Discussion

Chapter 4: Measure of Response: Elasticities from Microeconomics: Markets, Methods & Models by Douglas Curtis and Ian Irvine is available under a Creative Commons Attribution-NonCommercial- ShareAlike 3.0 Unported license. © Lyryx Learning Inc.

Chapter 4

Measures of response: elasticities

In this chapter we will explore:

4.1 Responsiveness as elasticities

4.2 Demand elasticities and expenditure

4.3 The short run, the long run and inflation

4.4 Cross-price and income elasticities

4.5 Income elasticity of demand

4.6 Supply side responses

4.7 Tax incidence

4.8 Identifying the elasticity

4.1 Price responsiveness of demand

Put yourself in the position of an entrepreneur. One of your many challenges is to price your

product appropriately. You may be Michael Dell choosing a price for your latest computer, or

the local restaurant owner pricing your table d’hôte, or you may be pricing your part-time snow-

shoveling service. A key component of the pricing decision is to know how responsive your market

is to variations in your pricing. How we measure responsiveness is the subject matter of this

chapter.

We begin by analyzing the responsiveness of consumers to price changes. For example, consumers

tend not to buy much more or much less food in response to changes in the general price level of

food. This is because food is a pretty basic item for our existence. In contrast, if the price of

textbooks becomes higher, students may decide to search for a second-hand copy, or make do with

lecture notes from their friends or downloads from the course web site. In the latter case students

have ready alternatives to the new text book, and so their expenditure patterns can be expected to

reflect these options, whereas it is hard to find alternatives to food. In the case of food consumers

are not very responsive to price changes; in the case of textbooks they are. The word ‘elasticity’

89

90 Measures of response: elasticities

that appears in this chapter title is just another term for this concept of responsiveness. Elasticity

has many different uses and interpretations, and indeed more than one way of being measured in

any given situation. Let us start by developing a suitable numerical measure.

The slope of the demand curve suggests itself as one measure of responsiveness: If we lowered

the price of a good by $1, for example, how many more units would we sell? The difficulty with

this measure is that it does not serve us well when comparing different products. One dollar may

be a substantial part of the price of your morning coffee and croissant, but not very important if

buying a computer or tablet. Accordingly, when goods and services are measured in different units

(croissants versus tablets), or when their prices are very different, it is often best to use a percentage

change measure, which is unit-free.

The price elasticity of demand is measured as the percentage change in quantity demanded, di-

vided by the percentage change in price. Although we introduce several other elasticity measures

later, when economists speak of the demand elasticity they invariably mean the price elasticity of

demand defined in this way.

The price elasticity of demand is measured as the percentage change in quantity demanded,

divided by the percentage change in price.

The price elasticity of demand can be written in different forms. We will use the Greek letter

epsilon, ε , as a shorthand symbol, with a subscript d to denote demand, and the capital delta, ∆, to denote a change. Therefore, we can write

Price elasticity of demand = Percentage change in quantity demanded

Percentage change in price

εd = %∆Q

%∆P (4.1a)

= ∆Q/Q

∆P/P (4.1b)

= ∆Q

∆P ×

P

Q (4.1c)

Calculating the value of the elasticity is not difficult. If we are told that a 10 percent price increase

reduces the quantity demanded by 20 percent, then the elasticity value is

εd = %∆Q

%∆P =

−20% 10%

=−2

The negative sign denotes that price and quantity move in opposite directions, but for brevity the

4.1. Price responsiveness of demand 91

negative sign is often omitted.

Consider now the data in Table 4.1 and the accompanying Figure 4.1. This data reflect the demand

equation for natural gas that we introduced in Chapter 3: P= 10−Q. Note first that, when the price and quantity change, we must decide what reference price and quantity to use in the percentage

change calculation in Equation 4.1. We could use the initial or final price-quantity combination, or

an average of the two. Each choice will yield a slightly different numerical value for the elasticity.

The best convention is to use the midpoint of the price values and the corresponding midpoint of

the quantity values. This ensures that the elasticity value is the same regardless of whether we start

at the higher price or the lower price. Using the subscript 1 to denote the initial value and 2 the

final value:

Average quantity = (Q1 +Q2)/2

Average price = (P1 +P2)/2

Price ($) Quantity Price elasticity Price elasticity Total

demanded (arc) (point) revenue ($)

(thousands

of cu ft.)

10.00 0 -9.0 −∞

8.00 2 -2.33 -4 16

6.00 4 -1.22 -1.5 24

5.00 5 -0.82 -1 25

4.00 6 -0.43 -0.67 24

2.00 8 -0.11 -0.25 16

0.00 10 0 0

Table 4.1: The demand for natural gas: elasticities and revenue

92 Measures of response: elasticities

P0 = 10 ε =−9

High elasticity range (elastic)

Low elasticity range (inelastic)

ε =−0.11

Q0=10

Price

Quantity

8

5

5

2

Mid point of D: ε =−1

Figure 4.1: Elasticity variation with linear demand

In the high-price region of the demand curve the elasticity takes on a high

value. At the mid-point of a linear demand curve the elasticity takes on a

value of one, and at lower prices the elasticity value continues to fall.

Using this rule, consider now the value of εd when price drops from $10.00 to $8.00. The change in price is $2.00 and the average price is therefore $9.00 [= ($10.00 + $8.00)/2]. On the quantity

side, demand goes from zero to 2 units (measured in thousands of cubic feet), and the average

quantity demanded is therefore (0 + 2)/2 = 1. Putting these numbers into the formula yields:

εd = %∆Q

%∆P =

−(2/1) (2/9)

=−9

Note that the price has declined in this instance and thus ∆P is negative. Continuing down the table in this fashion yields the full set of elasticity values in the third column.

