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Cross-Price Elasticity Formula

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Source: Image of Cross-Price Elasticity Graph for Substitutes created by Kate Eskra, Image of Cross-Price Elasticity Graph for Complements created by Kate Eskra

Hi, welcome to Economics. This is Kate. This tutorial is called Cross-Price Elasticity-- Compliments and Substitutes. As always my key terms are in red, and my examples are in green.

So the subject of this tutorial is pretty straightforward. I am going to be teaching you how to calculate cross-price elasticity when the price of a good changes. I'll show you the impact on both a substitute good and a complement good.

So you already know the law of demand. That as prices go up we tend to buy less. As prices fall we tend to buy more, but elasticity tells us how much more or less. So how responsive are we to these price changes?

OK. So let's use the example first of Adidas shoes. So we know that as the price of Adidas shoes rise people will probably buy fewer pair of Adidas shoes, but that's own-price elasticity. When the price of something goes up or down, how much more or less do they buy of the good itself? OK.

But for this tutorial, what we need to be talking about is how could that impact the sales of something related to it? So let's say, how could that impact Nike sales? When Adidas changes price, how could that impact Nike? That's what cross-price elasticity is all about. So let's take a look at it.

Cross-price elasticity is defined as, "The change of demand that occurs due to the change in price of a substitute or complement." OK. So here's the formula. And we derived this in a different tutorial. But all this is telling us is that elasticity is really a proportion. It is the percentage change in quantity, divided by the percentage change in price. OK?

We were able to eliminate this being divided by 2 with our midpoint formula, because there was a half in both the numerator and denominator. So simplified this was our formula. But with this one, this is going to be the quantity of one good, but the price of another good. OK.

So here's my specific example. Let's say on one model of shoes Adidas raises price from $110 up to $130. Certainly their quantity demanded will probably fall as they raise price. But what we're going to measure is the impact it has on Nike sales. And Nike, let's say, sees its sales go from 200 up to 240 this month at a certain store. So there is that.

So here we're looking at what happens to the quantity of Nike as the price of Adidas shoes changes. So this doesn't look like a normal demand curve, because we don't have the price of Nike and the quantity of Nike. We have the quantity we're looking at-- how responsive people are to the price of Adidas changing, but the impact on the quantity of Nike. OK?

So let's calculate it. Here we're going to plug-in the numbers, so here's the formula. And here the order of operations is definitely going to matter. OK. So what we take is the initial quantity minus the new quantity, and we divide by those added together. And then we divide all of that by the initial price minus the new price, divided by them added together. Remember this is us using the midpoint formula.

OK. So when we do all of that we end up with a cross-price elasticity coefficient of 1.125. And since the order did matter, it's important to note that this is a positive number. But what does that number actually tell us?

Well it tells us whether they're substitutes or complements. OK? So generally speaking quantity and price are going to move in opposite directions. As the price of quantity goes up, the quantity that people purchase goes down.

Both with cross-price elasticity here-- since our coefficient was positive, that was telling us that as the denominator went up, people actually bought more in quantity of the other good, so they must be substitutes. If the coefficient turns out to be negative, then they must be complements. And I'll be defining these for you in a few slides.

So again, remember that the order of the Quantity A, Quantity B, Price A, Price B that does matter. OK. So that elasticity of a positive 1.125 that tells us that Nike and Adidas must be substitutes since the coefficient was positive. So doesn't that make sense? When Adidas raises prices, people will buy more of a substitute like Nike. They're not perfect substitutes. Some people strongly prefer one to the other, but this tells us that is in fact the pattern that we saw happen.

So substitute goods defined are, "As the price of one good increases--" so remember as the price of Adidas went up-- "the demand for an alternative good meeting the same consumer needs--" like Nike-- "increased." That's what we saw happen.

All right. But let's look at a different situation. Let's look at the demand for apples, but we're not going to change the price of apples. We're going to change the price of caramel apple dip. Yum.

OK. So let's say that caramel apple dip goes on sale from $5 down to $3. What's going to happen to the demand for apples as that's the case? Well, let's say a grocery stores sees apple sales increase from a quantity of 200 up to 400.

OK. So now here we have a graph. Remember quantity is what we're looking at-- the effect on the quantity of apples purchased. But the price on this axis is not of apples, it's of caramel apple dip. So we're looking at the effect on the quantity of apples as the price of caramel apple dip changed.

OK. Here we go. So remember again the order does matter. So we're looking at the initial quantity of apples-- which was only 200-- minus the new quantity of apples being sold-- which is 400-- divided by them added together. And then we're dividing that whole thing by the percentage change in price. OK?

So the initial price of caramel apple dip was $5, now it's on sale for $3. Divided by them added together. When we do all of that, we do get a negative number in the numerator, which is important because that's going to make our coefficient negative.

So with that elasticity of a negative 1.32, it turns out that apples and caramel dip must be complements, since the coefficient is negative. So when caramel apple dip goes on sale, people buy more apples to go with it. You don't buy the dip just to eat on its own. Well some people might, but typically speaking you're going to buy something to go with it.

And that's what complements are. They are, "Goods for which the demand increases as the price of an associated good decreases." So as caramel apple dip went on sale, people bought the complement more. The demand for apples increased.

So what did you learn in this tutorial? We talked about how cross-price elasticity measures how consumers respond to a price change by buying a related good, not the actual good itself-- which was own-price elasticity. The coefficient-- if you're very careful about the order of the quantities and the prices-- that coefficient tells us how much consumers respond.

But it also tells us whether the goods are substitutes or complements. If you get a positive coefficient, it means they're substitutes. If you get a negative coefficient, it would indicate that the two goods are complements.

Thank you so much for listening. Have a great day.