Diiusion of Innovations under Supply Constraints

In this paper we present a canonical setting that illustrates the need for explicitly modeling interactions between manufacturing and marketing/sales decisions in a rm. We consider a single rm that sells an innovative product with a given market potential and incurs lost sales when unable to supply the product due to capacity constraints. For such rms, we present a new, deterministic model of demand modiied from the original model of Bass, that captures the eeect of sales lost due to supply constraints on future demand. We use this model to plan production and sales in the rm with the objective of maximizing total sales during the lifetime of the product. We show that the trivial, myopic sales plan that sells the maximal possible amount at each time instant is not necessarily optimal. Using Pontryagin's maximum principle we show that the structure of the optimal sales plan is of the \build-up" type in which the rm does not sell at all for a period of time (even though it incurs lost sales during this period), and builds up enough inventory to never lose sales once it begins selling. We compute the optimal build-up period. Finally, we investigate the eeects of changing model parameters, the beneets of initial inventory and delayed roll-out, the impact of the model on capacity planning, and the eeect of competition.