So, price elasticity is one kind of pricing metric that can be used to help optimize prices. But there are other types of price elasticities that can tell us very useful things, and I want to turn to one of them right now, and that's the cross-price elasticity. Now, the definition will look a little bit familiar, but it has a bit of a twist. So, if we look at the definition, it's percent change in quantity divided by the percent change in price. The difference here is that this percent change in price is not my price. It's the price of, maybe, one of my competitors, or another product that competes in the same space that I do. So, cross-price elasticity is fundamentally asking the question, when somebody else changes their price, what happens to my demand? You might imagine that is a very interesting thing that managers often want to know. And it can tell us a lot of things about the nature of competition, in whatever market we're working in. So, I've rewritten the answer. Percentage change of quantity divided by percentage change in price. Remember that that can be rewritten as just delta Q over delta P, that's the change, multiplied by P over Q. Looks very similar to price elasticities. Now, where do we get this information? We're going to go back to that regression output that we were looking at earlier. But this time, instead of grabbing the estimated coefficient on the chuck price, which is our known price, right? We're selling chuck. We are going to look at the estimated coefficient for chicken, because it is possible that as the price of chicken varies, that our demand could be affected by that. Now, recall something that I said earlier, that in this regression, I'm showing you real data here, and the chicken price is not quite significant. So, you would normally want to see something like 1.96 or greater for that t-statistic. But I'm going to use this information anyway, because I want to show you how to calculate a cross-price elasticity. So, I take that estimated coefficient, and then I use the definition. So, that estimated coefficient is what? That is the delta Q divided by delta P, right? The change in chuck quantity relative to the price change in chicken. So, I use that coefficient, then I multiply it by P over Q, okay? And those come from where? Remember, those come from the mean of the data. Okay, so, this is the price of chicken and our chuck quantity. We get the means of the data, and then we can put that in, and what comes out the other end of that is -0.84. So what does negative mean in a cross-price elasticity? It means that when their price goes up, what do you think happens to our demand? When their price goes up, our quantity goes down. And that means that somehow, they're almost compliments to each other, which may be a little counterintuitive to you. But think about that. For some products, let's say the price of Parmesan cheese went up. What might happen to the sales of spaghetti sauce? It might go down, right? Because people are looking at all the things they need to make, maybe, spaghetti and meatballs, with some Parmesan cheese on top, right? And if any one of those components, the price goes up, it may be that the sales of the other ones go down. And that means, they're compliments to each other, not substitutes to each other. Now, it is a little weird to me that chicken and beef are kind of viewed as complements in the marketplace. But that really is what the data is telling us here. And in particular, it says that a 10% increase in the price of chicken is going to decrease our demand for chuck by 8.4%. Okay, so what kinds of things can this tell us? Well, if you take a matrix cross-price elasticity, so this is a matrix that contains the cross-price elasticities between these four different brands of toothpaste. Along the diagonal, these are own price elasticities. They're sometimes what we just call the price elasticity. So, Aquafresh, relative to Aquafresh, that's just the price elasticity. But the off-diagonals in this matrix are cross-price elasticities. So, we can ask questions, if this is estimated from regression data, which brand is most sensitive to its own price? Well, by looking this, we can say, that's just the price elasticity. And it turns out that Aquafresh is the most sensitive to its own price. It's the most elastic. But, we can also ask bigger questions than that. Who's Crest's main competitor? All right, so let's look at this group of numbers right here. Well, Crest's main competitor looks to be Colgate. Why? Because the cross-price elasticity is the largest, and it's a positive cross-price elasticity. So, this means kind of what you would expect it to the mean. And that is, if Colgate increases their price, what's going to happen to demand for our product? It'll also go up, right? Conversely, it also means that if Colgate drops their price, we have something to worry about because our demand might go down. So, there's a strong effect here, where there is an effect between Crest and Aquafresh, but our demand is not as sensitive to Aquafresh's price as it is to Colgate's price. So, we can also sometimes detect underlying preferences by looking at something called asymmetric cross-price elasticities. For example, think about Audi and Volkswagen. You would expect what are called asymmetric cross-price elasticities there, in the following sense. You would expect that if Audi dropped their price a lot, that they would get some people to switch up from Volkswagen, right? Some people would look at that Audi price and say, wow, they've got a really good deal. I was going to buy a Volkswagen, but I'll buy an Audi. On the converse, if Volkswagen drops their price a lot, you might not attract as many people that want to get an Audi because they were willing to pay that anyway. So, that by looking at the magnitudes of the differences in the cross-price elasticities, and those asymmetries, sometimes you can really figure out which product is preferred in the marketplace to the other and by how much. And how much people are cross-shopping between those products. So, all of these things, which are pretty important, are something that can be elucidated by looking at cross-price elasticities.