The demand elasticity is said to be high if it is a large negative number; the large number denotes

a high degree of sensitivity. Conversely, the elasticity is low if it is a small negative number. High

and low refer to the size of the number, ignoring the negative sign. The term arc elasticity is also

used to define what we have just measured, indicating that it defines consumer responsiveness over

a segment or arc of the demand curve.

The arc elasticity of demand defines consumer responsiveness over a segment or arc of the

demand curve.

It is helpful to analyze this numerical example by means of the corresponding demand curve that

is plotted in Figure 4.1. It is a straight-line demand curve; but, despite this, the elasticity is not

4.1. Price responsiveness of demand 93

constant. At high prices the elasticity is high; at low prices it is low. The intuition behind this

pattern is as follows: When the price is high, a given price change represents a small percentage

change, whereas the resulting percentage quantity change will be large. The large percentage

quantity change results from the fact that, at the high price, the quantity consumed is small, and,

therefore, a small number goes into the denominator of the percentage quantity change. In contrast,

when we move to a lower price range on the demand function, a given absolute price change is

large in percentage terms, and the resulting quantity change is smaller in percentage terms.

Extreme cases

The elasticity decreases in going from high prices to low prices. This is true for most non-linear

demand curves also. Two exceptions are when the demand curve is horizontal and when it is

vertical.

When the demand curve is vertical, no quantity change results from a change in price from P1 to

P2, as illustrated in Figure 4.2. Therefore, the numerator in Equation 4.1 is zero, and the elasticity

has a zero value.

Q0

Dv

P1 Dh D′

Price

Quantity

P2

Infinite elasticity

Large elasticity

Zero elasticity

Figure 4.2: Limiting cases of price elasticity

When the demand curve is vertical (Dv), the elasticity is zero: a change

in price from P1 to P2 has no impact on the quantity demanded because

the numerator in the elasticity formula has a zero value. When D becomes

more horizontal the elasticity becomes larger and larger at Q0, eventually

becoming infinite.

In the horizontal case, we say that the elasticity is infinite, which means that any percentage price

change brings forth an infinite quantity change! This case is also illustrated in Figure 4.2 using

the demand curve Dh. As with the vertical demand curve, this is not immediately obvious. So

consider a demand curve that is almost horizontal, such as D′ instead of Dh. In this instance, we

94 Measures of response: elasticities

can achieve large changes in quantity demanded by implementing very small price changes. In

terms of Equation 4.1, the numerator is large and the denominator small, giving rise to a large

elasticity. Now imagine that this demand curve becomes ever more elastic (horizontal). The same

quantity response can be obtained with a smaller price change, and hence the elasticity is larger.

Pursuing this idea, we can say that, as the demand curve becomes ever more elastic, the elasticity

value tends towards infinity.

Using information on the slope of the demand curve

The elasticity formula, Equation 4.1 part (c), indicates that we could also compute the elasticity

values using information on the slope of the demand curve, ∆Q/∆P, multiplied by the appropriate price-quantity ratio. (Note that, even though we put price on the vertical axis, the slope of the

demand curve is ∆Q/∆P, as explained in Chapter 3; ∆P/∆Q is the inverse of this slope, or the slope of the inverse demand function.) Consider the price change from $10.00 to $8.00 again.

Columns 1 and 2 indicate that ∆Q/∆P = -2/$2.00, or by simply looking at the equation for the demand curve we can see that its slope is -1. Choosing again the midpoint values for price and

quantity yields P/Q = $9.00/1. Therefore the elasticity is

εd = (∆Q/∆P) =−1× (9.00/1.00) =−9

Knowing the slope of the demand curve can be very useful in establishing elasticity values when

the demand curve is not linear, or when price changes are miniscule, or when the curve intersects

the axes. Let us consider each of these cases in turn.

A non-linear demand curve is illustrated in Figure 4.3. If price increases from P0 to P1, then over

that range we can approximate the slope by the ratio (P1 −P0)/(Q1 −Q0). This is, essentially, an average slope over the range in question that can be used in the formula, in conjunction with an

average price and quantity of these values.

4.1. Price responsiveness of demand 95

Price

Quantity

P2 C

Q2

P0 A

Q0

P1 B

Q1

Figure 4.3: Non-linear demand curves

When the demand curve is non-linear the slope changes with the price.

Hence, equal price changes do not lead to equal quantity changes: The

quantity change associated with a change in price from P0 to P1 is smaller

than the change in quantity associated with the same change in price from

P0 to P2.

When a price change is infinitesimally small the resulting estimate is called a point elasticity

of demand. This differs slightly from the elasticity in column 3 of Table 4.1. In that case, we

computed the elasticity along different segments or arcs of the demand function. In Table 4.1, the

point elasticity at the point P = $8.00 is

εd = (∆Q/∆P) =−1× (8.00/2.00) =−4

The first term in this expression states that quantity changes by 1 unit for each $1 change in price,

and the second term states that the elasticity is being evaluated at the price-quantity combination

P = $8 and Q = 2. The value of the point elasticity at each price value listed in Table 4.1 is given in column 4. The arc elasticity values in column 3 span a price range, whereas the point elasticities

correspond exactly to each price value.

The point elasticity of demand is the elasticity computed at a particular point on the demand

curve.

This point elasticity formula can also be applied to the non-linear demand curve in Figure 4.3.

If we wished to compute this elasticity exactly at P2, we could draw a tangent to the function at

C and evaluate its slope. This slope could then be used in conjunction with the price-quantity

combination (P2,Q2) to evaluate εd at that point.

