Now let's move our discussion to elasticity, which is basically a measure of sensitivity between two variables such as quantity and price. So, when we're calculating elasticities, we're going to use percentage changes just like you see in this box here, and we're going to do that because it just makes it easier to compare across products. We get rid of units altogether; we're not talking about dollars or cents, so different prices of products, different sizes of markets can be compared using elasticity because it gets rid of those units. So let's go ahead and start here with our percentage change formula. In the numerator, we have the change in x, which is basically the new value minus the old value. Say, the price was $5, now it's $6, the change in x would be 6 minus 5, a $1 change, right, over the original value of x. In the denominator, we've got the original value, which would have been, say, in that case, the $5, right? Whatever we started at. So, we're going to be using this formula when we're calculating elasticities quite a bit.
So let's go ahead and define what elasticity is. It is a ratio between two variables. It's relating the changes between two, and it's specifically the percentage changes between two variables, right? That's what we're talking about here. So when we're calculating elasticities, most commonly, we are going to use these variables, right? The ones we've been using. We'll use quantity demanded, we'll use quantity supplied, we'll use price, and we're also going to use income in this chapter. So let's start with our first one, price elasticity of demand. This is the one we're going to spend the most time with. You get a lot of information out of it and it helps us answer the question: how does quantity demanded respond to a change in price? So, when the price goes up, does quantity demanded go down a little bit? Does quantity demanded go down a lot, right? It's how much is it going to change? That's the question we're answering here. So, look at our formula. We've got in the numerator, we've got percentage change and notice I'm using this percentage change, right? That triangle just means change. It's the Greek letter delta and we use it just for shorthand. It saves time. So, I'm going to be using that throughout the chapter and I'll just get you used to it. I'll usually write it out. So, percentage change in quantity demanded is going to be our numerator and in the denominator, notice we have another percentage change. So, we've got two percentage changes in one formula. We've got that percentage change in price and notice this real shorthand that I've got on the right here, right? We've got percentage change in quantity demanded over percentage change in price, so that's a really shorthand way we can write that out.
So let's go ahead and see what this means. Let's do an example with it. So we've got when the price of dog bills rise by 20%, you buy 10 percent fewer dog bills. What is your price elasticity of demand for dog bills? Look at that puppy right there. When would you ever buy one of these? I don't know, but maybe you would buy 1. So let's go ahead and do this problem. We've got the price of dog bills is going up 20 percent and notice here they made it easy. We're not actually calculating the percentage changes. We've just been given the percentage changes and that's okay. We'll be calculating them in a second. So, it tells us the price rises by 20%, right? So, this is going to be our percentage change in price and it tells us that we buy 10% fewer dog bills, so our percentage change in quantity demanded, is going to be that 10, right? So let's go ahead and do this in our formula. So our formula was percentage change in quantity demanded over percentage change in price right. So what was our percentage change in quantity demanded? We bought 10% fewer dog bills, so we're going to have -10%. I'm going to put -0.1 and then in the denominator, right, we bought 20% or excuse me, the price went up by 20%. So we're going to have a positive 0.2 there, right? 20% is 0.2 and notice we've got a negative in the numerator and positive in the denominator so we're going to get a negative answer here. We put this in our calculator and we're going to get an answer of -0.5. So negative half was our answer, right? But one thing I want to note about price elasticity of demand, remember the law of demand; whenever price goes up quantity demanded goes down or whenever price goes down, quantity demanded goes up, right? So there's always going to be this inverse relationship between the two and we're always going to get a negative number when we calculate price elasticity of demand. Just like we saw here, the price rose by 20% and we bought 10% fewer things. So, since we always get a negative answer, we're just going to ignore the negative altogether when we do price elasticity of demand. We're just always going to talk about positive numbers because it's going to help the analysis in the same way. So we use what's called the absolute value, right? Just the positive version of the answer because we're always going to get a negative number. So our answer here was half, right? 0.5. What does that mean, right? How do we analyze this 0.5? Let's go ahead and define some ranges where we're going to call demand elastic, inelastic, or unitelastic. So, demand is elastic when the elasticity of demand, what we just calculated and I'm going to be using this E with a little d for elasticity of demand, right? Price elasticity of demand is greater than 1. Alright? So, when we get a number greater than 1, we're going to call it elastic and when we get an elasticity of demand less than 1, that is when we're going to call it inelastic and remember this is the absolute value. It's always going to be less than 1 if you get a negative number every time. So we're talking about that absolute value being less than 1 and the last special case here is unit elastic when it equals 1.
So let's see this in the context of the problem and then we'll describe each of these situations. So it looks like since we got half, right, in our problem and elasticity of demand is less than 1, we're going to call our demand inelastic in this case. So what does inelastic mean? That means that you're not so sensitive to price changes, right? So what has happened here in the problem is the price went up 20%. It went up a whopping 20% here, but you only bought 10% fewer things, right? So that means that even though the price rose a lot, you didn't change your spending habits too much. You changed your spending habits less than the price change, right? So you're going to get a number less than 1, when the quantity demanded doesn't change as much as the price, right? So we got a situation where the numerator, the quantity demanded change, is smaller than the denominator, the price change, right? That's going to give us an inelastic demand, right? Just like we got here, which means that when the price goes up more than the quantity demanded which is the opposite of elastic, right? When we're elastic, that means the quantity demanded is going to change more than the price. So this will be a situation of say the price went up 20% and you bought 50% fewer dog bills, right? You bought way fewer. You were way more sensitive to price. So we're going to say here, more sensitive to price, I'll put p, and for inelastic, less sensitive to price. So the price can change and your—oh, let me get out of the way there. Cool, less sensitive to price. And this last one, unit elastic. The idea here is that the changes are going to be the same, right? For us to get an answer of 1, the numerator and the denominator would have had to be the same number. So that would have meant that a 20% rise in the price of dog bills would have been a 20% fewer quantity demanded, right. So we would have had the same in the numerator, and the denominator would give us unit elastic. Alright, so I want to go ahead and show you guys that there's a problem with our price elasticity of demand formula, and we're going to go ahead and do some examples using our percentage change and you'll see how we can actually get different answers using the same data. Alright, so let's do that on the next page.
