care must be exercised when choosing (P1, Q1) and (P2, Q2). To see why, consider Figure 3.2. As we will soon discover, the percent change in the quantity

Figure 3.2

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60 Chapter 3

demanded is greater than the percent change in price for all price-quantity combinations above the midpoint, such as points A and B, and vice versa for price-quantity combinations below the midpoint, such as points C and D. At the midpoint, the percent change in the quantity demand equals the percent change in price.

It is important for a manager to know whether the quantity demanded for the firm’s product is above or below the midpoint since a change in price will have predictable effects on a firm’s revenues and profits. Unfortunately, a manager may know whether the price charged is above or below the mid-point. For this reason, it is important when using the midpoint formula for managers to choose price-quantity combinations that are “close” together. This will minimize the possibility of spanning the midpoint, which would make it difficult to identify consumer sensitivity to price changes.

An alternative to the midpoint formula is the point price elasticity of demand (εx,x), which is unique at each and every point along a demand curve. The point price elasticity of demand is given by the equation

		ΔQ	x	P	P
e x , x	=			x	= bx	x	,	(3.12)

		ΔPx Qx	Qx

where βx is the slope of the demand curve.

Consider, for example, the demand curve given by Eq. (3.2). Using the midpoint formula, the price elasticity of demand over the line segment AB is Ex,x = −6.47. Since the slope of the demand curve is βx = −50, the point price elasticity of demand at point A is

A		ΔQx		Px		2.30
ex , x	=				= −50			= −9.58.	(3.13)
		ΔPx					12
			Qx

The point price elasticity of demand at point B is

B		ΔQxd	Px		2.10
ex , x	=				= −50		= −4.77.	(3.14)
				d		22
		ΔPx Qx

The reader should note that the value of the price elasticity of demand using the midpoint formula is between the point price elasticity at points A and B. The reader is also cautioned that the price elasticity of demand using the midpoint formula is not a simple average of the point price elasticities.

Solved Exercise

Suppose that the demand for a good is given by the equation Qx = 50 − 2.25Px.

Calculate the point price elasticity of demand when Px = $2.

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					Elasticity			61
Solution
	ΔQ	x		P		2	= −0.099.
ex , x =				x	= −2.25				(3.15)

ΔPx Qx		50
			− 2.25(2)

Midpoint Formula versus the Point Price Elasticity

The main advantage of the midpoint formula is its minimal data require-ments. Only two price-quantity combinations are needed to calculate its value. This is particularly important for small and medium sized firms that lack the financial and technical resources to conduct sophisticated market and statistical research. All a manager needs to do is change the price of the firm’s product by a small amount and use the resulting sales data to approximate the price elasticity of demand using the midpoint formula.

Unfortunately, the midpoint formula may not be a particularly good mea-sure of consumer sensitivity to price changes. The midpoint formula intro-duces distortions that become more acute when price changes are large, or when the underlying demand curve is nonlinear. The point price elasticity is a superior measure of consumer sensitivity to price changes, although manag-ers should limit its application to small changes in price.

Definitions

According to the law of demand, there is an inverse relationship between a percent change in price and the percent change in the quantity demanded. For this reason, the price elasticity of demand is between zero and −∞. It is sometimes convenient, however, to express the price elasticity of demand in absolute terms, which is always positive.

Price Elastic Demand

Demand is said to be price elastic when |εx,x| >1 (εx,x < −1). In this case, the absolute value of the percent change in quantity demanded is greater than the

absolute value of the percent change in price (|%∆Qx| > |%∆Px|). Suppose, for example, that a 2 percent increase in price results in a 4 percent decline in quantity demanded. The demand for this product is price elastic because |εx,x| = |−4/2| = 2 >1. Demand is described as perfectly elastic when |εx,x| = ∞ (εx,x = −∞). This occurs when ∆Qx/∆Px = −∞ or Px/Qx = ∞. These conditions are depicted for the linear demand curve in Figure 3.3.

