Shape of the Marketing Response Function

Summary

The shape of the response function (concave, convex, S-shaped, or linear) has profound implications for optimal budget allocation, pulsing strategies, and competitive dynamics. This note covers prior knowledge levels, the conditions under which each shape arises, and the budget implications.

Prior Knowledge Levels

Chapter 4 introduces a taxonomy of prior knowledge that constrains model specification:

Prior Knowledge Levels

• Level 0: Only the information set is known (which variables to include)
• Level 1: Causal ordering is known (which variables affect which)
• Level 2: Functional form and lag structure are fully specified

Most applied work operates at Level 1-2. The choice of functional form is a Level 2 specification that should be grounded in behavioral theory.

Concave Response (Diminishing Returns)

When Response is Concave

Response is concave when:

1. Awareness reaches saturation (most of the target audience is already aware)
2. Additional exposures yield diminishing incremental attitude change
3. The market is already heavily penetrated

Budget implication: Under concave response, the optimal strategy is continuous (even) spending — pulsing (concentrating spending) is suboptimal because it wastes money in periods of high spending where marginal returns are low.

Functional forms: power ($0<{\beta }_1<1$), semilog, modified exponential.

S-Shaped Response (Convex-Concave)

When Response is S-Shaped

Response is S-shaped when:

1. There is a threshold below which advertising has minimal effect (awareness builds slowly)
2. Above the threshold, response accelerates (word-of-mouth amplifies the message)
3. Eventually saturation is reached (concave portion)

Budget implication: Under S-shaped response, pulsing can be optimal — concentrating spending at levels above the threshold in some periods, and spending zero (or the minimum) in other periods.

Functional forms: log-reciprocal, ADBUDG (${\beta }_2>1$), Gompertz, logistic.

Convex Response (Increasing Returns)

Convex response (exponential form) implies increasing marginal returns — rare in mature markets but possible in new product introductions where word-of-mouth creates an accelerating adoption process (related to Product Adoption and Diffusion Models Bass model). Under convex response, it is optimal to concentrate all spending in one period (corner solution).

Asymmetric and Threshold Effects

Threshold Effect

A threshold level $X^{\ast }$ exists below which advertising has essentially zero effect. Above $X^{\ast }$, response becomes positive. Threshold models arise from:

• Awareness building (minimum exposures needed for recall)
• Media vehicle minimum reach requirements

Estimated via a kinked linear model: $Q={\beta }_0+{\beta }_1\max (X-X^{\ast },0)$

Hysteresis and Path Dependence

Hysteresis

Hysteresis in marketing means that temporary marketing actions can have permanent effects on sales. A brand that achieves high advertising levels builds consumer goodwill that persists even after spending returns to normal (fast learning/slow forgetting).

Formally: if brand performance is an evolving (unit-root) process, then even temporary shocks have persistent effects. This is operationalized through the multivariate persistence measures in Multivariate Persistence and Cointegration.

Ratchet models (Eq 4.47) capture a simple form of hysteresis: $Q_t={\beta }_0+{\beta }_1{X}_t+{\beta }_2{\max }_{i\le t}(X_i)$, where the maximum historical advertising level permanently anchors the baseline.

Implications for Pulsing Strategies

Response Shape	Optimal Strategy	Logic
Concave	Even spending (“maintenance”)	Avoid high-marginal-cost periods
S-shaped	Pulsing (on/off)	Exceed threshold in active periods
Convex	All-or-nothing	Increasing returns reward concentration
Linear	Indifferent	All spending levels equally efficient

Cross-Links

• Functional form catalogue: Functional Forms in Marketing
• ADBUDG for budget optimization: Optimal Marketing Decisions and Forecasting
• Hysteresis in time series: Multivariate Persistence and Cointegration
• Product adoption S-curves: Product Adoption and Diffusion Models
• Ratchet models: Carryover Effects and Distributed Lags