### 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 transformed into a linear model by applying what is referred to as the log-centring transformation (Cooper & Nakanishi, 1988). 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 offer several advantages. They effectively capture competitive effects. They meet logical consistency requirements — market shares for brands fall between 0 and 1, and the sum of all estimated market shares equals 1. Additionally, the response curves of market share models exhibit diminishing returns to scale at high levels of marketing activity. MCI models typically have a concave shape, while MNL models exhibit an S-shaped pattern.

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. These models do not account for potential carryover effects or changes in the market environment over time. Model parameters are assumed to remain fixed, which may not capture the dynamic nature of real-world marketing scenarios.

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### Market Mix Modelling - Solutions

Solutions for market mix modelling.