96 Measures of response: elasticities

Next, note that when a demand curve intersects the horizontal axis the elasticity value is zero,

regardless of the slope. Using Figure 4.1, we can see that this is because the price in the P/Q component of the elasticity formula equals zero at the intersection point Q0. Hence P/Q = 0 and the elasticity is therefore zero. Likewise, when approaching an intersection with the vertical axis,

defined by the point P0 in Figure 4.1, the denominator in the P/Q component becomes very small, making the P/Q ratio very large. As we get ever closer to the vertical axis, this ratio becomes correspondingly larger, and therefore we say that the elasticity approaches infinity.

Elastic and inelastic demands

While the elasticity value falls as we move down the demand curve, an important dividing line

occurs at the value of -1. This is illustrated in Table 4.1, and is a property of all straight-line

demand curves. Disregarding the negative sign, demand is said to be elastic if the price elasticity

is greater than unity, and inelastic if the value lies between unity and 0. It is unit elastic if the

value is exactly one.

Demand is elastic if the price elasticity is greater than unity. It is inelastic if the value lies between

unity and 0. It is unit elastic if the value is exactly one.

Economists frequently talk of goods as having a “high” or “low” demand elasticity. What does this

mean, given that the elasticity varies throughout the length of a demand curve? It signifies that, at

the price usually charged, the elasticity has a high or low value. For example, your weekly demand

for coffee at Starbucks might be unresponsive to variations in price around the value of $2.00, but

if the price were $4, you might be more responsive to price variations. Likewise, when we stated

at the beginning of this chapter that the demand for food tends to be inelastic, we really mean that

at the price we customarily face for food, demand is inelastic.

Determinants of price elasticity

Why is it that the price elasticities for some goods and services are high and for others low? One

answer lies in tastes: If a good or service is a basic necessity in one’s life, then price variations

have minimal effect and these products have a relatively inelastic demand.

A second answer lies in the ease with which we can substitute alternative goods or services for

the product in question. The local music school may find that the demand for its instruction is

responsive to the price charged for lessons if there are many independent music teachers who can

be hired directly by the parents of aspiring musicians. If Apple had no serious competition, it could

price the products higher than in the presence of Samsung, Google etc. The ease with which we

can substitute other goods or services is a key determinant. It follows that a critical role for the

4.1. Price responsiveness of demand 97

marketing department in a firm is to convince buyers of the uniqueness of the firm’s product.

Where product groups are concerned, the price elasticity of demand for one product is necessarily

higher than for the group as a whole: Suppose the price of one tablet brand alone falls. Buyers

would be expected to substitute towards this product in large numbers – its manufacturer would

find demand to be highly responsive. But if all brands are reduced in price, the increase in demand

for any one will be more muted. In essence, the one tablet whose price falls has several close

substitutes, but tablets in the aggregate do not.

Finally, there is a time dimension to responsiveness, and this is explored in Section 4.3.

Using price elasticities

Knowledge of elasticity values is useful in calculating the price change required to eliminate a

shortage or surplus. For example, shifts in the supply of agricultural products can create surpluses

and shortages. Because of variations in weather conditions, crop yields cannot be forecast accu-

rately. In addition, on account of the low elasticity of demand for such products, low crop yields

can increase prices radically, and bumper harvests can have the opposite impact.

Consider Figure 4.4. Econometricians tell us that the demand for foodstuffs is inelastic, so let

us operate in the lower (inelastic) part of this demand, D. A change in supply conditions (e.g. a

shortage of rain and a poorer harvest) shifts the supply from S1 to S2 with the consequence that the

price increases from P1 to P2. In this illustration the price increase is substantial. In contrast, with

a relatively flat, or elastic, demand, D′, through the initial point A, the shift in the supply curve

has a more moderate impact on the price (from P1 to P3), but a relatively larger impact on quantity

traded.

98 Measures of response: elasticities

D

D′

S2 S1 Price

Quantity

P1

A

Q1

P2

Q2

P3

Q3

Figure 4.4: The impact of elasticity on quantity fluctuations

In the lower part of the demand curve D, where demand is inelastic, e.g.

point A, a shift in supply from S1 to S2 induces a large percentage increase

in price, and a small percentage decrease in quantity demanded. In contrast,

for the demand curve D′ that goes through the original equilibrium, the

region A is now an elastic region, and the impact of the supply shift is

contrary: the %∆P is smaller and the %∆Q is larger.

4.2 Price elasticity and expenditure

In Figure 4.5, we examine the expenditure or revenue impact of a price reduction in two ranges

of a linear demand curve. Expenditure, or revenue, is the product of price times quantity. It is,

therefore, the area of a rectangle in a price/quantity diagram. From position A, a price reduction

from PA to PB has two impacts. It reduces the revenue that accrues from those QA units already

being sold; the negative sign between PA and PB marks this reduction. But it increases revenue

through additional sales from QA to QB. The area marked with a positive sign between QA and QB denotes this increase. Will the extra revenue caused by the quantity increase outweigh the loss in

revenue associated with each unit sold before the price was reduced? It turns out that at high prices

the positive impact outweighs the negative impact. The intuitive reason is that the existing sales

are small and, therefore, we lose a revenue margin on a very limited quantity. The net impact on

total expenditure of the price reduction is positive.

4.2. Price elasticity and expenditure 99

Price

0 Quantity

PE E

QE

PC C

QC

PB B

QB

PA A

QA

Elastic range

Inelastic range

(−)

(+)

(−)

(+)

Figure 4.5: Price elasticity and revenue

When the price falls from PA to PB, expenditure changes from PAAQA0 to

PBBQB0. In this elastic region expenditure increases, because the loss in

revenue on existing units (−) is less than the revenue gain (+) due to the additional units sold. The opposite occurs in the inelastic region CE.

