Percentage Change and Price Elasticity of Demand

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 elasticity is we're going to use percentage changes just like you see in this box here and we're gonna 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. Um So what we have here in the numerator, we have the change in X. Which is basically the new value minus the old value, say price was $5. Now it's $6. The change in X would be the six minus the five. A $1 change right over the original value of X. And 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 gonna be using this formula when we're calculating elasticity is quite a bit. Um So let's go ahead and define what is the elasticity. So it is a ratio between two variables. It's relating the changes between two variables and it's specifically the percentage changes between two variables. Right? That's what we're talking about here. So we're calculating elasticity ease 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 gonna use um income in this chapter when we're calculating these. So let's start with our first one price elasticity of demand. This is the one we're gonna spend the most time with. Um You get a lot of information out of it and it helps us answer to 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 This quantity demanded go down a lot. Right? It's how much is it going to be changing? That's 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 that triangle is um it just means change. It's the greek letter delta and we use it just for shorthand. It saves time. Um So I'm gonna be using that throughout the chapter and I'll just get you used to it. I'll usually write it out. Um So percentage change in quantity demanded is gonna be our numerator. And denominator notice we have another percentage change. So we've got 2% 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 short hand way we can write that out. So let's go ahead and see what this means. Let's do an example with it. Um so we've got when the price of dog bills rise by 20%. You buy 10% 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 one. So let's go ahead and do this problem. We've got the price of dog bills is going up 20%. 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 we'll be calculating them in a second. Um so it tells us the price rises by 20%. Right? So this is gonna be our percentage change in price. Is that 20%? And it tells us that we buy uh 10% fewer dog bills. So we're gonna have a percentage change in quantity demanded. Um is gonna 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 gonna have negative 10%. I'm gonna put negative 0.1. Um and then in the denominator right, we bought 20% or excuse me the price went up by 20%. So we're gonna have a positive 0.2 there, right? 20% is 0.2. Um and notice we've got a negative in the numerator and uh positive in the denominator. So we're gonna get a negative answer here. We put this in our calculator and we're gonna get an answer of negative 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 whatever price goes down, quantity demanded goes up, right? So there's always gonna be this inverse relationship between the two and we're always gonna 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 stuff. So since we always get a negative answer, we're just gonna ignore the negative altogether. When we do price elasticity of demand, we're just always gonna talk about positive numbers because it's gonna 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 um some ranges where we're gonna call demand elastic and elastic or unit elastic. So demand is alas when the elasticity of demand, what we just calculated and I'm gonna be using this e with a little D for elasticity of demand, right? Price elasticity of demand is greater than one. Alright, so when we get a number greater than one, we're gonna call it elastic. And when we get an elasticity of demand less than one, that is when we're gonna call it any elastic. And remember this is the absolute value, It's always going to be less than one if you get a negative number every time we're talking about that absolute value being less than one. And the last special case here is unit elastic when it equals one. 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 one. We're gonna call our demand any elastic in this case. So what does any elastic mean? That means that you're not so sensitive to price changes. Right? So what has happened here in the problem is price went up 20%, it went up a whopping 20% here. But you only bought 10% less stuff. Right? So that means that even though the price rose a lot, you didn't change your spending habits too much. You change your spending habits less than the price change, right? So you're gonna get a number less than one. Um 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 gonna give us and any elastic demand, right? Just like we got here which means that when yeah the price goes up more than the quantity demanded. Which is the opposite of 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% less dog bills, right? You bought way less, you were way more sensitive to price. So we're gonna say here more sensitive to price a puppy. And for any elastic less sensitive to price. So the price can change. And um your let me get out of the way there. Cool. Less sensitive to price. And this last one unit elastic. Um The idea here is that the changes are gonna be the same right for us to get an answer of one. 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. Um And we're gonna 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.

