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Price elasticity of demand using the midpoint method

Video transcript

What we're going to think about in this video is elasticity of demand-- tis-sit-tity, elasticity of demand. And what this is, is a measure of how does the quantity demanded change given a change in price? Or how does a change in price impact the quantity demanded? So change in price-- impact quantity-- want to be careful here-- quantity demanded. When you talk about demand, you're talking about the whole curve. Quantity demanded is a specific quantity-- quantity demanded. And the way that we, as economist-- I'm not really an economist, but since we're doing economics, we could pretend to be economists. The way that economists measure this is they measure it as a percent change in quantity over a percent-- over the percent change in price. And the reason why they do this, as opposed to just, say, change in quantity over change in price, is because if you did change in quantity over change in price you would have a number that's specific to the units you're using. So it would depend on whether you're doing quantity in terms of per hour, or per week, or per year. And so you would have different numbers based on the time frame, or the units, that you might use. But when you use a percentage it is a unitless number. Because the percentage-- you're taking a change in some quantity, divided by that quantity. So the units themselves actually cancel out. And the reason why it's called elasticity-- this might make some sense to you-- or the reason why I like to think it's called elasticity, is I imagine something that's the elastic. Like a elastic band or a rubber band. And in the rubber band, if you pull it, depending if something-- so let's say this one is inelastic. So if you pull, you're not going to able to pull it much. It's going to be fairly stiff. It's not going to stretch a lot. While something is elastic-- if something is elastic for a given amount of force-- so this is for a given amount of force-- you're not able to pull it much. And if something is elastic, maybe for the same amount of force, you're going to be able to pull it a lot. So this right over here is elastic. And so the analogy, maybe, might make a little bit sense-- relative to applied price and demand. Something is elastic-- so let me write this down. So let me write, very elastic. If a given change in price-- given price change you have-- and we'll talk about percentages in a little bit. But a given change in price, you have a large change in demand-- so large percentage change. And let me just speak in terms of percentage. Given a percentage change in P, you end up having a large percentage change in Q. That would be very elastic. So you could imagine the P is like the force, and the Q, the quantity demanded, is how far the thing can get stretched apart. And that's why we would call this very elastic. Just like a very elastic rubber band. And if something is very inelastic, if given a percent change in P, you have a small percent change in Q. So just like a rubber band-- for a given amount of force, if you're not able to pull it much at all, then it's inelastic. If you're able to pull a lot, it's elastic. Same thing with price and quantity. For a given change in price, if the percent quantity demanded changes a lot-- very elastic. If it doesn't change a lot-- very inelastic. Now, with that out of the way, let's actually calculate the elasticity for multiple points along this demand curve right over here. And I think that will give us a bit better grounding. Especially because there are a little slightly-- I would call them unusual ways of calculating the percent change in quantity and the percent change of price-- just so that we get the same number when we have a positive change in price. And the same as we get the negative change in price-- or a negative and a positive-- or a drop in price and an increase in price. So let me give myself some real estate over here because I want to do some actual mathematics. And actually all of this we will be reviewing in what I'm about do, and it will give me some real estate to work with. So let me clear all of that. And let me clear is that right over here. And what I'm going to do is I'm going to calculate the elasticity of demand at several points along this demand curve right over here. And so the first one, I will do it at point A to point B. So let me make another column right over here-- elasticity of demand. And actually, we're going to have one column that's elasticity of demand. So it's a big E with a little subscript D. And the other one, I'll just take its absolute value. Because, depending on-- sometimes people like to just think of the number, which will tend to be a negative number. And sometimes, people like to look at the absolute value of it. So we'll look at both and see what it actually means. So let's say our price drops from point A to point B. So from point A to point B we have a $1-- a negative $1 change in price. And we have a positive-- so this is a negative $1 change in price. And we have a positive $2-- sorry-- a positive two burger per hour change in quantity demanded. So what is the elasticity of demand there? So let's write it over here. I'll do it in A's color. So the elasticity of demand, remember, it's the percent change in quantity. So percent change in quantity-- I'll rewrite it. It's the percent change in quantity over percent change in price. And so we have-- what's our percent change in quantity? So it's going to be the change in quantity over some base quantity. So our change in quantity is two. So it's going to be equal to 2 over-- now