A study of present value maximization for the monopolist problem in time scales A STUDY OF PRESENT VALUE MAXIMIZATION FOR THE MONOPOLIST PROBLEM IN TIME SCALES PRESENT VALUE MAXIMIZATION OF MONOPOLIST IN TIME SCALES

Abstract

ABSTRACT The present value of the investment of the monopolist for the continuous case is given by P V = ∞ 0 e −rt q(t)p(t)dt with demand condition p(t) = f (t) − q(t) − a 1 q (t) − a 2 q (t),(C), and the present value for the discrete case is We will discuss various conditions and possibilities of maximization of present value of a monopolist. Basically we are focused on the Continuous (C) and Discrete (D) cases. In the (C), there is an exponential approach of growth if and only if a 2 = 0. The boundary conditions in (C) generate some mathematical issues. The first derivative of the quantity q (t) has finite jump at t = 0. If a 2 = 0 then the jump is similar to the jump of q(t) at t = 0. If the sufficient condition for the C problems are satisfied, then the demand equation is unstable. Finally, in (C) the maximum positive discount rate depends on a 1 and a 2 that yields finite maximum present value. In (D) we do not need to have any adjustment as long as α = 0. The sufficient conditions for maximum present value are satisfied for all t ∈ (0, 1). The optimal path is uniquely determined by the boundary condition and the choice of discount factor β. The stability of discount factor is a major player in (D) problem. The stable demand condition implies the existence of bounded finite maximum present value for all β ≤ 1. General results for time scales with right dense points are established. A study of issues that arise when unifying (C) and (D) is included in the analysis