5.3 Other Demand Elasticities
So far, we have focused on how much the quantity of a good demanded changes when its price changes. But what happens when the price of other goods changes? Or when your income changes? This is where the interdependence principle comes in—we need to consider how your decisions interact with each other.
We’ve learned one neat trick so far—how to measure the responsiveness of the quantity demanded to a change in the price. We can expand this idea, and measure the responsiveness of the quantity demanded to other factors, too.
Cross-Price Elasticity of Demand
Think back to Matilda’s demand for gas. When the price goes up, she buys less gas, forgoing a weekend road trip to spend time in town with friends instead. The interdependence principle reminds us to think about how her decision to buy less gas impacts her decisions in other markets. For instance, when she decides to buy less gas, she is also choosing to spend more time in town with friends. Spending more time in town means that her demand for public transportation increases. It’s quite typical to see that higher gas prices lead to increased demand for public transportation. Since, for many people, taking the bus is an alternative to driving.
The cross-price elasticity of demand measures how responsive the quantity demanded of one good is to price changes of another. Specifically, it measures the percent change in the quantity demanded following a 1% change in the price of another good. We measure the cross-price elasticity of demand as the ratio of the percent change in quantity demanded to the percent change in the price of another good. That is:
The sign of the cross-price elasticity of demand tells you something important: When it’s positive, it means that you buy more of a good when the price of another good goes up; when it’s negative, it means that you buy less. What determines whether you buy more or less? Whether the goods are substitutes or complements. Let’s see why.
The cross-price elasticity is positive for substitutes.
When goods are close substitutes, such as cars and public transportation, people tend to buy more of a good when the price of its substitute increases. Have you ever noticed how Coke and Pepsi tend to be priced the same? Imagine what would happen if you went to a restaurant and they told you that a Coke was $2, but a Pepsi was $3. I bet that they’d sell a lot more Coke than Pepsi. When you are close to indifferent between two goods, you tend to buy the lower-priced one. Even when you really prefer Pepsi over Coke, there likely comes a point where if the price of Pepsi is high enough compared to the price of Coke, you’ll make the switch.
The cross-price elasticity of substitutes is always positive, because you buy more Coke when the price of Pepsi goes up and you buy less Coke when the price of Pepsi goes down. The magnitude of the cross-price elasticity tells you how substitutable the two goods are. If a 10% rise in the price of Pepsi leads to a 50% increase in the quantity of Coke demanded, then the cross-price elasticity between these goods is 
Now consider how a price increase in Pepsi changes your demand for coffee. They are both caffeinated beverages, and both can be served cold. They are clearly substitutes, but many people don’t consider them to be close substitutes. In terms of the cross-price elasticity of demand, if a 10% rise in the price of Pepsi leads to a 3% increase in the quantity of coffee demanded, then the cross-price elasticity of demand between Pepsi and coffee is 0.3. This elasticity is positive, meaning that they are substitute goods. But, because the cross-price elasticity of demand is very small, you know that they are not close substitutes.
The cross-price elasticity is negative for complements.
Why is the printer so cheap, and the ink so expensive?
Let’s now turn to situations in which your quantity demanded falls in response to a rise in the price of another good. This happens when goods are complements—goods that go well together—such as printers and printer cartridges. When the price of a printer rises, people buy fewer printers, and because they are less likely to own a printer, they are also less likely to buy printer cartridges. When two goods are complements, like printers and printer cartridges, a higher price of one good leads people to buy less of another. This means that the cross-price elasticity between complementary goods is negative.
When your business produces goods that are strong complements, it is critically important that you pay close attention not only to the price elasticity of demand for each good, but also to the cross-price elasticity. In fact, this can be a critical element of a savvy pricing strategy. For instance, printer companies often set low prices for their printers—sometimes below cost!—because the cross-price elasticity of demand between printers and cartridges is extremely negative, and so a low price on printers yields a demand for a large quantity of cartridges. Moreover, because the price elasticity of demand for cartridges is quite inelastic—remember, there’s no good substitute for buying HP cartridges for an HP printer—they can charge a lot for these cartridges. Bottom line: The revenue they lose selling cheap printers is more than made up for by selling a lot of overpriced cartridges.
The cross-price elasticity is near zero for independent goods.
