Dynamic Pricing of a Status Symbol

We consider a market consisting of two populations, termed rich and poor for convenience. If a product is priced such that it is very expensive for the poor, but affordable to the rich, it becomes a status symbol for the poor and this makes it more desirable for the poor. At a lower price the product is affordable by both populations. However, as more of the poor buy the product, it becomes less of a status symbol for the poor, and simultaneously less appealing to the rich. We present a two-state non-linear optimal control problem and obtain profit-maximizing prices over time in this environment. We find that there are three categories of optimal price paths. One is status symbol pricing with high initial price, declining over time. The other two are mass market pricing, with price declining in one, and increasing and then decreasing in the other