Marketing mix models embody “sales response functions” to relate the effects of marketing activities on sales. The dependence of sales on the different element of the marketing mix is estimated from historical market data using econometric and time series analysis methods. The resulting sales response functions depict the effect of the different elements of the mix on sales.

The functional form specifies the relationship between the dependent variable (sales) and the independent variable (e.g. price, advertising, promotion etc.). It determines the shape of the sales response curve, and reflects the nature of the marketing activity.

Fundamentally there are four basic shapes — linear, concave, convex and S-shape — as shown in Exhibit 34.1.

The simplest functional form is the linear model which represents constant returns to scale:

$$ S=α+βX $$Where:

S is sales, the dependent variable, and

X is the independent variable representing the marketing effort for the marketing mix element.

α and β are model parameters. α is the intercept — the sales when there is no marketing effort. β is coefficient for X. A unit change in X results in a change of β units in sales.

The linear model is theoretically unsound because it suggests that sales increase indefinitely. However from a practical point of view, for a relatively small operating range, a linear model can provide a satisfactory approximation of the true relationship.

One of the most commonly used forms in market mix modelling is the concave shape, which is characterized by diminishing returns to scale as the marketing activity increases. This aligns well with the expectation that as the intensity of discounts, displays and advertising increases, the returns diminish.

The *semilog* (semi logarithmic) model is
an example of diminishing returns to scale function:

Another functional form that meets the diminishing return to
scale requirement is the *power* *model*:

The power model is also known as the *constant elasticity
model* due to its property that the power coefficient is the elasticity of
demand of the marketing mix variable *X*:

The most widely used marketing mix model is a variation of the power model, called the multiplicative model:

$$ S=e^α X_1^{β_1} X_2^{β_2} X_3^{β_3} ... X_j^{β_j}, $$Where X_{1} to X_{J} are variables
representing the marketing effort for various marketing mix elements, and X_{1}
to X_{J} > 0.

This nonlinear structural model can be transformed
into an estimation model that is *linear in parameters* by
taking logarithms on both sides. The advantage of this transformation, which is
shown below, is that the parameters of the original nonlinear model can be
estimated using linear-regression techniques.

The multiplicative form is widely used in marketing mix models such as Nielsen’s Scan*Pro, to evaluate promotions.

The *exponential* model (S=e^{α}e^{βX} is an
example of a convex shaped, increasing returns to scale model.
This may apply for price, provided it is represented as 1/P:

The model assumes that the sales response to decreases in price may exhibit increasing returns to scale.

In the S-shaped response function, sales exhibit increasing returns to scale at low levels and diminishing returns at high levels of marketing effort. This is plausible for advertising which at low levels gets drowned by the noise, and hits an upper limit at very high levels. Besides advertising, the S-shape response function is also used for modelling the effect of shelf space on sales in store.

The S-shape captures the notions of *threshold* and *saturation*. Below the threshold, marketing effort has no impact on
sales, and above saturation, there is no further increase in sales. Above/below
these bounds, consumers become insensitive to the marketing stimuli.

If it truly reflects the response of advertising
to sales, the S-Shape response function has implications on how advertising
should be flighted — drip versus burst or pulse. An advertising *burst* is
a heavy dose of advertising over a short interval. It would ensure that
advertising levels cross threshold levels. In contrast drip or continuous
advertising is much lighter weight of advertising spread over a much longer
time frame. If thresholds exist, marketers should use less *drip*
advertising and more burst or *pulse* advertising —
an approach that falls between burst and drip, with advertising going on and
off air over the weeks.

While conceptually appealing, there is not much empirical evidence to support the existence of an S-shape response to advertising effort. It is hard to prove or disprove considering that historical data tends to lie well within these theoretical upper and lower bounds, i.e. if they exist.

The *logistic* model depicted
below takes a functional form that conforms to the S-shape.

Where, S_{0} is the intercept and the threshold
level, and ^{0} is the saturation level.

The elasticity of demand, for a variable with an S-shaped response with sales, follows an inverted bell-shape, starting at 0 at threshold level to a maximum, and back to 0 at the saturation level.

A related but rare phenomenon is the notion of *supersaturation*. It refers to the excessive use of a marketing instrument, such as advertising,
that theoretical may repulse consumers, creating a negative response to sales. It
is rarely witnessed because marketers’ budgets are usually constrained; they
operate below saturation levels and well below supersaturation levels. The
supersaturation effect can be represented by the quadratic model:

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