So now let's take some of that psychology that we just learned and apply it to pricing. As you might imagine, that psychology really does apply to product line pricing and I want to go through how it's actually going to work. So if you think about product pricing, you can really decompose it into three questions. What am I going to price my most expensive product? What am I going to price my least expensive product? And then, what am I going to do with all of those middle products, those intermediate products? So, here's the way we're going to tackle it. We're going to start with the intermediate products. Now, just hold onto the minimum product and the maximum product. We're going to get there, but right now we're going to go through the intermediate products, because those are the products that most closely match the psychology that we were just learning. So, here's how you do it. First, you rank the products in ascending order of expected prices. So that literally means if you've got six products that you think are going to range from cheap to expensive, you kind of put them in order where the first product is the cheapest. The second is the next most expensive. The third is the next most expensive and so forth. Then you determine the low-end price and the high-end price. Now, just take that as a given, for a moment. We're absolutely going to talk about how to set those prices, but for right now, just take it as something that's been decided. And then we're going to use those prices, those pmin and pmax, along with the number of products in your line, to determine what those intermediate product prices should look like. And here's how you do it. We're going to calculate Pj. That's the fundamental thing that we're going to be calculating. Now, what is Pj? Well, that just means either P1, P2, P3, whatever product it happens to be along that quality spectrum where, as the numbers go up, the quality goes up and the prices go up at the same time. So this is what we call the jth ordered product. And Pj is going to be equal to P min, whatever that lowest price is, multiplied by K to the j minus 1. Now this K ends up being key. K is for key here, right? This is the thing that we're multiplying the prices by to get the next price. And where do we get that? Well, it's kind of mechanical, but it's mechanical in a way that matches the psychology and that's what makes it beautiful. To get K, this is what you've gotta solve. Log K equals 1/n-1. And remember, n is just the number of products in your product line. Multiplied by the log of (Pmax) minus the log of (Pmin) and remember, for purposes of our discussion right now, you have Pmax and Pmin. So, how do you really do this? Okay, I'm going to go through an example of this. But I do think it helps for you to get some intuition of why you do a messy equation like that. because you know, that's a messy equation, right? The reason you're doing that is because you want to develop product line prices that don't just increase by constant dollar amounts. It turns out that doesn't make sense. Just like the psychology we talked about, if you have very low priced products then very small changes in prices will be detected. It's like the lengths of those lines that we were just looking at. So low prices mean low changes up or down. But at very high prices in the product line, it takes a much larger stimulus, in this case the stimulus is a price, it takes much large price increases or decreases for those to be noticed. And what the mechanics will show you, that messy equation is the best way to set up the product line prices is not in constant dollars, constant dollar increases. No, we are not going to do that. We are going to do constant percent increases because that matches the psychology. And that messy equation that I just showed you is going to give you this constant percentages, so let's do an example. Suppose you have six products in the product line. And you set the maximum price to $150 and the minimum to 25, and remember, we'll talk about how to get those. What should the price of the other products be? Well, here is the mechanics. First, I've gotta get this K. And how do I do that? Remember, log K equals 1/n-1. It's from the equation. Here we have six products in the product line so the denominator is 5 and we have log of Pmax minus of log of Pmin. So log150- log25. That going to get you logK = .1556 and then to get the K, because this is a log. All you have to do is take ten to this number in the equation and that will give you k. So in this case, k is 1.43. And that k is important because that's the thing that we're going to be multiplying the prices by to get the next subsequent price. So in our example, P1 is $25, that's the lowest priced object, the lowest priced product. How do I get P2? Well, to get P2 I just take P1, I drop it down and I multiply it by this K which is 1.43. Now I bet you can see what's going on here. I bet you can. What we're doing is we're taking the price and we're marking it up by about 43%. That's what's going on here, right? 1.431 and that's giving me 35.78 for the next price. Now get to P3, what do I do? I take P2, I drop that down and I multiply that by that same K. So I'm taking the previous price, boosting it up by 43% to give me the new price, and so forth, all the way up to P6, which again is $150. So that's how we kind of do the mechanics of it. Now, let's do another example because I kind of really want to drive this home and make sure you get the intuition. So, let's say you're selling refrigerators and you've determined there is going to be seven in the product line. The lowest is going to be priced at $500 and the most expensive at 5,000, so this is a pretty big, broad product line. So how do you do it? There's our log(K). We've got n- 1, in this case n is 7. Okay, then we have log of Pmax. That's going to be my $5,000, minus log of Pmin and that is my $500, and if I take that and plug it in there's my nice refrigerators all in a row. From a cheapy to a very expensive one, I had log K = 1/6 because there's seven products in the line. Here I've substituted the max and the min prices, which reduces to log K = 1.67. And I do just what I did before. I just take 10 to that number which will give me K, which is 1.468. And what that means is to get the next price in the product line, we're going to be marking it up by about 47%, now I'm rounding a little bit, that's 46.8%. But there's my constant markup percentage. So let me apply that to the problem, so here's my K, 1.468. P1 is given at $500. To get P2, I take the 500, I drop it down to here. I multiply it by 1.468, I get $734 and then what do I do? 734, drop it down, multiply it by the K again. I get $1,077.51. And so forth, all the way up to P7. Now look, when I get to P7 it says $5,004.10. Well, didn't I say at the beginning that the maximum price was $5,000? Yes, I did. What's going on here? All this is is rounding, right? Because I just took the K out to three decimal places instead of like ten decimal places. So when you get to the end of this calculation, it could be that you're a little bit off from the max price because you're truncating this K. You're essentially rounding it. Now, this brings up a broader point, I think, in product line pricing and that is you get like P3 is $1,077.51. Are you really going to price that? Are you really going to set the third one to $1,077.51? You know? Probably not. That just doesn't look like a good price that you would see at the store. A better price might be $999 or $1099. So, you can't let the mathematics override your common sense, in terms of setting the price. But the mathematics can be a very good guide, because if you follow the mathematics reasonably closely, what you'll get is product line prices that consumers are happy with, that seem more normal, more reasonable to them because those prices match their underlying psychology.