In contrast, move now to point C and consider a further price reduction from PC to PE . There is

again a dual impact: a loss of revenue on existing sales, and a gain due to additional sales. But in

this instance the existing sales QC are large, and therefore the loss of a price margin on these sales

is more significant than the extra revenue that is generated by the additional sales. The net effect is

that total expenditure falls.

So if revenue increases in response to price declines at high prices, and falls at low prices, there

must be an intermediate region of the demand curve where the composite effects of the price

change just offset each other, and no change in revenue results from a price change. It transpires

that this occurs at the midpoint of the linear demand curve. Let us confirm this with the help of our

example in Table 4.1.

The fourth column of the table contains the point elasticities of demand, and the final column

defines the expenditure on the good at the corresponding prices. Point elasticities are very precise;

they are measured at a point rather than over a range or an arc. Note next that the point on this linear

demand curve where revenue is a maximum corresponds to its midpoint—where the elasticity is

unity. This is no coincidence. Price reductions increase revenue so long as demand is elastic,

but as soon as demand becomes inelastic such price declines reduce revenues. When does the

value become inelastic? Clearly, where the unit elasticity value is crossed. This is illustrated in

Figure 4.6, which defines the relationship between total revenue (T R), or total expenditure, and

quantity sold in Table 4.1. Total revenue increases initially with quantity, and this increasing

quantity of sales comes about as a result of lower prices. At a quantity of 5 units the price is $5.00.

This price-quantity combination corresponds to the mid-point of the demand curve.

100 Measures of response: elasticities

Revenue

Quantity

Rev= $16

8

Rev= $25

5

Revenue a maximum where elasticity is unity

Figure 4.6: Total revenue and elasticity

Based upon the data in Table 4.1, revenue increases with quantity sold up to

sales of 5 units. Beyond this output, the decline in price that must accom-

pany additional sales causes revenue to decline.

We now have a general conclusion: In order to maximize the possible revenue from the sale of a

good or service, it should be priced where the demand elasticity is unity.

Does this conclusion mean that every entrepreneur tries to find this magic region of the demand

curve in pricing her product? Not necessarily: Most businesses seek to maximize their profit rather

than their revenue, and so they have to focus on cost in addition to sales. We will examine this

interaction in later chapters. Secondly, not every firm has control over the price they charge; the

price corresponding to the unit elasticity may be too high relative to their competitors’ choices of

price. Nonetheless, many firms, especially in the early phase of their life-cycle, focus on revenue

growth rather than profit, and so, if they have any power over their price, the choice of the unit-

elastic price may be appropriate.

Elasticity values are sometimes more informative than diagrams and figures. To see why, consider

Figure 4.4 again. Since the demand curve, D, has a “vertical” profile, we tend to think of such a

demand as being less elastic than one with a more “horizontal” profile, D′. But that demand curve

could be redrawn with the scale of one or both of the axes changed. By using bigger spacing for

quantity units (or smaller spacing for the pricing units), a demand curve with a vertical profile could

be transformed into one with a horizontal profile! But elasticity calculations do not deceive. The

numerical values are always independent of how we mark off units in a diagram. Consequently,

when we see a demand curve with a vertical profile, we can indeed say that it is less elastic than

a “flatter” demand curve in the same region of the figure. But we cannot form such a conclusion

when comparing demand curves for different goods with different units and scales. The beauty of

elasticity lies in its honesty!

4.3. The time horizon and inflation 101

4.3 The time horizon and inflation

The price elasticity of demand is frequently lower in the short run than in the long run. For example,

a rise in the price of home heating oil may ultimately induce consumers to switch to natural gas or

electricity, but such a transition may require a considerable amount of time. Time is required for

decision-making and investment in new heating equipment. A further example is the elasticity of

demand for tobacco. Some adults who smoke may be seriously dependent and find quitting almost

impossible. But if young smokers, who are not yet addicted, decide to quit on account of the higher

price, then over a long period of time the percentage of the population that smokes will decline.

The full impact may take decades! So when we talk of the short run and the long run, there is no

simple rule for defining how long the long run actually is in terms of months or years. In some

cases, adjustment may be complete in weeks, in other cases years.

In Chapter 2 we distinguished between real and nominal variables. The former adjust for inflation;

the latter do not. Suppose all nominal variables double in value: Every good and service costs

twice as much, wage rates double, dividends and rent double, etc. This implies that whatever

bundle of goods was previously affordable is still affordable. Nothing has really changed. Demand

behaviour is unaltered by this doubling of all prices and all incomes.

How do we reconcile this with the idea that own-price elasticities measure changes in quantity

demanded as prices change? Keep in mind that elasticities measure the impact of changing one

variable alone, holding constant all of the others. But when all variables are changing simultane-

ously, it is incorrect to think that the impact on quantity of one price or income change is a true

measure of responsiveness or elasticity. The price changes that go into measuring elasticities are

therefore changes in relative prices.

4.4 Cross-price elasticities

The price elasticity of demand tells us about consumer responses to price changes in different

regions of the demand curve, holding constant all other influences. One of those influences is the

price of other goods and services. A cross-price elasticity indicates how demand is influenced by

changes in the prices of other products.

The cross-price elasticity of demand is the percentage change in the quantity demanded of a

product divided by the percentage change in the price of another.

In mathematical form we write the cross price elasticity of the demand for x due to a change in the

price of y as

102 Measures of response: elasticities

εd(x,y) = %∆Qx %∆Py

.

For example, if the price of cable-supply TV services declines, by how much will the demand for

satellite-supply TV services change? The cross-price elasticity may be positive or negative. When

the price of movie theatre tickets rises, the demand for Home Box Office movies rises, and vice

versa. In this example, we are measuring the cross-price elasticity of demand for HBO movies with

respect to the price of theatre tickets. These goods are clearly substitutable, and this is reflected in

a positive value of this cross-price elasticity: The percentage change in video rentals is positive in

response to the increase in movie theatre prices. The numerator and denominator in the equation

above have the same sign.

