Cross-Price Elasticity of Demand: Study with Video Lessons, Practice Problems & Examples

Cross price elasticity of demand measures how the quantity demanded of one good responds to the price change of another good, indicating whether they are substitutes, complements, or unrelated. The formula involves the quantity demanded of one good in the numerator and the price of another good in the denominator. A negative result indicates complements, while a positive result suggests substitutes. For example, if the price of tennis rackets rises and the quantity demanded for tennis balls falls, the cross price elasticity is negative, confirming they are complements.

Cross-Price Elasticity of Demand

Now let's talk about another elasticity of demand, the cross-price elasticity of demand. So the cross-price elasticity of demand helps us gauge whether goods are going to be substitutes or complements or just completely unrelated, right? Remember when we were talking about substitutes and complements in the demand shifts, the idea was that the price change in one good would affect the demand of another, right? And that's exactly what we see in our formula here, right? Look at our formula for cross-price elasticity of demand. We've got the quantity demanded of one good in the numerator, right? Notice how quantity demanded is still in the numerator just like always, but we've got the price of a different good here, Good Y, right? So quantity demanded of one good, price of another good, right? The idea here is we're going to see how quantity demanded of one good reacts to the change in the price of another good, and you can imagine that it's going to help us find substitutes and complements, right. So still just like before, we're going to keep using our midpoint method and luckily again the steps stay 99% the same, except in this case, we're using the price of a different good rather than the price of our own good. Just like with the income elasticity too, the only thing that changes we were using income instead of price, now we're using the price of another good instead of the price of our good. And once again, you can tell right here price and price in the denominator, they all kind of follow this flow. Whatever the name of it is, that's what's in the denominator, but it's also easy to just remember that quantity is always going to be in the numerator for all of our elasticities. Alright, so let's go ahead and do an example here for cross-price elasticity. You're going to see how similar our steps are and again with this one, we do have the positive and negative, differences, right? So we're going to end up making our conclusion based on it being positive or negative, so that's going to matter in this case as well. So, let's go ahead and do an example where we can use our steps, which are basically the same as we've been using. When the price of tennis rackets increased from $45 to $55 the quantity demanded of tennis balls dropped from 21,000 to 19,000. What is the cross-price elasticity of demand? So notice they gave us two quantities, right, but those are quantities of tennis balls and they gave us two prices. But those are prices of tennis racquets. Right? So, you can imagine that these are going to be complements, like I picked things that sound like complements, just for the sake of it, but, we can make sure by doing this calculation. But the idea here is we're seeing a price increase in one product and a quantity demanded decrease in the other product, right? So remember from demand shifts, that's exactly what happens with a complement. If the price goes up of a complement, the quantity demanded of the complement goes down, right? So in this case, that's exactly what we're seeing. Let's do our steps and do the analysis with elasticity. So we'll make two columns quantity demanded and that's quantity demanded of tennis balls and P that's our price column for the price of tennis racquets, and let's go ahead and do this. So first, we subtract 21,000 minus 19,000 equals 2,000. And price, the same thing. 55 minus 45 equals 10. Step 2, we're gonna add, 21,000 +19,000 equals 40,000. And on the other side, the same, 55 + 45 equals 100. Now step 3 is where we're gonna divide step 2 by 2. 40,000 divided by 2 equals 20,000. And on the other side, 100 divided by 2 equals 50. Step 4, this is where we're gonna get our actual percentage changes in each. So step 1 divided by step 3, 2,000 divided by 20,000 is going to equal 0.1. That's our percentage change in quantity demanded. Let's do the same thing with price. So we've got 10 divided by 50 and we're gonna have 0.2 as a percentage change in price, right? Now we haven't dealt with the positives or negatives yet, I always leave that for the last step where I go back to the problem. So first let's get the number of our cross-price elasticity which is just gonna be this quantity demanded 0.1, right? This is quantity divided by price, which was the 0.2 and we are going to get an answer of 0.5 right there, right. Half. So now we just have to check, is it positive or negative? So let's see for the prices, we had a price increase so the price is a positive and for tennis balls, the quantity decreased, right, dropped from 21 to 19. So that one's negative right there. So we have a negative and a positive, that means it's a negative. We have a negative 0.5 for our answer. So now how do we analyze this answer that we just got, negative 0.5? Well, right down here we've got our answers. So if it's positive, then we know they're substitutes. If it's negative, they're complements, and if it's zero, then you know they're unrelated. Okay? So in this case, we got a negative number which tells us that they are complements, but remember we can use some logic to kind of come to the same conclusion, right? We saw the price of tennis rackets go up and the quantity demanded of tennis balls go down, right? So the idea is if you remember from the demand shifts, when the price of one product goes up causing the other quantity demanded to go down, that means that they're going to be complements. Now if the price had gone up and the quantity demanded of the other one went up, that's when we're talking about substitutes and that's why you would see a positive answer in this case, up and up, right? So really in this case, you can make a lot of conclusions just from the problem without doing any math. Alright? So let's go ahead and do some practice problems about cross-price elasticity.

