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. 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. The model and its competitive and multi-echelon channel extensions have been used extensively in the literature.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.Prasad, A., Sethi, S.P., and Naik, P., "Understanding the Impact of Churn in Dynamic Oligopoly Markets," ''Automatica'', 48, 2012, 2882-2887.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
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, 615-64
SSRN 1898309
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
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
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, 1-1
SSRN 1088489
/ref>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.Rong, Z., & Qingzhong, R. (2013). Equivalence between sethi advertising model and a scalar LQ differential game. ''2013 25th Chinese Control and Decision Conference (CCDC)''. https://doi.org/10.1109/ccdc.2013.6561115
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:

: $dX_t =\left(rU_t\sqrt - \delta X_t\right)\,dt+\sigma(X_t)\,dz_t, \qquad X_0=x$.

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 coefficient
* $z_t$ 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_ \;E\left \int_0^\infty e^(\pi X_t-U_t^2)\,dt\right

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 beSethi, 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

: $V(x)=\bar\lambda x+ \frac,$

where

: $\bar\lambda=\frac.$

The optimal control for this problem is

: $U^*_t = u^*(X_t)=\frac = \begin > \bar & \text X_t < \bar, \\ = \bar & \text X_t = \bar, \\ < \bar & \text X_t > \bar, \end$

where

: $\bar x= \frac$

and

: $\bar u=\frac.$

Extensions of the Sethi model

* Competitive model: Nash differential games
* Multi-echelon Model
* Empirical testing of the Sethi model and extensions
* Cooperative advertising: Stackelberg differential games
* The Sethi durable goods modelKrishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. ''European Journal of Operational Research''.

See also

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

References

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Advertising
Mathematical economics
Optimal control
Stochastic models