Income and Cross Elasticity: Classify Goods With Confidence

You calculate an elasticity of -0.5. Does that mean the good is inferior, or that two goods are complements?

Both answers can be correct. The missing information is what changed in the denominator.

If income changed, you are investigating whether a good is normal or inferior. If another good’s price changed, you are investigating whether the two goods are substitutes or complements. The arithmetic looks similar, but the questions are different.

Let’s make that distinction feel natural, then work through the numbers without losing the signs.

Start with the variable that changed

Imagine a student buying drinks at a college canteen. Three separate events might change how many cups of tea she buys:

• Her monthly spending allowance increases.
• The price of coffee increases.
• The price of tea itself increases.

These are three different investigations. Treat each event separately, keeping other influences unchanged.

Cause being studied	Response measured	Elasticity needed
Consumer income changes	Quantity of tea demanded	Income elasticity
Coffee’s price changes	Quantity of tea demanded	Cross elasticity
Tea’s own price changes	Quantity of tea demanded	Price elasticity

Underline the changing variable in a numerical before writing a formula. This small habit prevents a surprising number of wrong classifications.

For the third case, our guide to price elasticity methods explains how to choose the appropriate calculation.

Income elasticity: how purchases respond to income

Income elasticity of demand measures the percentage response of quantity demanded to a percentage change in consumer income, with other relevant factors held constant.

Ey = percentage change in quantity demanded / percentage change in income

Here, Y stands for income and Q stands for quantity. Using initial values as the percentage bases:

Ey = [(Q1 - Q0) / Q0] / [(Y1 - Y0) / Y0]

The zeros mark the original values; the ones mark the new values. Multiplying both fractions by 100 produces percentages, but the two factors of 100 cancel in the ratio.

The formula asks how strongly purchases respond relative to the income change. It does not compare a number of units directly with a number of rupees.

What does the sign tell you?

Income elasticity	Classification	Meaning when income rises
Ey < 0	Inferior good	Quantity demanded falls
0 < Ey < 1	Normal good, usually called a necessity	Quantity rises proportionately less
Ey = 1	Normal good with unit income elasticity	Quantity rises by the same percentage
Ey > 1	Normal good, usually called a luxury	Quantity rises proportionately more
Ey = 0	Income-independent over the range studied	Quantity does not change

These categories describe behaviour in a particular setting. The USDA Economic Research Service elasticity glossary explains the distinction between responsiveness and the positive or negative income relationship.

A normal good is not necessarily a necessity. Both a necessity with Ey = 0.4 and a luxury with Ey = 1.8 have positive income elasticity.

Nor does “inferior” mean unsafe, badly made, or unworthy. A consumer may buy fewer low-cost bus journeys after an income increase because she can now afford a different travel option. For another consumer, bus journeys may rise with income. Let the stated behaviour decide the answer.

Example 1: a normal good that is a necessity

A household’s monthly income rises from Rs 40,000 to Rs 50,000. Its monthly purchase of a particular breakfast cereal rises from 20 packets to 22 packets. Assume prices and preferences remain unchanged.

Step 1: Find the quantity change.

Percentage change in Q = (22 - 20) / 20 x 100 = 10%.

Step 2: Find the income change.

Percentage change in Y = (50,000 - 40,000) / 40,000 x 100 = 25%.

Step 3: Divide and classify.

Ey = 10 / 25 = +0.4.

The cereal is a normal good for this household over the stated income range. Because the coefficient lies between zero and one, it is income inelastic and classified as a necessity in this example.

Notice the wording: a 25% income increase produced a 10% quantity increase. We are not claiming that every household treats this cereal in the same way.

Example 2: an inferior good

Monthly income increases from Rs 30,000 to Rs 36,000. Purchases of a particular low-cost ready meal fall from 25 packs to 20 packs per month.

• Percentage change in Q = (20 - 25) / 25 x 100 = -20%.
• Percentage change in Y = (36,000 - 30,000) / 30,000 x 100 = +20%.
• Ey = -20 / 20 = -1.

