Cross-price elasticity of demand
So far, we've been focused on the elasticity of demand for only one good. We've thought about how changes in the price of that good affect changes in its quantity. Now what we're going to explore is how we can go across goods. So we're going to talk about the cross elasticity of demand. And there's multiple different scenarios we could think about, but it's really thinking about how a price change in one good might affect the quantity demanded in another good. And to see an example of this, think about two airlines-- two competing airlines-- maybe it's the same exact route going at the exact same time, maybe between New York and London. So airline one, right over here-- airline two, very competitive, price right over here is $1,000 for a round trip. Quantity demanded is 200 tickets, let's say, in a given week. Airline two, price is $1,000 for the round trip, and the quantity demanded is 200 tickets as well. Now let's think about what will happen. What will happen if airline one raises its price from $1,000 to $1,100? In fact, we could even do something less dramatic than that, to $1,050-- so a relatively small increase in price. And remember, when we think about the percentage price increase, when we're thinking about elasticities in general, we don't just say, OK, $50 on top of $1,000, that's a 5% price increase. That's what we would do in everyday thinking. If you said you went from $1,000 to $1,050, you would say that's a $50 increase on a base of $1,000 or that is a 5% increase. But when you think about elasticities, because we want to have the same percent change between-- if you go from $1,000 to $1,050, or if you go from $1,050 down to 1,000-- we actually use the average point as a base. So the percent change in this scenario-- let me write it right over here. So our percent change-- and I'll write it in quotes, because it's a little bit different than what you do in traditional mathematics when you think about percent changes-- is you had a 50 change in price. Your price went up by 50, and on our base we will use 1,025, which is the average of 1,000 and 1,050. And so that gives us a change of 50 divided by 1,025 is equal to, let's say, roughly 4.9%. So this is approximately 4.9%, we'll say, "increase" in price, although we're going to put that increase in quotes, because we're using it on the average. And we do that so that if we said it was 1,050 to 1,000, it would still be a 4.9% decrease using this same idea-- using the midpoint as the base. Now, when that happens-- Everyone today, they use these travel sites where you can compare prices-- If these really are the exact same route, going from the exact same airport to the exact same other airport in London, leaving at the exact same time, everyone is going to gravitate to this one now, because it's only $1,000-- even just to save $50. Why would they ride on this airline? So this quantity demand is going to go to 0. And this quantity demanded is going to go to 400. And we're not going to think about the actual capacity of the planes and all that. We're going to have a very simple model here. So what was the percent change in quantity for airline two right over here? Well, once again, our change in quantity is 200, not 400. We went from 200 to 400. So we gained 200. And our base, we want to use the average of 200 and 400, which is 300. And so this is approximately 67%. So we have, all of a sudden, our cross elasticity of demand for airline two's tickets, relative to a1's price. And we get the percent change in the quantity demanded for a2's tickets, which is 67% over the percent change, not in a2's price change, but in a1's price change. That's why we call it cross elasticity. We're going from one good to another. So let's just say, for simplicity, roughly 5%. And so you do the math. So if you have 67% divided by 5%, you get to roughly 13.4. So this is approximately 13.4. So you have a very high cross elasticity of demand. In fact, if you even increase this, maybe by $5, you might have had the same effect. And so you would have had a very large number here. And that situation right here, for this cross elasticity of demand-- it's because these things are near perfect substitutes. The way that we set up this problem, we said, well, people don't care which one they take. They're just going to go for the cheapest one. And so when you have near substitutes, or nearly perfect substitutes, for each other, like this example right here, the cross elasticity of demand approaches infinity. It gets higher and higher and higher. In theory, if these are really, really, really identical, even if you raise this a penny, people will say, well, why would I waste a penny? I would just use airline two. And so this number would be even lower right over here. And so this thing might approach infinity. And notice this was a positive. When we just did regular price elasticity of demand, the only way that you would increase quantity for a traditional goods was by lowering price. But here, we raise price on a substitute competitive product, and we raise the demand for airline two's product, which actually made a lot of sense. So it wasn't a negative relationship. It's actually a positive value right over here. But you could have things in other-- you could have that negative relationship using cross elasticity of demand. This is an example of a substitute. We could think about the example of a complement. So what if we're talking about e-books? So let's say I have some type of an e-book, and the current quantity demanded in a given week is 1,000. And let's say that the price of an e-reader that you would need for my e-book is $100. But let's say that price of the e-reader goes down from $100 to $80. So you had a $20 decrease in price. Well, what's going to happen to my e-book, assuming its price does not change? Well, then the quantity demanded for my e-book will go up. So let's say the quantity demanded for my e-book goes up by 100, because more people are going to be able to afford this, or they're going to have money left over when they buy this to buy more e-books. And so I don't even know what the price for my e-book is, but at a given price point, the quantity demanded will go up. And so this goes to 1,100. And so I'll leave it to you to calculate this price elasticity of demand. But you will see that you will actually get a negative value, like we're used to seeing for regular price elasticity of demand. And when you do calculate it, remember, you want to do your percent price change in e-book quantity over percent change in e-reader price. And the other thing you have to remember, you don't just take negative 20 over 100. You take negative 20 over the average of these two, when you're thinking of it in the elasticity context. So this right over here-- actually, maybe we'll just work it through. Pause it, and try to do it yourself. So this value right over here is negative 20 over 90-- the average of those two-- and this value right over here is going to be plus 100 over the average of these two. So the average of those two is 1,050. And so this is 100 divided by 1,050, which gets you to about 0.95. So about 9 and 1/2% change in quantity demanded for my book. And then this denominator right here is negative 20 divided by 90. So you get a drop of 22%. And so if you divide the numerator by the denominator, you get 0.952 divided by negative 0.22222-- I'll just put couple of 2's there-- and you get a negative 0.43. So this is equal to negative 0.43. And this makes sense. If you lower the price of an e-reader-- this complement product, a product that goes along with my e-book-- it increases the demand. So just like you get with price elasticity of demand, you get a negative value over here. And what about completely two unrelated products? So let's say that I have basketballs, and the price of basketballs goes from, let's say, $20 to $30. What's going to happen to my e-book? Well, my e-book's not going to change. It's going to stay at $1,000. So my percent change in the quantity demanded of my e-book is going to be 0 in this example. So we're going to have 0, when we want to do this cross elasticity of demand, over my percent change in basketballs, which would be 30 over 25. So whatever that is-- 30 over 25 would be 10 over 25-- which is a 40% increase. So that would be 0 over 40%, which equals 0. So for unrelated products, products where the price of change in one of them does not affect the quantity demanded in the other, it makes complete sense that you have a 0 cross elasticity of demand. If they're complements, you would have a negative cross elasticity of demand. And if they're substitutes, you would have a positive one. And the closer the substitutes they are, the more positive your cross elasticity of demand is going to be.
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