Functional Forms in Marketing

Summary

Chapter 3 is the technical core of the book, cataloguing ten functional forms for the sales response function $Q=f(X)$, each with full derivation, elasticity formula, shape properties, and marketing interpretation. Also covers the SCAN*PRO multiplicative scanner model, random coefficients specifications, and switching models.

Response Sensitivity and Elasticity

Response Sensitivity and Elasticity

Response sensitivity (absolute): $\partial Q/\partial X$

Point elasticity (scale-free): $\eta =\frac{\partial Q}{\partial X}\cdot \frac{X}{Q}=\frac{d\ln Q}{d\ln X}$

Elasticity is preferred for cross-study comparison and meta-analysis because it is dimensionless. For marketing instruments, elasticities serve as the key empirical generalizations.

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1. Linear Form

Linear

$$Q={\beta }_0+{\beta }_{1}X$$

Elasticity:

$$\eta ={\beta }_1\frac{X}{Q}=\frac{{\beta }_{1}X}{{\beta }_0+{\beta }_{1}X}$$

Shape: Constant slope; no saturation. Increasing, decreasing, or flat depending on sign of ${\beta }_1$.

Use: First-order approximation in log-linear models; baseline for testing against nonlinear alternatives.

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2. Semilogarithmic Form

Semilogarithmic

$$Q={\beta }_0+{\beta }_1\ln X$$

Elasticity:

$$\eta =\frac{{\beta }_1}{Q}=\frac{{\beta }_1}{{\beta }_0+{\beta }_1\ln X}$$

Shape: Concave (diminishing returns) when ${\beta }_1>0$. Useful when a large range of $X$ is observed.

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3. Power / Log-Log Form (Constant Elasticity)

Power Form

$$Q=e^{{\beta }_0}{X}^{{\beta }_1}$$

Taking logs: $\ln Q={\beta }_0+{\beta }_1\ln X$

Elasticity:

$$\eta ={\beta }_1\text{(constant)}$$

Shape: Concave for $0<{\beta }_1<1$; convex (increasing returns) for ${\beta }_1>1$; linear for ${\beta }_1=1$.

Marketing use: Most common form for advertising and price response. The log-log specification fits by OLS directly on $\ln Q$ and $\ln X$.

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4. Multiplicative Form

Multiplicative (Multi-instrument)

$$Q=e^{{\beta }_0}{X}_1^{{\beta }_1}{X}_2^{{\beta }_2}\cdots X_J^{{\beta }_J}\text{\hspace{1em}(Eq 3.16)}$$

Taking logs: $\ln Q={\beta }_0+{\beta }_1\ln X_1+\cdots +{\beta }_J\ln X_J$

Elasticity of instrument $j$:

$${\eta }_j={\beta }_j\text{(constant, independent of other instruments)}$$

Properties: Cross-elasticities are zero (no interaction). Easily estimated by OLS on log-transformed data. The workhorse model for scanner data analysis (see SCAN*PRO below).

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5. Exponential Form (Increasing Returns)

Exponential

$$Q=e^{{\beta }_0}{e}^{{\beta }_{1}X}=e^{{\beta }_0+{\beta }_{1}X}$$

For pricing applications: $Q=Q^0{e}^{-{\beta }_{1}P}$, ${\beta }_1>0$

Elasticity:

$$\eta ={\beta }_{1}X$$

Shape: Convex — increasing marginal returns. Appropriate for threshold phenomena or situations where heavy spending compounds.

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6. Log-Reciprocal / Inverse Form (S-Shaped Saturation)

Log-Reciprocal

$$Q=\exp \left({\beta }_0-\frac{{\beta }_1}{X}\right)$$

Elasticity:

$$\eta =\frac{{\beta }_1}{X}$$

Shape: S-shaped with saturation at $e^{{\beta }_0}$; inflection point at $X={\beta }_1/2$. Increasing marginal returns at low $X$, diminishing returns at high $X$.

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7. Gompertz Form

Gompertz

$$Q={\beta }_0{\beta }_1^{{\beta }_2^{-{\beta }_{3}X}},0<{\beta }_1<1,0<{\beta }_2<1,{\beta }_3>0$$

Shape: Asymmetric S-curve with faster initial growth than logistic; upper asymptote ${\beta }_0$. Used for product adoption and life-cycle modeling. Related to Product Adoption and Diffusion Models.

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8. Modified Exponential (Saturation without S-Shape)

Modified Exponential

$$Q=Q^0(1-e^{-{\beta }_{1}X})$$

Elasticity:

$$\eta =\frac{{\beta }_{1}X{e}^{-{\beta }_{1}X}}{1-e^{-{\beta }_{1}X}}$$

Shape: Concave with upper asymptote $Q^0$. No inflection point. Appropriate when saturation is observed but there is no initial convex phase.