An increase in the demand for chicken, from 8,000 to 12,000, was caused by an increase in the price of beef from $4.50 to $5.50. Therefore, the cross-price elasticity for these two products is:

• A. 0.5
• B. -2.0
• C. 2.0
• D. -0.5

Here’s what students ask on this topic:

What is cross-price elasticity of demand and how is it calculated?

Cross-price elasticity of demand measures how the quantity demanded of one good responds to a price change in another good. It helps determine if goods are substitutes, complements, or unrelated. The formula is:

$\frac{\mathrm{\Delta Q_d}/\mathrm{Q_d}}{\mathrm{\Delta P_y}/\mathrm{P_y}}$

Where $\mathrm{\Delta Q_d}$ is the change in quantity demanded of Good X, $\mathrm{Q_d}$ is the initial quantity demanded of Good X, $\mathrm{\Delta P_y}$ is the change in price of Good Y, and $\mathrm{P_y}$ is the initial price of Good Y. A positive result indicates substitutes, a negative result indicates complements, and a zero result indicates unrelated goods.

How do you interpret the sign of cross-price elasticity of demand?

The sign of the cross-price elasticity of demand indicates the relationship between two goods. A positive cross-price elasticity suggests that the goods are substitutes, meaning that an increase in the price of one good leads to an increase in the quantity demanded of the other. A negative cross-price elasticity indicates that the goods are complements, meaning that an increase in the price of one good leads to a decrease in the quantity demanded of the other. If the cross-price elasticity is zero, the goods are unrelated, meaning that a price change in one good does not affect the quantity demanded of the other.

Can you provide an example of calculating cross-price elasticity of demand?

Sure! Let's say the price of tennis rackets increases from $45 to $55, and as a result, the quantity demanded of tennis balls decreases from 21,000 to 19,000. First, calculate the percentage change in quantity demanded and price:

$\frac{(\mathrm{21,000}-\mathrm{19,000})}{\mathrm{21,000}}=\; -0.095$

$\frac{(55-45)}{45}=\; 0.222$

Then, divide the percentage change in quantity demanded by the percentage change in price:

$\frac{-0.095}{0.222}=\; -0.43$

The cross-price elasticity of demand is -0.43, indicating that tennis rackets and tennis balls are complements.

What are some real-world examples of substitutes and complements?

Real-world examples of substitutes include tea and coffee, where an increase in the price of coffee may lead to an increase in the demand for tea. Another example is butter and margarine. For complements, consider smartphones and apps; an increase in the price of smartphones may decrease the demand for apps. Another example is cars and gasoline; if the price of cars increases, the demand for gasoline may decrease as fewer people buy cars.

Why is cross-price elasticity of demand important in economics?

Cross-price elasticity of demand is important because it helps businesses and policymakers understand the relationships between different goods. For businesses, knowing whether products are substitutes or complements can inform pricing strategies, marketing, and product development. For policymakers, understanding these relationships can help in assessing the impact of taxes, subsidies, and regulations on different markets. It also aids in predicting consumer behavior and market dynamics, which is crucial for making informed economic decisions.