Let's do another example with percentage changes. This time we're going to look at something called an elasticity. Now, an elasticity by definition is a percentage change. It's a ratio of percentage changes. The price elasticity of demand is the percentage change in quantity, relative to the percentage change in price. Now, recall that a percentage change is the new value of something minus the old value of something, as a ratio of the old value. So the percentage change in quantity is quantity 2 minus quantity 1, as a ratio of quantity 1. The percentage change in price is price 2 minus price 1, as a ratio of price 1. Now, I am going to simplify. By simplifying, what I mean is I've got this quantity divided by a set of prices, price 2 minus price 1, as a ratio of price 1. When I divide by something, I can actually multiply it by the reciprocal. So when I do that, that price ends up on top, price 1, and the price 2 minus price 1 ends up on the bottom. Then, what I do is I do a little bit of algebraic manipulation, and it turns into this last part of the equation here, price 1 divided by quantity 1 times Q2 minus Q1 divided by price 2 minus price 1. So these are percentage changes. In order to actually calculate elasticity, we're going to need to identify two points on this demand schedule and identify one of them as the starting price, the second as the second price, and a corresponding quantity 1 and a corresponding quantity 2. Let's choose these two points right here. Price 7 and price 6, and quantity 9 and quantity 12. Now, we have to identify which one is price 1 and price 2. So let's say this first one right here, that's price 1. Then this second price over here, that's going to be my price 2. Now, I've got my corresponding quantity 1 and quantity 2. This top quantity over here, that's going to be quantity 1, and then this second quantity down here, that's going to be quantity 2. Now, we're going to use those pieces of information to calculate our elasticity. So we're going to plug in the information here. My price one is 7, there's my price one right there. My quantity one is 9, here's my 9 right there. I multiply this by the change in my quantity. So I've got the 12 minus 9, and I've got a change in price. Here I've got a 6 minus 7. Then, I can simplify and solve. So I've got my 7 divided by my 9 times, here I'm going to have 3 over negative 1. So I'm going to end up with 21 divided by negative nine, which gives me an elasticity of negative 2.33. Pretty straightforward calculation. I want to show you that we get the same information if we leave it in the quantity 2 minus quantity 1, as ratio of quantity 1, and price 2 minus price 1, as a ratio of price 1. So let me just show you this calculation down here. So if we take quantity 2, 12 minus quantity 1, 9 divided by quantity 1, 9 divided by price 2, 6 minus price 1, 7 as a ratio price 1, 7. Up here in the numerator, we get 3-nines. Down here, we get negative one-seventh. Remember, when we divide, we can multiply by the reciprocal. So this ends up becoming 3-nines times negative 7 over 1, which equals negative 21-nines. It's the same as this. Percentage changes are not difficult. You do have to make sure that you're being very consistent. Make sure that you put the negative signs in as you need to. Don't be frightened or alarmed by the fact that you might be subtracting a larger number from a smaller number, that happens a lot. So your job is just to stay consistent and follow through with the patterns and the rules that we've discussed. It's not that challenging, and it'll be incredibly helpful.