Price Inelastic Demand

Demand is said to be price inelastic when |εx,x| <1 (−1 < εx,x < 0). This occurs when |%∆Qxd| < |%∆Px|. Suppose, for example, that a 2 percent increase in

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62 Chapter 3

price leads to a 1 percent decline in quantity demanded. The demand for this product is price inelastic because |εx,x| = |−1/2| < 1. Demand is perfectly inelastic when εx,x = 0, which occurs when ∆Qx/∆Px = 0 or Px/Qx = 0. These conditions are depicted in Figure 3.4.

Figure 3.3

Figure 3.4

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Elasticity

Unit Elastic Demand

Finally, demand is said to be unit elastic when |εx,x| =1 (εx,x = −1), which occurs when |%∆Qxd| = |%∆Px|. Suppose that a 2 percent increase in price leads to a 2 percent decline in quantity demanded. The demand for this prod-uct is unit elastic since |εx,x| = |−2/2| = 1.

Determinants of the Price Elasticity of Demand

Several factors that affect a consumer’s sensitivity to a price change. In this section we will discuss three factors that affect the price elasticity of demand, including the number of close substitutes, the proportion of the consumer’s income spent on the product, and the amount of time that a consumer has to adjust to the price change.

Substitutability

The price elasticity of demand is partly explained by the availability of close substitutes. Intuitively, the greater the number of close substitutes, the more price elastic is the demand for a product. This is because it is easier for buyers to find less expensive, albeit not perfect, alternatives. Conversely, the fewer the number of close substitutes, the less price elastic since substitutes are dif-ficult to locate.

The price elasticity of demand also depends on how narrowly we define classes of products. The demand for Coca-Cola, for example, is more price elastic than the demand for soft drinks in general. When the price of Coca-Cola increases, consumers can switch to Pepsi Cola, 7-Up, Dr. Pepper, Gatorade, and so on. On the other hand, there are fewer alternatives if there is an increase in the price of all soft drinks.

The demand curve for the product of a firm that has a large number of sub-stitutes is “flatter” than the demand curve for a firm with few substitutes. The extreme case is when a firm has a large number of rivals that sell identically the same product. In this case, the demand for the firm’s product is perfectly elastic (|εx,x| = ∞) and the demand curve horizontal. In this case, a manager that increases the price even slightly will discover that consumers will pur-chase nothing at all as they switch to the cheaper, identical products of rival firms. By contrast, when the demand for a firm’s good is perfectly inelastic (|εx,x| = 0), consumers do not alter their purchases at all in response to a price change. In this case, demand curve is vertical.

Of course, the demand for a good or service is rarely perfectly elastic or perfectly inelastic. The demand curve for most goods and services is down-ward sloping. In the next section, we will learn about the important relation-ship between the price elasticity of demand and the firm’s total revenue from

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64 Chapter 3

sales of its product. In Chapter 11, we will discuss how this information can be used to formulate an optimal pricing strategy.

Proportion of Income

When the purchase of a good constitutes a relatively large proportion of a person’s total expenditures increases, an increase in the price of a big-ticket item, such as a washer dryer or automobile, than an equivalent increase in the price of a good that constitutes a relatively small percentage of a family’s income, such as an ounce of table salt. In 2013, the median per family income in the U.S was around $51,000. Members of this group will sit up and take notice of a 10 percent increase in the price of a $20,000 Honda Civic, will hardly bat an eye-lash at an equivalent increase in the $1.40 price of a 2 liter bottle of Coke Classic.

Adjustment Period

In general, consumers tend to be less price sensitive in the short run than in the long run. For many goods and services it takes time for individuals and families to adjust their budgets expenditures to a price change. Consider, for example, an increase in the price of gasoline from, say, $3.50 to $5.00 per gallon. In the short run, many drivers have no choice but to pay the higher price since automobiles are essential to many aspects of a family’s daily lives, such as getting the kids to school, driving to work, grocery shopping, and so on. In the short run, many drivers may be able to make modest adjustments in their driving patterns to reduce the number of miles driven, perhaps by combining trips to the super market with the daily commute to work. Over a longer period of time, however, some people will trade in their gas guzzlers for more fuel efficient substitutes. The fact that consumer purchases are sensitive to the adjustment period also helps to explain why airlines charge higher fares for tickets purchased the day before a scheduled flight than when a ticket is purchased two months in advance.

