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Wikipedia / 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. 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:

  1. .

Where:

Explanation

, and a random effect using a diffusion or White noise term that can go either way.

  • is the coefficient of the effectiveness of advertising innovation.

  • is the decay constant.

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

  • brings in the random effect.

Example of an optimal advertising problem

, consider the following objective function:

denotes the discount rate.

is known as the value function for this problem, and it is shown to be

where

The optimal control for this problem is

where

and

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 model

See also

References