Income elasticity and cross elasticity of demand – Elasticity (Senior 4 Economics) | Ganzaa

Elasticity

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Senior 4 · Economics

Elasticity

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Lesson 3 of 4

Income elasticity and cross elasticity of demand

30 min

Who owns a television?

NISR's household survey of 2024 (EICV7) ranked Rwandan households into five equal groups, from the poorest fifth to the richest, by consumption per adult. In the poorest fifth, 74.1% owned a cellphone but only 2.0% a television; in the richest fifth, 95.2% owned a cellphone and 39.7% a television. As households become better off, they buy a little more of some things and a great deal more of others. Income elasticity of demand measures that difference. A second measure, cross elasticity, shows how the demand for one good responds to the price of another, such as bus tickets when moto fares rise.

Definition

Income elasticity of demand (YED): the responsiveness of quantity demanded to a change in consumers' income, ceteris paribus:

\text{YED} = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in income}}

Its sign matters: positive for normal goods, negative for inferior goods.

1. Types of income elasticity

YED	Type of good	Meaning	Example
$\text{YED} \lt 0$	Inferior good	Demand falls as income rises	A cheap staple replaced by preferred foods (illustrative)
$\text{YED} = 0$	Income has no effect	Demand does not change with income	Salt
$0 \lt \text{YED} \lt 1$	Normal necessity (income inelastic)	Demand rises, but less than in proportion to income	Basic food, soap, a basic phone
$\text{YED} \gt 1$	Normal luxury (income elastic)	Demand rises more than in proportion to income	Televisions, computers, travel abroad

Worked example 1: calculating YED (illustrative household)

A household's monthly income rises from Frw 200,000 to Frw 240,000, a rise of $\frac{40\,000}{200\,000} \times 100\% = 20\%$ . Its monthly purchases change as follows.

Good	Before	After	% change in Q	YED	Type
Meat	4 kg	5 kg	+25%	$\frac{25}{20} = 1.25$	Luxury
Beans	10 kg	11 kg	+10%	$\frac{10}{20} = 0.5$	Necessity
Cassava flour	8 kg	7 kg	$-12.5\%$	$\frac{-12.5}{20} = -0.625$	Inferior

2. What the survey data show

Reading the chart. The richest fifth are only about 1.3 times as likely as the poorest fifth to own a cellphone ( $\frac{95.2}{74.1} \approx 1.28$ ), but about 6 times as likely to own a smartphone ( $\frac{71.2}{11.9} \approx 5.98$ ) and about 20 times as likely to own a television ( $\frac{39.7}{2.0} \approx 19.9$ ). A basic cellphone behaves like a necessity; smartphones, televisions and computers behave like luxuries.

Limits of this evidence. These are shares of households owning one item, not quantities bought, and they compare different households at one moment rather than the same households as their income rises. Prices, electricity access and location (many televisions are in Kigali) also differ between the groups. The pattern points to high and low income elasticities but does not measure them exactly.

Definition

Cross elasticity of demand (XED): the responsiveness of the quantity demanded of good X to a change in the price of another good, Y, ceteris paribus:

\text{XED} = \frac{\%\ \text{change in quantity demanded of X}}{\%\ \text{change in price of Y}}

Substitutes: XED is positive (dearer moto rides, more bus passengers).
Complements: XED is negative (dearer fuel, fewer motorcycles bought).
Unrelated goods: XED is zero (the price of salt and the demand for school uniforms).

The larger the absolute value, the closer the relationship.

Worked example 2: calculating XED (illustrative)

(a) Moto fares on a route rise from Frw 1,000 to Frw 1,200, and daily bus passengers on the same route rise from 5,000 to 5,400.

\text{XED} = \frac{\frac{400}{5\,000} \times 100\%}{\frac{200}{1\,000} \times 100\%} = \frac{8\%}{20\%} = +0.4

Positive, so the two are substitutes, but weak ones: many riders keep using motos for door-to-door speed.

(b) The pump price of petrol rises by 25%, and motorcycle sales fall from 400 to 360 a month ( $- 10 %$ ).

\text{XED} = \frac{-10\%}{+25\%} = -0.4

Negative, so motorcycles and petrol are complements.

