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The '''Sethi model''' was developed by [[Suresh P. Sethi]] and describes the process of how sales evolve over time in response to [[advertising]].<ref name="sethi83">{{Cite journal |last=Sethi |first=S. P. |year=1983 |title=Deterministic and Stochastic Optimization of a Dynamic Advertising Model |journal=Optimal Control Application and Methods |volume=4 |issue=2 |pages=179–184 |doi=10.1002/oca.4660040207 }}</ref> 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.<ref name="sethi83" /> The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.<ref>{{Cite journal |last=Vidale |first=M. L. |last2=Wolfe |first2=H. B. |year=1957 |title=An Operations-Research Study of Sales Response to Advertising |journal=Operations Research |volume=5 |issue=3 |pages=370–381 |doi=10.1287/opre.5.3.370 }}</ref> The model and its competitive extensions have been used extensively in the literature.<ref name="Sorger89">{{Cite journal |last=Sorger |first=G. |year=1989 |title=Competitive Dynamic Advertising: A Modification of the Case Game |journal=Journal of Economic Dynamics and Control |volume=13 |issue=1 |pages=55–80 |doi=10.1016/0165-1889(89)90011-0 }}</ref><ref name="CV92">{{Cite journal |last=Chintagunta |first=P. K. |last2=Vilcassim |first2=N. J. |year=1992 |title=An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly |journal=Management Science |volume=38 |issue=9 |pages=1230–1244 |doi=10.1287/mnsc.38.9.1230 }}</ref><ref name="CJ95">{{Cite journal |last=Chintagunta |first=P. K. |authorlink2=Dipak C. Jain |last2=Jain |first2=D. C. |year=1995 |title=Empirical Analysis of a Dynamic Duopoly Model of Competition |journal=Journal of Economics & Management Strategy |volume=4 |issue=1 |pages=109–131 |doi=10.1111/j.1430-9134.1995.00109.x }}</ref><ref name="PS04">{{Cite journal |last=Prasad |first=A. |last2=Sethi |first2=S. P. |year=2004 |title=Competitive Advertising under Uncertainty: Stochastic Differential Game Approach |journal=Journal of Optimization Theory and Applications |volume=123 |issue=1 |pages=163–185 |doi=10.1023/B:JOTA.0000043996.62867.20 }}</ref><ref name="BKPS05">{{Cite journal |last=Bass |first=F. M. |last2=Krishamoorthy |first2=A. |last3=Prasad |first3=A. |last4=Sethi |first4=S. P. |year=2005 |title=Generic and Brand Advertising Strategies in a Dynamic Duopoly |journal=Marketing Science |volume=24 |issue=4 |pages=556–568 |doi=10.1287/mksc.1050.0119 }}</ref><ref name="NPS08">{{Cite journal |last=Naik |first=P. A. |last2=Prasad |first2=A. |last3=Sethi |first3=S. P. |year=2008 |title=Building Brand Awareness in Dynamic Oligopoly Markets |journal=Management Science |volume=54 |issue=1 |pages=129–138 |doi=10.1287/mnsc.1070.0755 }}</ref><ref name="EricEJOR">{{Cite journal |last=Erickson |first=G. M. |year=2009 |title=An Oligopoly Model of Dynamic Advertising Competition |journal=European Journal of Operations Research |volume= |issue= |pages= |doi= }}</ref><ref>{{Cite journal |last=Prasad |first=A. |last2=Sethi |first2=S. P. |year=2009 |title=Integrated Marketing Communications in Markets with Uncertainty and Competition |journal=Automatica |volume=45 |issue= 3|pages=601–610 |doi=10.1016/j.automatica.2008.09.018 }}</ref><ref name="EricOR">{{Cite journal |last=Erickson |first=G. M. |year=2009 |title=Advertising Competition in a Dynamic Oligopoly with Multiple Brands |journal=Operations Research |volume=57 |issue=5 |pages=1106–1113 |doi=10.1287/opre.1080.0663 }}</ref> Moreover, some of these extensions have been also tested empirically.<ref name="CV92" /><ref name="CJ95" /><ref name="NPS08" /><ref name="EricOR" />

==Model==

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

: <math> dX_t =\left(rU_t\sqrt{1-X_t} - \delta X_t\right)\,dt+\sigma(X_t)\,dz_t, \qquad X_0=x</math>.

