Sethi model

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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.[1][2] 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.[1] The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.[3] The model and its competitive and multi-echelon channel extensions have been used extensively in the literature[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][30][31][32][33][34][35] Moreover, some of these extensions have been also tested empirically.[6][7][10][13]

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]\displaystyle{ 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]\displaystyle{ X_t }[/math] is the market share at time [math]\displaystyle{ t }[/math]
  • [math]\displaystyle{ U_t }[/math] is the rate of advertising at time [math]\displaystyle{ t }[/math]
  • [math]\displaystyle{ r }[/math] is the coefficient of the effectiveness of advertising
  • [math]\displaystyle{ \delta }[/math] is the decay constant
  • [math]\displaystyle{ \sigma(X_t) }[/math] is the diffusion coefficient
  • [math]\displaystyle{ z_t }[/math] is the Wiener process (Standard Brownian motion); [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \delta }[/math], and a random effect using a diffusion or White noise term that can go either way.

  • The coefficient [math]\displaystyle{ r }[/math] is the coefficient of the effectiveness of advertising innovation.
  • The coefficient [math]\displaystyle{ \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.[1][5]
  • The diffusion term [math]\displaystyle{ \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]\displaystyle{ x }[/math], consider the following objective function:

[math]\displaystyle{ 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]\displaystyle{ \pi }[/math] denotes the sales revenue corresponding to the total market, i.e., when [math]\displaystyle{ x = 1 }[/math], and [math]\displaystyle{ \rho \gt 0 }[/math] denotes the discount rate.

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

[math]\displaystyle{ V(x)=\bar\lambda x+ \frac{\bar\lambda^2 r^2}{4 \rho}, }[/math]

where

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

The optimal control for this problem is[2]

[math]\displaystyle{ U^*_t = u^*(X_t)=\frac{r\bar\lambda \sqrt{1-\ X_t}}{2} = \begin{cases} {} \gt \bar{u} & \text{if } X_t \lt \bar{x}, \\ {} = \bar{u} & \text{if } X_t = \bar{x}, \\ {} \lt \bar{u} & \text{if } X_t \gt \bar{x}, \end{cases} }[/math]

where

[math]\displaystyle{ \bar x= \frac{r^2 \bar\lambda /2}{r^2 \bar\lambda /2+\delta} }[/math]

and

[math]\displaystyle{ \bar u=\frac{r\bar\lambda \sqrt{1-\bar x}}{2}. }[/math]

Extensions of the Sethi model

See also

References

  1. ↑ 1.0 1.1 1.2 Sethi, S. P. (1983). "Deterministic and Stochastic Optimization of a Dynamic Advertising Model". Optimal Control Applications and Methods 4 (2): 179–184. doi:10.1002/oca.4660040207. 
  2. ↑ 2.0 2.1 2.2 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. http://doi.org/10.1007/978-3-319-98237-3
  3. ↑ 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. 
  4. ↑ 4.0 4.1 4.2 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. doi:10.1111/poms.13487. ISSN 1059-1478. http://dx.doi.org/10.1111/poms.13487. 
  5. ↑ 5.0 5.1 5.2 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. 
  6. ↑ 6.0 6.1 6.2 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. 
  7. ↑ 7.0 7.1 7.2 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. 
  8. ↑ 8.0 8.1 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. 
  9. ↑ 9.0 9.1 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. 
  10. ↑ 10.0 10.1 10.2 10.3 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. 
  11. ↑ 11.0 11.1 Erickson, G. M. (2009). "An Oligopoly Model of Dynamic Advertising Competition". European Journal of Operational Research 197: 374–388. doi:10.1016/j.ejor.2008.06.023. 
  12. ↑ 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. 
  13. ↑ 13.0 13.1 13.2 13.3 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. 
  14. ↑ 14.0 14.1 Rong, Zhang; Qingzhong, Ren (May 2013). "Equivalence between Sethi advertising model and a scalar LQ differential game". 2013 25th Chinese Control and Decision Conference (CCDC). pp. 1244–1247. doi:10.1109/ccdc.2013.6561115. ISBN 978-1-4673-5534-6. 
  15. ↑ 15.0 15.1 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. 
  16. ↑ 16.0 16.1 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. 
  17. ↑ 17.0 17.1 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. 
  18. ↑ 18.0 18.1 18.2 Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. European Journal of Operational Research.
  19. ↑ 19.0 19.1 19.2 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.
  20. ↑ 20.0 20.1 20.2 20.3 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.
  21. ↑ 21.0 21.1 Chutani, A. and Sethi, S.P., "Cooperative Advertising in a Dynamic Retail Market Oligopoly," Dynamic Games and Applications, 2(4), 2012, 347-375.
  22. ↑ 22.0 22.1 Prasad, A., Sethi, S.P., and Naik, P., "Understanding the Impact of Churn in Dynamic Oligopoly Markets," Automatica, 48, 2012, 2882-2887.
  23. ↑ 23.0 23.1 23.2 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.
  24. ↑ 24.0 24.1 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.
  25. ↑ 25.0 25.1 He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., "Retail Competition and Cooperative Advertising," OR Letters, 39, 2011, 11-16. SSRN 1609854.
  26. ↑ 26.0 26.1 26.2 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.
  27. ↑ 27.0 27.1 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.
  28. ↑ 28.0 28.1 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
  29. ↑ 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.
  30. ↑ 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.
  31. ↑ Han, Jinhui; Sethi, Suresh P.; Siu, Chi Chung; Yam, Sheung Chi Phillip (2023-01-11). "Co-op advertising in randomly fluctuating markets" (in en). Production and Operations Management 32 (6): 1617–1635. doi:10.1111/poms.13929. ISSN 1059-1478. https://onlinelibrary.wiley.com/doi/10.1111/poms.13929. 
  32. ↑ Murray, A. (2016). "An Industry-focused Advertising Model". Proceedings of 5th the International Conference on Operations Research and Enterprise Systems. SCITEPRESS - Science and Technology Publications. pp. 87–91. doi:10.5220/0005653300870091. ISBN 978-989-758-171-7. http://dx.doi.org/10.5220/0005653300870091. 
  33. ↑ Chutani, Anshuman; Sethi, Suresh P. (2012-12-01). "Cooperative Advertising in a Dynamic Retail Market Oligopoly" (in en). Dynamic Games and Applications 2 (4): 347–375. doi:10.1007/s13235-012-0053-8. ISSN 2153-0793. https://doi.org/10.1007/s13235-012-0053-8. 
  34. ↑ He, Xiuli; Krishnamoorthy, Anand; Prasad, Ashutosh; Sethi, Suresh P. (2011-01-01). "Retail competition and cooperative advertising" (in en). Operations Research Letters 39 (1): 11–16. doi:10.1016/j.orl.2010.10.006. ISSN 0167-6377. https://www.sciencedirect.com/science/article/pii/S0167637710001446. 
  35. ↑ Kennedy, Adrian; Prasad, Ashutosh; Sethi, Suresh; Siu, Chi Chung; Yam, S. C. P. (2022-09-02). "Optimal Advertising and Product Durability Decisions in New Product Diffusion". Rochester, NY.




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