Okay, now let's move to the land of price optimization using some demand information, right? We've looked at those elasticities, but now we want to try to find the price that really is the best given the conditions that we're in. Now, in front of you is our good old demand function. You've seen this before, right? Q = 10- 2P and I'll just graft that below. And there's a marginal cost equal to 1. And we're going to take this demand and formation and this marginal cost information and come up with the price that would be best in this situation. Now, how do we do it? Well here's what we do. First we need to write down a profit function. And when I write down pi, I mean profit. And that's true in just about every textbook you'll ever look at, if it has price optimization they'll use pi for profit, it's just a convention. But what is profit in this setting? Well, it's the amount of money you make on each unit, kind of your marginal profit per unit, or your margin, which is just (P-MC), price minus marginal cost, and then you have to multiply that by the number that you sell, right, there's that Q right there. The number you sell multiplied how much you make on each unit, that'll give you the profit. Now, what do we do with this profit function? Well the first thing we do is we substitute in the demand information. And in our case that is 10- 2P, I'm getting that straight from the previous slide, the description of that demand function. So I substitute that in, and if you notice at this point because we know what MC is, this is a function, the profit function, all in terms of one unknown, and that one unknown is price. That's what we're trying to find. So all I've done here is multiply this through and simplify it a bit. Now, what do we do? Now we gotta do some of that calculus, nobody freak out. We're going to use that simplified profit function and a little bit of calculus to come up with the best price. So here's that simplified profit function again. What are we going to do? Well, we're going to maximize it related to price. So what does that mean? We're going to take the derivative, set it equal to zero, and that, when it's solved, should give us the best price in this particular situation. So that is what this is sitting right here. That is the derivative of this profit function with respect to price. And here is that price that we're trying to solve for. All of the rest is known, including marginal cost. So I then substitute in, here I've just rewritten it, 10 over 4 + MC over 2. What does that equal? Well, once I substitute in 1 for MC, because I told you that previously, simplify 10 over 4 is 2.5, this equals one-half, P* equals 3. What does that represent? The best price in this particular situation. And we can substitute that back into our demand function, 10- 2P. And this time we substitute in 3 for P, and we can see how much we're really going to sell at that price. So that is a basic price optimization. And of course, we can calculate the profit, that's quite straightforward. Now, that's kind of theoretically how you do it. Why is this harder in the real world? Well, in the real world nobody gives you demand. No one is ever going to hand you 10- 2P. Ron, here's 10- 2P, go optimize this thing. No, doesn't happen, right? You have to come up with the demand function so that can be tricky, coming up with what that looks like. You also might sell through someone who has their own incentives. That might be the demand function at the retail level, but what if you're selling through the retailer? You set your price, they set their price. Then you might have to consider retailer incentives too, not just the demand function that you have. And finally, I'll just add, some things are illegal. Some things you are just not allowed to do. Depending on whether you're pricing in the United States, pricing in the European Union, or pricing in other parts of the world, some things are restricted. So let's start with demand, because that's a very interesting place to start. How do we get demand? Well there's a few different ways. One is we can do surveys to try to get to the relationship between price and quantity. Are we going to do that? Yes, we're going to do a case called Zero Junk Mail and I'm going to show you how to get demand information from surveys. Another way to do it is something called conjoint analysis. This is done with specialized software. We will explain that, I'll explain it to you, right? And that's a little bit more of a sophisticated approach then just a paper and pen survey. But that can also produce very interesting and useful demand information. And finally, we can use purchased data itself, right? The data that comes off of the scanners at the grocery store. The data that comes out of the information systems at companies. To try to look at, if they change price how much did their quantity change? And can we estimate the demand relationship given that real purchase information that we have? Now, in the real world if I've got purchased data and it's really good purchased data, a lot of it, that'd be my first choice. That would be it, right? because that's real market data. I'm probably going to only use things like survey data, and to some extent conjoint data, if I don't have available good data to me, and sometimes, of course, that's the case. So I will just add that this kind of scanner data that we've been working with, I've shown you the beef data before when we did the price elasticities, almost any consumer packaged good firm, Proctor and Gamble, General Mills, all the big ones and even some of the small ones, are going to subscribe to this kind of service. So they are going to have the capability to look at demand functions using scanner data. So why don't we do that, right? Instead of doing 10- 2P, let me show you how you build up a demand model using the purchased data. You've seen this before, this is exactly what I showed you when we did the price elasticities. This is that beef data, where the quantity of beef is the dependent variable. And then that's a function of the price of the beef, the price of the chicken, the income of the individuals purchasing, and the trend in the data, the actual month that the quantity was purchased. Well, here's a theoretical demand model down here. Here's my quantity and then beta1 X1, beta2 X2, etc. Well, now let's plug this information into that theoretical model to see what it looks like. So in this slide I just make sure you have a slide so that you understand what each one of these variables mean, okay? And you can refer to it. So let's just go ahead and plug it in. So I've taken this theoretical model right here and then what is this thing sitting down right below it? All that is is that instead of writing alpha, I take the term that is related to that alpha. The alpha is just the intercept term, so I just grab 779.4 and pull it down. And then here, instead of beta1 X1 the first variable I'm looking at is price, that's here. What is price multiplied by? Its coefficient from the linear regression. And that coefficient is this, well, I rounded here, it's -50.2. And so forth. Each of these numbers that you see right here just corresponds to the coefficients right above it. And notice that there's no error term at the end. That's because when you do this kind of estimation you can assume that the error term goes away, really goes to zero. So that's how you take this kind of demand information coming out of a regression, coming out of purchased data, and build up a demand model.