Total and Marginal Revenue

The price elasticity of demand is a valuable business tool because it enables a manager to predict how a price will affect unit sales and revenues. Suppose, for example, that a manager is contemplating a 10 percent price increase and the demand for the firm’s product is price inelastic. If the price elasticity of

demand is εx,x = %∆Qx/%∆Px = −1/2, the result will be a 5 percent decline in unit sales. The effect on total revenues can be can be decomposed into two

effects. On the one hand, the price increase will push up revenues by about 10 percent. On the other hand, the decline in unit sales will push down rev-enues by about 5 percent. The net effect is an increase in total revenues of around 5 percent.

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Elasticity

Suppose, on the other hand, that demand is price elastic and εx,x = −2. In this case, a 10 percent increase in price will increase revenues by about 10 percent, but the fall in sales will reduce revenues by about 20 percent. In this case, the net effect is a decline in total revenues of around 10 percent. Finally, if demand is unit elastic, a 10 percent increase in price will increase total reve-nues by about 10 percent, which will be exactly offset by a 10 percent decline in revenues from lost unit sales. The net effect is no change in total revenues.

To make the discussion more concrete, suppose that the demand for good x is given by the equation

Q x = 80 −10Px .

(3.16)

Table 3.1 summarizes the price elasticity of demand and total revenue (TRx = Px Qx) for alternative price quantity combinations. At a price of $6, for example, the quantity demanded is 20 units. At this price-quantity combina-tion the point price elasticity of demand is −3 and the firm’s total revenue is $6(20) = $120. The price elasticity of demand tells us that in the neighbor-hood of this price-quantity combination, a 1 percent decline in price will result in 3 percent increase in the quantity demanded and an increase in total revenue. Suppose that the firm lowers price to $5, which increases sales to 30 units and total revenue to $5(30) = $150. At this price-quantity combina-tion the point price elasticity of demand is εx,x = −1.67. The price reduction from $6 to $5 results in a decline of total revenues of −$1 × 20 = −$20. The increase in sales at the lower price, however, results in increase in revenues of $5(10) = $50.

A $1 price reduction when demand is price elastic results in an increase in total revenue. What happens when price is reduced by $1 when demand is price inelastic? Suppose, for example, that the firm initially charges a price of $3 and sells 50 units. At this price-quantity combination the point price elasticity of demand is ε x,x = −0.6 and total revenue is $3(50) = $150. If the firm cuts price to $2, sales will increases to 60 units. At this price-quantity

Table 3.1

P	x	Q	x	∆Q	/∆P	x	P	/Q d	ε	x,x	Demand is	MR	TR
				x			x	x
0		80	−10		0		0		Perfectly inelastic	–8	0
1		70	−10		0.014	−0.14	Inelastic	–6	70
2		60	−10		0.033	−0.33	Inelastic	–4	120
3		50	−10		0.060	−0.60	Inelastic	–2	150
4		40	−10		0.100	−1.00	Unit elastic	0	160
5		30	−10		0.167	−1.67	Elastic	2	150
6		20	−10		0.300	−3.00	Elastic	4	120
7		10	−10		0.700	−7.00	Elastic	6	70
8		0	−10		∞		−∞		Perfectly elastic	8	0

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66 Chapter 3

combination the point price elasticity of demand is εx,x = −0.33 and total revenue is $2(60) = $120. Reducing price by $1 causes total revenue to decline from $150 to $120. The price reduction lowers total revenue by −$1(50) = −$5. The increase in revenues from greater units sales is only $2(10) = $20. Lowering price by $1 when demand is price elastic causes revenues to decline by $30.