Suppose that, in addition to going to fewer movies, we also eat less frequently in the restaurant

beside the movie theatre. In this case, the cross-price elasticity relating the demand for restaurant

meals to the price of movies is negative—an increase in movie prices reduces the demand for

meals. The numerator and denominator in the cross-price elasticity equation are opposite in sign.

In this instance, the goods are complements.

4.5 The income elasticity of demand

In Chapter 3 we stated that higher incomes tend to increase the quantity demanded at any price. To

measure the responsiveness of demand to income changes, a unit-free measure exists: the income

elasticity of demand. The income elasticity of demand is the percentage change in quantity

demanded divided by a percentage change in income.

The income elasticity of demand is the percentage change in quantity demanded divided by a

percentage change in income.

Let us use the Greek letter eta, η , to define the income elasticity of demand and I to denote income. Then,

ηd = %∆Q

%∆I

As an example, if monthly income increases by 10 percent, and the quantity of magazines pur-

chased increases by 15 percent, then the income elasticity of demand for magazines is 1.5 in value

(= 15%/10%). The income elasticity is generally positive, but not always – let us see why.

4.5. The income elasticity of demand 103

Normal, inferior, necessary, and luxury goods

The income elasticity of demand, in diagrammatic terms, is a percentage measure of how far the

demand curve shifts in response to a change in income. Figure 4.7 shows two possible shifts.

Suppose the demand curve is initially the one defined by D, and then income increases. If the

demand curve shifts to D1 as a result, the change in quantity demanded at the existing price is

(Q1−Q0). However, if instead the demand curve shifts to D2, that shift denotes a larger change in quantity (Q2 −Q0). Since the shift in demand denoted by D2 exceeds the shift to D1, the D2 shift is more responsive to income, and therefore implies a higher income elasticity.

P0

D D1 D2 Price

Quantity

A

Q0

B

Q1

C

Q2

Figure 4.7: Income elasticity and shifts in demand

At the price P0, the income elasticity measures the percentage horizontal

shift in demand caused by some percentage income increase. A shift from

A to B reflects a lower income elasticity than a shift to C. A leftward shift

in the demand curve in response to an income increase would denote a neg-

ative income elasticity – an inferior good.

In this example, the good is a normal good, as defined in Chapter 3, because the demand for it

increases in response to income increases. If the demand curve were to shift back to the left in

response to an increase in income, then the income elasticity would be negative. In such cases the

goods or services are inferior, as defined in Chapter 3.

Finally, we need to distinguish between luxuries, necessities, and inferior goods. A luxury good or

service is one whose income elasticity equals or exceeds unity. A necessity is one whose income

elasticity is greater than zero but less than unity. These elasticities can be understood with the

help of Equation 4.1 part (a). If quantity demanded is so responsive to an income increase that

the percentage increase in quantity demanded exceeds the percentage increase in income, then the

value is in excess of 1, and the good or service is called a luxury. In contrast, if the percentage

change in quantity demanded is less than the percentage increase in income, the value is less than

unity, and we call the good or service a necessity.

104 Measures of response: elasticities

A luxury good or service is one whose income elasticity equals or exceeds unity.

A necessity is one whose income elasticity is greater than zero and less than unity.

Luxuries and necessities can also be defined in terms of their share of a typical budget. An income

elasticity greater than unity means that the share of an individual’s budget being allocated to the

product is increasing. In contrast, if the elasticity is less than unity, the budget share is falling.

This makes intuitive sense—luxury cars are luxury goods by this definition because they take up a

larger share of the incomes of the rich than of the poor.

Inferior goods are those for which there exist higher-quality, more expensive, substitutes. For

example, lower-income households tend to satisfy their travel needs by using public transit. As

income rises, households normally reduce their reliance on public transit in favour of automobile

use. Inferior goods, therefore, have a negative income elasticity: in the income elasticity equation

definition, the numerator has a sign opposite to that of the denominator. As an example: in the

recession of 2008/09 McDonalds continued to remain profitable and increased its customer base –

in contrast to the more up-market Starbucks. This is a case where expenditure increased following

a decline in income, yielding a negative income elasticity of demand.

Inferior goods have negative income elasticity.

Lastly, note that while inferior products may be considered a special type of necessity, inferior

goods technically have a negative income elasticity, whereas necessities have positive elasticity

values.

Empirical research indicates that goods like food and fuel have income elasticities less than 1;

durable goods and services have elasticities slightly greater than 1; leisure goods and foreign holi-

days have elasticities very much greater than 1.

Income elasticities are useful in forecasting the demand for particular services and goods in a

growing economy. Suppose real income is forecast to grow by 15 percent over the next five years. If

we know that the income elasticity of demand for iPhones is 2.0, we could estimate the anticipated

growth in demand by using the income elasticity formula: since in this case η = 2.0 and %∆I = 15 it follows that 2.0 = %∆Q/15%. Therefore the predicted demand change must be 30%.

4.6 Elasticity of supply

Now that we have developed the various dimensions of elasticity on the demand side, the anal-

ysis of elasticities on the supply side is straightforward. The elasticity of supply measures the

4.7. Elasticities and tax incidence 105

responsiveness of the quantity supplied to a change in the price.

The elasticity of supply measures the responsiveness of quantity supplied to a change in the

price.

