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I would think about putting a couple of constraints on this pricing problem. In the paper I try to motivate these constraints from economic theory, but in this case, essentially I'm going to say that the average store price has to remain unchanged. And that the category sales have to remain unchanged. So if you think about this from a consumer perspective, or a competitor perspective, if my average prices aren't changing the way a competitor would look at it, it isn't the prices of each product but the overall prices that this competitor is charging. They would see that their average prices haven't changed so I don't really have an incentive to try to make a difference in my pricing strategy. From a consumer's standpoint, essentially the average price is $1.54 before I do the micro-marketing strategy, and after I do it, I'm still going to leave that price at about $1.50. In terms of category sales, I'm going to say that the category sales are not going to change. So if I've been selling 10 million dollars in orange juice before I implemented this micro-marketing strategy, I still want to sell 10 million dollars after I implement it. Essentially what I'm saying is that there's something of my model going on. I'm still trying to come up with these posterior profit functions, but what I'm going to do now is think about what's the posterior profit and what's the posterior distribution of these constrained profit functions.
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