The above example highlights several important features that are common to all linear demand curves. Although the slope of this linear demand curve is the same at every price-quantity combination, the price elasticity of demand can take on any value between 0 and −∞. Demand is perfectly inelastic (εx,x = 0) where the demand curve intersects the quantity (horizontal) axis. The demand curve is perfectly elastic (εx,x = −∞) where the demand curve inter-sects the price (vertical) axis. At the midpoint of the demand curve, demand is unit elastic (εx,x = −1). Demand is price elastic for any price-quantity com-bination above the midpoint. Finally, demand is price inelastic for any price quantity combination below the midpoint.

The relationship between total revenues and price elasticity for a linear demand curve depicted is summarized in Figure 3.5 and Table 3.2. Total rev-enue is maximized when demand is unit elastic, which occurs at the midpoint of the demand curve. As price is lowered (raised) in the elastic region, the quantity demanded increases (falls) and total revenue increases (decreases). When the price is lowered (raised) in the inelastic region, the quantity demanded increases (decreases) and total revenue falls (rises).

Figure 3.5 summarizes the relationships among the price elasticity of demand, total revenue and marginal revenue (MR), which is the change in total revenue (∆TR) from a change in the number of units sold. MR is positive for all price-quantity combinations along the elastic portion of the demand curve. Thus, lowering price will result in an increase in sales and revenues. MR is negative for all price-quantity combinations along the inelastic region of the demand curve. Reducing price in this region will result in an increase in sales, but a decrease in revenues.2

Formal Relationship between the Price Elasticity of Demand and Total Revenue

The tight relationship between price, the price elasticity of demand, total revenue and marginal revenue is summarized by the equation

			1		1 + ex , x
MR = Px 1	+			= Px				.	(3.17)
		e				e
							x , x
			x , x

We know that total revenue is maximized where MR = 0. Since Px > 0, the term in the parenthesis of Eq. (3.17) must be equal to zero, which occurs

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Elasticity

Figure 3.5

Table 3.2

\|εx,x\|	∆Px	∆Qx	∆TRx /∆Q = MRx
>1	−	+	+
>1	+	−	−
<1	−	+	−
<1	+	−	+
=1	−	+	−
=1	+	−	−

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68 Chapter 3

when εx,x = –1. When demand is elastic, (εx,x < −1) the term in the parenthe-sis and marginal revenue must be positive. Thus, lowering price increases

total revenue increases. When demand is inelastic (−1 < εx,x < 0), the term in parenthesis and marginal revenue must be negative. Lowering price reduces total revenue. These relationships are also summarized in Figure 3.5 and Table 3.2.

Solved Exercise

Suppose that the demand for a firm’s product is given by the equation

Qx = 50 − 2.5Px .

(3.18)

Calculate the point price elasticity of demand and total revenue for Px = $0, P x = $5, Px = $10, Px = $15, and Px = $20. What can you conclude about the relationship between the price elasticity of demand and total revenue?

Solution

Total revenue is TR = Px Qx. From Eq. (3.18) we get

ex , x			P
	= −2.5		x	.	(3.19)
		50
			− 2.5Px

Eq. (3.19) can be used to calculate the price elasticity of demand for each price. These calculations are summarized in Table 3.3.

According to Table 3.3, total revenue is zero when demand is perfectly elastic or perfectly inelastic. When demand is inelastic, total revenue increases (decreases) when price is raised (lowered). When demand is elas-tic, total revenue increases (decreases) when price is lowered (raised). When demand is unit elastic, total revenue is maximized at $250.