εs = %∆Q

%∆P

The subscript s denotes supply. This is exactly the same formula as for the demand curve, except

that the quantities now come from a supply curve. Furthermore, and in contrast to the demand

elasticity, the supply elasticity is generally a positive value because of the positive relationship

between price and quantity supplied. The more elastic, or the more responsive, is supply to a

given price change, the larger will be the elasticity value. In diagrammatic terms, this means

that “flatter” supply curves have a greater elasticity than more “vertical” curves at a given price

and quantity combination. Numerically the flatter curve has a larger value than the more vertical

supply – try drawing a supply diagram similar to Figure 4.2. Technically, a completely vertical

supply curve has a zero elasticity and a horizontal supply curve has an infinite elasticity – just as

in the demand cases.

As always we keep in mind the danger of interpreting too much about the value of this elasticity

from looking at the visual profiles of supply curves.

4.7 Elasticities and tax incidence

Elasticity values are critical in determining the impact of a government’s taxation policies. The

spending and taxing activities of the government influence the use of the economy’s resources. By

taxing cigarettes, alcohol and fuel, the government can restrict their use; by taxing income, the

government influences the amount of time people choose to work. Taxes have a major impact on

almost every sector of the Canadian economy.

To illustrate the role played by demand and supply elasticities in tax analysis, we take the example

of a sales tax. These can be of the specific or ad valorem type. A specific tax involves a fixed dollar

levy per unit of a good sold (e.g., $10 per airport departure). An ad valorem tax is a percentage

levy, such as Canada’s Goods and Services tax (e.g., 5 percent on top of the retail price of goods

and services). The impact of each type of tax is similar, and we will use the specific tax in our

example below.

A layperson’s view of a sales tax is that the tax is borne by the consumer. That is to say, if no sales

tax were imposed on the good or service in question, the price paid by the consumer would be the

106 Measures of response: elasticities

same net of tax price as exists when the tax is in place. Interestingly, this is not always the case.

The study of the incidence of taxes is the study of who really bears the tax burden, and this in turn

depends upon supply and demand elasticities.

Tax Incidence describes how the burden of a tax is shared between buyer and seller.

Consider Figures 4.8 and 4.9, which define an imaginary market for inexpensive wine. Let us

suppose that, without a tax, the equilibrium price of a bottle of wine is $5, and Q0 is the equilibrium

quantity traded. The pre-tax equilibrium is at the point A. The government now imposes a specific

tax of $4 per bottle. The impact of the tax is represented by an upward shift in supply of $4:

Regardless of the price that the consumer pays, $4 of that price must be remitted to the government.

As a consequence, the price paid to the supplier must be $4 less than the consumer price, and this is

represented by twin supply curves: one defines the price at which the supplier is willing to supply,

and the other is the tax-inclusive supply curve that the consumer faces.

D St

S

Price

Quantity

Pt = 8 B

Qt

P0 = 5 A

Q0

Pts = 4 C

$4=tax

Figure 4.8: Tax incidence with elastic supply

The imposition of a specific tax of $4 shifts the supply curve vertically by

$4. The final price at B (Pt) increases by $3 over the equilibrium price at

A. At the new quantity traded, Qt , the supplier gets $4 per unit (Pts), the

government gets $4 also and the consumer pays $8. The greater part of

the incidence is upon the buyer, on account of the relatively elastic supply

curve: his price increases by $3 of the $4 tax.

The introduction of the tax in Figure 4.8 means that consumers now face the supply curve St . The

new equilibrium is at point B. Note that the price has increased by less than the full amount of the

tax—in this example it has increased by $3. This is because the reduced quantity at B is provided

at a lower supply price: The supplier is willing to supply the quantity Qt at a price defined by C

($4), which is lower than A ($5).

4.7. Elasticities and tax incidence 107

So what is the incidence of the $4 tax? Since the market price has increased from $5 to $8, and the

price obtained by the supplier has fallen by $1, we say that the incidence of the tax falls mainly

on the consumer: the price to the consumer has risen by three dollars and the price received by the

supplier has fallen by just one dollar.

Consider now Figure 4.9, where the supply curve is less elastic, and the demand curve is un-

changed. Again the supply curve must shift upward with the imposition of the $4 specific tax.

But here the price received by the supplier is lower than in Figure 4.8, and the price paid by the

consumer does not rise as much – the incidence is different. The consumer faces a price increase

that is one-quarter, rather than three-quarters, of the tax value. The supplier faces a lower supply

price, and bears a higher share of the tax.

D

St S Price

Quantity

P0 = 5 A

Q0

Pt = 6 B

Qt

Pts = 2 C

$4=tax

Figure 4.9: Tax incidence with inelastic supply

The imposition of a specific tax of $4 shifts the supply curve vertically by

$4. The final price at B (Pt) increases by $1 over the no-tax price at A. At the

new quantity traded, Qt , the supplier gets $2 per unit (Pts), the government

gets $4 also and the consumer pays $6. The greater part of the incidence is

upon the supplier, on account of the relatively inelastic supply.

We can draw conclude from this example that, for any given demand, the more elastic is supply, the

greater is the price increase in response to a given tax. Furthermore, a more elastic supply curve

means that the incidence falls more on the consumer; while a less elastic supply curve means the

incidence falls more on the supplier. This conclusion can be verified by drawing a third version

of Figure 4.8 and 4.9, in which the supply curve is horizontal – perfectly elastic. When the tax is

imposed the price to the consumer increases by the full value of the tax, and the full incidence falls

on the buyer. While this case corresponds to the layperson’s intuition of the incidence of a tax,

economists recognize it as a special case of the more general outcome, where the incidence falls

on both the supply side and the demand side.

These are key results in the theory of taxation. It is equally the case that the incidence of the

tax depends upon the demand elasticity. In Figure 4.8 and 4.9 we used the same demand curve.

108 Measures of response: elasticities

However, it is not difficult to see that, if we were to redo the exercise with a demand curve of a

different elasticity, the incidence would not be identical. At the same time, the general result on

supply elasticities still holds. We will return to this material in Chapter 5.

