Sethi model

The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market via ${\displaystyle r}$, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market via ${\displaystyle \delta }$, and a random effect using a diffusion or White noise term that can go either way.

• The coefficient ${\displaystyle r}$ is the coefficient of the effectiveness of advertising innovation.
• The coefficient ${\displaystyle \delta }$ is the decay constant.
• The square-root term brings in the so-called word-of-mouth effect at least at low sales levels.[1][4]
• The diffusion term ${\displaystyle \sigma (X_{t})dz_{t}}$ brings in the random effect.

Subject to the Sethi model above with the initial market share ${\displaystyle x}$, consider the following objective function:

${\displaystyle V(x)=\max _{U_{t}\geq 0}\;E\left[\int _{0}^{\infty }e^{-\rho t}(\pi X_{t}-U_{t}^{2})\,dt\right],}$

where ${\displaystyle \pi }$ denotes the sales revenue corresponding to the total market, i.e., when ${\displaystyle x=1}$, and ${\displaystyle \rho >0}$ denotes the discount rate.

The function ${\displaystyle V(x)}$ is known as the value function for this problem, and it is shown to be[2]

${\displaystyle V(x)={\bar {\lambda }}x+{\frac {{\bar {\lambda }}^{2}r^{2}}{4\rho }},}$

where

${\displaystyle {\bar {\lambda }}={\frac {{\sqrt {(\rho +\delta )^{2}+r^{2}\pi }}-(\rho +\delta )}{r^{2}/2}}.}$

The optimal control for this problem is[2]

${\displaystyle U_{t}^{*}=u^{*}(X_{t})={\frac {r{\bar {\lambda }}{\sqrt {1-\ X_{t}}}}{2}}={\begin{cases}{}>{\bar {u}}&{\text{if }}X_{t}<{\bar {x}},\\{}={\bar {u}}&{\text{if }}X_{t}={\bar {x}},\\{}<{\bar {u}}&{\text{if }}X_{t}>{\bar {x}},\end{cases}}}$

where

${\displaystyle {\bar {x}}={\frac {r^{2}{\bar {\lambda }}/2}{r^{2}{\bar {\lambda }}/2+\delta }}}$

and

${\displaystyle {\bar {u}}={\frac {r{\bar {\lambda }}{\sqrt {1-{\bar {x}}}}}{2}}.}$