Dynamic Pricing of a Status Symbol

Abstract:  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, then 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 ceases to be a status symbol and becomes less appealing to the rich. We present a two-state nonlinear optimal control problem that aims to 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.