Statutory incidence

In the above example the tax is analyzed by means of shifting the supply curve. This implies

that the supplier is obliged to charge the consumer a tax and then return this tax revenue to the

government. But suppose the supplier did not bear the obligation to collect the revenue; instead

the buyer is required to send the tax revenue to the government. If this were the case we could

analyze the impact of the tax by reducing the market demand curve by the $4. This is because

the demand curve reflects what consumers are willing to pay, and when suppliers are paid in the

presence of the tax they will be paid the buyers’ demand price minus the tax that the buyers must

pay. It is not difficult to show that whether we move the supply curve upward (to reflect the

responsibility of the supplier to pay the government) or move the demand curve downward, the

outcome is the same – in the sense that the same price and quantity will be traded in each case.

Furthermore the incidence of the tax, measured by how the price change is apportioned between

the buyers and sellers is also unchanged.

Tax revenues and tax rates

It is useful to relate elasticity values to the policy question of the impact of higher or lower taxes on

government tax revenue. Consider a situation in which a tax is already in place and the government

considers increasing the rate of tax. Can an understanding of elasticities inform us on the likely

outcome? The answer is yes. Suppose that at the initial tax inclusive price demand is inelastic. We

know immediately that a tax rate increase that increases the price must increase total expenditure.

Hence the outcome is that the government will get a higher share of an increased total expenditure.

In contrast, if demand is elastic at the initial tax-inclusive price a tax rate increase that leads to a

higher price will decrease total expenditure. In this case the government will get a larger share of

a smaller pie – not as valuable from a tax revenue standpoint as a larger share of a larger pie.

4.8 Identifying demand and supply elasticities

Elasticities are very useful pieces of evidence on economic behaviour. But we need to take care in

making inferences from what we observe in market data. Upon observing price and expenditure

changes in a given market, it is tempting to infer that we can immediately calculate a demand

elasticity. But should we be thinking about supply elasticities? Let us look at the information

needed before rushing into calculations.

4.8. Identifying demand and supply elasticities 109

In order to identify a demand elasticity we need to be sure that we have price and quantity values

that lie on the same demand curve. And if we do indeed observe several price and quantity pairs

that reflect a market equilibrium on a demand curve, then it must be the case that those combi-

nations are caused by a shifting supply curve. Consider Figure 4.10. Suppose that we observe a

series of prices and accompanying quantities traded in three consecutive months, and we plot these

combinations to yield points A, B, C in panel (a) of the figure. If these points are market equilibria,

and if they lie on the same demand curve, it must be the case that the supply curve has shifted. That

is, if we can draw a single demand curve through these points, as in panel (b), the only way that

they each reflect demand conditions is for the supply curve to have shifted to create these points as

equilibria in the market.

Sa Sb

Sc

D

Price

Quantity

Price

Quantity (a) (b)

A A

B B

C C

Figure 4.10: Identifying elasticities

In order to establish that points such as A, B and C in Panel (a) lie on the

same demand curve, we must know that the supply curve alone has shifted

in such a way as to result in these equilibrium price-quantity combinations,

as illustrated in Panel (b).

Exactly the same logic holds if we can infer that market equilibrium points all lie on the same

supply curve. In that case the demand curve must have shifted in order to be able to identify the

points as belonging to the supply curve.

This challenge is what we call the identification problem in econometrics. Frequently new combi-

nations of price and quantity reflect shifts in both the supply curve and demand curve, and we need

to call upon the econometricians to tell us what shifts are taking place in the market.

110 Key Terms

KEY TERMS

Price elasticity of demand is measured as the percentage change in quantity demanded, di-

vided by the percentage change in price.

Arc elasticity of demand defines consumer responsiveness over a segment or arc of the de-

mand curve.

Point elasticity of demand is the elasticity computed at a particular point on the demand

curve.

Demand is elastic if the price elasticity is greater than unity. It is inelastic if the value lies

between unity and 0. It is unit elastic if the value is exactly one.

Cross-price elasticity of demand is the percentage change in the quantity demanded of a

product divided by the percentage change in the price of another.

Income elasticity of demand is the percentage change in quantity demanded divided by a

percentage change in income.

Luxury good or service is one whose income elasticity equals or exceeds unity.

Necessity is one whose income elasticity is greater than zero and is less than unity.

Inferior goods have a negative income elasticity.

Elasticity of supply is defined as the percentage change in quantity supplied divided by the

percentage change in price.

Tax Incidence describes how the burden of a tax is shared between buyer and seller.

Exercises 111

EXERCISES FOR CHAPTER 4

Exercise 4.1. Consider the information in the table below that describes the demand for movie

rentals from your on-line supplier Instant Flicks.

Price per movie ($) Quantity demanded Total revenue Elasticity of demand

2 1200

3 1100

4 1000

5 900

6 800

7 700

8 600

(a) Either on graph paper or a spreadsheet, map out the demand curve.

(b) In column 3, insert the total revenue generated at each price.

(c) At what price is total revenue maximized?

(d) In column 4, compute the elasticity of demand corresponding to each $1 price reduction,

using the average price and quantity at each state.

(e) Do you see a connection between your answers in parts (c) and (d)?

Exercise 4.2. Your fruit stall has 100 ripe bananas that must be sold today. Your supply curve is

therefore vertical. From past experience, you know that these 100 bananas will all be sold if the

price is set at 40 cents per unit.

(a) Draw a supply and demand diagram illustrating the market equilibrium price and quantity.

(b) The demand elasticity is -0.5 at the equilibrium price. But you now discover that 10 of

your bananas are rotten and cannot be sold. Draw the new supply curve and calculate the

percentage price increase that will be associate with the new equilibrium, on the basis of

your knowledge of the demand elasticity.

