# “Why does the supermarket only carry Tree Crap??”

I’ve often noticed that the supermarket next door to my parent’s home has very poor inventory control. They buy huge quantities of brands that not many people buy (seriously? half an aisle of Goya beans?), while they quickly run out of their meager supplies of the good stuff, and fail to replace it – obviously, if they sell not so much of something, they should not carry much of it – and so the cycle of unnecessarily low sales continues.

So why the heck does this happen? How are the store managers so bone-headed as to think that people actually want to eat Goya beans? Why can I only buy Tree Crap orange juice when I would like Tropicana? Don’t they want to make more money?

Perhaps we could explain the problem as one of an extensive-form game, in which the first period sees the consumer choosing whether or not to buy the product; the second period sees the seller keeping or replacing the product with a different brand; intuitively, the seller will choose to keep the same brand if the consumer buys in the previous round, and replace otherwise. We then repeat this indefinitely, with a discount factor . For the lay people, the discount factor indicates how much one cares about future times one will have to consume Tree Crap or Tropicana.

We don’t assume that the seller is all stupid, in that they know which one commands a greater price and still won’t sell it, but that they are only mostly stupid; and there is a big difference between mostly stupid, and all stupid. We assume that they can only tell how much people like the item by whether they are willing to buy it when it is sold. Now, if they actually had the brains to do a SURVEY or something (God, why can’t they do something so simple?), then this would be a moot point, as they could easily figure out which was more liked, but as it is, they can’t figure that out since they ain’t got no brains. (Did someone say brains? mmm….)

Let the net value of Tree Crap to the consumer be , and the net value of Tropicana to the consumer be . If

is sufficiently small, so that
,
the payoff to the consumer from buying the Tree Crap:

is greater in each period than from sitting it out and waiting for the next time for Tropicana:

– ya just gotta get yer orange juice, even if it’s Crappy.

Now of course, there is more than one consumer involved in the supermarket, so the model above isn’t quite right. Besides, if there were only one consumer, the supermarket wouldn’t be able to make much of a profit now, would it?

So let’s change what happens in the first period to adapt to these circumstances. We still consider two brands, Tree Crap and Tropicana, whose values,  and  (respectively) to consumers are the same across all N of them. We set it so the seller only provides the same brand in the next period if K of the consumers buy the item, where 0 < K < M. In non-technical terms, this means that the seller only keeps on selling Tree Crap if enough people keep on buying it.

Consider the following three cases:

(1) Suppose less than K – 1 other consumers are willing to purchase Tree Crap, and the rest refuse to stoop to that level. Then from the perspective of the individual consumer, he might as well purchase Tree Crap in that period, since his purchase will not lead to the seller providing Tree Crap in the next opportunity to buy orange juice.

(2) Suppose exactly K-1 other consumers are willing to purchase Tree Crap. Then the situation for the individual consumer is exactly the same as that in our initial case with only one consumer, since he makes the difference between the seller offering Tree Crap or Tropicana in the next period. Thus he will buy if and only if
.

(3) Suppose at least K other consumers are willing to purchase Tree Crap. Then the individual consumer might as well buy the Tree Crap, since he’s doomed to more Tree Crap next time, too.

Notice that situation (3) does not depend on . Thus, no matter how much one cares about future opportunities to buy orange juice, it is a Bayesian Nash equilibrium to buy Tree Crap in every period if everyone else is, too. Which of course means that the seller will keep on selling it. Why can’t they just do a frikkin’ survey???