Capacity Expansion and Probabilistic Growth

one is concerned with the interplay between economies of scale and an anticipated persistent growth in demand for capacity. The generalizations discussed here are of two types: (a) the use of probabilities in place of a constant rate of growth in demand; and (b) a study of the economies and the penalties involved in accumulating backlogs of unsatisfied demand. The possibility of accumulating such backlogs raises considerable doubt with respect to Chenery's excess capacity hypothesis. Surprisingly enough, generalization (b) leads to greater difficulties in analysis than (a). The use of probabilities to describe the growth process does little-if anything-to complicate matters. A probabilistic version of Chenery's model turns out to be closely related to the classical problem of gambler's ruin, and a powerful tool can be borrowed from that area-the Laplace transform for the duration of the game. Thanks to this transform, the zero-backlog probabilistic model becomes no more difficult to study than the corresponding deterministic one. A direct implication is that a probabilistic growth course makes it necessary to incur higher expected costs, and also makes it desirable to install plant capacity of a somewhat larger size than would be optimal if demand were growing at a steady rate equal to the expected value of the probabilistic increments. Uncertainty, in this sense, has a stimulating effect upon the magnitude of individual investments.