Sethi model explained

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 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. The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.^[2] The model and its competitive and multi-echelon channel extensions have been used extensively in the literature.^{[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]} Moreover, some of these extensions have been also tested empirically.

Model

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

dX _{t} = \left(rU _{t\sqrt{1-X} _{t}} - \delta X _{t\right)dt+\sigma(X} _{t)dz} _{t,} X _{0=x} .

Where:

X _{t} is the market share at time t U _{t} is the rate of advertising at time t r is the coefficient of the effectiveness of advertising\delta is the decay constant\sigma(X _{t)} is the diffusion coefficientz _{t} is the Wiener process (Standard Brownian motion); 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

r , the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market via \delta , and a random effect using a diffusion or White noise term that can go either way.

• The coefficient

r is the coefficient of the effectiveness of advertising innovation.

• The coefficient

\delta is the decay constant.

• The square-root term brings in the so-called word-of-mouth effect at least at low sales levels.
• The diffusion term

\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

x , consider the following objective function:V(x) = max _{U _{t} \geq 0} E\left[ \int ^{inf ty} _{0} e ^{-\rho} (\pi X _{t-U} ^{2 )dt\right],} ^{t}

where

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

The function

V(x) is known as the value function for this problem, and it is shown to be^[30] V(x) = \bar λ x + (\bar λ ^{2} r ^{2})/(4 \rho) ,

where

\bar λ= (\sqrt{(\rho + \delta) ^{2+r} ^{2 \pi})/(- (\rho + \delta)}{r) ^{2/2}.}

The optimal control for this problem is

U ^{*} _{t} = u ^{*(X} _{t) = (r\bar λ \sqrt{ 1- X _{t})/()} } = \begin > \bar & \text X_t < \bar, \\ = \bar & \text X_t = \bar, \\ < \bar & \text X_t > \bar,\end

where

\bar x = (r ^{2} \bar λ / 2)/(r ^{2} \bar λ / 2+ \delta)

and

\bar u = (r\bar λ \sqrt{ 1- \bar x)/() }.

Extensions of the Sethi model

• Competitive model: Nash differential games^[31]
• Multi-echelon Model^[3]
• Empirical testing of the Sethi model and extensions
• Cooperative advertising: Stackelberg differential games ^{[3] [32] [33]}
• The Sethi durable goods model^{[34] [35]}

See also

• Bass diffusion model
• differential games
• stochastic differential equation
• diffusion of innovations
• Stackleberg competition
• Nash equilibrium