Table 3.3

Px	Qx	TR	εx,x	Demand is
0	50	0	0	Perfectly inelastic
5	37.5	187.5	−0.33	Inelastic
10	25	250	−1.00	Unit elastic
15	12.5	187.5	−3.00	Elastic
20	0	0	−∞	Perfectly elastic

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Elasticity

INCOME ELASTICITY OF DEMAND

There are other important elasticity measures that are of concern to managers. The income elasticity of demand is a measure of consumer sensitivity to changes in money income, which reflects the ups and downs of the business cycle. The income elasticity of demand is the percent change in demand with respect to a percent change in money income, that is

Ex , M	=	%ΔQx	.	(3.20)

		%ΔM

As with the price elasticity of demand, how we estimate the income elastic-ity of demand depends on the information available. With pairs of values for income and quantity, the income elasticity of demand may be calculated as

		Q2 −Q1	M1+M2		ΔQx	M1+M2
Ex , M	=			=				.
		M2−M1	Q1 +Q2				Q1 +Q2
					ΔM

The corresponding point income elasticity of demand is

		ΔQd	M		M
e x , M	=	x				= bM		.
				d			d
		ΔM Qx	Qx

(3.21)

(3.22)

We learned that the demand for a normal good varies directly with con-sumers’ money income, in which case the value of βM is positive. Since money income and quantity must are also positive, for a normal good εx,M > 0, which says that a percent increase (decrease) in money income is accompanied by a percent increase (decrease) in demand. Normal goods may be further classi-fied as necessities or luxuries. The demand for a necessity is relatively insen-sitive to income changes. The percent change in the demand for a good is less than the percent change in money income, that is, 0 < εx,M < 1. Examples of necessities include electricity, rent, food, and health care. By contrast, the demand for luxuries tends to exaggerate swings consumers’ money income. A good is a luxury good when the percent change in demand is greater than the percent change in money income, that is εx,M > 1. Examples of luxuries include foreign travel, restaurant meals, and foreign imports.

Since the demand for an inferior good is inversely related to a consumer’s money income, the value of β M in Eq. (2.1) is negative, thus εx,M < 0. Inferior goods are fairly common for individual consumers, but difficult to identify at the market level. An oft cited example of an inferior good at the market level is the demand for bankruptcy services, which tends to increase during recessions.

In Chapter 2 we discussed how a change in a consumer’s real income. The income effect

the price of a good affects for normal goods is closely

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70 Chapter 3

related to the price elasticity of demand. Purchases of goods that are sensitive to price changes tend to be luxuries. Purchases of goods that are not sensitive to price changes tend to be necessities.

CROSS-PRICE ELASTICITY OF DEMAND

Another frequently used elasticity measure is the cross-price elasticity of demand, which measures the sensitivity of consumer purchases of a prod-uct with respect to a change in the price of a related good. Related goods in consumption may be complements or substitutes. The cross-price elasticity of demand measures the percent change in the demand for good x given a percent change in the price of related good y, that is

%ΔQ

ΔQ

(3.23)

e x , y

= by

%ΔP

ΔP

The cross-price elasticity of demand takes its sign from the value of βy. When εx,y > 0, goods x and y are substitutes. When εx,y < 0, the two goods are complements.

Up to this point, we have assumed that a firm produces a single product. Now, suppose that the firm produces multiple related products. What effect will a change in the price of one good have on a firm’s total revenues? Sup-pose, for example, that a company sells two goods, x and y. The company’s total revenue from the sale of these two products is

TR = TRx + TRy ,

(3.24)

where TRx = PxQx and TRy = PyQy. For very small changes in the price of good x, the change in the firm’s total revenue given is given by the equation

		%ΔPx
ΔTR = TRx (1	+ e x , x ) + TRy ( ey , x )	×		.	(3.25)
			100%

Solved Exercise

Suppose that a firm earns $5,000 a month from the sale of good x and $3,000 from the sale of good y. The price-elasticity of demand for good x

is εx,x = –2 and the cross-price elasticity of demand for good y is εy,x = –4. How will a one percent cut in the price of good x affect the firm’s total

revenue?

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			Elasticity						71
Solution
From Eq. (3.26) we obtain
ΔTR = $5,000 1 − 2	]	+ $3,000	(	−4	(	−0.01	= $170.	(3.26)
	[				)		)

This result says that a one percent cut in the price of good x will result in a $170 increase in total revenues.