3. Uses of YED and XED

Measure	Who uses it	How
YED	Firms planning output	As incomes grow, demand for income-elastic goods grows fastest, so firms invest there. NST2 aims to raise income per person from USD 1,040 in 2023 to USD 1,369 by 2029 (MINECOFIN), about 31.6% higher. A good with a YED of 1.5 would see demand grow by roughly $1.5 \times 31.6\% \approx 47\%$ , ceteris paribus.
YED	Government	Predicting tax revenue and planning infrastructure as incomes grow; deciding which industries to support
YED	Firms in a recession	Firms selling inferior goods may gain when incomes fall
XED	Firms	Pricing against rivals (substitutes) and pricing complements together (a cheap printer, dear ink)
XED	Competition authorities	Deciding which firms are in the same market before allowing a merger: a high positive XED means close competitors
XED	Government	Predicting how a tax on one good shifts demand to its substitutes

Warning

Do not drop the sign. For PED we often quote the absolute value, but for YED and XED the sign is the whole point: it tells you whether a good is normal or inferior, and whether two goods are substitutes or complements. "XED = 0.4" and "XED = −0.4" describe opposite relationships.

Tip

Signs to remember: YED negative = inferior; YED between 0 and 1 = necessity; YED above 1 = luxury. XED plus = substitutes ("plus, swap"); XED minus = complements ("minus, match").

Important

Discuss. If Rwanda's incomes rise as NST2 plans, which industries in Rwanda should grow fastest, and which might shrink? Use income elasticity and the EICV7 chart in your answer.

EconplusDal, "Income Elasticity of Demand (YED)". Watch for the three income categories (inferior, necessity, luxury) and the signs, then compare them with the Engel curves above.

✦

Practice

Define income elasticity of demand and cross elasticity of demand.
A family's income rises by 10% and its spending on restaurant meals rises by 30%. Calculate YED and classify the good.
When incomes rise from Frw 150,000 to Frw 180,000 a month, a household's purchases of sorghum fall from 6 kg to 5 kg. Calculate YED and classify the good.
The price of tea rises by 15% and the quantity of coffee demanded rises by 6%. Calculate XED and state the relationship.
The price of airtime falls by 20% and sales of mobile phones rise by 5%. Calculate XED and state the relationship.
Using the EICV7 data, explain why a television can be described as a luxury in Rwanda, and give two reasons why the data do not prove this.
Explain two uses of cross elasticity of demand to a firm.
"As Rwanda grows richer, agriculture's share of output will keep falling." Use income elasticity to explain why.

Show answersHide answers

YED: % change in quantity demanded ÷ % change in income. XED: % change in quantity demanded of X ÷ % change in price of Y.
$\frac{30}{10} = 3$ : a luxury (income elastic).
Income +20%; quantity $\frac{-1}{6} \approx -16.7\%$ ; YED $\approx \frac{-16.7}{20} \approx -0.83$ : an inferior good for this household.
$\frac{6}{15} = +0.4$ : substitutes.
$\frac{+5}{-20} = -0.25$ : complements.
Ownership rises from 2.0% in the poorest fifth to 39.7% in the richest, far faster than for cellphones. But ownership is not quantity bought; the groups differ in electricity access, location and prices; and the data compare different households, not the same household over time.
Any two: pricing against substitutes (a rival's price cut will hurt more if XED is high); pricing complements together; deciding which rivals or products to watch; arguing about mergers.
Food has a low income elasticity (a necessity), while manufactured goods and services such as travel, banking and entertainment have high income elasticities. As incomes rise, spending grows fastest on the latter, so their share of output rises and agriculture's falls. NISR's 2025 figures already show services at 52% of GDP.

Quick check

Match each elasticity value to what it tells you.

Tap one on the left, then its partner on the right.

Key points

YED = % change in quantity demanded ÷ % change in income: negative for inferior goods, 0 to 1 for necessities, above 1 for luxuries.
XED = % change in quantity of X ÷ % change in price of Y: positive for substitutes, negative for complements, zero for unrelated goods.
NISR's EICV7 (2024) shows ownership of TVs and computers rising far faster with living standards than ownership of basic cellphones.
Firms use YED to plan for growing incomes and XED to price against rivals and complements; authorities use XED to define markets.

Sources and credits

Ownership of durable goods by quintile: EICV7 Poverty Profile, Table 4.6, National Institute of Statistics of Rwanda (NISR), 2025. Ratios calculated by Ganzaa.
Income per person targets: National Strategy for Transformation (NST2) 2024–2029, abridged version, MINECOFIN.
Photo: Buses at Nyabugogo, Francisco Anzola, via Wikimedia Commons, CC BY 2.0.
Video: Y1 14) Income Elasticity of Demand (YED), EconplusDal, embedded from YouTube.

Written by Ganzaa: the text, the Engel-curve diagram, the bar chart (drawn from NISR's published table), the worked examples and the practice items. The household, moto and petrol figures are illustrative.

1Who owns a television?

21. Types of income elasticity

32. What the survey data show

43. Uses of YED and XED

Who owns a television?

1. Types of income elasticity

2. What the survey data show

3. Uses of YED and XED