# 1.2.Diminishing Returns Effect

Advertising can also have diminishing returns to scale or in other words the relationship between advertising and demand can be nonlinear. For example, the effect of 200 GRPs of advertising in a week on demand for a brand maybe less than twice that achieved with 100 GRPs of advertising. Typically, each incremental amount of advertising causes a progressively lesser effect on demand increase. This is a result of advertising saturation.

The usual approach to account for saturation is to transform the advertising variable to a non-linear scale for example log or negative exponential transformations. It is this transformed variable that is used in the sales response models.

T t c a n b e t r a n s f o r m e d t o a n a p p r o p r i a t e n o n l i n e a r f o r m l i k e t h e l o g i s t i c o negative exponential distribution, depending upon the type of diminishing returns or ‘saturation’ effect the response function is believed to follow. r

For example if advertising awareness followed a logarithmic distribution, then in a linear sales response model we would have:

S T = β L o g ( T T ) + ε T

(1)

Where S_{T }is sales at time T, T_{T }is the level of advertising GRPs at time T and ε_{T }is the random error component. In this case for 100 GRPs and assuming β =1, the sales effect of advertising would be 4.6 units and for 200 GRPs the sales effect would be 5.3 units. Therefore, for a 100% increase in advertising we would only have a 15% increase in sales. Advertising typically has a lower elasticity than other elements of the marketing-mix. This is considered acceptable by Brand Managers since advertising is also believed to have a long-term positive effect on Brand Equity, which is usually not captured by most econometric models.

Several versions of Adstock transformation are applied in the industry, we will examine some popular models in the following sections.

# 2. Adstock Models

2.1.Simple Decay-Effect Model Below is a simple formulation of the basic Adstock model of Broadbent (1979):

A t = T t + λ A t - 1

t=1,…., n

(2)

W h e r e A t i s t h e A d s t o c k a t t i m e t , T t i s t h e v a l u e o f t h e a d v e r t i s i n g v a r i a b l e a time t and λ is the ‘decay’ or lag weight parameter. Inclusion of the A_{t-1 }term, t

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