21.6 Market Response Model

Marketing Inputs:

• Selling effort
• advertising spending
• promotional spending

Marketing Outputs:

• sales
• share
• profit
• awareness

Give phenomena for a good model:

• P1: Dynamic sales response involves a sales growth rate and a sales decay rate that are different
• P2: Steady-state response can be concave or S-shaped. Positive sales at 0 adverting.
• P3: Competitive effects
• P4: Advertising effectiveness dynamics due to changes in media, copy, and other factors.
• P5: Sales still increase or fall off even as advertising is held constant.

Saunder (1987) phenomena

• P1: Output = 0 when Input = 0
• P2: The relationship between input and output is linear
• P3: Returns decrease as the scale of input increases (i.e., additional unit of input gives less output)
• P4: Output cannot exceed some level (i.e., saturation)
• P5: Returns increase as scale of input increases (i.e., additional unit of input gives more output)
• P6: Returns first increase and then decrease as input increases (i.e., S-shaped return)
• P7: Input must exceed some level before it produces any output (i.e., threshold)
• P8: Beyond some level of input, output declines (i.e., supersaturation point)

Aggregate Response Models

• Linear model: $Y = a + bX$

  • Through origin

  • can only handle constant returns to scale (i.e., can’t handle concave, convex, and S-shape)

• The Power Series/Polynomial model: $Y = a + bX + c X^2 + dX^3 + ...$

  • can’t handle saturation and threshold

• Fraction root model/ Power model: $Y = a+bX^c$ where c is prespecified

  • c = 1/2, called square root model

  • c = -1, called reciprocal model

  • c can be interpreted as elasticity if a = 0.

  • c = 1, linear

  • c <1, decreasing return

  • c>1, increasing returns

• Semilog model: $Y = a + b \ln X$

  • Good when constant percentage increase in marketing effort (X) result in constant absolute increase in sales (Y)

• Exponential model: $Y = ae^{bX}$ where X >0

  • b > 0, increasing returns and convex

  • b < 0, decreasing returns and saturation

• Modified exponential model: $Y = a(1-e^{-bX}) +c$

  • Decreasing returns and saturation

  • upper bound = a + c

  • lower bound = c

  • typically used in selling effort

• Logistic model: $Y = \frac{a}{a+ e^{-(b+cX)}}+d$

  • increasing return followed by decreasing return to scale, S-shape

  • saturation = a + d

  • good with saturation and s-shape

• Gompertz model

• ADBUDG model (Little 1970) : $Y = b + (a-b)\frac{X^c}{d + X^c}$

  • c > 1, S-shaped

  • 0 < c < 1

    • Concave

    • saturation effect

    • upper bound at a

    • lower bound at b

  • typically used in advertising and selling effort.

  • can handle, through origin, concave, saturation, S-shape

• Additive model for handling multiple Instruments: $Y = af(X_1) + bg(X_2)$

• Multiplicative model for handling multiple instruments: $Y = aX_1^b X_2^c$ where c and c are elasticities. More generally, $Y = af(X_1)\times bg(X_2)$

• Multiplicative and additive model: $Y = af(X_1) + bg(X_2) + cf(X_1) g(X_2)$

• Dynamic response model: $Y_t = a_0 + a_1 X_t + \lambda Y_{t-1}$ where $a_1$ = current effect, $\lambda$ = carry-over effect

Dynamic Effects

• Carry-over effect: current marketing expenditure influences future sales

  • Advertising adstock/ advertising carry-over is the same thing: lagged effect of advertising on sales

• Delayed-response effect: delays between when marketing investments and their impact

• Customer holdout effects

• Hysteresis effect

• New trier and wear-out effect

• Stocking effect

Simple Decay-effect model:

$$A_t = T_t + \lambda T_{t-1}, t = 1,...,$$

where

• $A_t$ = Adstock at time t
• $T_t$ = value of advertising spending at time t
• $\lambda$ = decay/ lag weight parameter

Response Models can be characterized by:

1. The number of marketing variables

2. whether they include competition or not

3. the nature of the relationship between the input variables

   1. Linear vs. S-shape

4. whether the situation is static vs. dynamic

5. whether the models reflect individual or aggregate response

6. the level of demand analyzed

   1. sales vs. market share

Market Share Model and Competitive Effects: $Y = M \times V$ where

• Y = Brand sales models

• V = product class sales models

• M = market-share models

Market share (attraction) models

$$M_i = \frac{A_i}{A_1 + ..+ A_n}$$

where $A_i$ attractiveness of brand i

Individual Response Model:

Multinomial logit model representing the probability of individual i choosing brand l is

$$P_{il} = \frac{e^{A_{il}}}{\sum_j e^{A_{ij}}}$$

where

• $A_{ij}$ = attractiveness of product j for individual i $A_{ij} = \sum_k w_k b_{ijk}$
• $b_{ijk}$ = individual i’s evaluation of product j on product attribute k, where the summation is over all the products that individual i is considering to purchase
• $w_k$ = importance weight associated with attribute k in forming product preferences.

References

Little, John D. C. 1970. “Models and Managers: The Concept of a Decision Calculus.” Management Science 16 (8): B-466-B-485. https://doi.org/10.1287/mnsc.16.8.b466.