ADVERTISING ELASTICITY OF DEMAND

A very useful management tool is the advertising elasticity of demand, which measures the percent change in unit sales arising from a percent change in advertising expenditures. Ceteris paribus, we would expect that an increase in advertising expenditures will increase in unit sales and revenues. Symbolically, the advertising elasticity of demand is

		%ΔQx		ΔQx	A
ex , A	=		=			.	(3.27)
		%ΔA
				ΔA Qx

Solved Exercise

A business consultant has estimated the following demand function for Rubi-con & Styx’s world-famous hot sauce “Sergeant Garcia’s Revenge.”

Qx = 62 − 2Px + 0.2M + 25A,

(3.28)

where Qx is the quantity demanded per month in thousands of units, Px is the price per 6-oz. bottle, M is an index of consumer income, and A is the com-pany’s monthly advertising expenditures per month in thousands of dollars. Assume that Px = $4, M = $150, and A = $4 (thousand).

a. Calculate the number of bottles of “Sergeant Garcia’s Revenge” demanded.

b. Calculate the price elasticity of demand. According to your calculations, is the demand for this product elastic, inelastic, or unit elastic? What, if anything, can you say about the demand for this product?

c. Calculate the income elasticity of demand. Is “Sergeant Garcia’s Revenge” a normal good or an inferior good? Is it a luxury or a necessity?

d. Calculate the advertising elasticity of demand. Explain your result.

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72 Chapter 3

Solution

a. Substituting the given information into the demand function we get

Qx = 62 − 2 ( 4) + 0.2 (150) + 25( 2) = 184 thousand.

(3.29)

b. The price elasticity of demand is

ΔQx

= −2

= −0.04.

ex , x

(3.30)

ΔPx

184

This result tells us that a 1 percent increase in the price of “Sergeant Garcia’s Revenge” results in a 0.04 percent decrease in quantity demanded. Since |εx,x| < 1, the demand for this product is price inelastic. This might suggest that “Sergeant Garcia’s Revenge” has no close substitutes.

c. The income elasticity of demand is

ΔQx

150

ex , M

= 0.2

= 0.16,

(3.31)

184

ΔM Qx

which says that a 1 percent increase in consumer income results in a 0.16 percent increase in the demand. Since 0 < εx,M < 1, this normal good is a necessity.

d. The advertising elasticity of demand is given as

ΔQx

ex , A

= 25

= 0.54.

(3.32)

184

ΔA Qx

This result tells us that a 1 percent increase in Rubicon & Styx’s advertis-ing expenditures results in a 0.54 percent increase in unit sales.

CHAPTER EXERCISES

3.1 Suppose that you are a portfolio manager for a large, diversified mutual fund. The fund’s chief economist is forecasting an economic slowdown. How will your knowledge of estimated income elasticities of demand to alter the composition of the portfolio?

3.2 Suppose that crime is positively related to profits from illegal drug sales and that the demand for drugs is price inelastic. The government’s primary weapon in the war on drugs is the interdiction of illegal drugs flowing into the country from outside its borders. What is the likely effect of this policy on domestic crime rates? Can you suggest an alter-native approach to the war on drugs?

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Elasticity

3.3 Explain why a monopolist would never price its product along the inelastic portion of a linear demand curve.

3.4 A consortium of the world’s leading oil producers form a cartel to control the supply of crude oil. The objective of the cartel is to raise price and increase revenues. What must be true about the demand for oil for this policy to be successful?

3.5 A consortium of the world’s leading coffee bean producers form a cartel to the supply of coffee beans. The objective of the cartel is to lower prices and increase cartel revenues. What must be true about the demand for coffee beans for this policy to be successful?

3.6 The Sylvan Corporation produces wood sorrels. The price elasticity of demand is −0.25.

a. What will happen to the quantity demanded if Sylvan raises its price by 10 percent?

b. What will happen to Sylvan’s revenues following this price increase?

