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To get started with the second part, the really interesting thing is people just don't think about going further -- now that we've thought about estimating these price elasticities or sensitivities, people usually stop at this point. They say, we see that we can estimate these things more efficiently, or the standard errors are smaller, so this is a better way to estimate things. The problem is that people want to think about what the decision is to be made with this stuff. In this case, the decision is going to be profits, or pricing.
What we really want to do is not think about the posterior of the parameters but about the posterior profit function.
During the next twenty minutes or so I'm going to concentrate on thinking about the profit -- the posterior profit function -- to see what implications that would have on pricing. First we need to just define some things, like profit. Profit is thinking about, let's sum up the profit for the chain; it's just going to be the sum of the profit for all the stores. Then the profit of each store is going to be sum billed for a year, in this case. We're going to think not just about the pricing point for one particular week, but let's take 52 weeks and think about this as a typical year.
Going back to what I originally said, the problem now is that you've got to make about a quarter of a million decisions, which is way too much for this model. It's not too difficult, it's just that we want to think more conventionally about how a retailer would try to implement these strategies. So, what a retailer would probably want to do is think about how could I use this stuff to try and set my average, everyday price? As I said, there are these feature prices, and feature prices tend to be distributed to everybody. You've got to offer everybody the same deal. What is different though, is this every-day pricing level. If I put up this plot about the prices that I'm going to be concerned about changing, essentially what I'm saying is that I'm going to take out a typical 52-week span, so I'm going to take the data set and it's going to look something like this. Then I'm going to say, well, what I'm interested in isn't changing these bottom prices here, but rather what level should the top prices be? And I'm going to think about, well, let's move all the prices up or down, and should we move them up by 10 percent or move them down by 10 percent? Obviously, you could think about the price for each individually, for each product separately. The problem is, if you do that, you've got a quarter of a million pricing decisions and it makes trying to set these things at a practical level next to impossible. What the retailer wants to do is start from a base pricing strategy and ask, now how do I try to customize each of these stores? The way to do that is to make this into a pricing decision about what this price index should be? I'm going to have this base price level and I've got to have some kind of modifier, and it's this modifier that we're really interested in. Should I move this modifier up by 10 percent from store one for product j or should I move it down? The point of doing it this way is that I really think about changing the everyday prices and if I were to leave this price index at one, it essentially means that everybody would have the same price. If I started changing this price index and making it smaller, then I could discount each individual product in each of the stores. So, I did away with the time aspect of the problem and tried to focus on the overall implications in terms of setting an everyday price.
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