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] [3] Moreover, some of these extensions have been also tested empirically. [6] [7] [10] [13]
The Sethi advertising model or simply the Sethi model provides a sales-advertising dynamics in the form of the following stochastic differential equation:
Where:
The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market via , the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market via , and a random effect using a diffusion or White noise term that can go either way.
Subject to the Sethi model above with the initial market share , consider the following objective function:
where denotes the sales revenue corresponding to the total market, i.e., when , and denotes the discount rate.
The function is known as the value function for this problem, and it is shown to be [2]
where
The optimal control for this problem is [2]
where
and