View source for Sethi model

The '''Sethi model''' was developed by Suresh P. Sethi and describes the process of how sales evolve over time in response to [[Finance:Advertising|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 Applications and Methods |volume=4 |issue=2 |pages=179–184 |doi=10.1002/oca.4660040207 |s2cid=123673289 }}</ref><ref name="Sethi00" /> The model assumes that 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 |last1=Vidale |first1=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 and multi-echelon channel extensions have been used extensively in the literature<ref name=":5">{{Cite journal|last1=Kennedy|first1=Adrian Patrick|last2=Sethi|first2=Suresh P.|last3=Siu|first3=Chi Chung|last4=Yam|first4=Sheung Chi Phillip|date=2021-07-26|title=Cooperative Advertising in a Dynamic Three-Echelon Supply Chain|url=http://dx.doi.org/10.1111/poms.13487|journal=Production and Operations Management|volume=30|issue=11|pages=3881–3905|doi=10.1111/poms.13487|s2cid=236272112|issn=1059-1478|via=}}</ref>.<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 |last1=Chintagunta |first1=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 |s2cid=153538282 }}</ref><ref name="CJ95">{{Cite journal |last1=Chintagunta |first1=P. K. |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 |last1=Prasad |first1=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 |s2cid=121005830 }}</ref><ref name="BKPS05">{{Cite journal |last1=Bass |first1=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 |s2cid=17054876 }}</ref><ref name="NPS08">{{Cite journal |last1=Naik |first1=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 |citeseerx=10.1.1.510.731 |s2cid=18908145 }}</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 Operational Research |volume= 197|pages= 374–388|doi= 10.1016/j.ejor.2008.06.023}}</ref><ref>{{Cite journal |last1=Prasad |first1=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 |s2cid=8385913 }}</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><ref name=":0" /><ref name=":1" /><ref name=":2" /><ref name=":3" /><ref name=":4" /><ref name=":6">Bensoussan, A., Chen, S., Chutani, A., Sethi, S.P., Siu, C.C., and Yam, S.C.P., "Feedback Stackelberg-Nash Equilibria in Mixed Leadership Games with an Application to Cooperative Advertising," ''SIAM Journal on Control and Optimization'', 2019, '''57'''(5), 3413-3444. [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3493916 SSRN 3493916.]</ref><ref name=":7">Chutani A. and Sethi, S.P., "A Feedback Stackelberg Game of Cooperative Advertising in a Durable Goods Oligopoly," ''Dynamic Games in Economics'', 13, J.L. Haunschmied, V. Veliov, and S. Wrzaczek (Eds.), Springer-Verlag Berlin Heidelberg, 2014, 89-114.</ref><ref name=":8">Chutani, A. and Sethi, S.P., "Cooperative Advertising in a Dynamic Retail Market Oligopoly," ''Dynamic Games and Applications'', '''2'''(4), 2012, 347-375.</ref><ref name=":9">Prasad, A., Sethi, S.P., and Naik, P., "Understanding the Impact of Churn in Dynamic Oligopoly Markets," ''Automatica'', '''48''', 2012, 2882-2887.</ref><ref name=":10">He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., "Co-Op Advertising in Dynamic Retail Oligopolies," ''Decision Sciences'', '''43'''(1), 2012, 73-105. [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1521239 SSRN 1521239].</ref><ref name=":11">Chutani, A. and Sethi, S.P., "Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain," ''Journal of Optimization Theory and Applications'', '''154'''(2), 2012, 615-643.[https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1898309 SSRN 1898309].</ref><ref name=":12">He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., "Retail Competition and Cooperative Advertising," ''OR Letters'', '''39''', 2011, 11-16. [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1609854 SSRN 1609854].</ref><ref name=":13">Krishnamoorthy, A., Prasad, A., and Sethi, S.P., "Optimal Pricing and Advertising in a Durable-Good Duopoly," ''European Journal of Operations Research'', '''200'''(2), 2010, 486-497. [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1114989 SSRN 1114989].</ref><ref name=":14">Prasad, A., Sethi, S.P., and Naik, P., "Optimal Control of an Oligopoly Model of Advertising," ''Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing (INCOM '09)'', Moscow, Russia, June 3–5, 2009. [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1376394 SSRN 1376394].</ref><ref name=":15">Bass, F.M., Krishnamoorthy, A., Prasad, A., and Sethi, S.P., "Advertising Competition with Market Expansion for Finite Horizon Firms," ''Journal of Industrial and Management Optimization'', '''1'''(1), February 2005, 1-19 [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1088489 SSRN 1088489]</ref><ref>Kennedy, A.P., Sethi, S.P., Siu, C.C., and Yam, S.C.P., “[https://onlinelibrary.wiley.com/doi/10.1111/poms.13487 Cooperative advertising in a dynamic three-echelon supply chain],” ''Production and Operations Management'', 30(11), 2021, 3881–3905. </ref><ref>Bensoussan, A., Chen, S., Chutani, A., Sethi, S.P., Siu, C.C., and Yam, S.C.P., “[https://epubs.siam.org/doi/10.1137/17M1153212 Feedback Stackelberg-Nash Equilibria in Mixed Leadership Games with an Application to Cooperative Advertising]”, ''SIAM Journal on Control and Optimization'', 57(5), 2019, 3413-3444.