Greetings. In a previous video, we introduced the concept of price elasticity. It's a way for us to think about this responsiveness of quantity demanded to changes in prices. Today we want to think about price elasticity along a linear demand curve. Why? Because linearity makes it easier to do, we don't have to worry about solving quadratic equations because we have some curve for functional form. Since we're dealing a lot with linear demand curves in this module, in this whole course, it'd be good to figure this out in more systematic way. I want to start by introducing another graph here. Remember last time I talked about something called inelastic demand curves and elastic demand curves. So I'm going to draw a couple of curves here that would fit that type of explanation, okay? So this is price. Price is vertical. Over here, we're going to have this and we're going to call this market 1, and over here we're going to have a different market, market 2. So one of them could be, for example, milk, and one of them could be gasoline, whatever. What I want you to think about is this idea that in this particular case, when you draw a steep demand curve, we'll call that the demand curve for market 1, versus a relatively flat demand curve, which we'll call this the demand curve for market 2, this picture gives you a very strong feeling that this is inelastic. Remember what the definition of elasticity was? Just to remind you in my little thought bubble, elasticity was the percentage change in quantity over the percentage change in price. In this particular first panel, will be called market 1, you can have wild changes in price with very little change in quantity. So the denominator is much stronger than the numerator, and that means that inelastic. The other hand over here on this picture, this picture since relatively flat, you know that you can have very small changes in prices, that's the denominator, can cause big swings in quantity demanded. So this looks to us like an elastic demand curve. That would be, I think, a very healthy supposition on your part to think of that. In fact I'm going to encourage you to do that by the time we end this video. But right now we have to step off the edge of the plank and say, "Let's really work this out and see if we can figure out what's happening here Larry." The only way we're going to do that is we're going to look at a picture. Let's assume that we have a linear demand. So assume we have a linear demand curve, that's fine, we've been doing that since we started, and so I'll draw that linear demand curve right here, got a price, we got quantity, and we've got some demand curve that looks like this, D_0. Now, let's think about our definition of elasticity. Wait a minute, I'm going to make it a little bigger. Elasticity was equal to the percentage change in quantity over the percentage change in price. That's the same thing as saying it's equal to the change in quantity divided by base quantity, over the change in price divided by base price, right? All I did is I expanded that numerator out. Percentage change in quantity would be what's the change in quantity divided by base, percentage change in price would be what's the change in price divided by base price. But that's the same thing as saying this is equal to change in Q over change in P times P over Q. Nothing fancy, no fancy algebra here, just simple simplification of this algebraic form. Well, now we have something to talk about. Let's think about this term. What is this term? Well, those of you who are looking at this will back me thinking of what I think that's the slope of that curve. Means that change in quantity over change in price. Well, if you remember what slope, depending how you might have learned slope, you may have done it strictly with a derivative out non-calculus or you may have learned the rise over the run, whatever, this is actually the inverse of the slope. Because slope would be change in P over change in Q. Change in P rise over the run, change in P is the rise, the run is how much it goes like this. Oops, I need to turn my pen on. This is the inverse of the slope of demand. Now, what's a certain property of the slope of the demand curve? Well, the slope of the demand curve, because it's linear, remember what we did, we assumed linearity, what's linearity mean? A linear function means that the slope of the function is a constant. That's what linear means, it has a constant slope. Well, if it has a constant slope but also has a constant inverse of the slope. Inverse of the slope is just one over the slope. So this is constant along the entire demand curve. That number, which is just part of our elasticity, this first term of our formula for elasticity is constant across at any point on the demand curve. But now let's think about this other point. What about this value? This was a ratio of the vertical number over the horizontal number because of the fact that the curve is downward sloping. This changes at every point on a linear demand curve. So our formula for elasticity tells us that elasticity for a linear demand curve, elasticity is a product of two numbers. One number is the inverse of the slope which is a constant, the other number changes at every point along the demand curve. What that means is that on a straight line demand curve, every single point on that demand curve has a different elasticity. Now, before you say, "Wait a minute, now you're really confusing me and why are we doing this?" Well, it's important that we get down every point along there. Every point along the demand curve has the same value for the first term, but has a different value for the second term. Instead of pointing with mice I should be pointing with my laxer. This is always constant along your demand curve, this is always changing along your demand curve. So elasticity as a whole is changing along a demand curve. But the good news is that we have a systematic way that has changes, and so I'm going to draw another graph curve, this would be the last one for here. I put this here and I put this here. Price on the vertical axis, quantity on the horizontal axis, we'll draw a straight line demand curve. This is a straight line demand curve, and along that linear demand curve, I could prove this to you, and if you want to do it yourself, I can show you a link of where to look it up in a textbook, how to do it, but we're not or you're not going to have to make this proof to proof is through what we call concurrent triangles. But if you were to pick the midpoint, let's call this point right here the midpoint of the demand curve, I can prove to you that every point in the upper half of this demand curve is elastic and every point in the lower half of this demand curve is inelastic. Again, it's not a very difficult proof, you don't even have to do calculus, you just do some analytical geometry, and you can make this proof work. But what this means is the following; it means that anytime you have a straight line demand curve, exactly half the points are going to be elastic and half the points are going to be inelastic. Now, I want you to know that because it's important that that's the fundamental rule and it's an easy proof. But the good part is, earlier, I showed you these two graphs and I said, "Hey, intuitively, when you see a steep curve, you should think that as inelastic." Like say this curve is pretty steep, and intuitively, when you see a relatively flat curve, you should think elastic. Now, was I tricking you? Your first response is well, Larry I know this is a linear demand curve, every point along that demand curve is different elasticity. In fact half of that demand curve is going to be elastic and half inelastic. So that's true. But see, think about what I really drew you here. In this particular case, I'll draw another picture just so I have a little bit more room. I drew that curve looking like this, okay? I drew that other curve, I should label this axis because otherwise I'll get mad at you when you don't label yours. I drew this, as this is the demand for market 1 and this was the demand for market 2. But see, really all I've done is show you the lower part of what would be a full linear demand curve. The upper half up here of course would be elastic and the lower half would be inelastic. Here, I've shown you that top half of a curve that would go way out over there. So all you're seeing is the upper part of that curve and as we showed earlier, the top part is elastic and the bottom part is inelastic. So don't worry, there's no reason to trying and trick you. I want you to keep in your mind the idea when you read a study that somebody produces that says that demand for gasoline is very inelastic, think steep curve. Even though you know that if it's a linear curve, there's always going to be a portion where there's elastic, it's also elastic not just always inelastic, for a linear demand curve. Finally, one of the things that we could do, and we won't in this course, but economists can devise lots of nonlinear demand curves. This form for elasticity is the same, it's just that if it's a non-linear curve, then this term, which is the inverse of the slope, will change throughout, and this term will change throughout. In fact depending on how you draw it, some of these curves can be done what we call constant elasticity demand curves. Because you can devise the slope of the curvature to be exactly offset this number so that anywhere you are on that curved demand curve has a constant elasticity. Okay. We'll do more about elasticity when we start thinking about generating revenue later in the course. Thanks.