So now let's see how we can actually get different elasticity. He's using the same data. The problem here ends up being with our percentage change formula. Remember that when we were using percentage change. Um By the way this is the shorthand right percentage and the delta, the triangle here is change. Uh We would do the change over the original which was basically the new minus the original. Right? That's the that's the numerator divided by the original. Right? So the change divided by the original, gives us the percentage shape and it's actually in this denominator that we end up having a problem. So let's go ahead and see uh through these examples. Let's see let's see it in action. We're gonna get a different elasticity when we're raising the price and when we're decreasing the price. So let's see this example. Pizza Companies lunch special Currently costs $5 at this price. The weekly demand is 2000 lunch specials. If they raised their price to $6, the weekly demand will drop to 1400 lunch specials. What is the price elasticity of demand. Alright so let's go ahead and start remember um our formula for elasticity of demand was our percentage change in quantity demanded over a percentage change in price. Right so let's go ahead and start with the quantity demanded. And I'm gonna go here and we're gonna use our percentage change formula just like I've written there above new minus original, divided by original. So the percentage change for quantity demanded. Let's see um First I'm gonna circle all our data we've got in blue, I'll circle our quantity demanded 2000 and it went down to 1400 and in red. I'll do the prices. Well I've been using red I'll use green for the prices here. Color of money. Five and six. Right? So let's start with our quantity demanded. Okay. And we had a demand of 2000 and it dropped to 1400. Right? So our new is 1400 minus the 2000 divided by the original of 2000. Right? So our original demand was 2000. Our new demand was 1400. What is going to be? The difference here? We're gonna get negative 600 over 2000. Right? We put that in our calculator and we're gonna get 0.3. Right? And this will be a negative 0.3. Right? But remember we're gonna drop all the negatives and positives because we're always gonna get one of them negative. So we're just gonna say 0.3 absolute value. And let's do the same thing for price. Right? For price, we had a price of $5 and it went up to $6. So the new was six minus the original of five divided by the original of five. Right? So six minus five is one divided by five. Put that in our calculator and we're gonna get 0.2 right? That. Oops, can you see that there? Alright six minus five divided by five. So it gives us 1/5 and we're gonna get 0.2 here for our percentage change in price. And we had 0.3 here for a percentage change in quantity demanded. Right, Okay, so let's go ahead. And sulfur elasticity in this case and I'll do it in right here. So elasticity of demand is going to equal that percentage change in quantity demanded. 0.3 divided by our percentage change in price, which was 0.2. Right. And what does that give us? It's going to give us 1.5. So our elasticity in demand in this case was 1.5. Right? And when we get an elasticity of demand greater than one, right? That means that it's elastic. So in this case We got an elasticity of demand greater than one and an elastic 1.5. So let's go ahead. In the next video, we're gonna do a similar example with similar data. Check it out

Alright so now let's check out this problem, A pizza companies lunch special currently cost $6 at this price. The weekly demand is 1400 lunch specials. If they lower their price to $5 the weekly demand will increase to 2000 lunch specials. What is price elasticity of demand? And you should be able to notice that we've got the same numbers here, right? Except in this case we started at a price of $6 and We're decreasing to $5 in the previous problem. Up here we started at a price of $5 and increase to $6. But the quantity demanded are all the same. Right? It's just which direction we're moving. So let's go ahead and solve our price elasticity of demand here. Right? We're gonna use still our percentage change formula. Remember that was new minus original divided by original. And just like I said um this is where we're gonna end up seeing the problem is going to be in this denominator. Um is the problem with this formula where we're gonna get different answers. So let's go ahead and do our uh our price elasticity. So elasticity of demand is going to equal percentage change in quantity demanded. Over percentage change in price. Right? So let's go ahead and do like we did before, we'll get our our quantities here and our prices we'll use green. Alright so let's go ahead and start with quantity demanded. Let's get our percentage change in quantity demanded. Alright um so in this case uh they started at 1400 lunch specials And demand will increase to 2000. So the new in this case, excuse me. The new is 2000 and the original was 1400. Right? They were at 1400. They're gonna decrease the price which will increase quantity demanded. So new minus original divided by the original which in this case was 1400 and we are going to get uh 600 in the numerator divided by 1400. We put that in our calculator and we're going to get 0.429. I'm gonna say okay I'm gonna cut it off at three decimals there, you can stop at two or three. We're just rounding and for price let's go ahead and do the same thing. So price, we had a new price of $5. Right? In this case they're lowering their price to $5. They had an original price of $6 and we're gonna divide by six there and we're gonna get five minus six is negative, 1/6. Um and remember we just get rid of the negative, right we're gonna be dealing with absolute values here. So I'm just gonna get rid of that negative there. We have 16 which we put that in and we're gonna get 0.1 and I'm gonna go to three decimals here as well. 1670.167. Right so there we go that is gonna be a percentage change in price over here, We have percentage change in quantity demanded. Right, whoops right there And right there, let's go ahead and put this in to our elasticity of demand formula and see what answer we get. So we get .429 is going to be in our numerator. I'll do it in blue just to keep it even. And in our denominator we will have .167. So let's go ahead and do that. Math. .429 divided by .167 gives us an elasticity of demand Equal to 2.569. So look at this. In this case we've got 2.569 before we got 1.5 right? We've gotten different answers in both cases and we used the same numbers, we used price of five and a price of six and a quantity demanded of 21,400, right? The numbers stayed the same. But we got a different answer whether we were going up in price or down in price and we don't want that. We want a consistent answer. Right? This problem came from what was in our, in our denominators, the original values. Notice in in this first problem for the quantity demanded um our denominator was 2000, right? Our numerator was 600 still but our denominator was 2000 and down here notice our numerator was 600. But our denominator was 1400 that's the problem, we're seeing the same thing is going to happen with price. We have different denominators in either case, so what we're gonna end up doing is taking an average and putting that in the denominator instead. So let's go ahead to the next video where I'm gonna show you a step by step method to do the averaging and get a consistent answer when we solve these problems. So let's go ahead and do that now.