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So, to get started with the second part, what I think is this is - the really interesting thing is people just don't think about taking, now that we've gone out and thought about estimating these price elasticities or these sensitivities, people usually stop at that point. They would say, we see that what happened is 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, well, what's the decision that's to be made with this stuff. So in this case, the decision is going to be profits, or pricing.
So what we really want to do, is lets think about not the posterior of the parameters but let's think about the posterior profit function.
So in the next twenty minutes or so what I'm going to do is concentrate on lets think about the profit - the posterior profit function, and let's see what implications that would have on pricing. Now, to get there, what we're going to do, is first just define some things, like profit. You know the idea behind profit is just to go back and look at - profit is just going to back and think about lets sum up the profit for the chain, is just going to be the sum of the profit for all the stores. And then the profit of each store is going to be sum billed for a year, in this case. We're going to think about not just the pricing point for one particular week, but let's go back and take 52 weeks and think about this, as okay this would be a typical year.
Now the problem is, going back, and what I originally said, is now you've got to make like quarter of a million decisions, which is way to much for this model. Not in terms that its too difficult, it's just lets go back and think more conventionally about how would a retailer try to implement these strategies. So, what a retailer would probably want to do is think about, well, how could I use this stuff to try and set my average, every-day price. So, as I said, there are these feature prices, and feature prices tend to be distributed to everybody. So everybody should xxxx what's being featured in price and you've got to offer everybody the same deal. What is different though, is this every-day pricing level. So if I go back and put up this plot about what the prices that I'm going to be concerned about changing, essentially what I'm saying is, okay, 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 what I'm going to do, is I'm going to say, well, what I'm interested in isn't changing these bottom prices here, but what level should the top prices be? And I'm just going to think about, well, let's move all the prices up or down, should we move them up by 10 percent or move them down by 10 percent? So, 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 decision and it makes trying to set these things at a practical level next to impossible. So, what the retailer just wants to do, is, okay, I've got my base pricing strategy, now how do I go out and try to customize each of these stores? So, the way to do that is to make this just into this pricing decision about, well, what should this price index be? So, I'm going to have this base price level and I've got to have some kind of modifier, and its 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? So, what the point of doing it this way is, 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, you know, then I could discount each individual product in each of the stores. So, I did away with time aspect of the problem and just tried to focus on the overall implications in terms of setting an everyday price.