Where:
* <math>X_t</math> is the market share at time <math>t</math>
* <math>U_t</math> is the rate of advertising at time <math>t</math>
* <math>r</math> is the coefficient of the effectiveness of advertising
* <math>\delta</math> is the decay constant
* <math>\sigma(X_t)</math> is the diffusion coefficient
* <math>z_t</math> is the [[Wiener process]] (Standard [[Brownian motion]]); <math>dz_t</math> 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 <math>r</math>, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market via <math>\delta</math>, and a random effect using a diffusion or White noise term that can go either way.

* The coefficient <math>r</math> is the coefficient of the effectiveness of advertising innovation.
* The coefficient <math>\delta</math> is the decay constant.
* The square-root term brings in the so-called word-of-mouth effect at least at low sales levels.<ref name="sethi83" /><ref name="Sorger89" />

* The diffusion term <math>\sigma(X_t)dz_t</math> brings in the random effect.

===Example of an optimal advertising problem===

Subject to the Sethi model above with the initial market share <math>x</math>, consider the following objective function:

: <math>V(x) = \max_{U_t \geq 0} \;E\left[ \int_0^\infty e^{-\rho t}(\pi X_t-U_t^2)\,dt\right],</math>

where <math>\pi</math> denotes the sales revenue corresponding to the total market, i.e., when <math>x = 1</math>, and <math>\rho > 0</math> denotes the discount rate.

The function <math>V(x)</math> is known as the value function for this problem, and it is shown to be<ref name="Sethi00">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</ref>

: <math>
V(x)=\bar\lambda x+ \frac{\bar\lambda^2 r^2}{4
\rho},
</math>

where

: <math>
\bar\lambda=\frac{\sqrt{(\rho+\delta)^2+r^2
\pi}-(\rho+\delta)}{r^2/2}.
</math>

The [[optimal control]] for this problem is<ref name="Sethi00" />

: <math> 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} </math>

where

: <math>
\bar x= \frac{r^2 \bar\lambda /2}{r^2 \bar\lambda /2+\delta}
</math>

and

: <math>
\bar u=\frac{r\bar\lambda \sqrt{1-\bar x}}{2}.
</math>

==Extensions of the Sethi model==

* Competitive extensions-Nash differential games<ref name="Sorger89" /><ref name="PS04" /><ref name="BKPS05" /><ref name="NPS08" /><ref name="EricEJOR" /><ref name="EricOR" />
* Empirical testing of the Sethi model and extensions<ref name="CV92" /><ref name="CJ95" /><ref name="NPS08" /><ref name="EricOR" />
* Stackelberg differential games <ref>{{Cite journal | last1 = He | first1 = X. | last2 = Prasad | first2 = A. | last3 = Sethi | first3 = S.P. | year = 2009 | title = Cooperative Advertising and Pricing in a Stochastic Supply Chain: Feedback Stackelberg Strategies | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1069063 | journal = Production and Operations Management | volume = 18 | issue = 1| pages = 78–94 }}</ref><ref>{{Cite journal | doi = 10.1007/s11518-007-5058-2 | last1 = He | first1 = X. | last2 = Prasad | first2 = A. | last3 = Sethi | first3 = S.P. | last4 = Gutierrez | first4 = G. | year = 2007| title = A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1069162 | journal = Journal of Systems Science and Systems Engineering | volume = 16 | issue = 4| pages = 385–413 }}</ref>

* The Sethi durable goods model<ref>{{Cite journal | doi = 10.1007/s10957-008-9472-5 | last1 = Sethi | first1 = S.P. | last2 = Prasad | first2 = A. | last3 = He | first3 = X. | year = 2008 | title = Optimal Advertising and Pricing in a New-Product Adoption Model | url = | journal = Journal of Optimization Theory and Applications | volume = 139 | issue = 2| pages = 351–360 }}</ref><ref>Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. ''European Journal of Operations Research''.</ref>

==See also==

* [[Bass diffusion model]]
* [[differential game]]s
* [[Stochastic differential equation]]
* [[Diffusion of innovations]]
* [[Stackleberg competition]]
* [[Nash equilibrium]]

==References==
{{Reflist|colwidth=30em}}

[[Category:Advertising]]
[[Category:Mathematical economics]]
[[Category:Optimal control]]
[[Category:Stochastic processes]]

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