# Market Mix Modelling — Competitive Effects and Market Share Models

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.

### Market Share Models

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 Mb and Sb are brand b’s market share and sales, and Ab is the effort expended over the brand’s marketing mix.

A commonly used form for Ab is the following multiplicative functional function:

$$A_b=e^{α_b}\prod_{k=1}^K X_{kb}^{β_{kb}}.e^{\delta_b}$$

Where Xki > 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 αb*=αb−x̄ and δb*=δb− 𝛿̄. Similarly Xkb/X̄k can be expressed in log-centred format as Xkb* (M̄ and X̄k 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. βki ln(Xki/X̄) for i ≠ b.

MCI model is essentially the normalized form of the multiplicative model. Similarly by specifying the attraction function (Ab) in terms of an exponential form (Ab = eαb eβ1bX1b eβ2bX2b eβ3bX3b ...) we derive what is referred to as the Multinomial Logit (MNL) model, which after applying the log centring transformation, takes the below form:

$$ln \left(\frac{M_b}{\bar{M}}\right)=α_b^*+\sum_{k=1}^K β_k (X_{kb}-\bar{X_k})+\delta_b^*,$$

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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### Market Mix Modelling - Solutions

Solutions for market mix modelling.