Exercise 4.3. University fees in the State of Nirvana have been frozen in real terms for 10 years.

During this period enrolments increased by 20 percent.

112 Exercises

(a) Draw a supply curve and two demand curves to represent the two equilibria described.

(b) Can you estimate a price elasticity of demand for university education in this market?

(c) In contrast, during the same time period fees in a neighbouring state increased by 60 percent

and enrolments increased by 15 percent. Illustrate this situation in a diagram.

Exercise 4.4. Consider the demand curve defined by the information in the table below.

Price of movies Quantity demanded Total revenue Elasticity of demand

2 200

3 150

4 120

5 100

(a) Plot the demand curve to scale and note that it is non-linear.

(b) Compute the total revenue at each price.

(c) Compute the arc elasticity of demand for each price segment.

Exercise 4.5. The demand curve for seats at the Drive-in Delight Theatre is given by P = 48− 0.2Q. The supply of seats is given by Q = 40.

(a) Plot the supply and demand curves to scale, and estimate the equilibrium price.

(b) At this equilibrium point, calculate the elasticities of demand and supply.

(c) The owner has additional space in his theatre, and is considering the installation of more

seats. He then remembers from his days as an economics student that this addition might not

necessarily increase his total revenue. If he hired you as a consultant, would you recommend

to him that he install additional seats or that he take out some of the existing seats and install

a popcorn concession instead? [Hint: You can use your knowledge of the elasticities just

estimated to answer this question.]

(d) For this demand curve, over what range of prices is demand inelastic?

Exercise 4.6. Waterson Power Corporation’s regulator has just allowed a rate increase from 9 to

11 cents per kilowatt hour of electricity. The short run demand elasticity is -0.6 and the long run

demand elasticity is -1.2.

(a) What will be the percentage reduction in power demanded in the short run?

Exercises 113

(b) What will be the percentage reduction in power demanded in the long run?

(c) Will revenues increase or decrease in the short and long runs?

Exercise 4.7. Consider the own- and cross-price elasticity data in the table below.

% change in price

CDs Magazines Cappuccinos

CDs -0.25 0.06 0.01

Magazines -0.13 -1.20 0.27% change in quantity

Cappuccinos 0.07 0.41 -0.85

(a) For which of the goods is demand elastic and for which is it inelastic?

(b) What is the effect of an increase in the price of CDs on the purchase of magazines and

cappuccinos? What does this suggest about the relationship between CDs and these other

commodities; are they substitutes or complements?

(c) In graphical terms, if the price of CDs or the price of cappuccinos increases, illustrate how

the demand curve for magazines shifts.

Exercise 4.8. You are responsible for running the Speedy Bus Company and have information

about the elasticity of demand for bus travel: The own-price elasticity is -1.4 at the current price.

A friend who works in the competing railway company also tells you that she has estimated the

cross-price elasticity of train-travel demand with respect to the price of bus travel to be 1.7.

(a) As an economic analyst, would you advocate an increase or decrease in the price of bus

tickets if you wished to increase revenue for Speedy?

(b) Would your price decision have any impact on train ridership?

Exercise 4.9. A household’s income and restaurant visits are observed at different points in time.

The table below describes the pattern.

114 Exercises

Income ($) Restaurant visits Income elasticity of demand

16,000 10

24,000 15

32,000 18

40,000 20

48,000 22

56,000 23

64,000 24

(a) Construct a scatter diagram showing quantity on the vertical axis and income on the hori-

zontal axis.

(b) Is there a positive or negative relationship between these variables?

(c) Compute the income elasticity for each income increase, using midpoint values.

(d) Are restaurant meals a normal or inferior good?

Exercise 4.10. Consider the following three supply curves: P = 2.25Q; P = 2+2Q; P = 6+1.5Q.

(a) Draw each of these supply curves to scale, and check that, at P = $18, the quantity supplied in each case is the same.

(b) Calculate the (point) supply elasticity for each curve at this price.

(c) Now calculate the same elasticities at P = $12.

(d) One elasticity value should be unchanged. Which one?

Exercise 4.11. The demand for bags of candy is given by P = 48−0.2Q, and the supply by P = Q.

(a) Illustrate the resulting market equilibrium in a diagram.

(b) If the government now puts a $12 tax on all such candy bags, illustrate on a diagram how the

supply curve will change.

(c) Compute the new market equilibrium.

(d) Instead of the specific tax imposed in part (b), a percentage tax (ad valorem) equal to 30

percent is imposed. Illustrate how the supply curve would change.

(e) Compute the new equilibrium.

Exercises 115

Exercise 4.12. Consider the demand curve P = 100−2Q. The supply curve is given by P = 30.

(a) Draw the supply and demand curves to scale and compute the equilibrium price and quantity

in this market.

(b) If the government imposes a tax of $10 per unit, draw the new equilibrium and compute the

new quantity traded and the amount of tax revenue generated.

(c) Is demand elastic or inelastic in this price range?

Exercise 4.13. In Exercise 4.12: As an alternative to shifting the supply curve, try shifting the

demand curve to reflect the $10 tax being imposed on the consumer.

(a) Solve again for the price that the consumer pays, the price that the supplier receives and the

tax revenue generated.

(b) Compare your answers with the previous question; they should be the same.

Exercise 4.14. The supply of Henry’s hamburgers is given by P = 2+0.5Q; demand is given by Q = 20.

(a) Illustrate and compute the market equilibrium.

(b) A specific tax of $3 per unit is subsequently imposed and that shifts the supply curve to

P = 5+0.5Q. Solve for the equilibrium price and quantity after the tax.

(c) Who bears the burden of the tax in parts (a) and (b)?