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9. Logistic Form

Logistic

$$Q=\frac{Q^0}{1+\exp \left(-({\beta }_0+{\sum }_j{\beta }_j{X}_j)\right)}$$

Shape: Symmetric S-curve, bounded in $(0,Q^0)$. Standard form for binary choice when $Q$ is market share (bounded in [0,1]). Related to Logit Purchase Decision Model (ABM consumer choice) and Discrete Choice Models (econometric random utility framework).

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10. ADBUDG Form (Little 1970)

ADBUDG

$$Q={\beta }_0+({\beta }_1-{\beta }_0)\frac{X^{{\beta }_2}}{{\beta }_3^{{\beta }_2}+X^{{\beta }_2}}$$

Parameters:

• ${\beta }_0$: sales at zero advertising (minimum)
• ${\beta }_1$: saturation sales (maximum)
• ${\beta }_2$: shape parameter (${\beta }_2<1$: concave; ${\beta }_2>1$: S-shaped)
• ${\beta }_3$: advertising level at midpoint $({\beta }_0+{\beta }_1)/2$

Elasticity at midpoint:

$$\eta =\frac{{\beta }_2}{2}({\beta }_1-{\beta }_0)/Q({\beta }_3)$$

Properties: Highly flexible; nests concave and S-shaped responses; parameterized by managerially interpretable quantities. Widely used in budget optimization (see Optimal Marketing Decisions and Forecasting).

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SCAN*PRO Scanner Model

SCAN*PRO

The SCAN*PRO model (Wittink et al.) applies the multiplicative form to weekly scanner data:

$$Q_{bst}=\exp ({\mu }_{bs})\cdot {\left(\frac{P_{bst}}{P_{bs}^{\ast }}\right)}^{{\beta }_1}\cdot \prod _k{F}_{kbst}^{{\gamma }_k}\cdot \prod _k{D}_{kbst}^{{\delta }_k}\cdot {ϵ}_{bst}$$

where $P_{bs}^{\ast }$ is a reference price, $F_k$ are feature advertising dummies, $D_k$ are display dummies. Estimated in log form by OLS or GLS at the store-brand-week level.

Key property: Allows brand-specific intercepts ${\mu }_{bs}$ capturing unobserved brand equity and store heterogeneity.

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Random Coefficients Model

Random Coefficients

When response varies across brands, markets, or time, parameters are treated as random:

$$Q_{it}={\beta }_{0i}+{\beta }_{1i}{X}_{it}+{ϵ}_{it}$$

where ${\beta }_{ji}={\bar{\beta }}_j+u_{ji}$, and $u_{ji}\sim (0,{\Sigma }_{\beta })$. Estimation via GLS (Swamy 1970) or hierarchical Bayes. Connects to Hierarchical Linear Models and the HB shrinkage estimator in Parameter Estimation in Market Response. (Eq 3.44 in book)

Systematically Varying Parameters

Parameters can be made functions of other variables (e.g., competitive context, season):

$${\beta }_1={\alpha }_0+{\alpha }_1{Z}_t$$

substituting into the model yields an interaction term ${\alpha }_1{Z}_t{X}_t$. This generalizes the constant-coefficient model and allows heterogeneous response across subgroups or time periods. See also Shape of the Marketing Response Function.

Switching Models

When the functional form itself is unknown a priori, a switching regression tests whether the data support different response regimes (e.g., pre/post competitive entry):

$$Q_t=\{\begin{array}{ll}{f}_1(X_t;{\beta }_1)+{ϵ}_t& \text{if regime 1}\\ f_2(X_t;{\beta }_2)+{ϵ}_t& \text{if regime 2}\end{array}$$

Regime assignment can be endogenous (Goldfeld-Quandt, Maddala-Nelson) or triggered by an observable event.

Functional Form Selection Summary

Form	Shape	Saturation	Constant $\eta$?	Marketing Context
Linear	Linear	No	No	Simple baseline
Semilog	Concave	No	No	Large advertising range
Power	Concave/convex	No	Yes	Advertising, price
Multiplicative	—	No	Yes	Multi-instrument
Exponential	Convex	No	No	Threshold advertising
Log-reciprocal	S-shaped	Yes	No	Threshold + saturation
Gompertz	Asymmetric S	Yes	No	Life cycle adoption
Modified exponential	Concave	Yes	No	Saturation without S
Logistic	Symmetric S	Yes	No	Market share
ADBUDG	Flexible	Yes	No	Budget optimization

Cross-Links

• Market share extensions: Market Share Models
• Dynamic extensions: Carryover Effects and Distributed Lags, Shape of the Marketing Response Function
• Estimation of nonlinear forms: Parameter Estimation in Market Response
• Empirical elasticities: Advertising and Promotion Effects, Price and Distribution Effects
• Product adoption S-curves: Product Adoption and Diffusion Models