The negative sign establishes an inferior good. Do not place -1 in the “unit income elasticity of a normal good” row: that row requires +1.

In magnitude, the response is one-for-one here, but its direction is opposite to the income change.

Example 3: falling income does not automatically mean an inferior good

Income falls from Rs 50,000 to Rs 45,000. Monthly spending occasions at a particular restaurant fall from 10 visits to 8 visits, with menu prices unchanged.

Percentage change in Q = -2 / 10 x 100 = -20%.

Percentage change in Y = -5,000 / 50,000 x 100 = -10%.

Ey = -20 / -10 = +2.

Both variables fell, so the signs cancel. These restaurant visits are a normal good and income elastic in the example. Quantity changed by twice the percentage change in income.

Cross elasticity: how another price affects this demand

Cross elasticity of demand, also called cross-price elasticity, links the quantity demanded of one good to the price of another.

Exy = percentage change in quantity demanded of X / percentage change in price of Y

With initial values as bases:

Exy = [(Qx1 - Qx0) / Qx0] / [(Py1 - Py0) / Py0]

In this article, Exy means the response of X’s quantity to Y’s price. Always state that relationship because textbooks may use different notation.

The own price of X, consumer income and other relevant influences are held constant. The OpenStax lesson on elasticity beyond own price sets out both income and cross-price measures.

Cross elasticity	Relationship	If Y becomes more expensive
Exy > 0	Substitutes	Demand for X rises
Exy < 0	Complements	Demand for X falls
Exy = 0	No measured cross-price response	Demand for X is unchanged

Substitutes offer alternative ways to satisfy a want. Complements are used together. Those descriptions help you anticipate the sign, but the actual response in the question is your evidence.

Example 4: substitute drinks

The price of drink Y rises from Rs 40 to Rs 50. Weekly quantity demanded of drink X rises from 100 cups to 120 cups. X’s price remains unchanged.

• Percentage change in Qx = 20 / 100 x 100 = +20%.
• Percentage change in Py = 10 / 40 x 100 = +25%.
• Exy = 20 / 25 = +0.8.

The positive coefficient indicates substitutes. A rise in Y’s price has shifted some demand towards X.

A complete interpretation is: a 25% increase in Y’s price is associated with a 20% increase in X’s quantity demanded, other things unchanged.

Do not write “X is a normal good”. Nothing here tells us how X responds to income.

Example 5: complementary purchases

The price of a particular printer rises from Rs 8,000 to Rs 10,000. Over the stated period, demand for its compatible ink packs falls from 200 to 170 packs. Assume the question isolates the printer-price effect.

Percentage change in ink quantity = -30 / 200 x 100 = -15%.

Percentage change in printer price = 2,000 / 8,000 x 100 = +25%.

Cross elasticity = -15 / 25 = -0.6.

The goods are complements in this example. The answer does not show that ink is inferior: the denominator contains the printer’s price, not income.

In a real market, replacement cycles and existing printer ownership would also matter. A classroom example holds those other influences fixed so that you can study one relationship clearly.

Example 6: a price fall with substitutes

The price of snack Y falls by 12%. Demand for snack X falls by 9%, while X’s own price stays constant.

Exy = -9 / -12 = +0.75.

These goods are substitutes. Y becomes cheaper, and some consumers switch away from X.

This is why “quantity fell, so the answer must be negative” is unreliable. You need the sign of the cause as well as the response.

The same good can carry two different labels

Suppose a consumer buys more of drink X after an income rise. In a separate observation, she also buys more X when drink Y becomes more expensive.

X can be normal with respect to income and a substitute for Y at the same time. There is no contradiction.

The first label describes X’s relationship with income. The second describes its relationship with another product’s price.

A product’s classification can also change across consumers or income ranges. The discussion of normal and inferior goods in NCERT’s Theory of Consumer Behaviour is useful background for understanding why the consumer and circumstances matter.

