Simultaneous and dynamic price, production, inventory and capacity decisions

Abstract

This paper concerns a profit maximizing firm that derives the optimal price for its output, level of output, level of inventory, and composition of productive capacity over time. First, it is assumed that capacity may be acquired to either increase or replace the firm's existing productive capacity. Second, it is assumed that the acquisition of new capacity causes a reduction in the firm's production cost. Lastly, it is assumed that demand, which is expressed as a function of time and price, must be met either from current production or from inventory. Clearly, the level of production cannot exceed the productive capacity at any time. The optimal control model formulated requires the solution of a two-point boundary value problem so that a discrete approximation numerical procedure is presented. Numerical results indicate that due to anticipated technological improvement, the acquisition of new capacity is postponed and occurs in greater magnitude later in the planning horizon. In addition, as a result of upgrading productive capacity through the acquisition of new technology, numerical results show that a firm lowers its per unit production cost and thereby reduces the optimal price charged for output which increases demand. Furthermore, since the level of capacity is increased due to the purchase of new technology, less reliance on inventory is necessary to meet peak demand. Lastly, it is shown that a larger firm optimally acquires a greater level of new capacity over the planning horizon to reduce its operating costs, and thereby further its competitive advantage.