References

Notes and References

1. Sethi . S. P. . 1983 . Deterministic and Stochastic Optimization of a Dynamic Advertising Model . Optimal Control Applications and Methods . 4 . 2 . 179–184 . 10.1002/oca.4660040207 . 123673289 .
2. Vidale . M. L. . Wolfe . H. B. . 1957 . An Operations-Research Study of Sales Response to Advertising . Operations Research . 5 . 3 . 370–381 . 10.1287/opre.5.3.370 .
3. Kennedy. Adrian Patrick. Sethi. Suresh P.. Siu. Chi Chung. Yam. Sheung Chi Phillip. 2021-07-26. Cooperative Advertising in a Dynamic Three-Echelon Supply Chain. Production and Operations Management. 30. 11. 3881–3905. 10.1111/poms.13487. 236272112. 1059-1478.
4. Sorger . G. . 1989 . Competitive Dynamic Advertising: A Modification of the Case Game . Journal of Economic Dynamics and Control . 13 . 1 . 55–80 . 10.1016/0165-1889(89)90011-0 .
5. Chintagunta . P. K. . Vilcassim . N. J. . 1992 . An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly . Management Science . 38 . 9 . 1230–1244 . 10.1287/mnsc.38.9.1230 . 153538282 .
6. Chintagunta . P. K. . Dipak C. Jain . Jain . D. C. . 1995 . Empirical Analysis of a Dynamic Duopoly Model of Competition . Journal of Economics & Management Strategy . 4 . 1 . 109–131 . 10.1111/j.1430-9134.1995.00109.x .
7. Prasad . A. . Sethi . S. P. . 2004 . Competitive Advertising under Uncertainty: Stochastic Differential Game Approach . Journal of Optimization Theory and Applications . 123 . 1 . 163–185 . 10.1023/B:JOTA.0000043996.62867.20 . 121005830 .
8. 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 . 10.1287/mksc.1050.0119 . 17054876 .
9. Naik . P. A. . Prasad . A. . Sethi . S. P. . 2008 . Building Brand Awareness in Dynamic Oligopoly Markets . Management Science . 54 . 1 . 129–138 . 10.1287/mnsc.1070.0755 . 10.1.1.510.731 . 18908145 .
10. Erickson . G. M. . 2009 . An Oligopoly Model of Dynamic Advertising Competition . European Journal of Operational Research . 197. 374–388. 10.1016/j.ejor.2008.06.023.
11. Prasad . A. . Sethi . S. P. . 2009 . Integrated Marketing Communications in Markets with Uncertainty and Competition . Automatica . 45 . 3. 601–610 . 10.1016/j.automatica.2008.09.018 . 8385913 .
12. Erickson . G. M. . 2009 . Advertising Competition in a Dynamic Oligopoly with Multiple Brands . Operations Research . 57 . 5 . 1106–1113 . 10.1287/opre.1080.0663 .
13. 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. SSRN 3493916.
14. 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.
15. Chutani, A. and Sethi, S.P., "Cooperative Advertising in a Dynamic Retail Market Oligopoly," Dynamic Games and Applications, 2(4), 2012, 347-375.
16. Prasad, A., Sethi, S.P., and Naik, P., "Understanding the Impact of Churn in Dynamic Oligopoly Markets," Automatica, 48, 2012, 2882-2887.
17. He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., "Co-Op Advertising in Dynamic Retail Oligopolies," Decision Sciences, 43(1), 2012, 73-105. SSRN 1521239.
18. 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.SSRN 1898309.
19. He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., "Retail Competition and Cooperative Advertising," OR Letters, 39, 2011, 11-16. SSRN 1609854.
20. 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. SSRN 1114989.
21. 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. SSRN 1376394.
22. 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 SSRN 1088489
23. Kennedy, A.P., Sethi, S.P., Siu, C.C., and Yam, S.C.P., “Cooperative advertising in a dynamic three-echelon supply chain,” Production and Operations Management, 30(11), 2021, 3881–3905.
24. 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, 57(5), 2019, 3413-3444.
25. Han . Jinhui . Sethi . Suresh P. . Siu . Chi Chung . Yam . Sheung Chi Phillip . 2023-01-11 . Co-op advertising in randomly fluctuating markets . Production and Operations Management . 32 . 6 . en . 1617–1635 . 10.1111/poms.13929 . 1059-1478.
26. Book: Murray, A. . Proceedings of 5th the International Conference on Operations Research and Enterprise Systems . An Industry-focused Advertising Model . 2016 . 87–91 . http://dx.doi.org/10.5220/0005653300870091 . SCITEPRESS - Science and Technology Publications . 10.5220/0005653300870091. 978-989-758-171-7 .
27. Chutani . Anshuman . Sethi . Suresh P. . 2012-12-01 . Cooperative Advertising in a Dynamic Retail Market Oligopoly . Dynamic Games and Applications . en . 2 . 4 . 347–375 . 10.1007/s13235-012-0053-8 . 256069615 . 2153-0793.
28. He . Xiuli . Krishnamoorthy . Anand . Prasad . Ashutosh . Sethi . Suresh P. . 2011-01-01 . Retail competition and cooperative advertising . Operations Research Letters . en . 39 . 1 . 11–16 . 10.1016/j.orl.2010.10.006 . 15113829 . 0167-6377.
29. Kennedy . Adrian P. . Sethi . Suresh P. . Siu . Chi Chung . Yam . Sheung Chi Phillip . Prasad . Ashutosh . 2022-09-02 . Optimal Advertising and Product Durability Decisions in New Product Diffusion . Rochester, NY. . en . 4202732 .
30. Sethi, S.P. (2021). Optimal Control Theory: Applications to Management Science and Economics. Fourth Edition. Springer. pp. 354-356. . http://doi.org/10.1007/978-3-319-98237-3
31. Book: Rong. Zhang. Qingzhong. Ren. May 2013. Equivalence between Sethi advertising model and a scalar LQ differential game. 1244–1247. 10.1109/ccdc.2013.6561115. 2013 25th Chinese Control and Decision Conference (CCDC). 978-1-4673-5534-6. 26614539.
32. He . X. . Prasad . A. . Sethi . S.P. . 2009 . Cooperative Advertising and Pricing in a Stochastic Supply Chain: Feedback Stackelberg Strategies . 1069063 . Production and Operations Management . 18 . 1. 78–94 . 10.1111/j.1937-5956.2009.01006.x. 15522449 .
33. 10.1007/s11518-007-5058-2 . He . X. . Prasad . A. . Sethi . S.P. . Gutierrez . G. . 2007. A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels . 1069162 . Journal of Systems Science and Systems Engineering . 16 . 4. 385–413 . 11443159 .
34. 10.1007/s10957-008-9472-5 . 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 . 16181059 .
35. Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. European Journal of Operational Research.