Finally, consider unrelated goods, such as Pepsi and shoes. Wearing shoes doesn’t make you more or less likely to feel like drinking Pepsi, and so shoes are neither a substitute nor a complement to Pepsi. Independent goods have a cross-price elasticity close to zero. If you keep in mind that unrelated goods have a cross-price elasticity near zero, that will help you think about why goods with a large positive cross-price elasticity are closely related substitutes, while goods with a large negative cross-price elasticity are closely related complements.
Interpreting the DATA
Is music streaming good for musicians?
In 2015, Taylor Swift pulled her music from Spotify, arguing that the streaming service wasn’t fairly compensating musicians. She believes that when fans stream music, they buy fewer CDs and downloads, leading musicians to earn less. Defenders of streaming services argue that Ms. Swift has it all wrong. They say that streaming is like the radio, and letting people hear the songs via streaming leads to more sales of CDs and paid downloads.
This argument is about the cross-price elasticity of demand between streaming services and sales of digital downloads and CDs. Ms. Swift thinks that this elasticity is positive—that streaming is a substitute for buying CDs and downloads—and large enough that streaming means a lot of lost sales and lost revenue. But if instead the two are complements—which is what those who liken streaming to the radio believe—then the cross-price elasticity of demand between the two is negative, and artists earn more because of streaming.
Taylor Swift on Spotify: That is not a complement.
So who’s right? Researchers examining Spotify concluded that Ms. Swift is right—the cross-price elasticity of demand between Spotify and paid downloads or CDs is positive. So they are substitutes—people buy fewer downloads and CDs when they have access to Spotify. However, she’s wrong about artists earning less. The cross-price elasticity of demand is small, meaning that the losses from displaced sales are small. It turns out that the losses are roughly offset by the fees that streaming services like Spotify pay to musicians, so overall earnings aren’t lower.
Income Elasticity of Demand
The interdependence principle reminds us to take into account not just your decisions about the consumption of other goods and services, but also to think about how your demand decisions are linked to other factors like your income. Since you only have a limited amount of income to spend, all of your demand decisions could change when your income changes. Recall from our study of demand that changes in your income shift your demand curve. Our next task is to measure how responsive demand is to income.
For example, my guess is that you will spend more on housing after you graduate and land a full-time job than you are spending today as a student. The reason is that housing is the kind of thing that people tend to spend more on when their income increases. This shouldn’t be a surprise. After all, housing is a normal good, and you’ve already learned that demand increases for normal goods when income increases. But how much does it go up? A lot or a little?
The income elasticity of demand measures how responsive your demand for a good is to changes in your income. Specifically, it measures by what percent the quantity demanded will change following a 1% change in income. We measure the income elasticity of demand as the ratio of the percent change in quantity demanded to the percent change in income. That is:
For instance, when average income rises in the United States by 10%, researchers have found that the quantity of restaurant meals rises by about 10%. This means the income elasticity of restaurant meals is about 
The income elasticity of demand is positive for normal goods.
Recall that normal goods are goods that you buy more of when your income goes up and less of when your income goes down. The income elasticity of demand for a normal good is positive, since the change in demand goes in the same direction as the change in income. The magnitude of income elasticity of demand tells us by how much. For example, restaurant meals are a normal good, and with an income elasticity of demand of 1, income is a major determinant of demand.
Necessities tend to have a small income elasticity for many of the same reasons that their price elasticity of demand is inelastic. You have to buy toilet paper, and just because your income goes up, you aren’t going to buy that much more.
Figure 6 shows estimates of the income elasticity of demand for various goods. As you might expect, demand for health expenditures, gas, and houses all increase with income, but their income elasticities are small. In contrast, demand for airplane tickets and new cars is highly sensitive to income, and both goods have a large income elasticity of demand.
Figure 6 | Income Elasticity of Demand for Various Goods
The income elasticity of demand is negative for inferior goods.
The goods you buy less of when your income goes up are inferior goods. These are often goods like ramen noodles: you buy them to “make do” when your income is low, but upgrade to something better when your income is higher. Because the quantity of inferior goods you consume moves in the opposite direction of your income, their income elasticity of demand is negative. Figure 6 shows that processed fruits and vegetables have a negative income elasticity of demand. As people’s incomes rise, they tend to switch to fresh fruits and vegetables instead.