5.9

## Darden School of Business, University of Virginia

Skip to 0 minutes and 0 secondsSo what is log. It's not this big block of wood that people carry, at least in our context it is not. But we just saw that log is this transformation that we do that's gonna give us, pay us some dividends when we calculate elasticity. We saw that if you use a log log model the coefficient directly gives us the elasticity. Now let's see why that is the case. So in the table here I have world population from year one to year 2000. And here is a population in millions. We went from 170 million to 6080 million in 2000. On this column here, we have log of population. So log of 170 is equal to 5.14.

Skip to 0 minutes and 56 secondsSo what log does is that it allows us to compute change. Right, rate at which something changes. That can be seen in this chart right here. What I have plotted is on the left side, I have population in millions and these blue dots plot the population in millions over time. So you can see that that is a nice hockey stick in world population. So around this time period here between 100, 1500 and 2000, world population actually took off. You get a nice hockey stick. There was something happening there. So what was it that was happening. Maybe the log transformation will help us understand. So if you see in the log transformation out here, this plots log of population in millions.

Skip to 1 minute and 52 secondsSo what you see here is something happened from around between 1500 and 2000. But even the rate of change in world population increased. That is the world population started growing at a faster rate between 1500 and 2000. So this is the kind of conceptual value that the log transformation provides, when you convert something into a log. Now what does that mean for regression and elasticity? So the key thing here is, the first difference of natural log gives you percent change. So what I have plotted in the graph here is in the blue line, it is percent change in dollars spent and the green is difference in log of dollars spent. OK.

Skip to 2 minutes and 47 secondsSo if you see here, it's not if, you are not able to make a difference between the blue and the green line here. That really tells you that percent change in dollar spent is the same as difference in log dollars spent. So if you take a log of dollar spent and take the difference between two values of log of dollar spent, that is equal into percent change in dollars spent. And why is that important for elasticity. Let's think about this. Elasticity is percent change in sales for a percent change in price. Now what is that thing?

Skip to 3 minutes and 35 secondsThat thing is pretty much the coefficient of price in a regression of log of sales on log of price and that is because the coefficient tells you, difference in y over difference in x. Now difference in log is percent change. So if you take the coefficient off a log log model it gives you percent change in y for a person change in x. Let's look at the numbers here. So let's recap. Elasticity can be obtained from Log/Log models. If you regress log of dollars spent on log of number of promotions. The coefficient here, 0.317 gives you the elasticity of sales to number of promotions.

Skip to 4 minutes and 32 secondsSo 0.317 here is change in Log spent, when log number of promotions increases by one unit or it is equal into percent change in dollar spent without that transformation what percent change in number of promo. Which is equal to the elasticity of promotion. Right. So what I have here is basically showing that the coefficient gives you this one. When log spent, what is log spent when promotion is zero. It's 2.2. Log of dollars spent when promotion is one it's 2.553. So if you take 2.553 minus 2.236. That is 2.553 - 2.236. It gives you 0.317. So I hope you are able to understand the value of the log transformation when you're calculating price elasticity.

Skip to 5 minutes and 52 secondsThis is a very important concept that would be very useful at least in marketing mix models.

# Understanding Log-Log Models

Learn how to derive elasticity from the co-efficient using a log-log model.

This is another tricky concept! Don’t hesitate to call on your fellow learners for help in the comments.