Optimal Performance of a Firm in a Dynamic Market

The problem of optimal dynamic transition of a firm with market power from a predetermined initial condition to a known statically optimal equilibrium state is formulated. The model of the firm is linear, discrete time, and behavioristic in nature with production, demand, inventory, and order and sales sectors. Prices and changes in the production rate constitute the decision variables which are assumed to be bounded. The optimal control sequence subject to bounded control efforts is derived to minimize a quadratic performance index by invoking the maximum principle. It consists of maximum effort transitions followed by a series of singular steps which ultimately drive the system to the equilibrium state. Existence and optimality of the solution are proved. Two typical numerical examples are included.