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For a formal definition of the profit gradient click here.
The place that I want to start is to go back and think about the profit function. So 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. Suppose I increase all the prices by 1 percent up, what would be the percentage effect on profits? So what these boxplots do are try to illustrate what's the posterior profit function. So in this case I just want to start off small, lets just forget about micro-marketing effects and lets just think about the overall pricing effects for this store. So the question here 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? Now 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. What that would imply is I'd get about a 40% increase in profits. Now if you start going across these, what's happening with the 96 is again, they are all increased prices, the national brands increase prices, all the stuff is saying increase prices. You know, 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. So the first thing is, let's just think about in terms of what's the implication for the retailer, well the point is that, it looks like the retailer is under-pricing and what they should do is systematically go out and raise their prices. Now the point is, that if we look at across each individual brands, some brands we want to increase the price more, other brands we might want to increase them just a small amount. Now, 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 just to try and visualize what's happening at this particular point, since this is how this retailer is going to see it. This incremental process. Now, the only thing that's being illustrated here, is to go back and let's 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.
Now, 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. So most of these are symmetric, but as we go on we might see some asymmetries. So the point is that first we see an overall increase, and next what we're seeing is that there's not a lot of differences at the store level between using the strong or this weak prior.
At the chain level - yes, we have to be sure that we state that.