# Sethi model

The Sethi model was developed by Suresh P. Sethi and describes the process of how sales evolve over time in response to advertising.[1] The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market, and a random effect that can go either way.

Suresh Sethi published his paper "Deterministic and Stochastic Optimization of a Dynamic Advertising Model" in 1983.[1] The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.[2] The model and its competitive extensions have been used extensively in the literature.[3][4][5][6][7][8][9][10][11] Moreover, some of these extensions have been also tested empirically.[4][5][8][11]

## Model

The Sethi advertising model or simply the Sethi model provides a sales-advertising dynamics in the form of the following stochastic differential equation:

${\displaystyle dX_{t}=\left(rU_{t}{\sqrt {1-X_{t}}}-\delta X_{t}\right)\,dt+\sigma (X_{t})\,dz_{t},\qquad X_{0}=x}$.

Where:

• ${\displaystyle X_{t}}$ is the market share at time ${\displaystyle t}$
• ${\displaystyle U_{t}}$ is the rate of advertising at time ${\displaystyle t}$
• ${\displaystyle r}$ is the coefficient of the effectiveness of advertising
• ${\displaystyle \delta }$ is the decay constant
• ${\displaystyle \sigma (X_{t})}$ is the diffusion coefficient
• ${\displaystyle z_{t}}$ is the Wiener process (Standard Brownian motion); ${\displaystyle dz_{t}}$ is known as White noise.

### Explanation

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][3]
• The diffusion term ${\displaystyle \sigma (X_{t})dz_{t}}$ brings in the random effect.

### Example of an optimal advertising problem

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[12]

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

where

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

The optimal control for this problem is[12]

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

where

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

and

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

## References

1. ^ a b c Sethi, S. P. (1983). "Deterministic and Stochastic Optimization of a Dynamic Advertising Model". Optimal Control Application and Methods. 4 (2): 179–184. doi:10.1002/oca.4660040207.
2. ^ Vidale, M. L.; Wolfe, H. B. (1957). "An Operations-Research Study of Sales Response to Advertising". Operations Research. 5 (3): 370–381. doi:10.1287/opre.5.3.370.
3. ^ a b c Sorger, G. (1989). "Competitive Dynamic Advertising: A Modification of the Case Game". Journal of Economic Dynamics and Control. 13 (1): 55–80. doi:10.1016/0165-1889(89)90011-0.
4. ^ a b c Chintagunta, P. K.; Vilcassim, N. J. (1992). "An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly". Management Science. 38 (9): 1230–1244. doi:10.1287/mnsc.38.9.1230.
5. ^ a b c Chintagunta, P. K.; Jain, D. C. (1995). "Empirical Analysis of a Dynamic Duopoly Model of Competition". Journal of Economics & Management Strategy. 4 (1): 109–131. doi:10.1111/j.1430-9134.1995.00109.x.
6. ^ a b Prasad, A.; Sethi, S. P. (2004). "Competitive Advertising under Uncertainty: Stochastic Differential Game Approach". Journal of Optimization Theory and Applications. 123 (1): 163–185. doi:10.1023/B:JOTA.0000043996.62867.20.
7. ^ a b Bass, F. M.; Krishamoorthy, A.; Prasad, A.; Sethi, S. P. (2005). "Generic and Brand Advertising Strategies in a Dynamic Duopoly". Marketing Science. 24 (4): 556–568. doi:10.1287/mksc.1050.0119.
8. ^ a b c d Naik, P. A.; Prasad, A.; Sethi, S. P. (2008). "Building Brand Awareness in Dynamic Oligopoly Markets". Management Science. 54 (1): 129–138. doi:10.1287/mnsc.1070.0755.
9. ^ a b Erickson, G. M. (2009). "An Oligopoly Model of Dynamic Advertising Competition". European Journal of Operations Research.
10. ^ Prasad, A.; Sethi, S. P. (2009). "Integrated Marketing Communications in Markets with Uncertainty and Competition". Automatica. 45 (3): 601–610. doi:10.1016/j.automatica.2008.09.018.
11. ^ a b c d Erickson, G. M. (2009). "Advertising Competition in a Dynamic Oligopoly with Multiple Brands". Operations Research. 57 (5): 1106–1113. doi:10.1287/opre.1080.0663.
12. ^ a b Sethi, S.P., Thompson, G.L. (2000). Optimal Control Theory: Applications to Management Science and Economics. Second Edition. Springer. ISBN 0-387-28092-8 and ISBN 0-7923-8608-6, pp. 352-355. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html
13. ^ He, X.; Prasad, A.; Sethi, S.P. (2009). "Cooperative Advertising and Pricing in a Stochastic Supply Chain: Feedback Stackelberg Strategies". Production and Operations Management. 18 (1): 78–94. doi:10.1111/j.1937-5956.2009.01006.x. SSRN .
14. ^ He, X.; Prasad, A.; Sethi, S.P.; Gutierrez, G. (2007). "A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels". Journal of Systems Science and Systems Engineering. 16 (4): 385–413. doi:10.1007/s11518-007-5058-2. SSRN .
15. ^ Sethi, S.P.; Prasad, A.; He, X. (2008). "Optimal Advertising and Pricing in a New-Product Adoption Model". Journal of Optimization Theory and Applications. 139 (2): 351–360. doi:10.1007/s10957-008-9472-5.
16. ^ Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. European Journal of Operations Research.