| Abstract |
The present value of the investment of the monopolist for the continuous case is given by
$$
PV=\int_0^\infty e^{-rt}q(t)p(t)\,dt
$$
with demand conditions
\begin{equation}
p(t)=f(t)-q(t)-a_1q'(t)-a_2q''(t),
\tag{C}
\end{equation}
and the present value for the discrete case is
$$
PV=\sum_{0}^{\infty}\,\beta^t q_tp_t
$$
with demand condition
\begin{equation}
p_t=f_q-q_t-\alpha q_{t-1},
\tag{D}
\end{equation}
provided \(0 < \beta\le 1\).
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 (C), there is an exponential approach o9f growth if and only if \(a_2\ne 0\). 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 conditions for the problems in (C) 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 \(\alpha\ne 0\). The sufficient conditions for maximum present value are satisfied for all \(t\in (0,1)\). The optimal path is uniquely determined by the boundary condition and the choice of discount factor \(\beta\). The stability of discount factor is a major player in problem (D). The stable demand condition implies the existence of bounded finite maximum present value for all \(\beta\le 1\). |