Initial-value percentages or the midpoint method?

All worked examples above use initial values. If a question supplies percentages, divide those signed percentages directly. If it specifies the midpoint or arc method, use averages as the bases instead.

For income elasticity under the midpoint method:

Ey = [(Q1 - Q0) / average Q] / [(Y1 - Y0) / average Y]

For cross elasticity, replace income with the other good’s price in the denominator.

The advantage is that reversing the journey between the same two observations leaves the elasticity unchanged. OpenStax explains the midpoint percentage method in its introductory elasticity lesson.

Example 7: the method changes the number

Income rises from Rs 20,000 to Rs 30,000, and quantity rises from 10 to 14 units.

Using initial values:

• Quantity change = 4 / 10 x 100 = 40%.
• Income change = 10,000 / 20,000 x 100 = 50%.
• Ey = 40 / 50 = 0.8.

Using midpoint values:

• Average quantity = (10 + 14) / 2 = 12.
• Average income = (20,000 + 30,000) / 2 = Rs 25,000.
• Ey = (4 / 12) / (10,000 / 25,000) = 5/6, or approximately 0.833.

Both calculations are internally consistent. They answer using different percentage bases. In an examination, follow the requested method or the convention taught for that exercise, and show your bases clearly.

Never calculate the numerator from an initial value and the denominator from an average value.

Finding an unknown quantity

You can rearrange the formula when the coefficient is supplied:

Percentage change in Q = elasticity x percentage change in the causal variable.

For a classroom exercise using a stated constant coefficient over the given change, apply the resulting percentage to the initial quantity. In real forecasting, applying a local elasticity to a large change is only an approximation unless the demand relationship justifies it.

Example 8: work backwards from income elasticity

Ey is -0.5. Income increases by 10%, and the initial quantity demanded is 80 units. Find the new quantity under the stated percentage method.

Percentage change in Q = -0.5 x 10% = -5%.

Change in quantity = -5% of 80 = -4 units.

New quantity = 80 - 4 = 76 units.

The direction makes sense: an income increase reduces demand for this inferior good.

For a cross-elasticity version, suppose Exy = +1.2, Y’s price rises by 5%, and initial demand for X is 200 units. The expected quantity change is +6%, or 12 units, giving 212 units under the same method.

What happens to the demand curve?

On an ordinary demand diagram for X, the vertical axis shows X’s own price and the horizontal axis shows quantity of X.

An income change or a change in Y’s price can shift X’s demand curve because neither is X’s own price. For example, higher income shifts demand for a normal good rightward; a dearer substitute can also shift it rightward. Higher income shifts demand for an inferior good leftward; a dearer complement can shift demand leftward.

If this distinction is still slippery, revisit movement along a demand curve versus a shift. Before drawing anything, label the price axis with the good’s name.

Mistakes to catch before submitting your answer

Dropping the negative sign. In some own-price exercises, students report absolute magnitude. Do not carry that habit into income or cross elasticity: the sign supplies the classification.

Dividing raw changes. A change of 4 packets divided by a change of Rs 10,000 is not an elasticity. Convert each change into a proportion or percentage of its appropriate base.

Using expenditure instead of quantity. A rise in the amount spent can reflect a price change. These formulae use quantity demanded. If a question gives expenditure, obtain quantity from expenditure divided by unit price when the necessary information is available.

Calling every positive answer “normal”. Positive income elasticity means normal; positive cross elasticity means substitutes.

Calling every normal good a necessity. Ey = +1.6 is normal and income elastic. The necessity range is 0 < Ey < 1.

Reading zero as a permanent universal fact. A zero response in an exercise means no response under its stated conditions. It does not establish that the goods can never be related or that income can never matter.

Assuming the reverse cross elasticity is identical. The response of X to Y’s price is not automatically the same number as the response of Y to X’s price. The numerator, denominator and percentage bases differ.