in traditional terms-- and this is what I want to, kind of, clarify-- is a little bit unusual in how we do it. But we do it, so that we get the same elasticity of demand whether we go from A to B or B to A. Or essentially, we get the same elasticity of demand along this whole part of the curve. Instead of just dividing the change in quantity divided by our starting point, what I want to do is I'm going to divide the change in quantity divided by the average of our starting and our ending, points. So that's going to be 2 over-- and I'll actually do the math explicitly. Actually, no, let's just think about it. What's the average between 2 and 4. Well, the average is just going to be 3. That's the average of 2 and 4. Let me write it down to, just so it's clear. That right over here is 2 plus 4 over 2. That's how you get 3. That's how you would calculate the average. So that is our proportionate change. And then you want to multiply by 100-- times 100-- to actually get a percentage. And then, what is our change in price? Well we're going to do the same thing, or the percent change in price. Our change in price is negative 1. It is negative 1 over-- and once again, we don't just do negative 1 divided by 9, we do it over the average of 8 and 9. And the average of 8 and 9 is 8.5. And then multiply by 100 to get your percentage. Now, these 100s, obviously, cancel out. These 100s cancel out. And so we are going to be left with-- when you divide by a fraction, it's the same thing as multiplying by its inverse. So we're going to get 2/3 times negative 8.5 over 1-- or times negative 8.5. I'll get out our calculator and it is-- well, multiply 2 times negative 8.5, and then divided by 3, which gives us negative 5.6667. It's really negative 5 2/3. So I'll just write it negative-- I'll round it-- it's negative 5.67. So this is approximately equal to negative 5.67. So right over here it's negative 5.67. And this absolute value is, obviously, just 5.67. And I'll leave it to you to verify, for yourself, that you'll get the same elasticity of demand using this technique-- where you use the average as your base in the percentage. Going from 9 to 8 as going-- going from 9 to 8 in price as going from 8 to 9 in price. Which is different than if you used the 9 as the base or the 8 as the base. So this right here is the elasticity of demand-- not just at point A. You can, kind of, view it is the average elasticity of demand over this little part of the curve, which is really a line in this example-- over this part of the arc. So we'll write that part right over here. I'll write the absolute value. The absolute value of our elasticity of demand is 5.67. Now let's do the other two sections right over here. So let's think about what happens when we go from C to D. So our elasticity of demand there. So from C to D we have a change in quantity, once again, of plus 2. And our change in price, once again, is negative 1. But we'll see, even though that the change in the quantity over-- the change of quantity is the same, and the change in price is the same, we're going to have a different elasticity of demand, because we have different starting points. Our starting points and our ending points for price are lower and our starting and ending points for quantity are higher. So it will actually change the percentage. So let's see what we get. So our percent change in quantity-- we have a change in quantity of 2. And then our average quantity is 9 plus 11, which is 20, divided by 2 is 10. All of that over percent change in price. So we have-- let me scroll down a little bit-- negative one divided by the average price. So negative 1 is the change in price. And we want to divide that by the average price. Well, $5.50 plus $4.50 is $10-- divided by 2 is $5.00. So the average is $5.00. And we can multiply the numerator by 100 and the denominator by 100, but that won't change anything, because we could just divide both by 100. And so this is equal to 2 over 10, times-- dividing by a fraction is the same thing as multiplying by its inverse-- times negative 5 over 1. And this is just because 2 over 10 is the same thing as 1/5. 1/5 times negative 5 over 1-- it is negative 1. So this right over here. So our elasticity of demand right over here is negative 1. Or it's absolute value is 1. So the absolute value of the elasticity of demand, right over here, is equal to 1. Now let's just do one more section, and maybe, the next video we can think a little bit about what it's telling us. So let's do this last section over here, just for some practice. I encourage you to pause it and try it yourself. And so we're going to think about this section right over here. So once again, our change in quantity is plus 2, and our change in price is negative 1. And our elasticity of demand-- change in quantity-- 2 over average quantity, which is 17. Change in price is negative 1 over average price-- 1 plus 2 divided by 2 is $1.50. Or $1.50 is right in between these two-- divided by $1.50. We don't have to multiply the numerator and the denominator by 100 because those just cancel out. So we get 2 over 17, times negative-- well, we could just write this as negative $1.50 over 1. And this is equal to-- getting our-- getting our calculator back out. So this is equal to-- I'll just write-- well, it's really just going to be negative 3 over 17, right? 2 times negative $1.50 is negative 3 over 17. So negative 3 divided by 17 is equal to, I'll just say, negative 0.18. So here it is, negative 0.18, and its absolute value is 0.18. So the elasticity of demand over here is 0.18. And I'll leave you there, and in the next video we'll think about these results a little bit.