</ref><ref>{{Cite journal |last1=Han |first1=Jinhui |last2=Sethi |first2=Suresh P. |last3=Siu |first3=Chi Chung |last4=Yam |first4=Sheung Chi Phillip |date=2023-01-11 |title=Co-op advertising in randomly fluctuating markets |url=https://onlinelibrary.wiley.com/doi/10.1111/poms.13929 |journal=Production and Operations Management |volume=32 |issue=6 |language=en |pages=1617–1635 |doi=10.1111/poms.13929 |issn=1059-1478}}</ref><ref>{{Cite book |last=Murray |first=A. |title=Proceedings of 5th the International Conference on Operations Research and Enterprise Systems |chapter=An Industry-focused Advertising Model |date=2016 |pages=87–91 |chapter-url=http://dx.doi.org/10.5220/0005653300870091 |publisher=SCITEPRESS - Science and Technology Publications |doi=10.5220/0005653300870091|isbn=978-989-758-171-7 }}</ref><ref>{{Cite journal |last1=Chutani |first1=Anshuman |last2=Sethi |first2=Suresh P. |date=2012-12-01 |title=Cooperative Advertising in a Dynamic Retail Market Oligopoly |url=https://doi.org/10.1007/s13235-012-0053-8 |journal=Dynamic Games and Applications |language=en |volume=2 |issue=4 |pages=347–375 |doi=10.1007/s13235-012-0053-8 |s2cid=256069615 |issn=2153-0793}}</ref><ref>{{Cite journal |last1=He |first1=Xiuli |last2=Krishnamoorthy |first2=Anand |last3=Prasad |first3=Ashutosh |last4=Sethi |first4=Suresh P. |date=2011-01-01 |title=Retail competition and cooperative advertising |url=https://www.sciencedirect.com/science/article/pii/S0167637710001446 |journal=Operations Research Letters |language=en |volume=39 |issue=1 |pages=11–16 |doi=10.1016/j.orl.2010.10.006 |s2cid=15113829 |issn=0167-6377}}</ref><ref>Kennedy, Adrian; Prasad, Ashutosh; Sethi, Suresh; Siu, Chi Chung; Yam, S. C. P. (2022-09-02). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4202732 "Optimal Advertising and Product Durability Decisions in New Product Diffusion]". Rochester, NY.</ref>
Moreover, some of these extensions have been also tested empirically.<ref name="CV92" /><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|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 [[Physics:White noise|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. (2021). ''Optimal Control Theory: Applications to Management Science and Economics''. Fourth Edition. Springer.  pp. 354-356. {{isbn|978-3-319-98236-6|978-3-319-98237-3}}. http://doi.org/10.1007/978-3-319-98237-3</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|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 model: Nash differential games<ref name="Sorger89" /><ref name="PS04" /><ref name="BKPS05" /><ref name="NPS08" /><ref name="EricEJOR" /><ref name="EricOR" /><ref name=":0">{{Cite book|last1=Rong|first1=Zhang|last2=Qingzhong|first2=Ren|date=May 2013|chapter=Equivalence between Sethi advertising model and a scalar LQ differential game|pages=1244–1247|doi=10.1109/ccdc.2013.6561115|title=2013 25th Chinese Control and Decision Conference (CCDC)|isbn=978-1-4673-5534-6|s2cid=26614539}}</ref><ref name=":4" /><ref name=":6" /><ref name=":7" /><ref name=":8" /><ref name=":9" /><ref name=":10" /><ref name=":13" /><ref name=":14" /><ref name=":15" />
* Multi-echelon Model<ref name=":5">{{Cite journal|last1=Kennedy|first1=Adrian Patrick|last2=Sethi|first2=Suresh P.|last3=Siu|first3=Chi Chung|last4=Yam|first4=Sheung Chi Phillip|date=2021-07-26|title=Cooperative Advertising in a Dynamic Three-Echelon Supply Chain|url=http://dx.doi.org/10.1111/poms.13487|journal=Production and Operations Management|volume=30|issue=11|pages=3881–3905|doi=10.1111/poms.13487|s2cid=236272112|issn=1059-1478|via=}}</ref>
*  Empirical testing of the Sethi model and extensions<ref name="CV92" /><ref name="CJ95" /><ref name="NPS08" /><ref name="EricOR" />
* Cooperative advertising: Stackelberg differential games <ref name=":5">{{Cite journal|last1=Kennedy|first1=Adrian Patrick|last2=Sethi|first2=Suresh P.|last3=Siu|first3=Chi Chung|last4=Yam|first4=Sheung Chi Phillip|date=2021-07-26|title=Cooperative Advertising in a Dynamic Three-Echelon Supply Chain|url=http://dx.doi.org/10.1111/poms.13487|journal=Production and Operations Management|volume=30|issue=11|pages=3881–3905|doi=10.1111/poms.13487|s2cid=236272112|issn=1059-1478|via=}}</ref><ref name=":1">{{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 | ssrn = 1069063 | journal = Production and Operations Management | volume = 18 | issue = 1| pages = 78–94 | doi=10.1111/j.1937-5956.2009.01006.x| s2cid = 15522449 }}</ref><ref name=":2">{{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 | ssrn = 1069162 | journal = Journal of Systems Science and Systems Engineering | volume = 16 | issue = 4| pages = 385–413 | s2cid = 11443159 }}</ref><ref name=":6" /><ref name=":7" /><ref name=":10" /><ref name=":12" />
* The Sethi durable goods model<ref name=":3">{{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 | journal = Journal of Optimization Theory and Applications | volume = 139 | issue = 2| pages = 351–360 | s2cid = 16181059 }}</ref><ref name=":4">Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. ''European Journal of Operational Research''.</ref><ref name=":7" /><ref name=":11" /><ref name=":13" />

==See also==

*  [[Bass diffusion model]]
*  [[Differential game|differential game]]s
*  [[Stochastic differential equation|stochastic differential equation]]
*  [[Physics:Diffusion of innovations|diffusion of innovations]]
*  Stackleberg competition
*  [[Nash equilibrium]]

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

<br />

[[Category:Mathematical economics]]
[[Category:Stochastic models]]

{{Sourceattribution|Sethi model}}