Income Elasticity and Cross-Price Elasticity (Normal, Inferior, Substitutes)

Say you get a raise. You eat out more, and you buy fewer packs of instant noodles. Module 2.1 called noodles an inferior good for you. But how inferior? And could you prove it with numbers? One number measures it, and its sign proves what kind of good it is. This is Module 2.5, income elasticity and cross-price elasticity.

In Module 2.3, price elasticity measured how strongly buyers react when a good's own price changes.

Quick refresher from Module 2.3: price elasticity of demand = % change in quantity demanded ÷ % change in price. Price up 10%, quantity down 20%: -20 ÷ 10 = -2. Drop the minus: its absolute value, 2, is bigger than 1, so demand is elastic.

But price is not the only thing that moves buyers. Elasticity can measure the response to anything that affects demand or supply. The recipe stays the same: percent change in quantity on top, percent change in whatever caused it on the bottom.

First cause: income. Income is one of the limits every buyer faces, so when it changes, buying changes. Income elasticity of demand = % change in quantity demanded ÷ % change in consumers' income. If the answer is positive, quantity moves the same way as income. That's a normal good: more money, more of it. If the answer is negative, quantity moves the opposite way. That's an inferior good: more money, less of it.

Quick refresher from Module 2.1: if the good's own price changes, you slide along its demand curve, so quantity demanded changes. If a TRIBE shifter changes, like income, the whole curve shifts, so demand changes.

TRIBE is tastes, prices of related goods, income, number of buyers, and expectations. A change in sellers' costs or technology is not on that list: it shifts supply, not demand.

On a graph, a raise shifts demand for a normal good to the right, and demand for an inferior good to the left.

Let's work one. Incomes in a town rise 8%. The quantity of instant noodles demanded falls 4%.

Quick refresher: percent change = change ÷ starting value × 100. Income goes from $500 to $540 a week: $40 ÷ $500 = 8%. Noodle sales go from 50 cases to 48, a change of -2: -2 ÷ 50 = -4%.

Formula: % change in quantity ÷ % change in income. Plug in: -4% ÷ +8%. That's -0.5. There are no units, because a percent divided by a percent is just a number. The minus sign proves noodles are inferior here. The 0.5 answers how inferior: each 1% rise in income cuts noodle buying by about half a percent.

So what just happened with the noodles? One: incomes rose 8%. Two: noodle buying fell 4%. Three: -4 ÷ 8 = -0.5. The minus sign means inferior, and the 0.5 says how strongly.

Same town, restaurant meals: quantity rises 12%. 12% ÷ 8% = +1.5. Positive, so restaurant meals are a normal good. Above 1, so buying grows faster than income.

So what happened with restaurant meals? One: the same 8% raise. Two: buying rose 12%. Three: 12 ÷ 8 = +1.5. Positive means normal, and above 1 means buying grew faster than income.

Given raw numbers instead of percents? Use the Module 2.3 rules: turn each change into a percent first, then divide.

Exam trap number one: inferior does not mean low quality, and it does not mean bad. It only describes behavior: when incomes rise, people buy less of it. Those noodles could taste great. If people who get a raise ride the bus less and drive more, bus rides are an inferior good for them. The word is about the math, not the product.

Second cause: the price of a different good. Cross-price elasticity of demand = % change in quantity demanded of one good ÷ % change in the price of another good. Positive means substitutes, goods you use in place of each other. The price of one and the quantity of the other move the same way.

Quick refresher before the numbers: percent change = change ÷ starting value. If butter goes from $4 to $4.40, that's $0.40 ÷ $4 = 10%.

Butter's price rises 10%, so margarine demand rises, and margarine buying rises 4%: 4 ÷ 10 = +0.4. Negative means complements, goods used together: the price of one and the quantity of the other move opposite ways. Consoles cost 10% more, so game buying falls 5%: -5 ÷ 10 = -0.5.

So what just happened? One: butter up 10%, margarine buying up 4%: 4 ÷ 10 = +0.4, substitutes. Two: consoles up 10%, game buying down 5%: -5 ÷ 10 = -0.5, complements. The sign names the pair.

Close to zero means not related: pencil prices don't change how much pizza you buy. On a graph, when the other good's price rises, this good's demand shifts right for a substitute and left for a complement.

So what just happened on the graphs? One: a raise shifts demand right for a normal good, left for an inferior one. Two: a pricier other good shifts demand right for a substitute, left for a complement.

Exam trap number two: using the wrong formula. The top is always a percent change in quantity demanded, never a plain change in quantity. For these three, only the bottom changes. Its own price: price elasticity. Income: income elasticity. Another good's price: cross-price elasticity. So before you divide, ask: what changed first? That goes on the bottom. And never flip it. If one good's price rises 5% and the other's quantity rises 10%, it's 10 ÷ 5 = +2. Flipped, you'd get +0.5, which is wrong.

Swap the story, price up 10% and quantity up 5%, and now +0.5 is right. Where each number sits decides it, not its size.

So what just happened? One: price up 5%, quantity up 10%, so 10 ÷ 5 = +2. Two: price up 10%, quantity up 5%, so 5 ÷ 10 = +0.5. Three: quantity goes on top, whatever the size.

Exam trap number three: the sign. With price elasticity of demand, you usually drop the minus sign. Here, never drop it. Negative income elasticity means inferior. Negative cross-price elasticity means complements. Match the sign to the formula you used. And only an income change can tell you normal or inferior. A negative price elasticity of demand doesn't make a good inferior. Almost every good has one. The size tells you how strong the reaction is. The sign tells you what kind. On the exam, give one value, sign included.

Quick check. Good X has a price elasticity of demand of -1.5 and an income elasticity of +0.8. Good X is: A) an inferior good B) a normal good C) a complement D) a substitute

B. Only the income elasticity decides normal or inferior. It's +0.8, positive, so X is normal. A, inferior, is the trap. The -1.5 is a price elasticity, and it's negative for almost every good. It can't tell you normal or inferior.

Next. A 5% rise in the price of good Y causes a 20% fall in the quantity demanded of good Z. The cross-price elasticity is: A) -0.25, complements B) +4, substitutes C) -4, complements D) -4, substitutes

C. Quantity of Z on top, price of Y on the bottom: -20 ÷ 5 = -4. Negative means complements. A, -0.25, is the trap: it flips the formula. Here the right answer is the bigger number, because quantity goes on top.

Recap: income elasticity, plus is normal and minus is inferior. Cross-price, plus is substitutes, minus is complements, near zero is unrelated. Percent change in quantity on top, and keep the sign. Next: Module 2.6, Consumer Surplus, Producer Surplus, and Market Equilibrium.