Ignoring other changes in real data. If income, prices and preferences all changed between two months, the raw ratio alone cannot isolate one causal effect. Classroom questions usually supply the other-things-equal assumption explicitly or implicitly.

Try these eight questions

Use initial-value percentages unless the question says otherwise. Assume other relevant influences are unchanged.

1. Income rises by 15% and quantity demanded rises by 6%. Calculate Ey and classify the good.
2. Income falls by 8% and quantity demanded rises by 4%. What is Ey?
3. Y’s price rises by 20% and demand for X falls by 5%. Calculate Exy and classify the relationship.
4. Y’s price falls by 10% and demand for X falls by 15%. What relationship does the answer indicate?
5. Income rises from Rs 24,000 to Rs 30,000. Quantity rises from 40 to 50 units. Find Ey.
6. Exy is -0.4. Y’s price rises by 15%, and initial demand for X is 500 units. Find the new quantity under the stated percentage method.
7. Income changes but quantity does not. What is Ey, assuming positive initial values and a nonzero income change?
8. Income rises from Rs 10,000 to Rs 15,000 and quantity rises from 20 to 30 units. Calculate income elasticity using midpoint percentages.

Answers with working

1. Ey = 6 / 15 = +0.4. Normal and income inelastic, classified as a necessity.
2. Ey = +4 / -8 = -0.5. Inferior: quantity and income moved in opposite directions.
3. Exy = -5 / 20 = -0.25. Complements.
4. Exy = -15 / -10 = +1.5. Substitutes. Both changes are negative, so the ratio is positive.
5. Income increased by 6,000 / 24,000 = 25%; quantity increased by 10 / 40 = 25%. Ey = +1, unit income elasticity.
6. Quantity change = -0.4 x 15% = -6%. Six percent of 500 is 30, so new quantity = 470 units.
7. Ey = 0. There is no quantity response over the stated change.
8. Average income is Rs 12,500 and average quantity is 25. Ey = (10 / 25) / (5,000 / 12,500) = 0.4 / 0.4 = +1.

When revising, cover the answers and say the classification aloud before calculating. Then check whether your arithmetic agrees with the direction you predicted.

Frequently asked questions

What is the main difference between income and cross elasticity?

Income elasticity studies a quantity response to consumer income. Cross elasticity studies a quantity response to another good’s price. The changed variable in the denominator identifies the measure.

Why must I retain the sign?

The sign shows whether the response and its cause move together or in opposite directions. Removing it erases the distinction between normal and inferior goods, or between substitutes and complements.

Can an inferior good have income elasticity of -2?

Yes. A negative coefficient establishes inferiority; its absolute size describes responsiveness. A value of -2 is possible and indicates a proportionately larger quantity response in the opposite direction.

Are luxury goods also normal goods?

Yes, in the income-elasticity classification. A luxury has Ey greater than one, which is positive. It therefore belongs within the broader normal-goods category.

Does negative cross elasticity mean that X is inferior?

No. It means that X and Y are complements in the relationship studied. To classify X as inferior, you need evidence about its response to income.

Can a good be normal and a complement at the same time?

Yes. Normal describes an income relationship, while complement describes a relationship with another good. Those labels answer different questions.

Does a larger positive cross elasticity suggest closer substitutes?

Other conditions being comparable, it indicates a stronger percentage response to the other price and can suggest closer substitution. Compare estimates for similar consumers, periods and methods rather than treating unrelated numbers as a universal ranking.

Is elasticity measured in rupees or units?

Neither. It is a ratio of percentage changes, so it has no unit. Write Ey = 0.4, not Ey = Rs 0.4 or Ey = 0.4%.

Which formula should I use when percentages are already given?

Divide the signed percentage quantity change by the signed percentage change in income or the related good’s price. There is no need to reconstruct original values.

What should the final sentence of a numerical answer say?

Name the coefficient, give its value and interpret it in context. For example: “Exy is +0.8, so X and Y are substitutes under the stated conditions.” That sentence connects the calculation to the economic idea.