So, just to get you the results, let's just start there.
The following table gives results.
So, lets start with a strong prior. Here again, my uniform pricing strategy, I'm still making profits of about 3.4 million. If I go to this optimal constrained strategy, what's going to happen to my yearly profits are going to be about 3.3 - about a .44 percent increase, if I'm saying that all of the stores have to follow the same pricing strategy. So remember, when I went from this uncon- when I put these constraints on I could think about having these constraints for the entire chain level, so could the store come up with a new pricing strategy for all the stores that still keeps the average price the same and still keeps the total sales the same. In that case, it's really difficult, but they can do a little better. Now suppose they go back and say that, look, all the stores can choose a different pricing strategy. So in that case, what I'd do is I'd get a 2.74 increase in profits, compare that to .44 percent increase and you see that, look, you can get a lot better with these micro-marketing strategies. You know, Dominic is concerned about a .66 percent increase in profit, and now you're talking about a 2.74 percent increase in profit, then this is really going to be important to them and you could really make a difference in their gross profits.
Again, if you look in terms of some kind of measureability, it looks like yes, these are definitely measurable. Since these constraints are - you may not think they are such a good idea, you can still think about can I come up with an optimal strategy without these constraints. Well, in this case the slope of the profit function is not exactly what I would have chosen, but to come up with it beforehand, I - there was not a lot of curvature in the profit function and you tend to move out and just increase all the prices. So, you come up with a 24 percent increase and profits would be uniform, if you do it at the micro-marketing level, you come up with almost a 26 percent increase in profits. So, what I did in this optimal unconstrained, sense that I don't have these average price, the total movement,the total revenue but I still have a constraint that says I'm only willing to pick about 10 percent, I'm going to allow my prices to changes by plus or minus 10 percent. So there still is a bound on pricing, it's just I haven't made it as great.
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Let's go back and think about, well, what exactly would this pricing strategy look like for an individual store and then lets think about what would it look like across all the stores. So, what I've done here, is I've thought about - here are my twelve brands, and let's think about the price changes. I can either decrease it by 10 percent or increase it by 10 percent. The x's denote what I would do for one individual store, so in this case, for this store I would want to increase my price by about 8 percent on Tropicana Premium, for Tropicana Premium 96 oz I would want to decrease it by about 5 percent. Then I would want to do Florida Natural about the same. I'd want to increase Tropicana xxxx by 10 percent, and on and on. So the idea is, if you average all these price effects, you'd see that about half are above and half are below. And if you weighted them by the movement you'd find out that the average prices remain unchanged. And on top of that, I can't illustrate it, but it's still there, if you were to take the revenue from having all the prices at the xxx level and then look at the revenue from having these price changes, the revenue would also remain unchanged. So, essentially what you've done, is you're trying to give people an incentive to switch for more profitable brands for you. Go back to these questions of fairness at the beginning, though it might be unfair just to increase all the prices by 10 percent, you know, it might have some dramatic effects, but why would it make any effect if I changed the price gap between Minute Maid and Tropicana in some stores and other stores I leave them together, you know, it doesn't seem to have a lot of difference. Now, what I've done to the box plots is, let's just illustrate what's the overall change across the all the stores. So, in this case, Tropicana, it can go anywhere from decreasing by a percent to increasing it by 10 percent. So the idea is that you see a lot of dispersion in - across the stores and what would be a good pricing strategy for them to follow.
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Real quickly, let me summarize some of the things that I think are really compelling about this. You started out thinking about trying to measure store elasticities, or these individual parameters, but I think this hierarchical model is really elegant. It really summarizes how we should do it, and from a frequentists perspective it really would have been difficult - not impossible - it would have just been very arbitrary - how would we have said what are these relationships. It would have been difficult to estimate them.
Some conclusions are given here.
The next problem is really thinking about taking this problem, this measurement perspective of price elasticities and then mapping into the profit function and then thinking about what's the consequences in terms of pricing policies. So it really puts it back into the retailer's decision framework. Some future directions are given here. So, I will now finish up and open it up to discussants.
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We're very fortunate in having two highly qualified discussants. The first discussant is Wagoner Kamakura and he is Professor of Business at the Katz School of Business here at Pitt. Wagner has actually pioneered the use of mixture models in marketing, so a particularly appropriate person -