A concept that's really important in marketing and that also has connections to regression is something called elasticity. Here we are going to look at price elasticity. What it means is, it is the percent change in sales for a percent change in price. So price elasticity is primarily change in sales over change in price multiplied by price over sales. And it is important to have this value over here, price over sales. That is different than just the coefficient. What we saw in regression was this set here, change in sales to change in price is your coefficient. Now if you take the coefficient and multiply that by average price over average sales, you would get price elasticity. Now, why are we so hung up on price elasticity? Why do we need elasticity? It is because elasticity has no units. It's unitless, which means every year you can measure elasticity, and track this elasticity over time so you can compare improvements or declines in the effectiveness of your market. So that's why it's a really useful concept to know. And we can see how it connects to regression through this value, the coefficient. But is there a way to modify, tweak the regression a little bit to just use the coefficient directly and it will be equal to elasticity? Let's see. So we're going to take the example of Belvedere Vodka. So far we have looked at made up numbers and you can say, hey, you're talking about made up numbers, I work with real people with real data give me some real examples. So here we go. What we have here is data from Belvedere Vodka over seven years in the US. So the data is from 2001 to 2007. We have sales of 9 liter cases of Belvedere Vodka and this is thousands. And what we are doing here is taking from this column to this column here, we're taking the log, which is the logarithmic transformation of sales, and we are going to look at logarithms and what they are in a short while. But for now, stay with me to understand that logarithm is a transformation that we make on the data. Now, we take a log transformation, so sales is 410, log of sales, is 6. This is price of 9 liter cases of Belvedere Vodka, $215. And log of that price is 5.3, and we have advertising how much advertising was done for Belvedere Vodka, and this is the log of that advertising value. And advertising is also in dollar. Thousands. So this is real data. So far, we have looked at made up data and you could be thinking, wait a minute you are showing me all this with made up data, but I am dealing with real people with real consumers in the real world. And does this regression apply in there. So here we have it, this is real data about Belvedere Vodka sales in the US and this is real data about the prices the managers at Belvedere Vodka set and how much they advertise in each of these seven years. So we're going to take all of this data and see the relationship between price and sales of Belvedere Vodka, and apply it into a regression model and come up with values that will then give us a relationship between price and sales. So let's see what we got here. So here is the output. Of the regression of log of price of Belvedere vodka on log of sales of Belvedere vodka. So let's see what the regression output gives us. So the R-squared is about 45%, this is how the data looks like. In the x axis, we have price and in the y axis, we have sales. And to be specific, we have log of price and log of sales. That's what we're plugging into the regression function. These green dots are the seven years of data and the black line is your regression equation. Just like we saw in the example. Now you know we then need to look at the coefficients and the p value. So the intercept is 12.68. So, if price was 0, that would be awesome. Free vodka, we all like it. If price was 0, then case sales, log of case sales of Belvedere Vodka is 12.6, the P-value is less than 0.05, which means, what does it mean? Think about it. Which means, if fee value is low, high confidence in the regression, right? So that's a good thing. Now next thing is Ln(Price). Coefficient is -1.25, P-value is less than 0.1, which is okay. It is still lower than the threshold of 10%. It's not great but it's good. It will do for now. But what do we have here, this is the coefficient, this the slope, this is the change in ln sale for change in ln price and that's the coefficient right here. Now here's the kicker, when you use ln of price as x and ln of sales as y the coefficient is the same as price elasticity. So now, by doing the log transformations on the x and y, and using that in the regression, you can actually just do the regression. Pick up the coefficient, and that gives you the elasticity, isn't that cool? Now we are going to see, very shortly, why that is the case. Why does doing what is called a log-log model give you elasticity?