Markets exist in a competitive environment. How consumers respond to a product is not only a function of what the product has to offer, but also a function of what competitors’ products have to offer.

Take price for instance. When a product’s price is
increased by a substantial amount, say 20%, it is likely to lose a large
proportion of its business to competitors that target the same consumers with
similar products. On the other hand if all market players take prices up by
20%, as might occur when excise duties or raw material prices soar, the impact
on product sales may be insignificant. For the majority of consumer sectors,
product categories *as a whole* are relatively inelastic to price
adjustments. If individual brands increase prices substantially, they lose
share … but if petroleum, milk, coffee, shampoo or cigarette prices increase by
even as much as 20%, the demand for these product categories is unlikely to
plunge.

In conclusion, rather than absolute, it is the relative price that has greatest significance on the demand for a product. This applies not only to price, but also the other elements of marketing mix.

The relative form of variables is often expressed
in terms of an index or share — price index for relative price, share of market
for sales, or share of voice for advertising (*Share of voice,* which is
the share of GRP or advertising expenditure within the product category, is a
measure that relates to the impact of advertising in a competitive setting).

*Cross elasticity* is the construct that
quantifies competitive effects. The cross elasticity of demand determines
the responsiveness in the sales of a product when a change in marketing effort
takes place in a competing product. When a product drops price, increases
advertising, improves product quality or expands distribution, it cannibalizes
competing products. The shift in business from one product to the other, on
account of a change in marketing effort, is captured by the cross elasticity of
demand.

One approach to capturing competitive effects is via market share models, known also as attraction models. These models are based on the notion that marketing effort generates "attraction" for the brand, and that the brand’s market share is a function of its share of total marketing effort. This supposition is captured as follows:

$$ Market\,Share, \, M_b=\frac{S_b}{\sum_j^{brands} S_j}=\frac{A_b}{\sum_j^{brands} A_j},$$Where M_{b} and S_{b} are brand b’s market
share and sales, and A_{b} is the effort expended over the brand’s
marketing mix.

A commonly used form for *A _{b}* is
the following multiplicative functional function:

Where X_{ki} > 0 are the K elements of the
marketing mix.

The model as a whole is referred to as the Multiplicative Competitive Interaction (MCI) Model. The MCI model without taking cross-effects into consideration becomes:

$$ M_b=\frac{\left( e^{α_b}\prod_{k=1}^K X_{kb}^{β_{kb}} \right)e^{\delta_b}}{\sum_{j=1}^B \left( e^{α_j}\prod_{k=1}^K X_{kj}^{β_{kj}} \right) e^{\delta_j}},$$This nonlinear model can be transform into a linear model by applying what is referred to as the log-centring transformation (Cooper & Nakanishi, 88). After applying this transformation the model takes the below form:

$$ ln \left(\frac{M_b}{\bar{M}}\right)=α_b^*+\sum_{k=1}^K β_k \,ln \left( \frac{X_{kb}}{\bar{X_k}} \right)+\delta_b^*,$$where α_{b}*=α_{b}−x̄ and
δ_{b}*=δ_{b}− 𝛿̄.
Similarly X_{kb}/X̄_{k} can be expressed in log-centred format as
X_{kb}* (M̄ and X̄_{k} are the
average market share and the average marketing effort for mix element k).

This model may be expanded to include terms that
capture the cross effect between variables, i.e. β_{ki} ln(X_{ki}/X̄) for i ≠ b.

MCI model is essentially the normalized form of
the multiplicative model.
Similarly by specifying the attraction function (A_{b}) in
terms of an exponential form (A_{b} = e^{αb} e^{β1bX1b}
e^{β2bX2b} e^{β3bX3b} ...) we
derive what is referred to as the *Multinomial Logit (MNL)* model, which
after applying the log centring transformation, takes the below form:

Market share models have several advantages. They capture competitive effects. They meet logical consistency requirements — brands’ market shares lies between 0 and 1, and the sum of their estimated market shares equals 1. Their response curves are characterized by diminishing returns to scale at high levels of marketing activity. The MCI models are concave in shape, whereas the MNL models are S-shaped.

One limitation of these models however, is that they are static in nature. Marketing efforts are assumed to impact only the time periods when they occur. And the market environment is assumed to be static — the model parameters remain fixed over time.

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