Analysis of a multi-period pricing problem

We consider the pricing decision of a firm over a planning horizon (T) in a market with a finite potential population (M). New consumers enter the market every period according to a diffusion process. Consumers differ in their willingness to pay and form expectations about future prices. The credible prices in this system have to be recursively (over time) profit maximizing prices or subgame perfect equilibrium prices. The problem of determining the perfect equilibrium prices is equivalent to the problem of determining a price path in a network of consumer states that satisfies a set of subgame properties. The problem of determining the exact subgame perfect prices involves a state space that expands exponentially in the planning horizon (T). We present an aggregation procedure that generates approximate price paths that satisfy the subgame properties in an aggregated state space. We provide computational results for the constant rate of entry and diffusion of customers to the system that justify our claim that the procedures are fast and provide good approximations to the actual subgame perfect prices in the system.