3.7 Suppose that the cross-price elasticity of demand for good x is 2.5 and the price of good y increases by 25 percent.

a. How would you characterize the relationship between goods x and y?

b. How will the increase in the price of good y affect unit sales of good x?

3.8 Suppose that the cross-price elasticity of demand for good m is −1.75 and the price of good n falls by 10 percent.

a. What is the relationship between goods m and good n?

b. How will the fall in the price of good n affect sales of good m?

3.9 Suppose that the income elasticity of demand for a good is 3.5.

a. How would you classify this good?

b. What increase in income is required for demand to increase by 21 percent?

3.10 Suppose that the estimated demand equation for a firm’s good is

Q x = 100 −10Px − 2Py + 0.1M + 0.2 A,

where Qxd is unit sales of good x, Px is the price of good x, Py is the price of good y, M is per-family money income in thousands of dollars, and A is the level of advertising expenditures in thousands of dollars. Suppose that Px = $2, Py = $3, M = $50 (thousand), and A = $20 (thousand). Calculate the price elasticity of demand.

3.11 Silkwood Enterprises specializes in gardening supplies. The demand for its new brand of fertilizer is given by the equation

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74 Chapter 3

Q = 120 − 4P.

a. Silkwood is currently charging $10 a pound. What is the price elasticity of demand?

b. At this price, what is Silkwood’s marginal revenue?

c. What price should Silkwood charge if it wants to maximize total revenue?

d. What is the price elasticity of demand at the revenue maximizing price?

3.12 Just-the-Fax, Max, Inc. has determined that the demand for its FAX machines is

Q = 3,000 −1.5P.

a. Calculate the point price elasticity of demand when P = $600.

b. At P = $600 what is Max’s marginal revenue?

c. Determine the total revenue maximizing price and quantity for the firm.

3.13 The market research department of Paradox Enterprises has deter-mined that the demand for its product is

Q x = 1, 000 − 5Px + 0.05M − 50Pz ,

where Px is the price, M median family income, and Pz is the price of bailiwicks. Suppose that Px = $5, M = $20,000, and Pz = $15. a. What is the price elasticity of demand?

b. Is the firm maximizing its total revenue at Px = $5. If not, what price should Paradox charge?

c. Calculate the income elasticity of demand at Px = $5.

d. Calculate the cross-price elasticity of demand at Px = $5.

3.14 Suppose that the demand equation for a firm’s product is

Q = 10 − 0.4P,

where Q is quantity and P is price.

a. Calculate the price elasticity of demand using the midpoint formula when P1 = $13 and P2 = $12.

b. Calculate the point price elasticity of demand at these prices. What, if anything, can you say about the relationship between the price elasticity of demand and total revenue at these prices?

c. What is the price elasticity of demand at the price that maximizes total revenue?

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Elasticity

3.15 Suppose that the demand equation for widgets is

Q = 10, 000 − 25P.

a. How many widgets will be sold for $100?

b. At what price will the demand for widgets zero?

c. What is the total revenue equation for widgets in terms of output? What is the marginal revenue equation in terms of output?

d. What is the price elasticity of demand when P = $200? What is the firm’s total and marginal revenue at this price?

e. What is the price elasticity of demand if the price of widgets falls to P = $150? At this price what is the firm’s total and marginal revenue? Explain your results.

f. What is the price elasticity of demand if the price of widgets rises to P = $250? At this price what is the firm’s total and marginal revenue? Explain your results.

g. Suppose that the supply of widgets is given by the equation

Q = −5, 000 + 50P.

What is the equilibrium price and quantity?

h. What is the relationship between quantity supplied and quantity demanded at P = $300?

3.16 The demand for a product is given by the equation

Q =50−2P.

a. What is the point price elasticity of demand at P = $20?

b. If the price falls to P = $15, what happens to total revenues? What does this imply about the price elasticity of demand?

c. Verify your answer to part b by computing the arc price elasticity over this price interval.

d. What, if anything, can you say about the relationship between the point price-elasticities of demand calculated in parts a and b, and the arc price elasticity of demand calculated in part c?

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