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For a formal definition of the profit gradient click here.
I want to start by thinking about the profit function. The profit function is just going to tell me that if I were to increase the prices, like in this case, by a small point -- if I increase all the prices by 1 percent up, what would be the percentage effect on profits? These boxplots are to illustrate what the posterior profit function is. In this case I just want to start off small, let's just forget about micro-marketing effects and think about the overall pricing effects for this store. The question that I'm asking is should we move our base price level up 10 percent, or 20 percent for a premium brand, or should we move it down? What this is going to say is that, suppose I were to have a 10 percent increase in the price of premium Tropicana 64 oz. That would imply that I'd get about a 40% increase in profits. Now if you start going across these, what's happening with the 96 is that again, they are all increased prices, the national brands increase prices, all the stuff is saying increase prices. Here we are getting some things that are close to zero, and here for Citrus Hill 96 oz. it looks like they are over-pricing it. They might want to drop the price there. The first thing is, let's just think in terms of the implication for the retailer; it looks like the retailer is under-pricing and what they should do is systematically raise their prices. The point is that if we look across each individual brand, for some brands we want to increase the price more, other brands we might want to increase them just a small amount. Again this is just a gradient, this is just thinking about it from a common point -- let's step away and move towards a better pricing strategy. We can move to the optimum pricing strategy, but before I get to the optimal, what's important is to visualize what's happening at this particular point, since this is how this retailer is going to see it. This incremental process. The only thing that's being illustrated here is to think about what's the impact of this k parameter or what's the impact of my prior on shrinkage in terms of the posterior profit function.
What you're seeing here is that the red ones denote the strong prior, the moderate prior, and the weak prior. I should also mention that the way I do my box plots, this is the 90 percentile, this is the 10 percentile, the extent of the whiskers, the extent of the boxes, the 25 and the 75 percentile, the median, the 50 percentile is the line and the dot represents the posterior mean. Most of these are symmetric, but as we go on we might see some asymmetries. First we see an overall increase, and next we're seeing that there's not a lot of differences at the store level between using the strong or the weak prior.
At the chain level - yes, we have to be sure that we state that.
That wasn't really the point of this exercise though. What we really wanted to do is think about the impact on a particular store. The chain level looks like everything should move in unison, but now let's think about the sensitivity for one individual store. What I've done here is that the x's denote what we just saw. So this is the mean of the moderate chain. Now what we are going to do is to look at one particular store. In this store, we would see that for Tropicana premium, if we increase it by 1 percent, we would see about half a percent increase in profits, so you see that in this one there's a good cross-balance between the chain decision and the individual store decision. In other cases, what you're seeing is that there is some disparity. For Citrus Hill, for the chain it looks like we want to decrease it slightly, but if we look at this individual store we see this big decrease, so here we would want to decrease the price of Citrus Hill 96 oz by more than what the chain would indicate. This is illustrating what I'm trying to get, and that is let's move from the chain decision down to this individual decision, and let's think about what some of the effects are.
What you're seeing is that here, the strong and weak didn't make a lot of difference in the chain level, but at the store level it started to make a big difference. Whether we're able to estimate these individual store levels' demand functions very well, and obviously what it's going to do is show that if we assume, or a priori think that it's very similar, that there's not a lot of random effects, obviously the profit function is going to move it together and tighten up this distribution. Whereas here, if we're not confident about whether there are a lot of random effects or not, then what's happening is this posterior profit is going to get pulled apart because we're not willing to make a lot of statements beforehand about how common the stores are.
Instead of just looking for an individual product in an individual store, what I'd like to do now is think about one product and take a cross of a lot of different stores. What we're looking at here is the chain, this was the first plot that I showed you, so if we were to increase Minute Maid by 1 percent we would expect a .4 percent increase in profit. Now let's pull out these different stores. You'll notice that there's general common tendency that everything wants to increase, but the point here is that how much to increase differs a lot across the stores. And my confidence is dramatically impacted by how strong I'm willing to make my prior assumptions about how many commonalities there are across the stores. In store number 40 I'd want to increase it a lot, in store number 53 I would increase it just a small amount. If I believed that there's a lot of commonalities, depending on what my risk preferences are, I'd -- maybe down at this tail end, just take a small step. If I'm neutral, I might want to go all the way to the mean; if I'm risk seeking, I might move all the way and follow the route into this confidence distribution.
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